Subgroups — C₂×#22586435 #31912344

Order 18176.

590 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556064	C₂	1	1	yes	maximal, characteristic
3	#22586435	C₂	1	4	yes	maximal, direct product factor
4	C₂×#427×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11765849	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	—	2	2	no	—
7	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
8	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
9	C₂²×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
10	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#14556064	—	2	1	no	—
12	C₂×#14556064	—	2	2	no	—
13	C₂×#14556064	C₄	1	4	yes	complemented
14	C₂×#14556064	C₂²	1	2	yes	—
15	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#117×C₇₁	—	2	1	no	—
17	#427×C₇₁	—	2	1	no	—
18	#427×C₇₁	—	2	1	no	—
19	#427×C₇₁	C₄	1	2	yes	—
20	#427×C₇₁	C₂²	1	2	yes	—
21	C₂×#107×C₇₁	C₄	1	2	yes	—
22	C₂×#106×C₇₁	—	2	1	no	—
23	C₂×#11765849	—	2	4	no	—
24	C₂×#11765849	—	2	2	no	—
25	C₂×#11765849	—	4	1	no	—
26	C₂×#11765849	—	4	2	no	—
27	C₂²×C₇₁⋊C₈	—	4	1	no	—
28	C₂×C₇₁⋊C₁₆	—	2	4	no	—
29	C₂×C₇₁⋊C₁₆	—	2	4	no	—
30	C₂×C₇₁⋊C₁₆	—	2	2	no	—
31	C₂×C₇₁⋊C₁₆	—	2	2	no	—
32	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	complemented
33	#14556064	—	2	4	no	—
34	#14556064	—	2	2	no	—
35	#14556064	C₂×C₄	1	4	yes	complemented
36	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	D₄	1	2	yes	—
43	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	#117×C₇₁	—	2	2	no	—
48	#117×C₇₁	—	2	2	no	—
49	#117×C₇₁	—	4	1	no	—
50	#117×C₇₁	—	4	1	no	—
51	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
53	C₂×C₄²×C₇₁	—	4	1	no	—
54	#107×C₇₁	—	2	2	no	—
55	#107×C₇₁	C₂×C₄	1	4	yes	—
56	#106×C₇₁	—	2	2	no	—
57	#106×C₇₁	—	2	2	no	—
58	#11765849	—	2	4	no	—
59	#11765849	—	4	4	no	—
60	#11765849	—	4	2	no	—
61	C₂×C₇₁⋊C₈	—	4	4	no	—
62	C₂×C₇₁⋊C₈	—	4	2	no	—
63	C₇₁⋊C₁₆	—	2	4	no	—
64	C₂⁴×C₇₁	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	—	4	2	no	—
66	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₁	C₄²	1	2	yes	—
77	C₂×C₈×C₇₁	C₄²	1	2	yes	—
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
94	C₄⋊C₄×C₇₁	—	2	4	no	—
95	C₄⋊C₄×C₇₁	—	4	2	no	—
96	C₂²⋊C₄×C₇₁	—	4	2	no	—
97	C₂²⋊C₄×C₇₁	—	4	2	no	—
98	C₄²×C₇₁	—	4	2	no	—
99	C₄²×C₇₁	—	4	2	no	—
100	C₂³×C₇₁	—	2	4	no	—
101	C₂³×C₇₁	#102	1	1	yes	characteristic
102	C₂³×C₇₁	#102	1	1	yes	characteristic
103	C₂³×C₇₁	#103	1	2	yes	—
104	C₂³×C₇₁	#106	1	2	yes	—
105	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₁	—	4	4	no	—
107	C₂×C₄×C₇₁	—	8	4	no	—
108	C₂×C₄×C₇₁	#95	1	2	yes	—
109	C₂×C₄×C₇₁	#95	1	2	yes	—
110	C₂×C₄×C₇₁	#102	1	2	yes	—
111	C₂×C₄×C₇₁	#102	1	2	yes	—
112	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#103	1	2	yes	—
115	C₂×C₄×C₇₁	#103	1	2	yes	—
116	C₂×C₄×C₇₁	#103	1	2	yes	—
117	C₂×C₄×C₇₁	#106	1	2	yes	—
118	C₂×C₄×C₇₁	#107	1	2	yes	—
119	C₂×C₄×C₇₁	#107	1	2	yes	—
120	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
126	C₈×C₇₁	#95	1	2	yes	—
127	C₈×C₇₁	#95	1	2	yes	—
128	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₇₁⋊C₈	—	4	4	no	—
133	C₂²×C₇₁	—	2	2	no	—
134	C₂²×C₇₁	—	2	2	no	—
135	C₂²×C₇₁	—	2	4	no	—
136	C₂²×C₇₁	—	2	4	no	—
137	C₂²×C₇₁	#340	1	2	yes	—
138	C₂²×C₇₁	#340	1	2	yes	—
139	C₂²×C₇₁	#359	1	4	yes	—
140	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
141	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
142	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
143	C₄×C₇₁	—	8	4	no	—
144	C₄×C₇₁	#340	1	2	yes	—
145	C₄×C₇₁	#340	1	2	yes	—
146	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
147	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
148	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
149	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
150	C₂×#1382	—	71	1	no	maximal, Sylow
151	C₂×C₇₁	—	2	2	no	—
152	C₂×C₇₁	—	2	1	no	—
153	C₂×C₇₁	—	2	1	no	—
154	C₂×C₇₁	#1382	1	4	yes	complemented
155	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
156	C₂×C₇₁	C₂×#359	1	2	yes	—
157	C₂³×C₁₆	—	71	1	no	—
158	C₂²×#108	—	71	1	no	—
159	C₂×#427	D₇₁	1	1	yes	characteristic
160	#1382	—	71	4	no	—
161	C₇₁	C₂×#1382	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	142	1	no	—
163	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
164	C₂×#117	—	2	1	no	—
165	C₂²×C₁₆	—	71	4	no	—
166	C₂²×C₁₆	—	71	2	no	—
167	C₂²×C₁₆	—	142	2	no	—
168	C₂²×C₁₆	—	142	1	no	—
169	C₂²×C₁₆	—	142	1	no	—
170	C₂×#108	—	71	4	no	—
171	C₂×#108	—	71	2	no	—
172	C₂×#108	—	142	2	no	—
173	C₂×#108	—	142	1	no	—
174	#427	—	2	1	no	—
175	#427	—	2	1	no	—
176	#427	Dic₇₁	1	2	yes	—
177	#427	C₂×D₇₁	1	2	yes	—
178	C₂×#107	Dic₇₁	1	2	yes	—
179	C₂×#106	—	2	1	no	—
180	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
181	C₂×M₁₆	—	284	1	no	—
182	C₂×M₁₆	—	284	2	no	—
183	C₂×M₁₆	—	142	4	no	—
184	C₂×M₁₆	—	142	2	no	—
185	C₂²×C₈	—	284	1	no	—
186	C₂²×C₈	Dic₁₄₂	1	2	yes	—
187	C₂²×C₈	C₄×D₇₁	1	2	yes	—
188	C₂²×C₈	C₄×D₇₁	1	2	yes	—
189	C₂²×C₈	D₂₈₄	1	2	yes	—
190	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	4	1	no	—
199	#117	—	4	1	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₄²	—	4	1	no	—
203	C₂×C₁₆	—	71	4	no	—
204	C₂×C₁₆	—	142	4	no	—
205	C₂×C₁₆	—	142	4	no	—
206	C₂×C₁₆	—	142	2	no	—
207	C₂×C₁₆	—	142	2	no	—
208	#108	—	71	4	no	—
209	#108	—	142	4	no	—
210	#108	—	142	2	no	—
211	#107	—	2	2	no	—
212	#107	C₂×Dic₇₁	1	4	yes	—
213	#106	—	2	2	no	—
214	#106	—	2	2	no	—
215	C₂⁴	#11765853	1	1	yes	characteristic
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
218	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
220	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
221	C₂²×C₄	#11765850	1	1	yes	characteristic
222	C₂²×C₄	#11765850	1	1	yes	characteristic
223	C₂²×C₄	#11765852	1	2	yes	—
224	C₂²×C₄	#11765853	1	1	yes	characteristic
225	C₂²×C₄	#11765855	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
227	M₁₆	—	284	4	no	—
228	M₁₆	—	284	2	no	—
229	M₁₆	—	142	4	no	—
230	C₂×C₈	—	284	4	no	—
231	C₂×C₈	—	284	2	no	—
232	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
233	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
234	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
235	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
236	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
237	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
239	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
240	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
241	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
242	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
243	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
244	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
248	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
249	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₁₆	—	142	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	#14556049	1	1	yes	characteristic
259	C₂³	#14556049	1	1	yes	characteristic
260	C₂³	#14556051	1	2	yes	—
261	C₂³	#14556059	1	2	yes	—
262	C₂³	C₂×#11765853	1	1	yes	characteristic
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₇₁⋊#95	1	2	yes	—
266	C₂×C₄	C₇₁⋊#95	1	2	yes	—
267	C₂×C₄	#14556048	1	2	yes	—
268	C₂×C₄	#14556048	1	2	yes	—
269	C₂×C₄	#14556049	1	1	yes	characteristic
270	C₂×C₄	#14556049	1	1	yes	characteristic
271	C₂×C₄	#14556051	1	2	yes	—
272	C₂×C₄	#14556052	1	2	yes	—
273	C₂×C₄	#14556052	1	2	yes	—
274	C₂×C₄	#14556059	1	2	yes	—
275	C₂×C₄	#14556061	1	2	yes	—
276	C₂×C₄	#14556061	1	2	yes	—
277	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
281	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
282	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
283	C₈	—	284	4	no	—
284	C₈	C₇₁⋊#95	1	2	yes	—
285	C₈	C₇₁⋊#95	1	2	yes	—
286	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
287	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
288	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
289	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	#17985761	1	2	yes	—
295	C₂²	#17985761	1	2	yes	—
296	C₂²	#17985806	1	4	yes	—
297	C₂²	C₂×#14556049	1	1	yes	characteristic
298	C₂²	C₂×#14556051	1	1	yes	characteristic
299	C₂²	C₂×#14556059	1	1	yes	characteristic
300	C₄	—	8	4	no	—
301	C₄	#17985761	1	2	yes	—
302	C₄	#17985761	1	2	yes	—
303	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
304	C₄	C₂×#14556048	1	1	yes	characteristic
305	C₄	C₂×#14556052	1	1	yes	characteristic
306	C₄	C₂×#14556061	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	#22586435	1	4	yes	direct product factor
311	C₂	C₂×#17985761	1	1	yes	characteristic
312	C₂	C₂×#17985806	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11765850
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#427×C₇₁	C₂
socle	C₂³×C₇₁	#102
prime radical	C₂⁴×C₇₁	SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#427	D₇₁
hypocenter	C₇₁	C₂×#1382
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #22586435	C₂	direct (N × H)
9. C₂²×C₇₁⋊C₁₆	C₄	semidirect
13. C₂×#14556064	C₄	semidirect
32. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
35. #14556064	C₂×C₄	semidirect
154. C₂×C₇₁	#1382	semidirect
161. C₇₁	C₂×#1382	semidirect
310. C₂	#22586435	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₁⋊C₁₆	2
C₂²×#14556064	2
#22586435	2
C₂×#427×C₇₁	2
C₂×#1382	71

Subgroups — C2×#22586435 #31912344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586435 #31912344