Subgroups — #32155274

Order 18176.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601350	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590578	C₂	1	2	yes	maximal, complemented
4	#22590564	C₂	1	1	yes	maximal, characteristic
5	#22590492	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590476	C₂	1	2	yes	maximal, complemented
7	#22590404	C₂	1	2	yes	maximal, complemented
8	#22590380	C₂	1	1	yes	maximal, characteristic
9	#22590248	C₂	1	1	yes	maximal, characteristic, complemented
10	#22590207	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986734	—	2	1	no	—
13	Q₈×Dic₁₄₂	—	2	1	no	—
14	C₂×#14556071	C₂²	1	2	yes	—
15	#17986404	C₂²	1	1	yes	characteristic
16	#17986403	C₂²	1	2	yes	—
17	#17986384	C₂²	1	2	yes	—
18	#17986376	C₂²	1	1	yes	characteristic
19	#17986294	—	2	1	no	—
20	#17986292	—	2	2	no	—
21	#17986217	—	2	2	no	—
22	#17986199	—	2	1	no	—
23	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₇₁	C₂²	1	2	yes	—
25	#17986081	C₂²	1	2	yes	—
26	#17985824	C₂²	1	2	yes	—
27	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
28	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
29	#17985803	C₂²	1	2	yes	—
30	#17985801	C₂²	1	2	yes	—
31	#17985801	C₂²	1	2	yes	—
32	#17985799	C₂²	1	2	yes	—
33	#17985774	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
37	#366×C₇₁	C₂²	1	1	yes	characteristic
38	#358×C₇₁	C₂²	1	1	yes	characteristic
39	#358×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₂×#11765852	—	2	2	no	—
42	C₂×D₅₆₈	—	2	1	no	—
43	C₂×D₅₆₈	C₂³	1	2	yes	—
44	#14556128	—	2	1	no	—
45	#14556128	—	4	1	no	—
46	#14556127	—	2	2	no	—
47	#14556127	—	4	2	no	—
48	#14556125	—	2	1	no	—
49	#14556117	—	2	2	no	—
50	#14556116	—	4	1	no	—
51	#14556113	—	4	2	no	—
52	Q₈×Dic₇₁	—	2	2	no	—
53	C₄×Dic₁₄₂	—	4	1	no	—
54	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
55	C₄×D₂₈₄	—	4	1	no	—
56	#14556090	—	2	2	no	—
57	#14556071	—	2	4	no	—
58	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
59	#14556062	—	2	1	no	—
60	#14556062	C₂³	1	2	yes	—
61	#14556061	—	2	2	no	—
62	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
63	Dic₇₁⋊C₈	D₄	1	2	yes	—
64	#14556052	—	2	2	no	—
65	#14556048	—	2	2	no	—
66	C₈×Dic₇₁	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₇₁	—	2	1	no	—
70	C₄×Q₈×C₇₁	—	2	1	no	—
71	Q₃₂×C₇₁	—	2	2	no	—
72	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
74	#107×C₇₁	C₂³	1	1	yes	characteristic
75	#107×C₇₁	C₂³	1	1	yes	characteristic
76	#103×C₇₁	—	2	1	no	—
77	#103×C₇₁	—	2	1	no	—
78	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₂	—	4	1	no	—
80	C₂×Dic₁₄₂	—	4	2	no	—
81	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
82	C₂×D₂₈₄	—	2	2	no	—
83	C₂×D₂₈₄	—	4	1	no	—
84	C₂×D₂₈₄	—	4	1	no	—
85	C₂×C₄×D₇₁	—	4	2	no	—
86	#11765852	—	8	2	no	—
87	D₅₆₈	—	2	2	no	—
88	D₅₆₈	—	2	2	no	—
89	D₅₆₈	—	8	1	no	—
90	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₇₁	—	2	2	no	—
92	C₄⋊Dic₇₁	—	4	1	no	—
93	C₄⋊Dic₇₁	—	4	2	no	—
94	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
95	Dic₇₁⋊C₄	—	4	2	no	—
96	Dic₇₁⋊C₄	D₈	1	2	yes	—
97	(C₂×D₇₁)⋊C₄	—	4	2	no	—
98	(C₂×D₇₁)⋊C₄	—	4	2	no	—
99	(C₂×D₇₁)⋊C₄	—	4	2	no	—
100	C₄×Dic₇₁	—	4	2	no	—
101	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
102	C₇₁⋊C₁₆	—	2	2	no	—
103	C₂×Q₈×C₇₁	—	2	1	no	—
104	C₂×Q₈×C₇₁	—	2	1	no	—
105	Q₁₆×C₇₁	—	2	1	no	—
106	Q₁₆×C₇₁	—	2	1	no	—
107	Q₁₆×C₇₁	—	2	1	no	—
108	Q₁₆×C₇₁	#41	1	2	yes	—
109	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₁	—	2	1	no	—
112	C₄⋊C₄×C₇₁	—	2	1	no	—
113	C₄⋊C₄×C₇₁	—	4	1	no	—
114	C₄⋊C₄×C₇₁	—	4	1	no	—
115	C₄²×C₇₁	—	4	1	no	—
116	C₄²×C₇₁	—	4	1	no	—
117	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₁	—	2	1	no	—
119	C₁₆×C₇₁	—	2	1	no	—
120	C₂²×D₇₁	—	4	2	no	—
121	Q₈×C₇₁	—	2	2	no	—
122	Q₈×C₇₁	—	4	1	no	—
123	Q₈×C₇₁	—	4	1	no	—
124	Q₈×C₇₁	—	4	1	no	—
125	C₂×C₄×C₇₁	—	4	1	no	—
126	C₂×C₄×C₇₁	—	4	1	no	—
127	C₂×C₄×C₇₁	—	4	1	no	—
128	C₂×C₄×C₇₁	—	4	1	no	—
129	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₇₁	—	4	2	no	—
133	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
134	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
135	D₂₈₄	—	4	2	no	—
136	D₂₈₄	—	4	2	no	—
137	D₂₈₄	—	8	2	no	—
138	D₂₈₄	—	16	1	no	—
139	C₄×D₇₁	—	16	2	no	—
140	Dic₁₄₂	—	2	2	no	—
141	Dic₁₄₂	—	16	2	no	—
142	Dic₁₄₂	—	16	1	no	—
143	C₈×C₇₁	—	2	1	no	—
144	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
145	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
146	C₇₁⋊C₈	—	2	2	no	—
147	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₇₁	—	8	2	no	—
149	C₂×D₇₁	—	8	2	no	—
150	C₂×D₇₁	—	8	2	no	—
151	C₄×C₇₁	—	2	1	no	—
152	C₄×C₇₁	—	4	2	no	—
153	C₄×C₇₁	—	8	1	no	—
154	C₄×C₇₁	—	8	1	no	—
155	C₄×C₇₁	—	8	1	no	—
156	C₄×C₇₁	#510	1	2	yes	—
157	C₄×C₇₁	#548	1	1	yes	characteristic
158	C₄×C₇₁	#577	1	1	yes	characteristic
159	Dic₇₁	—	2	2	no	—
160	Dic₇₁	—	2	2	no	—
161	Dic₇₁	—	8	2	no	—
162	#33942	—	71	1	no	maximal, Sylow
163	C₂×C₇₁	#3358	1	1	yes	characteristic
164	C₂×C₇₁	C₂×#510	1	1	yes	characteristic
165	C₂×C₇₁	#3415	1	1	yes	characteristic
166	D₇₁	—	16	2	no	—
167	#3396	—	71	1	no	—
168	Q₈×Q₁₆	—	71	1	no	—
169	#2243	—	71	2	no	—
170	#2240	—	71	1	no	—
171	#2227	—	71	1	no	—
172	#2224	—	71	2	no	—
173	#2211	—	71	2	no	—
174	#2207	—	71	1	no	—
175	#2183	—	71	1	no	—
176	#2175	—	71	2	no	—
177	C₄×Q₃₂	D₇₁	1	1	yes	characteristic, complemented
178	C₇₁	#33942	1	1	yes	characteristic, complemented, Sylow
179	#559	—	142	1	no	—
180	Q₈²	—	142	1	no	—
181	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
182	C₂×SD₃₂	—	71	2	no	—
183	#500	—	71	2	no	—
184	#500	—	71	1	no	—
185	#495	—	71	2	no	—
186	#493	—	71	1	no	—
187	#477	—	142	2	no	—
188	#477	—	142	1	no	—
189	#463	—	142	2	no	—
190	#460	—	142	1	no	—
191	C₈×Q₈	—	71	1	no	—
192	C₄×Q₁₆	—	71	2	no	—
193	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
194	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
195	C₄×D₈	—	71	2	no	—
196	#366	—	71	2	no	—
197	#366	C₂×D₇₁	1	1	yes	characteristic
198	C₄⋊C₁₆	—	71	2	no	—
199	C₄⋊C₁₆	—	71	1	no	—
200	#358	—	71	2	no	—
201	#358	C₂×D₇₁	1	1	yes	characteristic
202	#358	C₂×D₇₁	1	1	yes	characteristic
203	#357	—	71	2	no	—
204	#357	—	71	2	no	—
205	#357	—	71	2	no	—
206	#346	—	71	2	no	—
207	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
208	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
209	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
210	C₂×SD₁₆	—	142	2	no	—
211	C₂×D₈	—	71	2	no	—
212	C₂×D₈	—	142	1	no	—
213	C₄⋊Q₈	—	284	1	no	—
214	C₄⋊Q₈	—	284	2	no	—
215	C₄⋊Q₈	—	142	2	no	—
216	C₄⋊Q₈	—	142	1	no	—
217	#127	—	142	1	no	—
218	#124	—	284	1	no	—
219	#124	—	142	2	no	—
220	#123	—	284	2	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	284	1	no	—
224	C₄×Q₈	—	71	1	no	—
225	C₄×Q₈	—	142	2	no	—
226	C₄×D₄	—	284	1	no	—
227	C₄×D₄	—	142	2	no	—
228	Q₃₂	—	2	2	no	—
229	SD₃₂	—	142	4	no	—
230	C₂×C₁₆	—	71	2	no	—
231	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
233	#107	—	71	2	no	—
234	#107	—	142	2	no	—
235	#107	—	142	1	no	—
236	#107	C₂²×D₇₁	1	1	yes	characteristic
237	#107	C₂²×D₇₁	1	1	yes	characteristic
238	C₄⋊C₈	—	71	2	no	—
239	C₄⋊C₈	—	71	1	no	—
240	#103	—	2	1	no	—
241	#103	—	2	1	no	—
242	#103	—	142	2	no	—
243	#102	—	142	2	no	—
244	C₄×C₈	—	71	2	no	—
245	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	284	2	no	—
249	C₂×Q₈	—	284	1	no	—
250	C₂×Q₈	—	71	1	no	—
251	C₂×D₄	—	284	1	no	—
252	C₂×D₄	—	284	1	no	—
253	C₂×D₄	—	142	2	no	—
254	C₂²×C₄	—	284	2	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	Q₁₆	#11765865	1	2	yes	—
259	SD₁₆	—	568	2	no	—
260	D₈	—	568	1	no	—
261	D₈	—	142	2	no	—
262	D₈	—	142	2	no	—
263	C₂×C₈	—	71	2	no	—
264	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	284	2	no	—
271	C₄⋊C₄	—	284	2	no	—
272	C₄⋊C₄	—	284	1	no	—
273	C₄⋊C₄	—	71	2	no	—
274	C₄⋊C₄	—	71	1	no	—
275	C₄⋊C₄	—	142	2	no	—
276	C₂²⋊C₄	—	284	2	no	—
277	C₂²⋊C₄	—	284	2	no	—
278	C₂²⋊C₄	—	284	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	284	2	no	—
282	C₄²	—	71	2	no	—
283	C₄²	D₄×D₇₁	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	142	2	no	—
287	C₂³	—	284	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	1136	1	no	—
293	Q₈	—	1136	2	no	—
294	Q₈	—	142	2	no	—
295	D₄	—	284	2	no	—
296	D₄	—	284	2	no	—
297	D₄	—	568	2	no	—
298	D₄	—	1136	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	284	2	no	—
304	C₂×C₄	—	71	2	no	—
305	C₂×C₄	—	71	2	no	—
306	C₂×C₄	—	1136	2	no	—
307	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
308	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	142	2	no	—
312	C₈	C₂×#11765865	1	1	yes	characteristic
313	C₈	#14556175	1	1	yes	characteristic
314	C₂²	—	568	2	no	—
315	C₂²	—	568	2	no	—
316	C₂²	—	568	2	no	—
317	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	568	2	no	—
324	C₄	—	142	2	no	—
325	C₄	—	142	2	no	—
326	C₄	#17986441	1	2	yes	—
327	C₄	#17986658	1	1	yes	characteristic
328	C₄	#17986806	1	1	yes	characteristic
329	C₂	—	1136	2	no	—
330	C₂	#22600973	1	1	yes	characteristic
331	C₂	C₂×#17986441	1	1	yes	characteristic
332	C₂	#22601478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₁
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₄×Q₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	#17986376	C₂²
hypercenter	C₄×Q₃₂	D₇₁
hypocenter	C₇₁	#33942
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₄₂	C₂	semidirect
3. #22590578	C₂	semidirect
5. #22590492	C₂	semidirect
6. #22590476	C₂	semidirect
7. #22590404	C₂	semidirect
9. #22590248	C₂	semidirect
10. #22590207	C₂	semidirect
11. C₄×Q₃₂×C₇₁	C₂	semidirect
177. C₄×Q₃₂	D₇₁	semidirect
178. C₇₁	#33942	semidirect

Maximal subgroups

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

subgroup	index
#22601350	2
Q₁₆×Dic₁₄₂	2
#22590578	2
#22590564	2
#22590492	2
#22590476	2
#22590404	2
#22590380	2
#22590248	2
#22590207	2
C₄×Q₃₂×C₇₁	2
#33942	71