Subgroups — #31888500

Order 18176.

392 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597315	C₂	1	1	yes	maximal, characteristic
2	#22588091	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586929	C₂	1	1	yes	maximal, characteristic
4	#22586827	C₂	1	1	yes	maximal, characteristic
5	#22586240	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986533	C₂²	1	1	yes	characteristic
9	#17986129	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic, complemented
11	#17986078	C₄	1	1	yes	characteristic, complemented
12	#17986070	—	2	1	no	—
13	C₈×D₂₈₄	C₄	1	1	yes	characteristic, complemented
14	C₈×C₂²⋊D₇₁	—	2	1	no	—
15	#17986058	—	2	1	no	—
16	#17986056	C₂²	1	1	yes	characteristic
17	#17985953	—	2	1	no	—
18	C₂×#14556039	C₂²	1	1	yes	characteristic
19	#17985845	C₂²	1	1	yes	characteristic
20	#17985753	—	2	1	no	—
21	#17985753	—	2	1	no	—
22	#17985740	C₄	1	2	yes	complemented
23	#17985738	C₄	1	2	yes	complemented
24	#17985737	C₄	1	2	yes	—
25	#17985734	C₄	1	2	yes	—
26	#431×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	#322×C₇₁	—	2	1	no	—
30	#322×C₇₁	—	2	1	no	—
31	C₂×#11765848	—	2	1	no	—
32	C₂²×C₇₁⋊C₈	—	2	1	no	—
33	C₂×C₈×D₇₁	—	2	1	no	—
34	#14556122	—	2	1	no	—
35	#14556120	C₂×C₄	1	1	yes	characteristic
36	#14556116	C₂×C₄	1	1	yes	characteristic
37	#14556111	—	2	1	no	—
38	#14556107	C₂×C₄	1	1	yes	characteristic
39	#14556102	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₇₁	—	2	1	no	—
43	#14556085	—	2	1	no	—
44	C₂×C₄×Dic₇₁	—	2	1	no	—
45	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	Dic₇₁⋊C₈	—	2	1	no	—
48	Dic₇₁⋊C₈	—	2	1	no	—
49	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
50	(C₂×D₇₁)⋊C₈	—	2	1	no	—
51	(C₂×D₇₁)⋊C₈	—	2	1	no	—
52	#14556039	—	2	1	no	—
53	#14556039	—	2	1	no	—
54	#14556039	C₂×C₄	1	2	yes	—
55	#14556038	—	2	1	no	—
56	C₈×Dic₇₁	—	2	1	no	—
57	C₇₁⋊#95	—	2	1	no	—
58	C₇₁⋊#95	C₂×C₄	1	2	yes	—
59	C₂×M₁₆×C₇₁	—	2	1	no	—
60	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₇₁	—	2	1	no	—
64	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
65	#97×C₇₁	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₇₁	—	2	1	no	—
67	C₄×C₈×C₇₁	—	2	1	no	—
68	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
72	C₂×Dic₁₄₂	C₄²	1	1	yes	characteristic
73	C₂×D₂₈₄	C₄²	1	1	yes	characteristic
74	C₂×C₂²⋊D₇₁	—	2	1	no	—
75	C₂²×Dic₇₁	—	2	2	no	—
76	C₂×C₄×D₇₁	—	2	1	no	—
77	#11765848	—	4	2	no	—
78	C₂×C₇₁⋊C₈	—	2	1	no	—
79	C₂×C₇₁⋊C₈	—	2	1	no	—
80	C₂×C₇₁⋊C₈	—	2	2	no	—
81	C₂×C₇₁⋊C₈	—	4	1	no	—
82	C₂×C₇₁⋊C₈	—	4	1	no	—
83	C₈×D₇₁	—	4	2	no	—
84	C₄⋊Dic₇₁	C₄²	1	1	yes	characteristic
85	C₄⋊Dic₇₁	C₄²	1	1	yes	characteristic
86	Dic₇₁⋊C₄	—	2	1	no	—
87	Dic₇₁⋊C₄	—	2	1	no	—
88	Dic₇₁⋊C₄	—	2	1	no	—
89	Dic₇₁⋊C₄	—	2	1	no	—
90	C₂²⋊Dic₇₁	—	2	1	no	—
91	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	(C₂×D₇₁)⋊C₄	—	2	1	no	—
93	(C₂×D₇₁)⋊C₄	—	2	1	no	—
94	(C₂×D₇₁)⋊C₄	—	2	1	no	—
95	C₄×Dic₇₁	—	2	1	no	—
96	C₄×Dic₇₁	—	4	1	no	—
97	C₄×Dic₇₁	—	4	1	no	—
98	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
101	M₁₆×C₇₁	—	4	2	no	—
102	C₂×C₈×C₇₁	—	2	2	no	—
103	C₂×C₈×C₇₁	—	2	2	no	—
104	C₂×C₈×C₇₁	—	4	1	no	—
105	C₂×C₈×C₇₁	—	4	1	no	—
106	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
110	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇₁	#41	1	1	yes	characteristic
114	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	—	4	1	no	—
120	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
126	C₂²⋊D₇₁	—	4	2	no	—
127	C₂×Dic₇₁	—	2	1	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	—	2	1	no	—
130	C₂×Dic₇₁	—	4	2	no	—
131	C₂×Dic₇₁	—	4	2	no	—
132	C₂×Dic₇₁	—	4	2	no	—
133	D₂₈₄	—	4	1	no	—
134	C₄×D₇₁	—	4	1	no	—
135	C₄×D₇₁	—	4	1	no	—
136	Dic₁₄₂	—	4	1	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₈×C₇₁	—	4	2	no	—
142	C₈×C₇₁	—	4	2	no	—
143	C₇₁⋊C₈	—	4	2	no	—
144	C₇₁⋊C₈	—	4	1	no	—
145	C₇₁⋊C₈	—	4	1	no	—
146	C₂²×C₇₁	—	2	1	no	—
147	C₂²×C₇₁	—	2	1	no	—
148	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₇₁	#444	1	2	yes	—
150	C₂×D₇₁	—	4	1	no	—
151	C₂×D₇₁	—	4	1	no	—
152	C₂×D₇₁	—	4	1	no	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	#421	1	1	yes	characteristic
159	C₄×C₇₁	#421	1	1	yes	characteristic
160	Dic₇₁	—	4	1	no	—
161	Dic₇₁	—	4	1	no	—
162	Dic₇₁	—	4	1	no	—
163	Dic₇₁	—	8	2	no	—
164	#8248	—	71	1	no	maximal, Sylow
165	C₂×C₇₁	—	2	2	no	—
166	C₂×C₇₁	#1757	1	1	yes	characteristic
167	C₂×C₇₁	#1759	1	1	yes	characteristic
168	C₂×C₇₁	#1765	1	1	yes	characteristic
169	D₇₁	—	8	1	no	—
170	#2966	—	71	1	no	—
171	#1746	—	71	1	no	—
172	#1524	—	71	1	no	—
173	#1500	—	71	1	no	—
174	#1451	D₇₁	1	1	yes	characteristic, complemented
175	#1302	—	71	1	no	—
176	#1300	—	71	1	no	—
177	C₇₁	#8248	1	1	yes	characteristic, complemented, Sylow
178	#529	—	71	1	no	—
179	#446	—	71	1	no	—
180	C₈×Q₈	—	71	1	no	—
181	#436	—	71	1	no	—
182	#435	—	142	1	no	—
183	C₈×D₄	—	71	1	no	—
184	C₈×D₄	—	142	1	no	—
185	#433	—	142	1	no	—
186	#432	—	71	1	no	—
187	#431	C₂×D₇₁	1	1	yes	characteristic
188	#405	—	142	1	no	—
189	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
190	C₂×#97	—	71	1	no	—
191	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
192	#376	—	71	1	no	—
193	#336	—	142	1	no	—
194	#336	—	142	1	no	—
195	#330	—	71	2	no	—
196	#329	—	71	2	no	—
197	#328	—	71	2	no	—
198	#327	—	71	2	no	—
199	#322	—	2	1	no	—
200	#322	—	2	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	142	1	no	—
203	C₂²×C₈	—	142	1	no	—
204	C₂²×C₈	—	142	1	no	—
205	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
206	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
207	#126	—	142	1	no	—
208	#125	—	71	1	no	—
209	#124	—	71	1	no	—
210	#123	—	142	1	no	—
211	C₂²⋊Q₈	—	71	1	no	—
212	#121	—	71	1	no	—
213	C₄×Q₈	—	71	1	no	—
214	C₄×D₄	—	71	1	no	—
215	C₄×D₄	—	142	1	no	—
216	#117	—	142	1	no	—
217	C₂×C₄²	—	142	1	no	—
218	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	71	1	no	—
221	C₄⋊C₈	—	71	1	no	—
222	C₄⋊C₈	—	142	1	no	—
223	C₄⋊C₈	—	142	1	no	—
224	C₂²⋊C₈	—	142	1	no	—
225	C₂²⋊C₈	—	142	1	no	—
226	C₂²⋊C₈	—	142	1	no	—
227	C₂²⋊C₈	C₄×D₇₁	1	2	yes	—
228	#97	—	71	2	no	—
229	#97	—	142	1	no	—
230	#97	—	142	1	no	—
231	#97	—	142	1	no	—
232	#97	C₄×D₇₁	1	2	yes	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	142	1	no	—
236	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
237	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
240	#95	—	71	2	no	—
241	#95	—	142	1	no	—
242	C₂×Q₈	—	71	1	no	—
243	C₂×D₄	—	71	1	no	—
244	C₂×D₄	—	142	1	no	—
245	C₂²×C₄	—	142	2	no	—
246	C₂²×C₄	—	142	1	no	—
247	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
249	C₂²×C₄	#11765865	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	284	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	284	2	no	—
257	C₂×C₈	—	284	1	no	—
258	C₂×C₈	—	284	1	no	—
259	C₂×C₈	—	142	1	no	—
260	C₂×C₈	—	142	1	no	—
261	C₂×C₈	—	142	2	no	—
262	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
263	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
265	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
266	C₄⋊C₄	—	71	1	no	—
267	C₄⋊C₄	—	71	1	no	—
268	C₄⋊C₄	—	142	1	no	—
269	C₄⋊C₄	—	142	1	no	—
270	C₄⋊C₄	—	142	1	no	—
271	C₄⋊C₄	—	142	1	no	—
272	C₂²⋊C₄	—	142	1	no	—
273	C₂²⋊C₄	—	142	1	no	—
274	C₂²⋊C₄	—	142	1	no	—
275	C₂²⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₄²	—	284	1	no	—
278	C₄²	—	284	1	no	—
279	C₄²	—	142	1	no	—
280	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
283	C₄²	#11765865	1	1	yes	characteristic
284	C₂³	—	142	1	no	—
285	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
286	Q₈	—	284	1	no	—
287	D₄	—	284	1	no	—
288	D₄	—	284	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	284	1	no	—
294	C₂×C₄	—	284	1	no	—
295	C₂×C₄	—	284	2	no	—
296	C₂×C₄	—	284	2	no	—
297	C₂×C₄	—	284	2	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
302	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	#14556085	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
305	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	284	2	no	—
314	C₈	—	284	1	no	—
315	C₈	—	284	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	284	1	no	—
319	C₂²	—	284	1	no	—
320	C₂²	—	284	1	no	—
321	C₂²	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
322	C₂²	#17986122	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	284	1	no	—
329	C₄	—	284	1	no	—
330	C₄	—	284	1	no	—
331	C₄	—	568	2	no	—
332	C₄	#17986015	1	1	yes	characteristic
333	C₄	#17986015	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	568	1	no	—
336	C₂	#22588141	1	1	yes	characteristic
337	C₂	#22588153	1	1	yes	characteristic
338	C₂	#22588184	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1451	C₂
socle	C₂²×C₇₁	C₄×C₂²⋊C₄
prime radical	#14556102	C₂×C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1451	D₇₁
hypocenter	C₇₁	#8248
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #22588091	C₂	semidirect
5. #22586240	C₂	semidirect
6. #22586234	C₂	semidirect
7. C₇₁×#1451	C₂	semidirect
9. #17986129	C₄	semidirect
10. C₈×Dic₁₄₂	C₄	semidirect
11. #17986078	C₄	semidirect
13. C₈×D₂₈₄	C₄	semidirect
22. #17985740	C₄	semidirect
23. #17985738	C₄	semidirect
174. #1451	D₇₁	semidirect
177. C₇₁	#8248	semidirect

Maximal subgroups

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

subgroup	index
#22597315	2
#22588091	2
#22586929	2
#22586827	2
#22586240	2
#22586234	2
C₇₁×#1451	2
#8248	71