Subgroups — #31914322

Order 18176.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597241	C₂	1	1	yes	maximal, characteristic
2	#22589969	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985783	C₂	1	1	yes	maximal, characteristic, complemented
4	#22589955	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586430	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586425	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986797	C₄	1	1	yes	characteristic
9	C₂×#14556134	C₂²	1	1	yes	characteristic
10	C₂×#14556068	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17986119	—	2	2	no	—
13	#17986114	—	2	2	no	—
14	#17986047	C₄	1	1	yes	characteristic
15	#17986016	C₄	1	1	yes	characteristic
16	#17986015	C₄	1	1	yes	characteristic
17	#17985954	C₂²	1	1	yes	characteristic
18	#17985784	—	2	2	no	—
19	#17985783	—	2	2	no	—
20	(C₂×D₇₁)⋊C₁₆	—	2	2	no	—
21	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
22	#504×C₇₁	—	2	2	no	—
23	#504×C₇₁	—	2	2	no	—
24	C₂×M₃₂×C₇₁	—	2	2	no	—
25	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556159	—	2	1	no	—
29	#14556158	C₈	1	2	yes	—
30	#14556154	—	2	1	no	—
31	#14556153	C₈	1	2	yes	—
32	#14556148	—	2	2	no	—
33	#14556134	—	2	1	no	—
34	#14556134	—	2	1	no	—
35	#14556134	—	2	1	no	—
36	#14556134	—	2	1	no	—
37	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
38	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
39	C₂×#11765848	—	2	1	no	—
40	C₂×C₈×D₇₁	—	2	1	no	—
41	#14556087	C₂×C₄	1	1	yes	characteristic
42	#14556068	—	2	1	no	—
43	#14556068	—	2	1	no	—
44	C₂×C₇₁⋊C₁₆	—	2	1	no	—
45	C₂×C₇₁⋊C₁₆	—	2	1	no	—
46	C₂×C₇₁⋊C₁₆	—	2	1	no	—
47	C₂×C₇₁⋊C₁₆	—	2	1	no	—
48	#14556063	—	2	2	no	—
49	#14556055	—	2	2	no	—
50	#14556054	—	2	2	no	—
51	#14556038	—	2	1	no	—
52	C₈×Dic₇₁	—	2	1	no	—
53	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
54	#131×C₇₁	—	2	1	no	—
55	#131×C₇₁	—	2	1	no	—
56	#131×C₇₁	D₄	1	2	yes	—
57	#131×C₇₁	D₄	1	2	yes	—
58	C₂×M₁₆×C₇₁	—	2	1	no	—
59	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₁	—	2	1	no	—
61	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
62	M₃₂×C₇₁	—	2	4	no	—
63	M₃₂×C₇₁	—	2	1	no	—
64	M₃₂×C₇₁	—	2	1	no	—
65	M₃₂×C₇₁	—	2	1	no	—
66	M₃₂×C₇₁	D₄	1	2	yes	—
67	C₂×C₁₆×C₇₁	—	2	4	no	—
68	C₂×C₁₆×C₇₁	—	2	1	no	—
69	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
70	#11765865	—	2	1	no	—
71	#11765865	—	2	1	no	—
72	#11765865	—	2	1	no	—
73	#11765865	—	2	1	no	—
74	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
75	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
76	C₂×C₂²⋊D₇₁	—	2	1	no	—
77	C₂×C₄×D₇₁	—	2	1	no	—
78	#11765855	—	2	2	no	—
79	#11765853	—	2	2	no	—
80	#11765852	—	2	2	no	—
81	#11765850	—	2	2	no	—
82	#11765849	—	2	1	no	—
83	#11765849	—	2	1	no	—
84	#11765849	—	2	1	no	—
85	#11765849	C₂×C₈	1	2	yes	—
86	#11765848	—	4	1	no	—
87	#11765848	—	4	1	no	—
88	C₂×C₇₁⋊C₈	—	2	1	no	—
89	C₂×C₇₁⋊C₈	—	2	1	no	—
90	C₈×D₇₁	—	4	1	no	—
91	C₈×D₇₁	—	4	1	no	—
92	C₄⋊Dic₇₁	—	2	1	no	—
93	C₂²⋊Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	—	2	1	no	—
95	C₇₁⋊C₁₆	—	4	1	no	—
96	C₇₁⋊C₁₆	—	4	1	no	—
97	#41×C₇₁	—	2	1	no	—
98	#41×C₇₁	—	2	1	no	—
99	#41×C₇₁	C₂²⋊C₄	1	2	yes	—
100	#41×C₇₁	C₂×C₈	1	2	yes	—
101	C₂×Q₈×C₇₁	M₁₆	1	1	yes	characteristic
102	C₂×D₄×C₇₁	—	2	1	no	—
103	C₂×D₄×C₇₁	M₁₆	1	1	yes	characteristic
104	C₂²×C₄×C₇₁	—	2	1	no	—
105	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₇₁	—	2	1	no	—
107	M₁₆×C₇₁	—	2	2	no	—
108	M₁₆×C₇₁	—	2	2	no	—
109	M₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
110	C₂×C₈×C₇₁	—	2	1	no	—
111	C₂×C₈×C₇₁	—	2	2	no	—
112	C₂×C₈×C₇₁	—	2	2	no	—
113	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
119	C₁₆×C₇₁	—	2	2	no	—
120	C₁₆×C₇₁	—	2	2	no	—
121	C₂³×C₇₁	—	2	1	no	—
122	C₂³×C₇₁	#131	1	1	yes	characteristic
123	C₂²×D₇₁	—	2	1	no	—
124	Q₈×C₇₁	—	2	1	no	—
125	Q₈×C₇₁	C₂²⋊C₈	1	2	yes	—
126	D₄×C₇₁	—	2	2	no	—
127	D₄×C₇₁	—	2	2	no	—
128	D₄×C₇₁	—	2	1	no	—
129	D₄×C₇₁	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₇₁	—	2	1	no	—
131	C₂×C₄×C₇₁	—	2	2	no	—
132	C₂×C₄×C₇₁	—	2	2	no	—
133	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
139	C₂²⋊D₇₁	—	4	1	no	—
140	C₂²⋊D₇₁	—	4	1	no	—
141	C₂×Dic₇₁	—	2	1	no	—
142	C₂×Dic₇₁	—	4	1	no	—
143	D₂₈₄	—	2	1	no	—
144	D₂₈₄	—	2	1	no	—
145	C₄×D₇₁	—	4	1	no	—
146	C₄×D₇₁	—	4	1	no	—
147	Dic₁₄₂	—	2	1	no	—
148	Dic₁₄₂	—	2	1	no	—
149	C₈×C₇₁	—	2	2	no	—
150	C₈×C₇₁	#120	1	1	yes	characteristic
151	C₈×C₇₁	#121	1	1	yes	characteristic
152	C₈×C₇₁	#121	1	1	yes	characteristic
153	C₈×C₇₁	#123	1	1	yes	characteristic
154	C₇₁⋊C₈	—	2	2	no	—
155	C₇₁⋊C₈	—	4	1	no	—
156	C₂²×C₇₁	—	2	2	no	—
157	C₂²×C₇₁	—	4	1	no	—
158	C₂²×C₇₁	—	8	1	no	—
159	C₂²×C₇₁	#408	1	1	yes	characteristic
160	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
161	C₂²×C₇₁	#435	1	1	yes	characteristic
162	C₂×D₇₁	—	4	2	no	—
163	C₂×D₇₁	—	4	1	no	—
164	C₄×C₇₁	—	2	2	no	—
165	C₄×C₇₁	#386	1	1	yes	characteristic
166	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
168	C₄×C₇₁	#435	1	1	yes	characteristic
169	Dic₇₁	—	4	1	no	—
170	Dic₇₁	—	8	1	no	—
171	#12979	—	71	1	no	maximal, Sylow
172	C₂×C₇₁	—	2	1	no	—
173	C₂×C₇₁	—	2	1	no	—
174	C₂×C₇₁	—	2	1	no	—
175	C₂×C₇₁	—	4	2	no	—
176	C₂×C₇₁	#1869	1	1	yes	characteristic
177	D₇₁	—	8	1	no	—
178	#3407	D₇₁	1	1	yes	characteristic, complemented
179	#2955	—	71	1	no	—
180	#2117	—	71	1	no	—
181	C₂×#350	—	71	1	no	—
182	#2113	—	71	1	no	—
183	#1379	—	71	1	no	—
184	#1378	—	71	1	no	—
185	C₇₁	#12979	1	1	yes	characteristic, complemented, Sylow
186	#575	—	71	1	no	—
187	C₂×#131	—	71	1	no	—
188	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
189	#504	—	2	2	no	—
190	#504	—	2	2	no	—
191	C₂×M₃₂	—	2	2	no	—
192	C₂×M₃₂	—	71	1	no	—
193	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
194	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
195	C₂²×C₁₆	—	71	1	no	—
196	#444	—	142	2	no	—
197	#443	—	142	2	no	—
198	#430	—	71	1	no	—
199	#421	—	71	1	no	—
200	#421	—	71	1	no	—
201	#405	—	71	1	no	—
202	#350	—	142	2	no	—
203	#350	—	142	2	no	—
204	C₂²⋊C₁₆	—	142	2	no	—
205	C₂×#41	—	71	1	no	—
206	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
207	#137	—	71	2	no	—
208	#137	—	142	1	no	—
209	AGL(1,ℤ/8)	—	71	2	no	—
210	AGL(1,ℤ/8)	—	142	1	no	—
211	#135	—	142	2	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	142	1	no	—
215	#131	—	142	1	no	—
216	#131	—	142	1	no	—
217	#131	—	142	1	no	—
218	#131	D₂₈₄	1	2	yes	—
219	#131	C₂²⋊D₇₁	1	2	yes	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	71	1	no	—
222	C₂×M₁₆	—	71	1	no	—
223	C₂×M₁₆	—	142	1	no	—
224	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	142	1	no	—
227	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
228	#117	—	71	1	no	—
229	M₃₂	—	2	4	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	142	1	no	—
234	M₃₂	—	142	1	no	—
235	M₃₂	C₂²⋊D₇₁	1	2	yes	—
236	C₂×C₁₆	—	2	4	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	142	1	no	—
239	C₂×C₁₆	—	142	1	no	—
240	C₂×C₁₆	—	142	1	no	—
241	C₂×C₁₆	—	142	1	no	—
242	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
243	#108	—	142	2	no	—
244	#104	—	142	2	no	—
245	#104	—	142	2	no	—
246	#97	—	142	1	no	—
247	C₄×C₈	—	142	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	142	1	no	—
251	#41	—	142	1	no	—
252	#41	—	142	1	no	—
253	#41	—	142	1	no	—
254	#41	(C₂×D₇₁)⋊C₄	1	2	yes	—
255	#41	C₈×D₇₁	1	2	yes	—
256	C₂×Q₈	—	71	1	no	—
257	C₂×Q₈	#11765848	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	71	1	no	—
260	C₂×D₄	—	142	1	no	—
261	C₂×D₄	#11765848	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	142	1	no	—
264	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
265	Q₁₆	—	142	2	no	—
266	SD₁₆	—	142	2	no	—
267	SD₁₆	—	142	2	no	—
268	D₈	—	142	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	284	1	no	—
273	M₁₆	—	284	1	no	—
274	M₁₆	—	71	2	no	—
275	M₁₆	—	142	1	no	—
276	M₁₆	—	142	1	no	—
277	M₁₆	—	142	1	no	—
278	M₁₆	(C₂×D₇₁)⋊C₄	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	284	1	no	—
283	C₂×C₈	—	284	1	no	—
284	C₂×C₈	—	142	1	no	—
285	C₂×C₈	—	142	1	no	—
286	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
289	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
290	C₂×C₈	#11765865	1	1	yes	characteristic
291	C₂×C₈	#11765865	1	1	yes	characteristic
292	C₄⋊C₄	—	142	1	no	—
293	C₂²⋊C₄	—	142	1	no	—
294	C₄²	—	142	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	284	1	no	—
298	C₁₆	—	284	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	142	1	no	—
301	C₂³	#14556134	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	142	1	no	—
304	Q₈	—	142	1	no	—
305	Q₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	284	1	no	—
310	D₄	—	284	1	no	—
311	D₄	—	142	1	no	—
312	D₄	—	142	1	no	—
313	D₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	284	1	no	—
318	C₂×C₄	—	284	1	no	—
319	C₂×C₄	—	284	1	no	—
320	C₂×C₄	—	142	1	no	—
321	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
323	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
324	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
325	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
326	C₂×C₄	#14556134	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	284	1	no	—
329	C₈	—	142	2	no	—
330	C₈	#14556097	1	1	yes	characteristic
331	C₈	#14556102	1	1	yes	characteristic
332	C₈	#14556102	1	1	yes	characteristic
333	C₈	#14556111	1	1	yes	characteristic
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	284	2	no	—
338	C₂²	—	284	1	no	—
339	C₂²	#17985965	1	1	yes	characteristic
340	C₂²	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
341	C₂²	#17986070	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	284	1	no	—
344	C₄	—	568	1	no	—
345	C₄	#17985878	1	1	yes	characteristic
346	C₄	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
347	C₄	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
348	C₄	#17986070	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	568	1	no	—
354	C₂	#22588632	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556097
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#3407	C₂
socle	C₂×C₇₁	#1869
prime radical	#17986797	C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#3407	D₇₁
hypocenter	C₇₁	#12979
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

Complements

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

N	H ≅ G/N	type
2. #22589969	C₂	semidirect
3. C₂×#17985783	C₂	semidirect
4. #22589955	C₂	semidirect
5. #22586430	C₂	semidirect
6. #22586425	C₂	semidirect
7. C₇₁×#3407	C₂	semidirect
178. #3407	D₇₁	semidirect
185. C₇₁	#12979	semidirect

Maximal subgroups

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

subgroup	index
#22597241	2
#22589969	2
C₂×#17985783	2
#22589955	2
#22586430	2
#22586425	2
C₇₁×#3407	2
#12979	71