Subgroups — #32155379

Order 18176.

484 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅₆₈	C₂	1	1	yes	maximal, characteristic
2	#22590627	C₂	1	2	yes	maximal, complemented
3	#22590613	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590494	C₂	1	2	yes	maximal
5	#22590492	C₂	1	2	yes	maximal, complemented
6	#22590482	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590474	C₂	1	1	yes	maximal, characteristic
8	#22590249	C₂	1	1	yes	maximal, characteristic, complemented
9	#22590202	C₂	1	2	yes	maximal
10	C₄×#14556071	C₂	1	1	yes	maximal, characteristic
11	C₇₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₂₈₄	—	2	1	no	—
13	C₂×#14556071	C₂²	1	1	yes	characteristic
14	#17986404	C₂²	1	1	yes	characteristic
15	#17986404	C₂²	1	2	yes	—
16	#17986280	—	2	1	no	—
17	#17986280	—	2	2	no	—
18	C₄×D₅₆₈	C₂²	1	1	yes	characteristic
19	C₄×D₅₆₈	C₂²	1	2	yes	—
20	#17985826	C₂²	1	2	yes	—
21	#17985826	C₂²	1	1	yes	characteristic
22	#17985824	C₂²	1	2	yes	—
23	#17985824	C₂²	1	2	yes	—
24	#17985824	C₂²	1	2	yes	—
25	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
27	#17985803	C₂²	1	2	yes	—
28	#17985803	C₂²	1	1	yes	characteristic
29	#17985799	C₂²	1	2	yes	—
30	#17985799	C₂²	1	1	yes	characteristic
31	#17985775	C₂²	1	2	yes	—
32	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	#557×C₇₁	—	2	1	no	—
34	#500×C₇₁	C₂²	1	1	yes	characteristic
35	#499×C₇₁	C₂²	1	2	yes	—
36	#479×C₇₁	—	2	2	no	—
37	#475×C₇₁	—	2	1	no	—
38	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
41	C₂×Q₈×D₇₁	—	4	1	no	—
42	C₂×D₅₆₈	C₂³	1	1	yes	characteristic
43	#14556128	—	2	2	no	—
44	#14556128	—	2	1	no	—
45	#14556108	—	4	1	no	—
46	#14556108	—	4	2	no	—
47	C₄×D₂₈₄	—	2	2	no	—
48	C₄×D₂₈₄	—	2	1	no	—
49	#14556071	Q₈	1	4	yes	complemented
50	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
53	#14556062	—	2	2	no	—
54	#14556062	—	2	1	no	—
55	#14556062	C₂³	1	2	yes	—
56	#14556062	C₂³	1	1	yes	characteristic
57	#14556050	—	2	2	no	—
58	#14556050	—	2	1	no	—
59	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₇₁	—	2	1	no	—
61	C₄⋊Q₈×C₇₁	—	4	1	no	—
62	#125×C₇₁	—	2	2	no	—
63	#125×C₇₁	—	4	2	no	—
64	C₄×Q₈×C₇₁	—	2	1	no	—
65	C₄×Q₈×C₇₁	—	2	2	no	—
66	C₄×Q₈×C₇₁	—	4	1	no	—
67	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
68	#107×C₇₁	—	2	1	no	—
69	#107×C₇₁	C₂³	1	2	yes	—
70	#107×C₇₁	C₂³	1	1	yes	characteristic
71	#106×C₇₁	—	2	2	no	—
72	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
73	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
74	#103×C₇₁	—	2	2	no	—
75	#103×C₇₁	—	2	1	no	—
76	C₄×C₈×C₇₁	C₂³	1	2	yes	—
77	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₄₂	—	4	2	no	—
79	C₂×Dic₁₄₂	—	4	1	no	—
80	Q₈×D₇₁	—	8	4	no	—
81	C₂×D₂₈₄	—	2	1	no	—
82	C₂×C₄×D₇₁	—	4	2	no	—
83	C₂×C₄×D₇₁	—	4	1	no	—
84	D₅₆₈	—	2	1	no	—
85	D₅₆₈	C₂×Q₈	1	2	yes	—
86	C₄⋊Dic₇₁	—	2	2	no	—
87	C₄⋊Dic₇₁	—	2	1	no	—
88	C₄⋊Dic₇₁	—	4	2	no	—
89	C₄⋊Dic₇₁	—	4	1	no	—
90	(C₂×D₇₁)⋊C₄	—	4	2	no	—
91	(C₂×D₇₁)⋊C₄	—	4	1	no	—
92	C₇₁⋊C₁₆	—	2	1	no	—
93	C₇₁⋊C₁₆	—	2	2	no	—
94	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	complemented
95	C₂×Q₈×C₇₁	—	2	1	no	—
96	C₂×Q₈×C₇₁	—	4	1	no	—
97	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
98	Q₁₆×C₇₁	—	2	1	no	—
99	Q₁₆×C₇₁	C₂×Q₈	1	2	yes	complemented
100	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	—	2	1	no	—
104	C₄⋊C₄×C₇₁	—	2	2	no	—
105	C₄⋊C₄×C₇₁	—	4	1	no	—
106	C₄⋊C₄×C₇₁	—	4	2	no	—
107	C₄⋊C₄×C₇₁	—	4	2	no	—
108	C₄⋊C₄×C₇₁	—	4	2	no	—
109	C₄⋊C₄×C₇₁	—	4	1	no	—
110	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
111	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
112	C₄²×C₇₁	—	4	2	no	—
113	C₄²×C₇₁	—	4	1	no	—
114	C₄²×C₇₁	C₂×D₄	1	2	yes	—
115	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₇₁	—	4	1	no	—
117	Q₈×C₇₁	—	2	2	no	—
118	Q₈×C₇₁	—	2	2	no	—
119	Q₈×C₇₁	—	4	1	no	—
120	Q₈×C₇₁	—	8	2	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	—	4	1	no	—
123	C₂×C₄×C₇₁	—	4	2	no	—
124	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₇₁	—	4	1	no	—
130	C₂×Dic₇₁	—	4	2	no	—
131	D₂₈₄	—	2	2	no	—
132	D₂₈₄	—	4	1	no	—
133	C₄×D₇₁	—	8	4	no	—
134	C₄×D₇₁	—	8	2	no	—
135	Dic₁₄₂	—	8	4	no	—
136	Dic₁₄₂	—	8	2	no	—
137	C₈×C₇₁	—	2	2	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
140	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₇₁	—	4	2	no	—
143	C₂×D₇₁	—	8	1	no	—
144	C₂×D₇₁	—	8	1	no	—
145	C₄×C₇₁	—	2	2	no	—
146	C₄×C₇₁	—	2	2	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	4	2	no	—
149	C₄×C₇₁	—	8	2	no	—
150	C₄×C₇₁	—	8	1	no	—
151	C₄×C₇₁	#509	1	2	yes	—
152	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
153	C₄×C₇₁	#578	1	1	yes	characteristic
154	Dic₇₁	—	8	1	no	—
155	Dic₇₁	—	8	2	no	—
156	#33953	—	71	1	no	maximal, Sylow
157	C₂×C₇₁	Q₈×D₈	1	1	yes	characteristic
158	C₂×C₇₁	C₂×#509	1	1	yes	characteristic
159	C₂×C₇₁	#3417	1	1	yes	characteristic
160	D₇₁	—	8	2	no	—
161	#3383	D₇₁	1	1	yes	characteristic, complemented
162	Q₈×D₈	—	71	1	no	—
163	#2255	—	71	2	no	—
164	#2252	—	71	1	no	—
165	#2228	—	71	2	no	—
166	#2227	—	71	2	no	—
167	#2225	—	71	1	no	—
168	#2224	—	71	1	no	—
169	#2183	—	71	1	no	—
170	#2175	—	71	2	no	—
171	C₄×SD₃₂	—	71	1	no	—
172	C₇₁	#33953	1	1	yes	characteristic, complemented, Sylow
173	#557	—	2	1	no	—
174	D₄×Q₈	—	142	1	no	—
175	C₂×SD₃₂	—	71	1	no	—
176	#500	—	71	1	no	—
177	#500	—	71	2	no	—
178	#500	C₂×D₇₁	1	1	yes	characteristic
179	#499	C₂×D₇₁	1	2	yes	—
180	#479	—	2	2	no	—
181	#475	—	2	1	no	—
182	#474	—	142	1	no	—
183	#474	—	142	2	no	—
184	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
185	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
186	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
187	C₄×D₈	—	71	1	no	—
188	C₄×D₈	—	71	2	no	—
189	#367	—	71	1	no	—
190	#367	—	71	2	no	—
191	#366	—	71	2	no	—
192	#366	—	71	2	no	—
193	#366	—	71	2	no	—
194	C₄⋊C₁₆	—	71	1	no	—
195	C₄⋊C₁₆	—	71	2	no	—
196	#358	—	71	1	no	—
197	#358	—	71	2	no	—
198	#357	—	71	1	no	—
199	#357	—	71	2	no	—
200	#346	—	71	2	no	—
201	C₄×C₁₆	—	71	1	no	—
202	C₂²×Q₈	—	284	1	no	—
203	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
204	C₂×D₈	—	71	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	4	1	no	—
207	C₄⋊Q₈	—	142	2	no	—
208	C₄⋊Q₈	—	142	1	no	—
209	#125	—	2	2	no	—
210	#125	—	4	2	no	—
211	C₂²⋊Q₈	—	284	1	no	—
212	C₂²⋊Q₈	—	284	2	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
217	C₄×D₄	—	142	2	no	—
218	C₄×D₄	—	142	1	no	—
219	SD₃₂	—	71	4	no	—
220	C₂×C₁₆	—	71	1	no	—
221	C₂×C₁₆	—	71	1	no	—
222	C₂×C₁₆	—	71	2	no	—
223	#107	—	2	1	no	—
224	#107	—	71	1	no	—
225	#107	—	71	2	no	—
226	#107	—	142	2	no	—
227	#107	—	142	1	no	—
228	#107	C₂²×D₇₁	1	2	yes	—
229	#107	C₂²×D₇₁	1	1	yes	characteristic
230	#106	—	2	2	no	—
231	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
232	C₄⋊C₈	C₂²⋊D₇₁	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	2	1	no	—
235	#102	—	142	2	no	—
236	#102	—	142	1	no	—
237	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₇₁	1	2	yes	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	284	2	no	—
242	C₂×Q₈	—	284	1	no	—
243	C₂×Q₈	—	568	4	no	—
244	C₂×Q₈	#11765850	1	1	yes	characteristic
245	C₂×D₄	—	142	1	no	—
246	C₂²×C₄	—	284	2	no	—
247	C₂²×C₄	—	284	1	no	—
248	Q₁₆	—	2	1	no	—
249	Q₁₆	Q₈×D₇₁	1	2	yes	complemented
250	D₈	—	71	2	no	—
251	D₈	—	142	1	no	—
252	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
253	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	284	2	no	—
263	C₄⋊C₄	—	284	1	no	—
264	C₄⋊C₄	—	142	2	no	—
265	C₄⋊C₄	—	142	1	no	—
266	C₄⋊C₄	#11765850	1	1	yes	characteristic
267	C₄⋊C₄	#11765850	1	2	yes	—
268	C₂²⋊C₄	—	284	2	no	—
269	C₂²⋊C₄	—	284	1	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
273	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
274	C₁₆	—	71	2	no	—
275	C₁₆	—	142	1	no	—
276	C₁₆	—	142	2	no	—
277	C₂³	—	284	1	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	8	2	no	—
282	Q₈	—	568	4	no	—
283	Q₈	—	568	2	no	—
284	D₄	—	284	1	no	—
285	D₄	—	142	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	284	2	no	—
290	C₂×C₄	—	284	1	no	—
291	C₂×C₄	—	568	4	no	—
292	C₂×C₄	—	568	2	no	—
293	C₂×C₄	C₂×#11765850	1	2	yes	—
294	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
295	C₂×C₄	C₂×#11765850	1	2	yes	—
296	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
301	C₈	#14556176	1	1	yes	characteristic
302	C₂²	—	284	2	no	—
303	C₂²	—	568	1	no	—
304	C₂²	—	568	1	no	—
305	C₂²	C₂²×#11765850	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	568	2	no	—
313	C₄	—	568	1	no	—
314	C₄	#17986434	1	2	yes	—
315	C₄	Q₈×C₂²⋊D₇₁	1	1	yes	characteristic
316	C₄	#17986814	1	1	yes	characteristic
317	C₂	—	568	2	no	—
318	C₂	Q₈×#11765850	1	1	yes	characteristic
319	C₂	C₂×#17986434	1	1	yes	characteristic
320	C₂	#22601491	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3383	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂×D₅₆₈	C₂³
hypercenter	#3383	D₇₁
hypocenter	C₇₁	#33953
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
2. #22590627	C₂	semidirect
3. #22590613	C₂	semidirect
5. #22590492	C₂	semidirect
6. #22590482	C₂	semidirect
8. #22590249	C₂	semidirect
11. C₇₁×#3383	C₂	semidirect
49. #14556071	Q₈	semidirect
94. C₇₁⋊C₁₆	C₂×Q₈	semidirect
99. Q₁₆×C₇₁	C₂×Q₈	semidirect
161. #3383	D₇₁	semidirect
172. C₇₁	#33953	semidirect
249. Q₁₆	Q₈×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₅₆₈	2
#22590627	2
#22590613	2
#22590494	2
#22590492	2
#22590482	2
#22590474	2
#22590249	2
#22590202	2
C₄×#14556071	2
C₇₁×#3383	2
#33953	71