Subgroups — #542×D₂₇₁ #46849734

Order 34688.

1132 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774521	C₂	1	1	yes	maximal, characteristic, complemented
2	#31774520	C₂	1	1	yes	maximal, characteristic, complemented
3	#31774519	C₂	1	1	yes	maximal, characteristic, complemented
4	#31774518	C₂	1	1	yes	maximal, characteristic, complemented
5	#31774517	C₂	1	2	yes	maximal, complemented
6	#31774516	C₂	1	1	yes	maximal, characteristic, complemented
7	#31774515	C₂	1	2	yes	maximal, complemented
8	#31774514	C₂	1	1	yes	maximal, characteristic, complemented
9	#31774513	C₂	1	2	yes	maximal, complemented
10	#31774512	C₂	1	1	yes	maximal, characteristic, complemented
11	#31774511	C₂	1	2	yes	maximal, complemented
12	C₄⋊Q₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#126×D₂₇₁	C₂	1	2	yes	maximal, complemented
14	#124×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#124×D₂₇₁	C₂	1	2	yes	maximal, complemented
16	#123×D₂₇₁	C₂	1	2	yes	maximal, complemented
17	C₂²⋊Q₈×D₂₇₁	C₂	1	2	yes	maximal, complemented
18	C₂²⋊Q₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
19	#121×D₂₇₁	C₂	1	1	yes	maximal, characteristic
20	C₄×Q₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×D₄×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
22	#117×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
23	#542×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×Q₈×D₂₇₁	C₂²	1	1	yes	characteristic
25	C₂×Q₈×D₂₇₁	C₂²	1	2	yes	—
26	C₂×D₄×D₂₇₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₂₇₁	C₂²	1	2	yes	—
28	C₂²×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₂₇₁	C₂²	1	2	yes	—
30	#22576160	C₂²	1	1	yes	characteristic, complemented
31	#22576159	C₂²	1	1	yes	characteristic, complemented
32	#22576159	C₂²	1	2	yes	complemented
33	#22576159	C₂²	1	1	yes	characteristic, complemented
34	#22576158	C₂²	1	2	yes	complemented
35	#22576155	C₂²	1	2	yes	complemented
36	#22576155	C₂²	1	2	yes	complemented
37	#22576155	C₂²	1	2	yes	complemented
38	#22576154	C₂²	1	2	yes	complemented
39	#22576153	C₂²	1	2	yes	complemented
40	#22576153	C₂²	1	2	yes	complemented
41	#22576153	C₂²	1	2	yes	complemented
42	#22576149	C₂²	1	2	yes	complemented
43	#22576149	C₂²	1	1	yes	characteristic, complemented
44	#22576148	C₂²	1	2	yes	complemented
45	#22576148	C₂²	1	1	yes	characteristic, complemented
46	#22576147	C₂²	1	1	yes	characteristic, complemented
47	#22576147	C₂²	1	2	yes	complemented
48	#22576146	C₂²	1	2	yes	complemented
49	#22576146	C₂²	1	2	yes	complemented
50	#22576146	C₂²	1	2	yes	complemented
51	#22576146	C₂²	1	2	yes	complemented
52	#22576146	C₂²	1	2	yes	complemented
53	#22576146	C₂²	1	2	yes	complemented
54	#22576145	C₂²	1	2	yes	complemented
55	#22576145	C₂²	1	2	yes	complemented
56	#22576144	C₂²	1	2	yes	complemented
57	#22576143	C₂²	1	2	yes	complemented
58	#22576142	C₂²	1	2	yes	complemented
59	#22576142	C₂²	1	2	yes	complemented
60	#22576141	C₂²	1	2	yes	complemented
61	#22576140	C₂²	1	1	yes	characteristic, complemented
62	#22576140	C₂²	1	2	yes	complemented
63	#22576139	C₂²	1	2	yes	complemented
64	#22576139	C₂²	1	1	yes	characteristic, complemented
65	#22576138	C₂²	1	2	yes	complemented
66	#22576138	C₂²	1	2	yes	complemented
67	#22576138	C₂²	1	2	yes	complemented
68	#22576137	C₂²	1	2	yes	complemented
69	#22576137	C₂²	1	2	yes	complemented
70	#22576137	C₂²	1	2	yes	complemented
71	#22576136	C₂²	1	2	yes	complemented
72	#22576136	C₂²	1	1	yes	characteristic, complemented
73	#22576135	C₂²	1	1	yes	characteristic
74	#22576134	C₂²	1	1	yes	characteristic
75	#22576133	C₂²	1	2	yes	—
76	#22576132	C₂²	1	2	yes	—
77	#22576131	C₂²	1	1	yes	characteristic
78	Q₈×Dic₂₇₁	C₂²	1	1	yes	characteristic, complemented
79	C₄×Dic₅₄₂	C₂²	1	2	yes	complemented
80	C₄×Dic₅₄₂	C₂²	1	1	yes	characteristic, complemented
81	#22576125	C₂²	1	1	yes	characteristic, complemented
82	#22576125	C₂²	1	2	yes	complemented
83	C₄×D₁₀₈₄	C₂²	1	1	yes	characteristic, complemented
84	D₄×Dic₂₇₁	C₂²	1	1	yes	characteristic, complemented
85	#22576122	C₂²	1	1	yes	characteristic, complemented
86	C₄×C₂²⋊D₂₇₁	C₂²	1	2	yes	complemented
87	(C₂²⋊D₂₇₁)⋊C₄	C₂²	1	2	yes	complemented
88	#22576119	C₂²	1	1	yes	characteristic, complemented
89	#22576118	C₂²	1	1	yes	characteristic, complemented
90	#22576118	C₂²	1	1	yes	characteristic, complemented
91	#22576117	C₂²	1	1	yes	characteristic, complemented
92	#22576117	C₂²	1	1	yes	characteristic, complemented
93	#22576116	C₂²	1	1	yes	characteristic, complemented
94	#22576116	C₂²	1	1	yes	characteristic, complemented
95	C₄⋊C₄×D₂₇₁	C₂²	1	1	yes	characteristic
96	C₄⋊C₄×D₂₇₁	C₂²	1	1	yes	characteristic
97	C₄⋊C₄×D₂₇₁	C₂²	1	1	yes	characteristic
98	C₄⋊C₄×D₂₇₁	C₂²	1	1	yes	characteristic
99	C₄⋊C₄×D₂₇₁	C₂²	1	2	yes	—
100	C₄⋊C₄×D₂₇₁	C₂²	1	2	yes	—
101	C₄⋊C₄×D₂₇₁	C₂²	1	2	yes	—
102	C₂²⋊C₄×D₂₇₁	C₂²	1	1	yes	characteristic
103	C₂²⋊C₄×D₂₇₁	C₂²	1	1	yes	characteristic
104	C₂²⋊C₄×D₂₇₁	C₂²	1	2	yes	—
105	C₂²⋊C₄×D₂₇₁	C₂²	1	2	yes	—
106	C₂²⋊C₄×D₂₇₁	C₂²	1	2	yes	—
107	C₂²⋊C₄×D₂₇₁	C₂²	1	2	yes	—
108	C₂²⋊C₄×D₂₇₁	C₂²	1	2	yes	—
109	C₄²×D₂₇₁	C₂²	1	1	yes	characteristic
110	C₄²×D₂₇₁	C₂²	1	1	yes	characteristic
111	C₄²×D₂₇₁	C₂²	1	2	yes	—
112	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
113	#126×C₂₇₁	C₂²	1	2	yes	complemented
114	#124×C₂₇₁	C₂²	1	2	yes	complemented
115	#124×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
116	#123×C₂₇₁	C₂²	1	2	yes	complemented
117	C₂²⋊Q₈×C₂₇₁	C₂²	1	2	yes	complemented
118	C₂²⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
119	#121×C₂₇₁	C₂²	1	1	yes	characteristic
120	C₄×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
122	#117×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₂₇₁	C₂³	1	1	yes	characteristic
124	C₂³×D₂₇₁	C₂³	1	2	yes	—
125	C₂×Dic₅₄₂	C₂³	1	2	yes	—
126	C₂×Dic₅₄₂	C₂³	1	1	yes	characteristic
127	C₂×Dic₅₄₂	C₂³	1	2	yes	—
128	C₂×Dic₅₄₂	C₂³	1	2	yes	—
129	C₂×Dic₅₄₂	C₂³	1	2	yes	—
130	Q₈×D₂₇₁	—	2	4	no	—
131	Q₈×D₂₇₁	—	4	4	no	—
132	C₂×D₁₀₈₄	C₂³	1	1	yes	characteristic
133	C₂×D₁₀₈₄	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₂₇₁	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₂₇₁	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₂₇₁	C₂³	1	2	yes	—
137	D₄×D₂₇₁	—	2	4	no	—
138	D₄×D₂₇₁	—	4	4	no	—
139	C₂²×Dic₂₇₁	C₂³	1	1	yes	characteristic
140	C₂²×Dic₂₇₁	C₂³	1	2	yes	—
141	C₂×C₄×D₂₇₁	—	2	4	no	—
142	C₂×C₄×D₂₇₁	—	4	4	no	—
143	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
147	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
149	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
151	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
153	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
154	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
155	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
156	C₄⋊Dic₂₇₁	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₂₇₁	C₂³	1	2	yes	—
158	C₄⋊Dic₂₇₁	C₂³	1	1	yes	characteristic
159	C₄⋊Dic₂₇₁	C₂³	1	1	yes	characteristic
160	C₄⋊Dic₂₇₁	C₂³	1	2	yes	—
161	C₄⋊Dic₂₇₁	C₂³	1	2	yes	—
162	C₄⋊Dic₂₇₁	C₂³	1	1	yes	characteristic
163	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
164	Dic₂₇₁⋊C₄	C₂³	1	1	yes	characteristic
165	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
166	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
167	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
168	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
169	Dic₂₇₁⋊C₄	C₂³	1	1	yes	characteristic
170	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
171	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
172	Dic₂₇₁⋊C₄	C₂³	1	2	yes	—
173	Dic₂₇₁⋊C₄	C₂³	1	1	yes	characteristic
174	Dic₂₇₁⋊C₄	C₂³	1	1	yes	characteristic
175	C₂²⋊Dic₂₇₁	C₂³	1	2	yes	—
176	C₂²⋊Dic₂₇₁	C₂³	1	2	yes	—
177	C₂²⋊Dic₂₇₁	C₂³	1	2	yes	—
178	C₂²⋊Dic₂₇₁	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₂₇₁	C₂³	1	2	yes	—
180	C₂²⋊Dic₂₇₁	C₂³	1	1	yes	characteristic
181	C₂²⋊Dic₂₇₁	C₂³	1	2	yes	—
182	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
183	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
184	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
185	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
186	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
187	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
188	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
189	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
190	(C₂×D₂₇₁)⋊C₄	C₂³	1	1	yes	characteristic
191	(C₂×D₂₇₁)⋊C₄	C₂³	1	1	yes	characteristic
192	(C₂×D₂₇₁)⋊C₄	C₂³	1	1	yes	characteristic
193	(C₂×D₂₇₁)⋊C₄	C₂³	1	1	yes	characteristic
194	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
195	(C₂×D₂₇₁)⋊C₄	C₂³	1	2	yes	—
196	C₄×Dic₂₇₁	C₂³	1	2	yes	—
197	C₄×Dic₂₇₁	C₂³	1	2	yes	—
198	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
199	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
200	C₄×Dic₂₇₁	C₂³	1	2	yes	—
201	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
202	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
203	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
204	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
205	C₂×Q₈×C₂₇₁	C₂³	1	1	yes	characteristic
206	C₂×Q₈×C₂₇₁	C₂³	1	2	yes	—
207	C₂×D₄×C₂₇₁	C₂³	1	1	yes	characteristic
208	C₂×D₄×C₂₇₁	C₂³	1	2	yes	—
209	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
210	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
211	C₄⋊C₄×C₂₇₁	C₂³	1	1	yes	characteristic
212	C₄⋊C₄×C₂₇₁	C₂³	1	1	yes	characteristic
213	C₄⋊C₄×C₂₇₁	C₂³	1	2	yes	—
214	C₄⋊C₄×C₂₇₁	C₂³	1	2	yes	—
215	C₄⋊C₄×C₂₇₁	C₂³	1	1	yes	characteristic
216	C₄⋊C₄×C₂₇₁	C₂³	1	2	yes	—
217	C₄⋊C₄×C₂₇₁	C₂³	1	1	yes	characteristic
218	C₂²⋊C₄×C₂₇₁	C₂³	1	2	yes	—
219	C₂²⋊C₄×C₂₇₁	C₂³	1	2	yes	—
220	C₂²⋊C₄×C₂₇₁	C₂³	1	2	yes	—
221	C₂²⋊C₄×C₂₇₁	C₂³	1	1	yes	characteristic
222	C₂²⋊C₄×C₂₇₁	C₂³	1	2	yes	—
223	C₂²⋊C₄×C₂₇₁	C₂³	1	1	yes	characteristic
224	C₂²⋊C₄×C₂₇₁	C₂³	1	2	yes	—
225	C₄²×C₂₇₁	C₂³	1	2	yes	—
226	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
227	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
228	C₂²×D₂₇₁	—	2	2	no	—
229	C₂²×D₂₇₁	—	2	4	no	—
230	C₂²×D₂₇₁	—	2	4	no	—
231	C₂²×D₂₇₁	—	2	2	no	—
232	C₂²×D₂₇₁	—	2	2	no	—
233	C₂²×D₂₇₁	—	2	4	no	—
234	C₂²×D₂₇₁	C₂⁴	1	1	yes	characteristic
235	C₂²×D₂₇₁	C₂⁴	1	2	yes	—
236	C₂²×D₂₇₁	C₂⁴	1	1	yes	characteristic
237	Dic₅₄₂	—	2	2	no	—
238	Dic₅₄₂	—	2	4	no	—
239	Dic₅₄₂	—	4	2	no	—
240	Dic₅₄₂	—	4	2	no	—
241	Dic₅₄₂	—	4	2	no	—
242	D₁₀₈₄	—	2	2	no	—
243	D₁₀₈₄	—	4	2	no	—
244	C₂²⋊D₂₇₁	—	2	4	no	—
245	C₂²⋊D₂₇₁	—	4	2	no	—
246	C₂²⋊D₂₇₁	—	4	2	no	—
247	C₂×Dic₂₇₁	—	2	2	no	—
248	C₂×Dic₂₇₁	—	4	2	no	—
249	C₂×Dic₂₇₁	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₂₇₁	C₂⁴	1	1	yes	characteristic
251	C₂×Dic₂₇₁	C₂⁴	1	2	yes	—
252	C₂×Dic₂₇₁	C₂⁴	1	1	yes	characteristic
253	C₂×Dic₂₇₁	C₂⁴	1	1	yes	characteristic
254	C₂×Dic₂₇₁	C₂⁴	1	2	yes	—
255	C₂×Dic₂₇₁	C₂⁴	1	1	yes	characteristic
256	C₂×Dic₂₇₁	C₂⁴	1	2	yes	—
257	C₂×Dic₂₇₁	C₂⁴	1	1	yes	characteristic
258	C₄×D₂₇₁	—	2	2	no	—
259	C₄×D₂₇₁	—	2	4	no	—
260	C₄×D₂₇₁	—	2	2	no	—
261	C₄×D₂₇₁	—	2	4	no	—
262	C₄×D₂₇₁	—	2	4	no	—
263	C₄×D₂₇₁	—	2	2	no	—
264	C₄×D₂₇₁	—	2	2	no	—
265	C₄×D₂₇₁	—	2	2	no	—
266	C₄×D₂₇₁	—	2	2	no	—
267	C₄×D₂₇₁	—	2	2	no	—
268	C₄×D₂₇₁	—	4	2	no	—
269	C₄×D₂₇₁	—	4	2	no	—
270	C₄×D₂₇₁	#41	1	4	yes	—
271	C₂³×C₂₇₁	C₂⁴	1	1	yes	characteristic
272	C₂³×C₂₇₁	C₂⁴	1	2	yes	—
273	Q₈×C₂₇₁	—	2	2	no	—
274	Q₈×C₂₇₁	—	4	2	no	—
275	D₄×C₂₇₁	—	2	2	no	—
276	D₄×C₂₇₁	—	4	2	no	—
277	C₂×C₄×C₂₇₁	—	2	2	no	—
278	C₂×C₄×C₂₇₁	—	4	2	no	—
279	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
280	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
281	C₂×C₄×C₂₇₁	C₂⁴	1	2	yes	—
282	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
283	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
284	C₂×C₄×C₂₇₁	C₂⁴	1	2	yes	—
285	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
286	C₂×C₄×C₂₇₁	C₂⁴	1	2	yes	—
287	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
288	C₂²×C₂₇₁	—	2	1	no	—
289	C₂²×C₂₇₁	—	2	2	no	—
290	C₂²×C₂₇₁	—	2	2	no	—
291	C₂²×C₂₇₁	—	2	1	no	—
292	C₂²×C₂₇₁	—	2	1	no	—
293	C₂²×C₂₇₁	—	2	2	no	—
294	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
295	C₂×D₂₇₁	—	2	1	no	—
296	C₂×D₂₇₁	—	2	2	no	—
297	C₂×D₂₇₁	—	2	2	no	—
298	C₂×D₂₇₁	—	2	1	no	—
299	C₂×D₂₇₁	—	2	1	no	—
300	C₂×D₂₇₁	—	2	2	no	—
301	C₂×D₂₇₁	—	4	4	no	—
302	C₂×D₂₇₁	—	4	8	no	—
303	C₂×D₂₇₁	C₂×#41	1	2	yes	—
304	C₂×D₂₇₁	H(5,𝔽₂)	1	2	yes	—
305	C₂×D₂₇₁	2₋¹⁺⁴	1	2	yes	—
306	C₄×C₂₇₁	—	2	1	no	—
307	C₄×C₂₇₁	—	2	2	no	—
308	C₄×C₂₇₁	—	2	1	no	—
309	C₄×C₂₇₁	—	2	2	no	—
310	C₄×C₂₇₁	—	2	2	no	—
311	C₄×C₂₇₁	—	2	1	no	—
312	C₄×C₂₇₁	—	2	1	no	—
313	C₄×C₂₇₁	—	2	1	no	—
314	C₄×C₂₇₁	C₂×#41	1	2	yes	—
315	Dic₂₇₁	—	2	1	no	—
316	Dic₂₇₁	—	2	2	no	—
317	Dic₂₇₁	—	2	1	no	—
318	Dic₂₇₁	—	2	2	no	—
319	Dic₂₇₁	—	2	2	no	—
320	Dic₂₇₁	—	2	1	no	—
321	Dic₂₇₁	—	2	1	no	—
322	Dic₂₇₁	—	2	1	no	—
323	Dic₂₇₁	C₂×#41	1	2	yes	—
324	C₂×C₂₇₁	—	4	1	no	—
325	C₂×C₂₇₁	—	4	2	no	—
326	C₂×C₂₇₁	C₂²×#41	1	1	yes	characteristic
327	C₂×C₂₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂×C₂₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
329	D₂₇₁	—	4	1	no	—
330	D₂₇₁	—	4	2	no	—
331	D₂₇₁	#542	1	4	yes	direct product factor
332	C₂₇₁	C₂×#542	1	1	yes	characteristic, complemented, Sylow
333	C₂×#542	—	271	1	no	maximal, Sylow
334	#542	—	271	1	no	—
335	#542	—	271	2	no	—
336	#542	—	271	2	no	—
337	#542	—	271	1	no	—
338	#542	—	271	2	no	—
339	#542	—	271	2	no	—
340	#542	—	271	1	no	—
341	#542	—	271	1	no	—
342	#542	—	271	1	no	—
343	#542	—	271	1	no	—
344	#542	—	271	1	no	—
345	#542	D₂₇₁	1	1	yes	characteristic, direct product factor
346	C₂×C₄⋊Q₈	—	271	1	no	—
347	C₂×#126	—	271	2	no	—
348	C₂×#124	—	271	1	no	—
349	C₂×#124	—	271	2	no	—
350	C₂×#123	—	271	2	no	—
351	C₂×C₂²⋊Q₈	—	271	2	no	—
352	C₂×C₂²⋊Q₈	—	271	1	no	—
353	C₂×#121	—	271	1	no	—
354	C₂×C₄×Q₈	—	271	1	no	—
355	C₂×C₄×D₄	—	271	1	no	—
356	C₂×#117	—	271	1	no	—
357	C₂²×Q₈	—	271	1	no	—
358	C₂²×Q₈	—	271	2	no	—
359	C₂²×D₄	—	271	1	no	—
360	C₂²×D₄	—	271	2	no	—
361	C₂³×C₄	—	271	1	no	—
362	C₂³×C₄	—	271	2	no	—
363	C₄⋊Q₈	—	271	2	no	—
364	C₄⋊Q₈	—	271	2	no	—
365	C₄⋊Q₈	—	271	1	no	—
366	C₄⋊Q₈	—	271	1	no	—
367	C₄⋊Q₈	—	271	1	no	—
368	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic, complemented
369	#126	—	271	2	no	—
370	#126	—	271	2	no	—
371	#126	—	271	2	no	—
372	#126	—	271	2	no	—
373	#126	—	271	2	no	—
374	#126	—	271	2	no	—
375	#126	—	271	2	no	—
376	#126	C₂×D₂₇₁	1	2	yes	complemented
377	#124	—	271	1	no	—
378	#124	—	271	2	no	—
379	#124	—	271	1	no	—
380	#124	—	271	2	no	—
381	#124	—	271	2	no	—
382	#124	—	271	2	no	—
383	#124	—	271	2	no	—
384	#124	—	271	2	no	—
385	#124	—	271	2	no	—
386	#124	—	271	2	no	—
387	#124	—	271	1	no	—
388	#124	—	271	2	no	—
389	#124	C₂×D₂₇₁	1	1	yes	characteristic, complemented
390	#124	C₂×D₂₇₁	1	2	yes	complemented
391	#123	—	271	2	no	—
392	#123	—	271	2	no	—
393	#123	—	271	2	no	—
394	#123	—	271	2	no	—
395	#123	—	271	2	no	—
396	#123	—	271	2	no	—
397	#123	—	271	2	no	—
398	#123	C₂×D₂₇₁	1	2	yes	complemented
399	C₂²⋊Q₈	—	271	2	no	—
400	C₂²⋊Q₈	—	271	1	no	—
401	C₂²⋊Q₈	—	271	2	no	—
402	C₂²⋊Q₈	—	271	1	no	—
403	C₂²⋊Q₈	—	271	2	no	—
404	C₂²⋊Q₈	—	271	2	no	—
405	C₂²⋊Q₈	—	271	2	no	—
406	C₂²⋊Q₈	—	271	2	no	—
407	C₂²⋊Q₈	—	271	1	no	—
408	C₂²⋊Q₈	—	271	2	no	—
409	C₂²⋊Q₈	—	271	2	no	—
410	C₂²⋊Q₈	—	271	2	no	—
411	C₂²⋊Q₈	C₂×D₂₇₁	1	2	yes	complemented
412	C₂²⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic, complemented
413	#121	—	271	1	no	—
414	#121	—	271	2	no	—
415	#121	—	271	1	no	—
416	#121	—	271	1	no	—
417	#121	—	271	2	no	—
418	#121	C₂×D₂₇₁	1	1	yes	characteristic
419	C₄×Q₈	—	271	1	no	—
420	C₄×Q₈	—	271	2	no	—
421	C₄×Q₈	—	271	1	no	—
422	C₄×Q₈	—	271	2	no	—
423	C₄×Q₈	—	271	1	no	—
424	C₄×Q₈	C₂×D₂₇₁	1	1	yes	characteristic, complemented
425	C₄×D₄	—	271	1	no	—
426	C₄×D₄	—	271	2	no	—
427	C₄×D₄	—	271	1	no	—
428	C₄×D₄	—	271	2	no	—
429	C₄×D₄	—	271	1	no	—
430	C₄×D₄	C₂×D₂₇₁	1	1	yes	characteristic, complemented
431	#117	—	271	1	no	—
432	#117	—	271	1	no	—
433	#117	—	271	1	no	—
434	#117	—	271	1	no	—
435	#117	—	271	1	no	—
436	#117	—	271	1	no	—
437	#117	—	271	1	no	—
438	#117	C₂×D₂₇₁	1	1	yes	characteristic, complemented
439	C₂×C₄⋊C₄	—	271	1	no	—
440	C₂×C₄⋊C₄	—	271	1	no	—
441	C₂×C₄⋊C₄	—	271	1	no	—
442	C₂×C₄⋊C₄	—	271	1	no	—
443	C₂×C₄⋊C₄	—	271	2	no	—
444	C₂×C₄⋊C₄	—	271	2	no	—
445	C₂×C₄⋊C₄	—	271	2	no	—
446	C₂×C₂²⋊C₄	—	271	1	no	—
447	C₂×C₂²⋊C₄	—	271	1	no	—
448	C₂×C₂²⋊C₄	—	271	2	no	—
449	C₂×C₂²⋊C₄	—	271	2	no	—
450	C₂×C₂²⋊C₄	—	271	2	no	—
451	C₂×C₂²⋊C₄	—	271	2	no	—
452	C₂×C₂²⋊C₄	—	271	2	no	—
453	C₂×C₄²	—	271	1	no	—
454	C₂×C₄²	—	271	1	no	—
455	C₂×C₄²	—	271	2	no	—
456	C₂⁴	—	271	1	no	—
457	C₂⁴	—	271	2	no	—
458	C₂×Q₈	—	271	2	no	—
459	C₂×Q₈	—	271	1	no	—
460	C₂×Q₈	—	271	2	no	—
461	C₂×Q₈	—	271	2	no	—
462	C₂×Q₈	—	271	2	no	—
463	C₂×Q₈	—	542	4	no	—
464	C₂×Q₈	—	1084	4	no	—
465	C₂×Q₈	C₂²×D₂₇₁	1	1	yes	characteristic
466	C₂×Q₈	C₂²×D₂₇₁	1	2	yes	—
467	C₂×D₄	—	271	2	no	—
468	C₂×D₄	—	271	2	no	—
469	C₂×D₄	—	271	2	no	—
470	C₂×D₄	—	271	1	no	—
471	C₂×D₄	—	271	2	no	—
472	C₂×D₄	—	542	4	no	—
473	C₂×D₄	—	1084	4	no	—
474	C₂×D₄	C₂²×D₂₇₁	1	1	yes	characteristic
475	C₂×D₄	C₂²×D₂₇₁	1	2	yes	—
476	C₂²×C₄	—	271	1	no	—
477	C₂²×C₄	—	271	1	no	—
478	C₂²×C₄	—	271	2	no	—
479	C₂²×C₄	—	271	1	no	—
480	C₂²×C₄	—	271	1	no	—
481	C₂²×C₄	—	271	2	no	—
482	C₂²×C₄	—	271	1	no	—
483	C₂²×C₄	—	271	1	no	—
484	C₂²×C₄	—	271	2	no	—
485	C₂²×C₄	—	271	1	no	—
486	C₂²×C₄	—	271	2	no	—
487	C₂²×C₄	—	271	1	no	—
488	C₂²×C₄	—	271	1	no	—
489	C₂²×C₄	—	271	2	no	—
490	C₂²×C₄	—	271	2	no	—
491	C₂²×C₄	—	542	4	no	—
492	C₂²×C₄	—	1084	4	no	—
493	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
494	C₂²×C₄	C₂²×D₂₇₁	1	2	yes	—
495	C₄⋊C₄	—	271	1	no	—
496	C₄⋊C₄	—	271	1	no	—
497	C₄⋊C₄	—	271	1	no	—
498	C₄⋊C₄	—	271	1	no	—
499	C₄⋊C₄	—	271	1	no	—
500	C₄⋊C₄	—	271	1	no	—
501	C₄⋊C₄	—	271	2	no	—
502	C₄⋊C₄	—	271	1	no	—
503	C₄⋊C₄	—	271	2	no	—
504	C₄⋊C₄	—	271	1	no	—
505	C₄⋊C₄	—	271	2	no	—
506	C₄⋊C₄	—	271	2	no	—
507	C₄⋊C₄	—	271	2	no	—
508	C₄⋊C₄	—	271	2	no	—
509	C₄⋊C₄	—	271	2	no	—
510	C₄⋊C₄	—	271	2	no	—
511	C₄⋊C₄	—	271	2	no	—
512	C₄⋊C₄	—	271	2	no	—
513	C₄⋊C₄	—	271	2	no	—
514	C₄⋊C₄	C₂²×D₂₇₁	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₂₇₁	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₂₇₁	1	1	yes	characteristic
517	C₄⋊C₄	C₂²×D₂₇₁	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×D₂₇₁	1	2	yes	—
519	C₄⋊C₄	C₂²×D₂₇₁	1	2	yes	—
520	C₄⋊C₄	C₂²×D₂₇₁	1	2	yes	—
521	C₂²⋊C₄	—	271	1	no	—
522	C₂²⋊C₄	—	271	1	no	—
523	C₂²⋊C₄	—	271	2	no	—
524	C₂²⋊C₄	—	271	2	no	—
525	C₂²⋊C₄	—	271	2	no	—
526	C₂²⋊C₄	—	271	2	no	—
527	C₂²⋊C₄	—	271	2	no	—
528	C₂²⋊C₄	—	271	1	no	—
529	C₂²⋊C₄	—	271	1	no	—
530	C₂²⋊C₄	—	271	1	no	—
531	C₂²⋊C₄	—	271	1	no	—
532	C₂²⋊C₄	—	271	2	no	—
533	C₂²⋊C₄	—	271	2	no	—
534	C₂²⋊C₄	—	271	2	no	—
535	C₂²⋊C₄	—	271	2	no	—
536	C₂²⋊C₄	—	271	2	no	—
537	C₂²⋊C₄	—	271	2	no	—
538	C₂²⋊C₄	—	271	2	no	—
539	C₂²⋊C₄	—	271	2	no	—
540	C₂²⋊C₄	—	271	2	no	—
541	C₂²⋊C₄	—	271	2	no	—
542	C₂²⋊C₄	C₂²×D₂₇₁	1	1	yes	characteristic
543	C₂²⋊C₄	C₂²×D₂₇₁	1	1	yes	characteristic
544	C₂²⋊C₄	C₂²×D₂₇₁	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₂₇₁	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₂₇₁	1	2	yes	—
547	C₂²⋊C₄	C₂²×D₂₇₁	1	2	yes	—
548	C₂²⋊C₄	C₂²×D₂₇₁	1	2	yes	—
549	C₄²	—	271	1	no	—
550	C₄²	—	271	1	no	—
551	C₄²	—	271	1	no	—
552	C₄²	—	271	1	no	—
553	C₄²	—	271	1	no	—
554	C₄²	—	271	1	no	—
555	C₄²	—	271	2	no	—
556	C₄²	—	271	2	no	—
557	C₄²	—	271	2	no	—
558	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
559	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
560	C₄²	C₂²×D₂₇₁	1	2	yes	—
561	C₂³	—	271	1	no	—
562	C₂³	—	271	1	no	—
563	C₂³	—	271	2	no	—
564	C₂³	—	542	2	no	—
565	C₂³	—	542	4	no	—
566	C₂³	—	542	4	no	—
567	C₂³	—	542	2	no	—
568	C₂³	—	542	2	no	—
569	C₂³	—	542	4	no	—
570	C₂³	C₂³×D₂₇₁	1	1	yes	characteristic
571	C₂³	C₂³×D₂₇₁	1	2	yes	—
572	Q₈	—	2	2	no	—
573	Q₈	—	4	2	no	—
574	Q₈	—	542	4	no	—
575	Q₈	—	542	2	no	—
576	Q₈	—	1084	2	no	—
577	Q₈	—	1084	2	no	—
578	Q₈	—	1084	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	4	2	no	—
581	D₄	—	542	4	no	—
582	D₄	—	542	2	no	—
583	D₄	—	1084	2	no	—
584	D₄	—	1084	2	no	—
585	D₄	—	1084	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	271	1	no	—
589	C₂×C₄	—	271	4	no	—
590	C₂×C₄	—	271	1	no	—
591	C₂×C₄	—	271	1	no	—
592	C₂×C₄	—	271	2	no	—
593	C₂×C₄	—	271	1	no	—
594	C₂×C₄	—	271	1	no	—
595	C₂×C₄	—	271	2	no	—
596	C₂×C₄	—	271	1	no	—
597	C₂×C₄	—	271	2	no	—
598	C₂×C₄	—	542	2	no	—
599	C₂×C₄	—	542	2	no	—
600	C₂×C₄	—	542	2	no	—
601	C₂×C₄	—	542	4	no	—
602	C₂×C₄	—	542	4	no	—
603	C₂×C₄	—	542	4	no	—
604	C₂×C₄	—	542	2	no	—
605	C₂×C₄	—	542	2	no	—
606	C₂×C₄	—	542	2	no	—
607	C₂×C₄	—	542	2	no	—
608	C₂×C₄	—	542	2	no	—
609	C₂×C₄	—	1084	2	no	—
610	C₂×C₄	—	1084	2	no	—
611	C₂×C₄	—	1084	2	no	—
612	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
613	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
614	C₂×C₄	C₂³×D₂₇₁	1	2	yes	—
615	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
616	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
617	C₂×C₄	C₂³×D₂₇₁	1	2	yes	—
618	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
619	C₂×C₄	C₂³×D₂₇₁	1	2	yes	—
620	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
621	C₂²	—	2	1	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	271	2	no	—
628	C₂²	—	271	2	no	—
629	C₂²	—	271	2	no	—
630	C₂²	—	542	1	no	—
631	C₂²	—	542	2	no	—
632	C₂²	—	542	2	no	—
633	C₂²	—	542	1	no	—
634	C₂²	—	542	1	no	—
635	C₂²	—	542	2	no	—
636	C₂²	—	1084	4	no	—
637	C₂²	—	1084	8	no	—
638	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
639	C₄	—	2	1	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	1	no	—
645	C₄	—	2	1	no	—
646	C₄	—	2	1	no	—
647	C₄	—	271	2	no	—
648	C₄	—	542	1	no	—
649	C₄	—	542	2	no	—
650	C₄	—	542	1	no	—
651	C₄	—	542	2	no	—
652	C₄	—	542	2	no	—
653	C₄	—	542	1	no	—
654	C₄	—	542	1	no	—
655	C₄	—	542	1	no	—
656	C₄	#41×D₂₇₁	1	2	yes	—
657	C₂	—	4	1	no	—
658	C₂	—	4	2	no	—
659	C₂	—	271	4	no	—
660	C₂	—	1084	1	no	—
661	C₂	—	1084	2	no	—
662	C₂	C₂×#41×D₂₇₁	1	1	yes	characteristic
663	C₂	H(5,𝔽₂)×D₂₇₁	1	1	yes	characteristic
664	C₂	2₋¹⁺⁴×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₇₁
commutator, socle, cosocle	C₂²×C₂₇₁	C₂⁵
Fitting	#542×C₂₇₁	C₂
prime radical	#121×D₂₇₁	C₂
hypercenter	#542	D₂₇₁
hypocenter	C₂₇₁	C₂×#542
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ #542

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #31774521	C₂	semidirect
2. #31774520	C₂	semidirect
3. #31774519	C₂	semidirect
4. #31774518	C₂	semidirect
5. #31774517	C₂	semidirect
6. #31774516	C₂	semidirect
7. #31774515	C₂	semidirect
8. #31774514	C₂	semidirect
9. #31774513	C₂	semidirect
10. #31774512	C₂	semidirect
11. #31774511	C₂	semidirect
12. C₄⋊Q₈×D₂₇₁	C₂	semidirect
13. #126×D₂₇₁	C₂	semidirect
14. #124×D₂₇₁	C₂	semidirect
15. #124×D₂₇₁	C₂	semidirect
16. #123×D₂₇₁	C₂	semidirect
17. C₂²⋊Q₈×D₂₇₁	C₂	semidirect
18. C₂²⋊Q₈×D₂₇₁	C₂	semidirect
20. C₄×Q₈×D₂₇₁	C₂	semidirect
21. C₄×D₄×D₂₇₁	C₂	semidirect
22. #117×D₂₇₁	C₂	semidirect
23. #542×C₂₇₁	C₂	semidirect
30. #22576160	C₂²	semidirect
31. #22576159	C₂²	semidirect
32. #22576159	C₂²	semidirect
33. #22576159	C₂²	semidirect
34. #22576158	C₂²	semidirect
35. #22576155	C₂²	semidirect
36. #22576155	C₂²	semidirect
37. #22576155	C₂²	semidirect
38. #22576154	C₂²	semidirect
39. #22576153	C₂²	semidirect
40. #22576153	C₂²	semidirect
41. #22576153	C₂²	semidirect
42. #22576149	C₂²	semidirect
43. #22576149	C₂²	semidirect
44. #22576148	C₂²	semidirect
45. #22576148	C₂²	semidirect
46. #22576147	C₂²	semidirect
47. #22576147	C₂²	semidirect
48. #22576146	C₂²	semidirect
49. #22576146	C₂²	semidirect
50. #22576146	C₂²	semidirect
51. #22576146	C₂²	semidirect
52. #22576146	C₂²	semidirect
53. #22576146	C₂²	semidirect
54. #22576145	C₂²	semidirect
55. #22576145	C₂²	semidirect
56. #22576144	C₂²	semidirect
57. #22576143	C₂²	semidirect
58. #22576142	C₂²	semidirect
59. #22576142	C₂²	semidirect
60. #22576141	C₂²	semidirect
61. #22576140	C₂²	semidirect
62. #22576140	C₂²	semidirect
63. #22576139	C₂²	semidirect
64. #22576139	C₂²	semidirect
65. #22576138	C₂²	semidirect
66. #22576138	C₂²	semidirect
67. #22576138	C₂²	semidirect
68. #22576137	C₂²	semidirect
69. #22576137	C₂²	semidirect
70. #22576137	C₂²	semidirect
71. #22576136	C₂²	semidirect
72. #22576136	C₂²	semidirect
78. Q₈×Dic₂₇₁	C₂²	semidirect
79. C₄×Dic₅₄₂	C₂²	semidirect
80. C₄×Dic₅₄₂	C₂²	semidirect
81. #22576125	C₂²	semidirect
82. #22576125	C₂²	semidirect
83. C₄×D₁₀₈₄	C₂²	semidirect
84. D₄×Dic₂₇₁	C₂²	semidirect
85. #22576122	C₂²	semidirect
86. C₄×C₂²⋊D₂₇₁	C₂²	semidirect
87. (C₂²⋊D₂₇₁)⋊C₄	C₂²	semidirect
88. #22576119	C₂²	semidirect
89. #22576118	C₂²	semidirect
90. #22576118	C₂²	semidirect
91. #22576117	C₂²	semidirect
92. #22576117	C₂²	semidirect
93. #22576116	C₂²	semidirect
94. #22576116	C₂²	semidirect
112. C₄⋊Q₈×C₂₇₁	C₂²	semidirect
113. #126×C₂₇₁	C₂²	semidirect
114. #124×C₂₇₁	C₂²	semidirect
115. #124×C₂₇₁	C₂²	semidirect
116. #123×C₂₇₁	C₂²	semidirect
117. C₂²⋊Q₈×C₂₇₁	C₂²	semidirect
118. C₂²⋊Q₈×C₂₇₁	C₂²	semidirect
120. C₄×Q₈×C₂₇₁	C₂²	semidirect
121. C₄×D₄×C₂₇₁	C₂²	semidirect
122. #117×C₂₇₁	C₂²	semidirect
331. D₂₇₁	#542	direct (N × H)
332. C₂₇₁	C₂×#542	semidirect
345. #542	D₂₇₁	direct (N × H)
368. C₄⋊Q₈	C₂×D₂₇₁	semidirect
376. #126	C₂×D₂₇₁	semidirect
389. #124	C₂×D₂₇₁	semidirect
390. #124	C₂×D₂₇₁	semidirect
398. #123	C₂×D₂₇₁	semidirect
411. C₂²⋊Q₈	C₂×D₂₇₁	semidirect
412. C₂²⋊Q₈	C₂×D₂₇₁	semidirect
424. C₄×Q₈	C₂×D₂₇₁	semidirect
430. C₄×D₄	C₂×D₂₇₁	semidirect
438. #117	C₂×D₂₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#31774521	2
#31774520	2
#31774519	2
#31774518	2
#31774517	2
#31774516	2
#31774515	2
#31774514	2
#31774513	2
#31774512	2
#31774511	2
C₄⋊Q₈×D₂₇₁	2
#126×D₂₇₁	2
#124×D₂₇₁	2
#123×D₂₇₁	2
C₂²⋊Q₈×D₂₇₁	2
#121×D₂₇₁	2
C₄×Q₈×D₂₇₁	2
C₄×D₄×D₂₇₁	2
#117×D₂₇₁	2
#542×C₂₇₁	2
C₂×#542	271

Subgroups — #542×D271 #46849734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #542×D₂₇₁ #46849734