Subgroups — C₂×C₇₁×#3328 #31882020

Order 18176.

2082 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#3328	C₂	1	16	yes	maximal, direct product factor
4	C₂×#487×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#485×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×#475×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#470×C₇₁	C₂	1	2	yes	maximal, complemented
8	C₂×#462×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#451×C₇₁	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#436×C₇₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#419×C₇₁	C₂	1	2	yes	maximal, complemented
13	C₂³×Q₈×C₇₁	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
17	#556×C₇₁	—	2	4	no	—
18	#556×C₇₁	C₂²	1	8	yes	—
19	D₄×Q₈×C₇₁	—	2	4	no	—
20	D₄×Q₈×C₇₁	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×#126×C₇₁	—	2	2	no	—
25	C₂×#124×C₇₁	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₇₁	—	2	2	no	—
27	C₂×C₂²⋊Q₈×C₇₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₇₁	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
30	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₇₁	—	2	1	no	—
33	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
34	C₂×#117×C₇₁	C₂²	1	2	yes	—
35	#487×C₇₁	C₂²	1	8	yes	complemented
36	#485×C₇₁	C₂²	1	16	yes	complemented
37	#475×C₇₁	C₂²	1	8	yes	complemented
38	#470×C₇₁	C₂²	1	16	yes	complemented
39	#462×C₇₁	C₂²	1	8	yes	complemented
40	#451×C₇₁	C₂²	1	16	yes	complemented
41	C₄×Q₁₆×C₇₁	C₂²	1	8	yes	complemented
42	#436×C₇₁	C₂²	1	8	yes	—
43	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
44	C₂×#106×C₇₁	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
46	#419×C₇₁	C₂²	1	16	yes	complemented
47	C₂×#103×C₇₁	C₂²	1	1	yes	characteristic
48	C₂×#103×C₇₁	C₂²	1	2	yes	—
49	C₂×#103×C₇₁	C₂²	1	2	yes	—
50	C₂×#103×C₇₁	C₂²	1	2	yes	—
51	C₂×#103×C₇₁	C₂²	1	2	yes	—
52	C₂×#103×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₇₁	—	2	1	no	—
56	C₂²×Q₈×C₇₁	—	2	2	no	—
57	C₂²×Q₈×C₇₁	—	2	4	no	—
58	C₂²×Q₈×C₇₁	—	2	8	no	—
59	C₂²×Q₈×C₇₁	—	4	4	no	—
60	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
61	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₇₁	C₂³	1	8	yes	—
63	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₇₁	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₇₁	—	2	2	no	—
66	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₇₁	—	2	4	no	—
68	C₂×Q₁₆×C₇₁	—	4	4	no	—
69	C₂×Q₁₆×C₇₁	C₂³	1	4	yes	—
70	C₂×Q₁₆×C₇₁	C₂³	1	8	yes	—
71	C₂×M₁₆×C₇₁	—	4	4	no	—
72	C₂×M₁₆×C₇₁	C₂³	1	8	yes	—
73	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₇₁	—	2	8	no	—
77	C₄⋊Q₈×C₇₁	—	2	2	no	—
78	C₄⋊Q₈×C₇₁	—	2	2	no	—
79	C₄⋊Q₈×C₇₁	C₂³	1	4	yes	—
80	C₄⋊Q₈×C₇₁	C₂³	1	4	yes	—
81	#126×C₇₁	—	2	16	no	—
82	#124×C₇₁	—	2	8	no	—
83	C₂²⋊Q₈×C₇₁	—	2	4	no	—
84	C₂²⋊Q₈×C₇₁	—	2	4	no	—
85	C₂²⋊Q₈×C₇₁	—	2	16	no	—
86	C₂²⋊Q₈×C₇₁	C₂³	1	8	yes	—
87	C₂²⋊Q₈×C₇₁	C₂³	1	8	yes	—
88	C₄×Q₈×C₇₁	—	2	8	no	—
89	C₄×Q₈×C₇₁	—	2	2	no	—
90	C₄×Q₈×C₇₁	—	2	2	no	—
91	C₄×Q₈×C₇₁	C₂³	1	4	yes	—
92	C₄×Q₈×C₇₁	C₂³	1	4	yes	—
93	C₄×D₄×C₇₁	—	2	8	no	—
94	C₄×D₄×C₇₁	D₄	1	4	yes	—
95	C₄×D₄×C₇₁	C₂³	1	4	yes	—
96	#117×C₇₁	—	2	4	no	—
97	#117×C₇₁	C₂³	1	8	yes	—
98	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
113	C₂×C₄²×C₇₁	—	2	2	no	—
114	C₂×C₄²×C₇₁	—	2	1	no	—
115	C₂×C₄²×C₇₁	—	2	1	no	—
116	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
117	#107×C₇₁	C₂³	1	4	yes	—
118	#106×C₇₁	C₂³	1	8	yes	—
119	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
120	#103×C₇₁	C₂³	1	4	yes	—
121	#103×C₇₁	C₂³	1	8	yes	—
122	#103×C₇₁	C₂³	1	8	yes	—
123	#103×C₇₁	C₂³	1	8	yes	—
124	#103×C₇₁	C₂³	1	4	yes	—
125	#103×C₇₁	C₂³	1	8	yes	—
126	C₂²⋊C₈×C₇₁	C₂³	1	8	yes	—
127	C₄×C₈×C₇₁	C₂³	1	4	yes	—
128	C₂⁴×C₇₁	C₂×D₄	1	2	yes	—
129	C₂×Q₈×C₇₁	—	2	1	no	—
130	C₂×Q₈×C₇₁	—	2	1	no	—
131	C₂×Q₈×C₇₁	—	2	4	no	—
132	C₂×Q₈×C₇₁	—	2	8	no	—
133	C₂×Q₈×C₇₁	—	2	2	no	—
134	C₂×Q₈×C₇₁	—	2	2	no	—
135	C₂×Q₈×C₇₁	—	2	16	no	—
136	C₂×Q₈×C₇₁	—	2	2	no	—
137	C₂×Q₈×C₇₁	—	2	4	no	—
138	C₂×Q₈×C₇₁	—	2	2	no	—
139	C₂×Q₈×C₇₁	—	2	4	no	—
140	C₂×Q₈×C₇₁	—	4	8	no	—
141	C₂×Q₈×C₇₁	—	4	8	no	—
142	C₂×Q₈×C₇₁	—	4	16	no	—
143	C₂×Q₈×C₇₁	—	4	4	no	—
144	C₂×Q₈×C₇₁	#41	1	4	yes	—
145	C₂×Q₈×C₇₁	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₇₁	C₂⁴	1	2	yes	—
147	C₂×Q₈×C₇₁	C₂⁴	1	2	yes	—
148	C₂×D₄×C₇₁	—	2	4	no	—
149	C₂×D₄×C₇₁	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₇₁	—	2	1	no	—
151	C₂²×C₄×C₇₁	—	2	4	no	—
152	C₂²×C₄×C₇₁	—	2	4	no	—
153	C₂²×C₄×C₇₁	—	2	8	no	—
154	C₂²×C₄×C₇₁	—	2	1	no	—
155	C₂²×C₄×C₇₁	—	2	1	no	—
156	C₂²×C₄×C₇₁	—	2	1	no	—
157	C₂²×C₄×C₇₁	—	2	2	no	—
158	C₂²×C₄×C₇₁	—	2	2	no	—
159	C₂²×C₄×C₇₁	—	4	8	no	—
160	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₇₁	—	2	8	no	—
167	Q₁₆×C₇₁	—	4	8	no	—
168	M₁₆×C₇₁	—	4	8	no	—
169	C₂×C₈×C₇₁	—	2	2	no	—
170	C₂×C₈×C₇₁	—	2	2	no	—
171	C₂×C₈×C₇₁	—	2	4	no	—
172	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
173	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
174	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₇₁	—	2	4	no	—
176	C₄⋊C₄×C₇₁	—	2	8	no	—
177	C₄⋊C₄×C₇₁	—	2	4	no	—
178	C₄⋊C₄×C₇₁	—	2	4	no	—
179	C₄⋊C₄×C₇₁	—	2	4	no	—
180	C₄⋊C₄×C₇₁	—	2	8	no	—
181	C₄⋊C₄×C₇₁	—	2	4	no	—
182	C₄⋊C₄×C₇₁	—	2	1	no	—
183	C₄⋊C₄×C₇₁	—	2	2	no	—
184	C₄⋊C₄×C₇₁	—	2	1	no	—
185	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
186	C₄⋊C₄×C₇₁	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₇₁	C₂⁴	1	2	yes	—
188	C₄⋊C₄×C₇₁	C₂⁴	1	2	yes	—
189	C₂²⋊C₄×C₇₁	—	2	8	no	—
190	C₂²⋊C₄×C₇₁	—	2	8	no	—
191	C₂²⋊C₄×C₇₁	—	2	8	no	—
192	C₂²⋊C₄×C₇₁	C₂×D₄	1	8	yes	—
193	C₄²×C₇₁	—	2	8	no	—
194	C₄²×C₇₁	—	2	4	no	—
195	C₄²×C₇₁	—	2	4	no	—
196	C₄²×C₇₁	C₂×D₄	1	2	yes	—
197	C₄²×C₇₁	C₂⁴	1	2	yes	—
198	C₂³×C₇₁	—	2	4	no	—
199	C₂³×C₇₁	—	2	4	no	—
200	C₂³×C₇₁	—	2	4	no	—
201	C₂³×C₇₁	C₂²×D₄	1	4	yes	—
202	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₇₁	—	2	4	no	—
204	Q₈×C₇₁	—	2	2	no	—
205	Q₈×C₇₁	—	2	2	no	—
206	Q₈×C₇₁	—	2	4	no	—
207	Q₈×C₇₁	—	2	2	no	—
208	Q₈×C₇₁	—	2	4	no	—
209	Q₈×C₇₁	—	4	16	no	—
210	Q₈×C₇₁	—	4	2	no	—
211	Q₈×C₇₁	—	4	4	no	—
212	Q₈×C₇₁	—	4	8	no	—
213	Q₈×C₇₁	C₂×#41	1	4	yes	—
214	D₄×C₇₁	—	2	8	no	—
215	C₂×C₄×C₇₁	—	2	4	no	—
216	C₂×C₄×C₇₁	—	2	4	no	—
217	C₂×C₄×C₇₁	—	2	2	no	—
218	C₂×C₄×C₇₁	—	2	2	no	—
219	C₂×C₄×C₇₁	—	2	8	no	—
220	C₂×C₄×C₇₁	—	2	4	no	—
221	C₂×C₄×C₇₁	—	2	4	no	—
222	C₂×C₄×C₇₁	—	2	2	no	—
223	C₂×C₄×C₇₁	—	2	2	no	—
224	C₂×C₄×C₇₁	—	2	2	no	—
225	C₂×C₄×C₇₁	—	2	4	no	—
226	C₂×C₄×C₇₁	—	2	4	no	—
227	C₂×C₄×C₇₁	—	4	16	no	—
228	C₂×C₄×C₇₁	—	4	4	no	—
229	C₂×C₄×C₇₁	—	4	2	no	—
230	C₂×C₄×C₇₁	—	4	4	no	—
231	C₂×C₄×C₇₁	—	4	2	no	—
232	C₂×C₄×C₇₁	—	4	2	no	—
233	C₂×C₄×C₇₁	#137	1	4	yes	—
234	C₂×C₄×C₇₁	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
239	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
240	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
241	C₈×C₇₁	—	2	2	no	—
242	C₈×C₇₁	—	2	2	no	—
243	C₈×C₇₁	—	2	4	no	—
244	C₂²×C₇₁	—	2	4	no	—
245	C₂²×C₇₁	—	2	4	no	—
246	C₂²×C₇₁	—	2	4	no	—
247	C₂²×C₇₁	—	4	8	no	—
248	C₂²×C₇₁	#546	1	2	yes	—
249	C₂²×C₇₁	C₂×#137	1	2	yes	—
250	C₂²×C₇₁	#578	1	2	yes	—
251	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₇₁	—	2	4	no	—
253	C₄×C₇₁	—	2	4	no	—
254	C₄×C₇₁	—	2	2	no	—
255	C₄×C₇₁	—	4	4	no	—
256	C₄×C₇₁	—	4	2	no	—
257	C₄×C₇₁	—	4	4	no	—
258	C₄×C₇₁	—	4	2	no	—
259	C₄×C₇₁	—	4	2	no	—
260	C₄×C₇₁	#546	1	2	yes	—
261	C₄×C₇₁	C₂×#137	1	4	yes	—
262	C₄×C₇₁	C₂²×#41	1	1	yes	characteristic
263	C₄×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂×#3328	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂×C₇₁	—	4	4	no	—
266	C₂×C₇₁	#3328	1	4	yes	direct product factor
267	C₂×C₇₁	C₂×#546	1	1	yes	characteristic
268	C₂×C₇₁	C₂²×#137	1	1	yes	characteristic
269	C₂×C₇₁	C₂×#578	1	1	yes	characteristic
270	C₂×#556	C₂×C₇₁	1	1	yes	characteristic
271	C₂×D₄×Q₈	C₂×C₇₁	1	1	yes	characteristic
272	#3328	C₂×C₇₁	1	16	yes	direct product factor
273	C₂×#487	C₂×C₇₁	1	1	yes	characteristic, complemented
274	C₂×#485	C₂×C₇₁	1	2	yes	complemented
275	C₂×#475	C₂×C₇₁	1	1	yes	characteristic, complemented
276	C₂×#470	C₂×C₇₁	1	2	yes	complemented
277	C₂×#462	C₂×C₇₁	1	1	yes	characteristic, complemented
278	C₂×#451	C₂×C₇₁	1	2	yes	complemented
279	C₂×C₄×Q₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
280	C₂×#436	C₂×C₇₁	1	1	yes	characteristic
281	C₂×#419	C₂×C₇₁	1	2	yes	complemented
282	C₇₁	C₂×#3328	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³×Q₈	C₂²×C₇₁	1	2	yes	—
284	C₂²×Q₁₆	C₂²×C₇₁	1	1	yes	characteristic
285	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	—
286	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
287	#556	—	2	4	no	—
288	#556	C₂²×C₇₁	1	8	yes	—
289	D₄×Q₈	—	2	4	no	—
290	D₄×Q₈	C₂²×C₇₁	1	8	yes	—
291	C₂×C₄⋊Q₈	—	2	1	no	—
292	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
293	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
294	C₂×#126	—	2	2	no	—
295	C₂×#124	—	2	1	no	—
296	C₂×C₂²⋊Q₈	—	2	2	no	—
297	C₂×C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
298	C₂×C₂²⋊Q₈	C₂²×C₇₁	1	2	yes	—
299	C₂×C₄×Q₈	—	2	1	no	—
300	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
301	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
302	C₂×C₄×D₄	—	2	1	no	—
303	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
304	C₂×#117	C₂²×C₇₁	1	2	yes	—
305	#487	C₂²×C₇₁	1	8	yes	complemented
306	#485	C₂²×C₇₁	1	16	yes	complemented
307	#475	C₂²×C₇₁	1	8	yes	complemented
308	#470	C₂²×C₇₁	1	16	yes	complemented
309	#462	C₂²×C₇₁	1	8	yes	complemented
310	#451	C₂²×C₇₁	1	16	yes	complemented
311	C₄×Q₁₆	C₂²×C₇₁	1	8	yes	complemented
312	#436	C₂²×C₇₁	1	8	yes	—
313	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
314	C₂×#106	C₂²×C₇₁	1	2	yes	—
315	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
316	#419	C₂²×C₇₁	1	16	yes	complemented
317	C₂×#103	C₂²×C₇₁	1	2	yes	—
318	C₂×#103	C₂²×C₇₁	1	2	yes	—
319	C₂×#103	C₂²×C₇₁	1	2	yes	—
320	C₂×#103	C₂²×C₇₁	1	2	yes	—
321	C₂×#103	C₂²×C₇₁	1	1	yes	characteristic
322	C₂×#103	C₂²×C₇₁	1	1	yes	characteristic
323	C₂×C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
324	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
325	C₂²×Q₈	—	2	1	no	—
326	C₂²×Q₈	—	2	2	no	—
327	C₂²×Q₈	—	2	4	no	—
328	C₂²×Q₈	—	2	8	no	—
329	C₂²×Q₈	—	4	4	no	—
330	C₂²×Q₈	C₂³×C₇₁	1	1	yes	characteristic
331	C₂²×Q₈	C₂³×C₇₁	1	2	yes	—
332	C₂²×Q₈	C₂³×C₇₁	1	8	yes	—
333	C₂²×Q₈	C₂³×C₇₁	1	1	yes	characteristic
334	C₂²×D₄	D₄×C₇₁	1	1	yes	characteristic
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
337	C₂×Q₁₆	—	2	4	no	—
338	C₂×Q₁₆	—	4	4	no	—
339	C₂×Q₁₆	C₂³×C₇₁	1	4	yes	—
340	C₂×Q₁₆	C₂³×C₇₁	1	8	yes	—
341	C₂×M₁₆	—	4	4	no	—
342	C₂×M₁₆	C₂³×C₇₁	1	8	yes	—
343	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
345	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
346	C₄⋊Q₈	—	2	8	no	—
347	C₄⋊Q₈	—	2	2	no	—
348	C₄⋊Q₈	—	2	2	no	—
349	C₄⋊Q₈	C₂³×C₇₁	1	4	yes	—
350	C₄⋊Q₈	C₂³×C₇₁	1	4	yes	—
351	#126	—	2	16	no	—
352	#124	—	2	8	no	—
353	C₂²⋊Q₈	—	2	4	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	16	no	—
356	C₂²⋊Q₈	C₂³×C₇₁	1	8	yes	—
357	C₂²⋊Q₈	C₂³×C₇₁	1	8	yes	—
358	C₄×Q₈	—	2	8	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	C₂³×C₇₁	1	4	yes	—
362	C₄×Q₈	C₂³×C₇₁	1	4	yes	—
363	C₄×D₄	—	2	8	no	—
364	C₄×D₄	D₄×C₇₁	1	4	yes	—
365	C₄×D₄	C₂³×C₇₁	1	4	yes	—
366	#117	—	2	4	no	—
367	#117	C₂³×C₇₁	1	8	yes	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₂²⋊C₄	—	2	2	no	—
381	C₂×C₂²⋊C₄	—	2	2	no	—
382	C₂×C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
383	C₂×C₄²	—	2	2	no	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
387	#107	C₂³×C₇₁	1	4	yes	—
388	#106	C₂³×C₇₁	1	8	yes	—
389	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
390	#103	C₂³×C₇₁	1	4	yes	—
391	#103	C₂³×C₇₁	1	8	yes	—
392	#103	C₂³×C₇₁	1	8	yes	—
393	#103	C₂³×C₇₁	1	8	yes	—
394	#103	C₂³×C₇₁	1	4	yes	—
395	#103	C₂³×C₇₁	1	8	yes	—
396	C₂²⋊C₈	C₂³×C₇₁	1	8	yes	—
397	C₄×C₈	C₂³×C₇₁	1	4	yes	—
398	C₂⁴	C₂×D₄×C₇₁	1	2	yes	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	4	no	—
402	C₂×Q₈	—	2	8	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	16	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	4	8	no	—
411	C₂×Q₈	—	4	8	no	—
412	C₂×Q₈	—	4	16	no	—
413	C₂×Q₈	—	4	4	no	—
414	C₂×Q₈	#41×C₇₁	1	4	yes	—
415	C₂×Q₈	C₂⁴×C₇₁	1	4	yes	—
416	C₂×Q₈	C₂⁴×C₇₁	1	2	yes	—
417	C₂×Q₈	C₂⁴×C₇₁	1	2	yes	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	C₂×D₄×C₇₁	1	4	yes	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	4	8	no	—
430	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
436	Q₁₆	—	2	8	no	—
437	Q₁₆	—	4	8	no	—
438	M₁₆	—	4	8	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	4	no	—
442	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
443	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
444	C₂×C₈	C₂⁴×C₇₁	1	4	yes	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
456	C₄⋊C₄	C₂⁴×C₇₁	1	4	yes	—
457	C₄⋊C₄	C₂⁴×C₇₁	1	2	yes	—
458	C₄⋊C₄	C₂⁴×C₇₁	1	2	yes	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	C₂×D₄×C₇₁	1	8	yes	—
463	C₄²	—	2	8	no	—
464	C₄²	—	2	4	no	—
465	C₄²	—	2	4	no	—
466	C₄²	C₂×D₄×C₇₁	1	2	yes	—
467	C₄²	C₂⁴×C₇₁	1	2	yes	—
468	C₂³	—	2	4	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	C₂²×D₄×C₇₁	1	4	yes	—
472	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
473	Q₈	—	2	4	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	2	2	no	—
476	Q₈	—	2	4	no	—
477	Q₈	—	2	2	no	—
478	Q₈	—	2	4	no	—
479	Q₈	—	4	16	no	—
480	Q₈	—	4	4	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	8	no	—
483	Q₈	C₂×#41×C₇₁	1	4	yes	—
484	D₄	—	2	8	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	#137×C₇₁	1	4	yes	—
504	C₂×C₄	C₂²×D₄×C₇₁	1	4	yes	—
505	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
508	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
509	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
510	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
511	C₈	—	2	2	no	—
512	C₈	—	2	2	no	—
513	C₈	—	2	4	no	—
514	C₂²	—	2	4	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	2	4	no	—
517	C₂²	—	4	8	no	—
518	C₂²	#546×C₇₁	1	2	yes	—
519	C₂²	C₂×#137×C₇₁	1	2	yes	—
520	C₂²	#578×C₇₁	1	2	yes	—
521	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
522	C₄	—	2	2	no	—
523	C₄	—	2	4	no	—
524	C₄	—	2	4	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	2	no	—
530	C₄	#546×C₇₁	1	2	yes	—
531	C₄	C₂×#137×C₇₁	1	4	yes	—
532	C₄	C₂²×#41×C₇₁	1	1	yes	characteristic
533	C₄	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
534	C₂	—	4	4	no	—
535	C₂	C₇₁×#3328	1	4	yes	direct product factor
536	C₂	C₂×#546×C₇₁	1	1	yes	characteristic
537	C₂	C₂²×#137×C₇₁	1	1	yes	characteristic
538	C₂	C₂×#578×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
prime radical	C₂²×D₄×C₇₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄×D₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#3328	C₂	direct (N × H)
4. C₂×#487×C₇₁	C₂	semidirect
5. C₂×#485×C₇₁	C₂	semidirect
6. C₂×#475×C₇₁	C₂	semidirect
7. C₂×#470×C₇₁	C₂	semidirect
8. C₂×#462×C₇₁	C₂	semidirect
9. C₂×#451×C₇₁	C₂	semidirect
10. C₂×C₄×Q₁₆×C₇₁	C₂	semidirect
12. C₂×#419×C₇₁	C₂	semidirect
35. #487×C₇₁	C₂²	semidirect
36. #485×C₇₁	C₂²	semidirect
37. #475×C₇₁	C₂²	semidirect
38. #470×C₇₁	C₂²	semidirect
39. #462×C₇₁	C₂²	semidirect
40. #451×C₇₁	C₂²	semidirect
41. C₄×Q₁₆×C₇₁	C₂²	semidirect
46. #419×C₇₁	C₂²	semidirect
264. C₂×#3328	C₇₁	direct (N × H)
266. C₂×C₇₁	#3328	direct (N × H)
272. #3328	C₂×C₇₁	direct (N × H)
273. C₂×#487	C₂×C₇₁	semidirect
274. C₂×#485	C₂×C₇₁	semidirect
275. C₂×#475	C₂×C₇₁	semidirect
276. C₂×#470	C₂×C₇₁	semidirect
277. C₂×#462	C₂×C₇₁	semidirect
278. C₂×#451	C₂×C₇₁	semidirect
279. C₂×C₄×Q₁₆	C₂×C₇₁	semidirect
281. C₂×#419	C₂×C₇₁	semidirect
282. C₇₁	C₂×#3328	direct (N × H)
305. #487	C₂²×C₇₁	semidirect
306. #485	C₂²×C₇₁	semidirect
307. #475	C₂²×C₇₁	semidirect
308. #470	C₂²×C₇₁	semidirect
309. #462	C₂²×C₇₁	semidirect
310. #451	C₂²×C₇₁	semidirect
311. C₄×Q₁₆	C₂²×C₇₁	semidirect
316. #419	C₂²×C₇₁	semidirect
535. C₂	C₇₁×#3328	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#556×C₇₁	2
C₂×D₄×Q₈×C₇₁	2
C₇₁×#3328	2
C₂×#487×C₇₁	2
C₂×#485×C₇₁	2
C₂×#475×C₇₁	2
C₂×#470×C₇₁	2
C₂×#462×C₇₁	2
C₂×#451×C₇₁	2
C₂×C₄×Q₁₆×C₇₁	2
C₂×#436×C₇₁	2
C₂×#419×C₇₁	2
C₂×#3328	71

Subgroups — C2×C71×#3328 #31882020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#3328 #31882020