Subgroups — #31890651

Order 18176.

386 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556064	C₂	1	1	yes	maximal, characteristic, complemented
2	#22587099	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586779	C₂	1	2	yes	maximal
4	#22586231	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2965	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	—	2	1	no	—
7	#17986023	C₂²	1	2	yes	—
8	C₄×#11765849	C₂²	1	2	yes	—
9	#17985764	—	2	1	no	—
10	#17985752	—	2	1	no	—
11	#17985752	—	2	1	no	—
12	Q₈⋊(C₇₁⋊C₈)	—	2	2	no	—
13	D₄⋊(C₇₁⋊C₈)	—	2	2	no	—
14	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
15	#446×C₇₁	C₄	1	1	yes	characteristic
16	#446×C₇₁	C₄	1	1	yes	characteristic
17	#436×C₇₁	C₄	1	1	yes	characteristic
18	#436×C₇₁	C₄	1	1	yes	characteristic
19	#435×C₇₁	—	2	1	no	—
20	#435×C₇₁	—	2	1	no	—
21	#431×C₇₁	—	2	1	no	—
22	#431×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₇₁	—	2	1	no	—
24	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#11765849	—	2	2	no	—
26	C₂×#11765849	—	2	2	no	—
27	#14556064	—	2	4	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
29	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
30	#14556039	—	2	2	no	—
31	C₄×C₇₁⋊C₈	—	2	2	no	—
32	C₂×#41×C₇₁	Q₈	1	1	yes	characteristic
33	C₂×M₁₆×C₇₁	—	2	1	no	—
34	C₂×M₁₆×C₇₁	—	2	1	no	—
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
39	C₄×D₄×C₇₁	—	2	1	no	—
40	C₄×D₄×C₇₁	—	2	1	no	—
41	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
43	#117×C₇₁	—	2	1	no	—
44	#117×C₇₁	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	—	2	1	no	—
46	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₁	—	2	1	no	—
48	C₄⋊C₈×C₇₁	—	2	1	no	—
49	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₇₁	—	2	1	no	—
52	C₂²⋊C₈×C₇₁	—	2	1	no	—
53	C₂²⋊C₈×C₇₁	—	2	1	no	—
54	#97×C₇₁	—	2	1	no	—
55	#97×C₇₁	—	2	1	no	—
56	C₄×C₈×C₇₁	—	2	1	no	—
57	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
61	#11765849	—	2	4	no	—
62	#11765849	—	4	2	no	—
63	#11765849	—	8	2	no	—
64	C₂×C₇₁⋊C₈	—	4	2	no	—
65	C₂×C₇₁⋊C₈	—	4	2	no	—
66	#41×C₇₁	—	4	2	no	—
67	C₂×Q₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₇₁	—	2	1	no	—
69	C₂×D₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	—	2	1	no	—
71	C₂²×C₄×C₇₁	—	2	1	no	—
72	C₂²×C₄×C₇₁	—	2	1	no	—
73	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
76	M₁₆×C₇₁	—	4	2	no	—
77	M₁₆×C₇₁	—	4	2	no	—
78	C₂×C₈×C₇₁	—	2	2	no	—
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	1	no	—
81	C₂×C₈×C₇₁	—	4	1	no	—
82	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₇₁	—	2	1	no	—
87	C₄⋊C₄×C₇₁	—	2	1	no	—
88	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₇₁	—	2	1	no	—
91	C₂²⋊C₄×C₇₁	—	2	1	no	—
92	C₂²⋊C₄×C₇₁	—	2	1	no	—
93	C₄²×C₇₁	—	2	1	no	—
94	C₄²×C₇₁	—	2	1	no	—
95	C₄²×C₇₁	—	2	1	no	—
96	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₁	#41	1	1	yes	characteristic
100	C₂³×C₇₁	—	2	1	no	—
101	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
102	Q₈×C₇₁	—	2	2	no	—
103	D₄×C₇₁	—	2	2	no	—
104	D₄×C₇₁	—	4	2	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	4	2	no	—
114	C₂×C₄×C₇₁	—	4	1	no	—
115	C₂×C₄×C₇₁	—	4	1	no	—
116	C₂×C₄×C₇₁	—	4	1	no	—
117	C₂×C₄×C₇₁	—	4	1	no	—
118	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₇₁	—	2	1	no	—
125	C₈×C₇₁	—	2	1	no	—
126	C₈×C₇₁	—	2	1	no	—
127	C₈×C₇₁	—	4	1	no	—
128	C₈×C₇₁	—	4	1	no	—
129	C₈×C₇₁	#104	1	2	yes	—
130	C₇₁⋊C₈	—	4	4	no	—
131	C₇₁⋊C₈	—	8	2	no	—
132	C₂²×C₇₁	—	2	1	no	—
133	C₂²×C₇₁	—	2	1	no	—
134	C₂²×C₇₁	—	2	1	no	—
135	C₂²×C₇₁	—	4	1	no	—
136	C₂²×C₇₁	—	4	1	no	—
137	C₂²×C₇₁	—	4	1	no	—
138	C₂²×C₇₁	#380	1	1	yes	characteristic
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	—	4	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
146	C₄×C₇₁	#421	1	1	yes	characteristic
147	C₄×C₇₁	#430	1	2	yes	—
148	C₄×C₇₁	#472	1	1	yes	characteristic
149	C₄×C₇₁	#473	1	1	yes	characteristic
150	#8673	—	71	1	no	maximal, Sylow
151	C₂×C₇₁	—	4	1	no	—
152	C₂×C₇₁	—	8	1	no	—
153	C₂×C₇₁	#1806	1	1	yes	characteristic
154	C₂×C₇₁	#1811	1	1	yes	characteristic
155	C₂×C₇₁	#1816	1	1	yes	characteristic
156	#2965	D₇₁	1	1	yes	characteristic
157	C₄×#108	—	71	1	no	—
158	#1567	—	71	1	no	—
159	#1489	—	71	2	no	—
160	#1299	—	71	2	no	—
161	C₇₁	#8673	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
163	#446	Dic₇₁	1	1	yes	characteristic
164	#446	Dic₇₁	1	1	yes	characteristic
165	#436	Dic₇₁	1	1	yes	characteristic
166	#436	Dic₇₁	1	1	yes	characteristic
167	#435	—	2	1	no	—
168	#435	—	2	1	no	—
169	#431	—	2	1	no	—
170	#431	C₂×D₇₁	1	1	yes	characteristic
171	C₂×#108	—	142	1	no	—
172	#423	—	71	2	no	—
173	C₄×M₁₆	—	2	1	no	—
174	C₄×M₁₆	—	71	2	no	—
175	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
176	#341	—	142	1	no	—
177	#335	—	142	1	no	—
178	#335	—	142	1	no	—
179	Q₈⋊C₈	—	142	2	no	—
180	D₄⋊C₈	—	142	2	no	—
181	C₂×#41	Dic₁₄₂	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	142	2	no	—
185	C₂×M₁₆	—	142	2	no	—
186	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
188	C₄×Q₈	D₂₈₄	1	1	yes	characteristic
189	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	D₂₈₄	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₇₁	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	Dic₁₄₂	1	1	yes	characteristic
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
198	#108	—	142	4	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	142	2	no	—
202	C₄⋊C₈	—	142	2	no	—
203	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
204	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	142	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	142	2	no	—
213	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
217	#41	—	4	2	no	—
218	C₂×Q₈	C₄⋊Dic₇₁	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊Dic₇₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
226	C₂²×C₄	#11765865	1	1	yes	characteristic
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	284	2	no	—
230	M₁₆	—	568	2	no	—
231	M₁₆	—	142	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	284	2	no	—
237	C₂×C₈	—	284	2	no	—
238	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
245	C₄⋊C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
254	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
255	C₄²	#11765865	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	#14556107	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
276	C₂×C₄	#14556087	1	1	yes	characteristic
277	C₂×C₄	#14556102	1	1	yes	characteristic
278	C₂×C₄	#14556102	1	1	yes	characteristic
279	C₂×C₄	#14556107	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	284	4	no	—
286	C₈	—	568	2	no	—
287	C₈	#14556055	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	#17985859	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×#14556055	1	1	yes	characteristic
302	C₄	#17986016	1	1	yes	characteristic
303	C₄	#17986048	1	2	yes	—
304	C₄	#17986269	1	1	yes	characteristic
305	C₄	#17986274	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	8	1	no	—
308	C₂	#22588360	1	1	yes	characteristic
309	C₂	#22588382	1	1	yes	characteristic
310	C₂	#22588404	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556102
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2965	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂×#41×C₇₁	Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#2965	D₇₁
hypocenter	C₇₁	#8673
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
1. C₄×#14556064	C₂	semidirect
2. #22587099	C₂	semidirect
4. #22586231	C₂	semidirect
161. C₇₁	#8673	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556064	2
#22587099	2
#22586779	2
#22586231	2
C₇₁×#2965	2
#8673	71