Subgroups — #31912704

Order 18176.

498 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	C₂×#17986081	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985813	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985811	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586448	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×#14556090	—	2	1	no	—
10	C₂×#14556068	C₂²	1	1	yes	characteristic
11	#17986081	—	2	1	no	—
12	#17986081	—	2	1	no	—
13	#17986081	C₄	1	2	yes	—
14	#17986081	C₄	1	2	yes	—
15	C₂×#14556064	C₂²	1	1	yes	characteristic
16	C₂×#14556048	—	2	1	no	—
17	M₁₆×Dic₇₁	—	2	1	no	—
18	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
19	#17985813	C₄	1	4	yes	complemented
20	#17985811	C₄	1	4	yes	complemented
21	#17985774	—	2	1	no	—
22	C₁₆×Dic₇₁	—	2	1	no	—
23	#17985763	—	2	1	no	—
24	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
27	C₂²×D₂₈₄	—	2	1	no	—
28	C₂²×C₄×D₇₁	—	4	1	no	—
29	C₂×D₅₆₈	—	2	1	no	—
30	C₂×D₅₆₈	—	2	1	no	—
31	C₂×D₅₆₈	—	2	1	no	—
32	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
33	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
34	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
35	C₂×#11765849	—	2	1	no	—
36	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
37	#14556090	—	2	2	no	—
38	#14556090	—	2	2	no	—
39	#14556090	—	4	1	no	—
40	#14556090	—	4	1	no	—
41	C₂×(C₂×D₇₁)⋊C₄	—	4	1	no	—
42	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
43	#14556068	—	2	1	no	—
44	#14556068	C₂×C₄	1	2	yes	complemented
45	C₂×C₇₁⋊C₁₆	—	2	1	no	—
46	#14556064	—	2	1	no	—
47	#14556064	C₂×C₄	1	2	yes	complemented
48	#14556048	—	2	2	no	—
49	#14556048	—	2	2	no	—
50	#14556038	—	4	1	no	—
51	C₈×Dic₇₁	—	4	1	no	—
52	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₇₁	D₄	1	1	yes	characteristic
55	C₈×Dic₇₁	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₇₁	—	2	1	no	—
57	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
59	M₃₂×C₇₁	—	2	1	no	—
60	M₃₂×C₇₁	C₂×C₄	1	2	yes	complemented
61	C₂×C₁₆×C₇₁	—	2	1	no	—
62	#108×C₇₁	—	2	1	no	—
63	#108×C₇₁	C₂×C₄	1	2	yes	complemented
64	#107×C₇₁	—	2	1	no	—
65	#107×C₇₁	C₂×C₄	1	2	yes	—
66	C₂³×D₇₁	—	4	1	no	—
67	C₂×D₂₈₄	—	2	2	no	—
68	C₂×D₂₈₄	—	2	2	no	—
69	C₂×D₂₈₄	—	2	2	no	—
70	C₂×D₂₈₄	—	4	1	no	—
71	C₂×D₂₈₄	—	4	1	no	—
72	C₂×D₂₈₄	—	4	1	no	—
73	C₂²×Dic₇₁	SD₁₆	1	1	yes	characteristic
74	C₂²×Dic₇₁	SD₁₆	1	1	yes	characteristic
75	C₂×C₄×D₇₁	—	4	2	no	—
76	C₂×C₄×D₇₁	—	4	2	no	—
77	C₂×C₄×D₇₁	—	8	4	no	—
78	D₅₆₈	—	2	2	no	—
79	D₅₆₈	—	2	2	no	—
80	D₅₆₈	—	2	2	no	—
81	D₅₆₈	C₄²	1	4	yes	—
82	#11765849	—	2	2	no	—
83	#11765849	—	4	1	no	—
84	C₂×C₇₁⋊C₈	—	2	1	no	—
85	C₂×C₇₁⋊C₈	—	2	1	no	—
86	C₂×C₇₁⋊C₈	—	4	1	no	—
87	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	(C₂×D₇₁)⋊C₄	—	4	2	no	—
90	(C₂×D₇₁)⋊C₄	—	4	2	no	—
91	C₄×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×Dic₇₁	D₈	1	1	yes	characteristic
94	C₄×Dic₇₁	D₈	1	1	yes	characteristic
95	C₇₁⋊C₁₆	—	2	2	no	—
96	C₂²×C₄×C₇₁	—	4	1	no	—
97	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
98	M₁₆×C₇₁	—	2	2	no	—
99	M₁₆×C₇₁	—	4	1	no	—
100	C₂×C₈×C₇₁	—	4	1	no	—
101	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
106	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
107	C₄⋊C₄×C₇₁	—	2	2	no	—
108	C₄⋊C₄×C₇₁	—	4	1	no	—
109	C₁₆×C₇₁	—	2	2	no	—
110	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
111	C₂²×D₇₁	—	4	2	no	—
112	C₂²×D₇₁	—	4	2	no	—
113	C₂²×D₇₁	—	4	2	no	—
114	C₂²×D₇₁	—	8	2	no	—
115	C₂²×D₇₁	—	8	2	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	8	2	no	—
118	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
123	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
124	C₂×Dic₇₁	—	2	2	no	—
125	C₂×Dic₇₁	—	4	1	no	—
126	C₂×Dic₇₁	#102	1	1	yes	characteristic
127	C₂×Dic₇₁	#102	1	1	yes	characteristic
128	C₂×Dic₇₁	#102	1	1	yes	characteristic
129	C₂×Dic₇₁	#102	1	1	yes	characteristic
130	D₂₈₄	—	2	4	no	—
131	D₂₈₄	—	4	2	no	—
132	D₂₈₄	—	4	2	no	—
133	D₂₈₄	—	4	2	no	—
134	C₄×D₇₁	—	8	4	no	—
135	C₄×D₇₁	—	8	4	no	—
136	C₈×C₇₁	—	4	2	no	—
137	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
138	C₈×C₇₁	#117	1	1	yes	characteristic
139	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
140	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
141	C₇₁⋊C₈	—	2	1	no	—
142	C₇₁⋊C₈	—	2	1	no	—
143	C₇₁⋊C₈	—	4	2	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	2	1	no	—
146	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
147	C₂²×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
148	C₂²×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
149	C₂×D₇₁	—	4	4	no	—
150	C₂×D₇₁	—	8	4	no	—
151	C₂×D₇₁	—	8	4	no	—
152	C₂×D₇₁	—	8	2	no	—
153	C₂×D₇₁	—	8	2	no	—
154	C₄×C₇₁	—	8	2	no	—
155	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₇₁	#416	1	1	yes	characteristic
157	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
158	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
159	Dic₇₁	—	2	1	no	—
160	Dic₇₁	—	2	1	no	—
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	#361	1	2	yes	complemented
163	C₄×#361	—	71	1	no	maximal, Sylow
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	—	2	1	no	—
166	C₂×C₇₁	C₄×#102	1	1	yes	characteristic
167	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
168	C₂×C₇₁	#2148	1	1	yes	characteristic
169	D₇₁	—	8	4	no	—
170	C₂×C₄×D₈	—	71	1	no	—
171	C₂×#361	—	71	1	no	—
172	C₂×#361	—	71	1	no	—
173	C₄×M₃₂	—	71	1	no	—
174	C₄×#108	—	71	1	no	—
175	#1387	—	71	1	no	—
176	#1387	D₇₁	1	1	yes	characteristic, complemented
177	C₇₁	C₄×#361	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	71	1	no	—
179	C₂×C₄×D₄	—	142	1	no	—
180	C₂×M₃₂	—	71	1	no	—
181	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
182	C₄×D₈	—	71	2	no	—
183	C₄×D₈	—	71	2	no	—
184	C₄×D₈	—	142	1	no	—
185	C₄×D₈	—	142	1	no	—
186	C₂×#108	—	71	1	no	—
187	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
188	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
189	C₂×#102	—	142	1	no	—
190	C₄×M₁₆	—	142	1	no	—
191	C₂×C₄×C₈	—	71	1	no	—
192	#361	—	71	4	no	—
193	#361	—	71	4	no	—
194	#346	—	142	1	no	—
195	C₄×C₁₆	—	142	1	no	—
196	#341	—	142	1	no	—
197	C₂²×D₄	—	142	1	no	—
198	C₂³×C₄	—	284	1	no	—
199	C₂×D₈	—	71	2	no	—
200	C₂×D₈	—	71	2	no	—
201	C₂×D₈	—	71	2	no	—
202	C₂×D₈	—	142	1	no	—
203	C₂×D₈	—	142	1	no	—
204	C₂×D₈	—	142	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	142	1	no	—
207	C₂²×C₈	—	71	1	no	—
208	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
209	C₄×D₄	—	284	1	no	—
210	C₄×D₄	—	284	1	no	—
211	C₄×D₄	—	142	2	no	—
212	C₄×D₄	—	142	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₂²⋊C₄	—	284	1	no	—
215	C₂×C₄²	—	71	1	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	71	2	no	—
218	M₃₂	—	142	1	no	—
219	M₃₂	C₄×D₇₁	1	2	yes	complemented
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	142	1	no	—
222	#108	—	2	1	no	—
223	#108	—	71	2	no	—
224	#108	—	142	1	no	—
225	#108	C₄×D₇₁	1	2	yes	complemented
226	#107	—	2	1	no	—
227	#107	C₄×D₇₁	1	2	yes	—
228	#102	—	142	2	no	—
229	#102	—	142	2	no	—
230	#97	—	284	1	no	—
231	C₄×C₈	—	284	1	no	—
232	C₄×C₈	—	71	1	no	—
233	C₄×C₈	—	71	1	no	—
234	C₄×C₈	—	71	1	no	—
235	C₄×C₈	—	71	1	no	—
236	C₂⁴	—	284	1	no	—
237	C₂×D₄	—	284	1	no	—
238	C₂×D₄	—	284	1	no	—
239	C₂×D₄	—	284	1	no	—
240	C₂×D₄	—	142	2	no	—
241	C₂×D₄	—	142	2	no	—
242	C₂×D₄	—	142	2	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	284	2	no	—
245	C₂²×C₄	—	284	2	no	—
246	C₂²×C₄	—	568	4	no	—
247	C₂²×C₄	—	71	1	no	—
248	C₂²×C₄	—	71	1	no	—
249	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
250	D₈	—	71	4	no	—
251	D₈	—	142	2	no	—
252	D₈	—	142	2	no	—
253	D₈	—	142	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	284	1	no	—
257	M₁₆	—	142	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	284	1	no	—
260	C₂×C₈	—	71	1	no	—
261	C₂×C₈	—	71	1	no	—
262	C₂×C₈	—	142	1	no	—
263	C₂×C₈	—	142	1	no	—
264	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
267	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
268	C₂×C₈	#11765864	1	1	yes	characteristic
269	C₂×C₈	#11765866	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	284	2	no	—
273	C₂²⋊C₄	—	284	2	no	—
274	C₄²	—	71	1	no	—
275	C₄²	—	71	1	no	—
276	C₄²	—	71	1	no	—
277	C₄²	—	71	1	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	142	2	no	—
280	C₂³	—	284	2	no	—
281	C₂³	—	284	2	no	—
282	C₂³	—	284	2	no	—
283	C₂³	—	568	2	no	—
284	C₂³	—	568	2	no	—
285	C₂³	SD₁₆×D₇₁	1	1	yes	characteristic
286	D₄	—	284	2	no	—
287	D₄	—	284	2	no	—
288	D₄	—	284	2	no	—
289	D₄	—	142	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	284	1	no	—
293	C₂×C₄	—	568	4	no	—
294	C₂×C₄	—	568	4	no	—
295	C₂×C₄	—	71	1	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	142	2	no	—
300	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
301	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
302	C₂×C₄	#14556090	1	1	yes	characteristic
303	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
304	C₂×C₄	#14556149	1	1	yes	characteristic
305	C₂×C₄	#14556151	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	284	2	no	—
308	C₈	—	142	1	no	—
309	C₈	—	142	1	no	—
310	C₈	C₄²×D₇₁	1	1	yes	characteristic
311	C₈	#14556086	1	1	yes	characteristic
312	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
313	C₈	#14556090	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	284	4	no	—
317	C₂²	—	568	4	no	—
318	C₂²	—	568	4	no	—
319	C₂²	—	568	2	no	—
320	C₂²	—	568	2	no	—
321	C₂²	#102×D₇₁	1	1	yes	characteristic
322	C₂²	#17986084	1	1	yes	characteristic
323	C₂²	#17986086	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	71	2	no	—
326	C₄	—	142	1	no	—
327	C₄	—	142	1	no	—
328	C₄	—	142	1	no	—
329	C₄	#17985846	1	1	yes	characteristic
330	C₄	#17985994	1	1	yes	characteristic
331	C₄	#17986079	1	1	yes	characteristic
332	C₄	#17986081	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	568	4	no	—
336	C₂	#22588154	1	1	yes	characteristic
337	C₂	#361×D₇₁	1	1	yes	characteristic
338	C₂	#22590088	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1387	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂²×D₅₆₈	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1387	D₇₁
hypocenter	C₇₁	C₄×#361
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₇₁	C₂	semidirect
5. #108×Dic₇₁	C₂	semidirect
6. #22586448	C₂	semidirect
7. C₇₁×#1387	C₂	semidirect
19. #17985813	C₄	semidirect
20. #17985811	C₄	semidirect
44. #14556068	C₂×C₄	semidirect
47. #14556064	C₂×C₄	semidirect
60. M₃₂×C₇₁	C₂×C₄	semidirect
63. #108×C₇₁	C₂×C₄	semidirect
162. Dic₇₁	#361	semidirect
176. #1387	D₇₁	semidirect
177. C₇₁	C₄×#361	semidirect
219. M₃₂	C₄×D₇₁	semidirect
225. #108	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986081	2
C₂×#17985813	2
C₂×#17985811	2
M₃₂×Dic₇₁	2
#108×Dic₇₁	2
#22586448	2
C₇₁×#1387	2
C₄×#361	71