Subgroups — #31885545

Order 18176.

406 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588483	C₂	1	1	yes	maximal, characteristic
2	#22588457	C₂	1	1	yes	maximal, characteristic
3	#22586166	C₂	1	2	yes	maximal
4	#22586163	C₂	1	2	yes	maximal
5	C₇₁×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	#17985754	—	2	1	no	—
8	#17985754	—	2	1	no	—
9	#17985754	—	2	1	no	—
10	#17985754	C₂²	1	2	yes	—
11	#17985754	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
13	#391×C₇₁	C₄	1	1	yes	characteristic
14	#391×C₇₁	C₂²	1	1	yes	characteristic
15	#389×C₇₁	C₄	1	1	yes	characteristic
16	#389×C₇₁	C₂²	1	1	yes	characteristic
17	#387×C₇₁	—	2	1	no	—
18	#387×C₇₁	—	2	1	no	—
19	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
20	C₄×C₄⋊C₄×C₇₁	C₄	1	1	yes	characteristic
21	#376×C₇₁	—	2	1	no	—
22	C₄³×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₇₁⋊C₈	—	2	2	no	—
24	C₂²×C₇₁⋊C₈	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
26	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Q₈×C₇₁	—	2	2	no	—
28	C₄×Q₈×C₇₁	C₈	1	4	yes	complemented
29	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	—	2	1	no	—
36	C₂×C₄²×C₇₁	—	2	1	no	—
37	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
44	#95×C₇₁	—	2	1	no	—
45	#95×C₇₁	—	2	1	no	—
46	#95×C₇₁	—	2	1	no	—
47	#95×C₇₁	—	2	1	no	—
48	#95×C₇₁	—	2	1	no	—
49	#95×C₇₁	—	2	1	no	—
50	#95×C₇₁	—	2	1	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₂×C₇₁⋊C₈	—	4	2	no	—
54	C₂×C₇₁⋊C₈	—	4	2	no	—
55	C₂×C₇₁⋊C₈	—	4	2	no	—
56	C₂×C₇₁⋊C₈	—	4	2	no	—
57	C₂×Q₈×C₇₁	—	2	1	no	—
58	C₂×Q₈×C₇₁	—	4	1	no	—
59	C₂×Q₈×C₇₁	—	4	1	no	—
60	C₂×Q₈×C₇₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇₁	—	2	1	no	—
62	C₂²×C₄×C₇₁	—	2	1	no	—
63	C₂²×C₄×C₇₁	—	2	1	no	—
64	C₂²×C₄×C₇₁	—	2	1	no	—
65	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₇₁	—	2	4	no	—
73	C₄⋊C₄×C₇₁	—	2	1	no	—
74	C₄⋊C₄×C₇₁	—	4	1	no	—
75	C₄⋊C₄×C₇₁	—	4	1	no	—
76	C₄⋊C₄×C₇₁	—	4	2	no	—
77	C₄⋊C₄×C₇₁	—	4	1	no	—
78	C₄⋊C₄×C₇₁	—	4	1	no	—
79	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
80	C₄²×C₇₁	—	2	4	no	—
81	C₄²×C₇₁	—	2	1	no	—
82	C₄²×C₇₁	—	2	1	no	—
83	C₄²×C₇₁	—	2	2	no	—
84	C₄²×C₇₁	—	4	1	no	—
85	C₄²×C₇₁	—	4	1	no	—
86	C₄²×C₇₁	—	4	1	no	—
87	C₄²×C₇₁	—	4	1	no	—
88	C₄²×C₇₁	—	4	2	no	—
89	C₄²×C₇₁	C₂×C₈	1	2	yes	—
90	C₄²×C₇₁	M₁₆	1	2	yes	—
91	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	Q₈×C₇₁	—	4	2	no	—
93	Q₈×C₇₁	—	4	2	no	—
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	—	2	1	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	2	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	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	—	4	1	no	—
113	C₂×C₄×C₇₁	—	4	1	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	—	4	1	no	—
119	C₂×C₄×C₇₁	—	4	1	no	—
120	C₂×C₄×C₇₁	—	4	1	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
123	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
128	C₂×C₄×C₇₁	#137	1	2	yes	—
129	C₇₁⋊C₈	—	8	2	no	—
130	C₇₁⋊C₈	—	8	2	no	—
131	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₇₁	#410	1	1	yes	characteristic
133	C₂²×C₇₁	#413	1	1	yes	characteristic
134	C₂²×C₇₁	#417	1	1	yes	characteristic
135	C₂²×C₇₁	#419	1	1	yes	characteristic
136	C₂²×C₇₁	#421	1	1	yes	characteristic
137	C₂²×C₇₁	#421	1	1	yes	characteristic
138	C₄×C₇₁	—	2	1	no	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	2	2	no	—
141	C₄×C₇₁	—	2	2	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	2	1	no	—
144	C₄×C₇₁	—	2	2	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	—	4	2	no	—
147	C₄×C₇₁	—	4	2	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	—	8	1	no	—
150	C₄×C₇₁	—	8	1	no	—
151	#7596	—	71	1	no	maximal, Sylow
152	C₂×C₇₁	#1458	1	1	yes	characteristic
153	C₂×C₇₁	#1477	1	1	yes	characteristic
154	C₂×C₇₁	#1479	1	1	yes	characteristic
155	C₂×C₇₁	#1512	1	1	yes	characteristic
156	C₂×C₇₁	#1515	1	1	yes	characteristic
157	C₂×C₇₁	#1535	1	1	yes	characteristic
158	C₂×C₇₁	#1559	1	1	yes	characteristic
159	#2295	D₇₁	1	1	yes	characteristic
160	#1839	—	71	1	no	—
161	#1831	—	71	1	no	—
162	#1272	—	71	2	no	—
163	#1271	—	71	2	no	—
164	C₇₁	#7596	1	1	yes	characteristic, complemented, Sylow
165	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	142	1	no	—
167	#391	Dic₇₁	1	1	yes	characteristic
168	#391	C₂×D₇₁	1	1	yes	characteristic
169	#389	Dic₇₁	1	1	yes	characteristic
170	#389	C₂×D₇₁	1	1	yes	characteristic
171	#387	—	2	1	no	—
172	#387	—	2	1	no	—
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	Dic₇₁	1	1	yes	characteristic
175	#376	—	2	1	no	—
176	C₄³	C₂×D₇₁	1	1	yes	characteristic
177	#336	—	71	2	no	—
178	#336	—	71	2	no	—
179	#336	—	142	1	no	—
180	#336	—	142	1	no	—
181	#336	—	142	1	no	—
182	C₂²×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
183	C₂²×C₈	—	142	2	no	—
184	C₂²×C₈	—	142	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	C₇₁⋊C₈	1	4	yes	complemented
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
196	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
202	C₄⋊C₈	—	284	2	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	#95	—	2	1	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	C₂×C₇₁⋊C₈	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
225	C₂×C₈	—	284	2	no	—
226	C₂×C₈	—	284	2	no	—
227	C₂×C₈	—	284	2	no	—
228	C₂×C₈	—	284	2	no	—
229	C₂×C₈	—	284	2	no	—
230	C₂×C₈	—	284	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
249	C₄²	#11765849	1	2	yes	—
250	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	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	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
282	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
283	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
284	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
285	C₂×C₄	#14556153	1	1	yes	characteristic
286	C₂×C₄	#14556158	1	1	yes	characteristic
287	C₂×C₄	#14556159	1	2	yes	—
288	C₈	—	568	2	no	—
289	C₈	—	568	2	no	—
290	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
291	C₂²	#17985973	1	1	yes	characteristic
292	C₂²	#17985983	1	1	yes	characteristic
293	C₂²	#17986000	1	1	yes	characteristic
294	C₂²	#17986008	1	1	yes	characteristic
295	C₂²	#17986016	1	1	yes	characteristic
296	C₂²	#17986016	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	1	no	—
310	C₂	#22586654	1	1	yes	characteristic
311	C₂	#22586727	1	1	yes	characteristic
312	C₂	#22586735	1	1	yes	characteristic
313	C₂	#22586878	1	1	yes	characteristic
314	C₂	#22586891	1	1	yes	characteristic
315	C₂	#22586971	1	1	yes	characteristic
316	C₂	#22587070	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2295	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#2295	D₇₁
hypocenter	C₇₁	#7596
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
28. C₄×Q₈×C₇₁	C₈	semidirect
164. C₇₁	#7596	semidirect
186. C₄×Q₈	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22588483	2
#22588457	2
#22586166	2
#22586163	2
C₇₁×#2295	2
#7596	71