Subgroups — #31913373

Order 18176.

566 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal
3	#22586418	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
6	C₄⋊(C₇₁⋊C₁₆)	—	2	4	no	—
7	C₂²⋊(C₇₁⋊C₁₆)	—	2	4	no	—
8	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₇₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	—	2	4	no	—
12	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₇₁⋊C₁₆	—	2	4	no	—
17	C₂×C₇₁⋊C₁₆	—	2	4	no	—
18	C₂×C₇₁⋊C₁₆	—	2	4	no	—
19	C₂³×C₄×C₇₁	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₇₁	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	—	2	2	no	—
23	C₂²×C₈×C₇₁	—	2	2	no	—
24	C₂²×C₈×C₇₁	—	2	2	no	—
25	C₂²×C₈×C₇₁	—	2	2	no	—
26	C₂²×C₈×C₇₁	—	2	2	no	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	D₄	1	2	yes	—
31	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	—	2	2	no	—
34	C₂×C₄²×C₇₁	—	2	2	no	—
35	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	—	2	2	no	—
40	C₄×C₈×C₇₁	—	2	2	no	—
41	C₄×C₈×C₇₁	—	2	2	no	—
42	C₄×C₈×C₇₁	D₄	1	4	yes	—
43	C₇₁⋊C₁₆	—	4	4	no	—
44	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	—	2	2	no	—
46	C₂²×C₄×C₇₁	—	2	2	no	—
47	C₂²×C₄×C₇₁	—	2	2	no	—
48	C₂²×C₄×C₇₁	—	2	2	no	—
49	C₂²×C₄×C₇₁	—	2	2	no	—
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	—	2	1	no	—
52	C₂²×C₄×C₇₁	—	2	1	no	—
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	C₁₆	1	4	yes	complemented
55	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
62	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	—	2	8	no	—
64	C₂×C₈×C₇₁	—	2	1	no	—
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	1	no	—
67	C₂×C₈×C₇₁	—	2	1	no	—
68	C₂×C₈×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	—	2	1	no	—
70	C₂×C₈×C₇₁	—	2	4	no	—
71	C₂×C₈×C₇₁	—	2	4	no	—
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
75	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₇₁	#41	1	2	yes	—
79	C₄²×C₇₁	—	2	2	no	—
80	C₄²×C₇₁	—	2	2	no	—
81	C₄²×C₇₁	—	2	1	no	—
82	C₄²×C₇₁	—	2	1	no	—
83	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₇₁	M₁₆	1	2	yes	—
85	C₂³×C₇₁	—	2	1	no	—
86	C₂³×C₇₁	—	2	1	no	—
87	C₂³×C₇₁	—	2	1	no	—
88	C₂³×C₇₁	—	2	1	no	—
89	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
90	C₂³×C₇₁	C₂×C₁₆	1	2	yes	—
91	C₂³×C₇₁	M₃₂	1	2	yes	—
92	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₇₁	—	2	8	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	8	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	—	2	2	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
111	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
112	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
113	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
114	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
120	C₈×C₇₁	—	2	2	no	—
121	C₈×C₇₁	—	2	2	no	—
122	C₈×C₇₁	—	2	2	no	—
123	C₈×C₇₁	#121	1	2	yes	—
124	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
125	C₂²×C₇₁	—	2	2	no	—
126	C₂²×C₇₁	—	2	2	no	—
127	C₂²×C₇₁	—	2	2	no	—
128	C₂²×C₇₁	—	2	2	no	—
129	C₂²×C₇₁	—	2	2	no	—
130	C₂²×C₇₁	—	2	2	no	—
131	C₂²×C₇₁	C₄⋊C₁₆	1	4	yes	complemented
132	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₇₁	#431	1	1	yes	characteristic
135	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
136	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
137	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
138	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
139	C₄×C₇₁	—	2	2	no	—
140	C₄×C₇₁	—	2	2	no	—
141	C₄×C₇₁	—	2	2	no	—
142	C₄×C₇₁	—	2	2	no	—
143	C₄×C₇₁	C₂²⋊C₁₆	1	4	yes	complemented
144	C₄×C₇₁	#380	1	1	yes	characteristic
145	C₄×C₇₁	#407	1	1	yes	characteristic
146	C₄×C₇₁	#423	1	1	yes	characteristic
147	C₄×C₇₁	#432	1	1	yes	characteristic
148	#12783	—	71	1	no	maximal, Sylow
149	C₂×C₇₁	—	2	4	no	—
150	C₂×C₇₁	#1797	1	1	yes	characteristic
151	C₂×C₇₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
152	C₂×C₇₁	#2112	1	1	yes	characteristic
153	C₂×C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
154	C₂×C₇₁	#2150	1	1	yes	characteristic
155	C₂×C₇₁	#2162	1	1	yes	characteristic
156	C₂×C₇₁	#2163	1	1	yes	characteristic
157	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
158	C₂×C₄⋊C₁₆	—	71	2	no	—
159	C₂×C₂²⋊C₁₆	—	71	2	no	—
160	#1374	—	71	2	no	—
161	C₇₁	#12783	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
163	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
164	C₂²×C₄²	Dic₇₁	1	1	yes	characteristic
165	C₂²×C₁₆	—	71	4	no	—
166	C₂×C₄×C₈	—	2	4	no	—
167	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
168	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
171	C₄⋊C₁₆	—	142	4	no	—
172	C₂²⋊C₁₆	—	142	4	no	—
173	C₂³×C₄	C₇₁⋊C₈	1	1	yes	characteristic
174	C₂³×C₄	C₇₁⋊C₈	1	1	yes	characteristic
175	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	Dic₁₄₂	1	2	yes	—
182	C₂²×C₈	D₂₈₄	1	2	yes	—
183	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
184	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₇₁⋊C₈	1	1	yes	characteristic
190	C₂×C₄²	C₇₁⋊C₈	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
193	C₂×C₁₆	—	142	4	no	—
194	C₂×C₁₆	—	142	4	no	—
195	C₂×C₁₆	—	142	4	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂²⋊D₇₁	1	4	yes	—
200	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₇₁⋊C₁₆	1	4	yes	complemented
211	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
212	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
217	C₂²×C₄	#11765849	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
229	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
230	C₂×C₈	#11765849	1	2	yes	—
231	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
232	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
233	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
234	C₂×C₈	#11765865	1	2	yes	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
240	C₄²	#11765849	1	2	yes	—
241	C₁₆	—	284	4	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
247	C₂³	C₂×C₇₁⋊C₁₆	1	2	yes	—
248	C₂³	#14556068	1	2	yes	—
249	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	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	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
265	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
266	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
267	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
268	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
269	C₂×C₄	#14556068	1	2	yes	—
270	C₂×C₄	#14556068	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
273	C₂×C₄	#14556087	1	1	yes	characteristic
274	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
275	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
276	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	#14556102	1	2	yes	—
281	C₈	#14556107	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₄⋊(C₇₁⋊C₁₆)	1	4	yes	complemented
289	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
290	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
291	C₂²	#17986052	1	1	yes	characteristic
292	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
293	C₂²	C₂×#14556068	1	1	yes	characteristic
294	C₂²	C₂×#14556068	1	1	yes	characteristic
295	C₂²	C₂×#14556068	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	C₂²⋊(C₇₁⋊C₁₆)	1	4	yes	complemented
301	C₄	#17985859	1	1	yes	characteristic
302	C₄	#17985963	1	1	yes	characteristic
303	C₄	#17986023	1	1	yes	characteristic
304	C₄	#17986056	1	1	yes	characteristic
305	C₂	—	2	4	no	—
306	C₂	#22588320	1	1	yes	characteristic
307	C₂	C₂×C₂²⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
308	C₂	#22589951	1	1	yes	characteristic
309	C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
310	C₂	#22590096	1	1	yes	characteristic
311	C₂	#22590139	1	1	yes	characteristic
312	C₂	#22590143	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₈
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	#12783
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
3. #22586418	C₂	semidirect
54. C₂²×C₄×C₇₁	C₁₆	semidirect
131. C₂²×C₇₁	C₄⋊C₁₆	semidirect
143. C₄×C₇₁	C₂²⋊C₁₆	semidirect
161. C₇₁	#12783	semidirect
210. C₂²×C₄	C₇₁⋊C₁₆	semidirect
288. C₂²	C₄⋊(C₇₁⋊C₁₆)	semidirect
300. C₄	C₂²⋊(C₇₁⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×C₂²⋊(C₇₁⋊C₁₆)	2
#22586418	2
C₂²×C₄×C₈×C₇₁	2
#12783	71