Subgroups — #31912313

Order 18176.

586 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985954	C₂	1	1	yes	maximal, characteristic
3	#22586423	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11765849	C₄	1	1	yes	characteristic
6	C₂×#14556087	C₄	1	1	yes	characteristic
7	C₂×#14556068	—	2	2	no	—
8	C₂×#14556068	—	2	1	no	—
9	C₂×#14556068	C₄	1	4	yes	complemented
10	C₂×#14556068	C₂²	1	2	yes	—
11	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
12	#17985954	—	2	1	no	—
13	#17985954	—	2	1	no	—
14	#17985954	—	2	1	no	—
15	#17985954	—	2	1	no	—
16	C₂×#14556038	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
18	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₇₁	—	2	2	no	—
20	C₂×M₃₂×C₇₁	—	2	1	no	—
21	C₂×M₃₂×C₇₁	C₄	1	4	yes	complemented
22	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₇₁	—	2	1	no	—
24	C₂×#11765849	—	2	1	no	—
25	C₂×#11765849	—	2	1	no	—
26	C₂×#11765849	—	2	2	no	—
27	C₂×#11765849	C₈	1	4	yes	—
28	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
29	#14556087	—	2	1	no	—
30	#14556087	—	2	1	no	—
31	#14556087	—	2	1	no	—
32	#14556087	—	2	1	no	—
33	C₂×C₄⋊Dic₇₁	C₈	1	2	yes	—
34	C₂×C₂²⋊Dic₇₁	C₈	1	2	yes	—
35	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
36	#14556068	—	2	4	no	—
37	#14556068	—	2	2	no	—
38	#14556068	C₂×C₄	1	4	yes	—
39	C₂×C₇₁⋊C₁₆	—	2	4	no	—
40	C₂×C₇₁⋊C₁₆	—	2	2	no	—
41	#14556038	—	2	2	no	—
42	#14556038	—	2	2	no	—
43	C₈×Dic₇₁	—	2	2	no	—
44	C₈×Dic₇₁	—	2	2	no	—
45	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₇₁	D₄	1	2	yes	—
50	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
55	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
56	M₃₂×C₇₁	—	2	4	no	—
57	M₃₂×C₇₁	—	2	2	no	—
58	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
59	C₂×C₁₆×C₇₁	—	2	4	no	—
60	C₂×C₁₆×C₇₁	—	2	2	no	—
61	C₂²×Dic₇₁	C₂×C₈	1	4	yes	—
62	#11765849	—	2	4	no	—
63	#11765849	—	2	4	no	—
64	C₂×C₇₁⋊C₈	—	2	2	no	—
65	C₂×C₇₁⋊C₈	—	2	2	no	—
66	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
67	C₄⋊Dic₇₁	—	2	2	no	—
68	C₄⋊Dic₇₁	—	2	2	no	—
69	C₂²⋊Dic₇₁	—	2	2	no	—
70	C₂²⋊Dic₇₁	—	2	2	no	—
71	C₄×Dic₇₁	—	2	2	no	—
72	C₄×Dic₇₁	—	2	2	no	—
73	C₇₁⋊C₁₆	—	2	4	no	—
74	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
76	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
82	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₄²	1	2	yes	—
86	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
93	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
95	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
96	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
97	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
103	C₁₆×C₇₁	—	2	4	no	—
104	C₂³×C₇₁	—	4	2	no	—
105	C₂³×C₇₁	C₄×C₈	1	2	yes	—
106	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
109	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#95	1	2	yes	—
111	C₂×C₄×C₇₁	#95	1	2	yes	—
112	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₇₁	#97	1	2	yes	—
114	C₂×C₄×C₇₁	#97	1	2	yes	—
115	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
118	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
128	C₂×Dic₇₁	—	2	4	no	—
129	C₂×Dic₇₁	—	4	4	no	—
130	C₈×C₇₁	#95	1	2	yes	—
131	C₈×C₇₁	#95	1	2	yes	—
132	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₇₁⋊C₈	—	2	4	no	—
137	C₂²×C₇₁	—	2	2	no	—
138	C₂²×C₇₁	—	2	2	no	—
139	C₂²×C₇₁	—	2	2	no	—
140	C₂²×C₇₁	—	4	2	no	—
141	C₂²×C₇₁	—	4	2	no	—
142	C₂²×C₇₁	#336	1	2	yes	—
143	C₂²×C₇₁	#336	1	2	yes	—
144	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
145	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₇₁	#336	1	2	yes	—
148	C₄×C₇₁	#336	1	2	yes	—
149	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
150	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
151	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
153	Dic₇₁	—	4	4	no	—
154	#12551	—	71	1	no	maximal, Sylow
155	C₂×C₇₁	—	2	2	no	—
156	C₂×C₇₁	—	2	2	no	—
157	C₂×C₇₁	—	2	1	no	—
158	C₂×C₇₁	—	2	1	no	—
159	C₂×C₇₁	C₂×#336	1	1	yes	characteristic
160	C₂×C₇₁	#2115	1	2	yes	—
161	C₂²×M₃₂	—	71	1	no	—
162	C₂²×M₃₂	D₇₁	1	1	yes	characteristic
163	C₂×#405	—	71	1	no	—
164	#1377	—	71	4	no	—
165	C₇₁	#12551	1	1	yes	characteristic, complemented, Sylow
166	C₂²×M₁₆	—	71	1	no	—
167	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
168	C₂×#117	—	71	1	no	—
169	C₂×M₃₂	—	2	2	no	—
170	C₂×M₃₂	—	2	1	no	—
171	C₂×M₃₂	—	71	4	no	—
172	C₂×M₃₂	—	71	2	no	—
173	C₂×M₃₂	—	142	1	no	—
174	C₂×M₃₂	—	142	2	no	—
175	C₂×M₃₂	Dic₇₁	1	4	yes	complemented
176	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
177	C₂²×C₁₆	—	2	1	no	—
178	C₂²×C₁₆	—	142	1	no	—
179	#405	—	142	1	no	—
180	#405	—	142	1	no	—
181	#405	—	142	1	no	—
182	#405	—	142	1	no	—
183	C₂×#97	—	71	2	no	—
184	C₂×C₄×C₈	—	71	2	no	—
185	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
186	C₂×M₁₆	—	71	4	no	—
187	C₂×M₁₆	—	142	1	no	—
188	C₂×M₁₆	—	142	1	no	—
189	C₂×M₁₆	—	142	2	no	—
190	C₂²×C₈	—	71	2	no	—
191	C₂²×C₈	Dic₁₄₂	1	2	yes	—
192	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
193	C₂²×C₈	C₄×D₇₁	1	2	yes	—
194	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
195	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
196	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
197	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
201	#117	—	142	1	no	—
202	#117	—	142	1	no	—
203	#117	—	142	1	no	—
204	#117	—	142	1	no	—
205	C₂×C₄⋊C₄	—	71	2	no	—
206	C₂×C₂²⋊C₄	—	71	2	no	—
207	C₂×C₄²	—	71	2	no	—
208	M₃₂	—	2	4	no	—
209	M₃₂	—	2	2	no	—
210	M₃₂	—	71	4	no	—
211	M₃₂	—	142	4	no	—
212	M₃₂	—	142	2	no	—
213	M₃₂	C₂×Dic₇₁	1	4	yes	—
214	C₂×C₁₆	—	2	4	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	142	4	no	—
217	C₂×C₁₆	—	142	2	no	—
218	#97	—	142	2	no	—
219	#97	—	142	2	no	—
220	C₄×C₈	—	142	2	no	—
221	C₄×C₈	—	142	2	no	—
222	C₂⁴	C₈×D₇₁	1	1	yes	characteristic
223	C₂²×C₄	—	71	4	no	—
224	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
225	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
228	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
229	C₂²×C₄	C₈×D₇₁	1	2	yes	—
230	C₂²×C₄	#11765848	1	2	yes	—
231	C₂²×C₄	#11765848	1	1	yes	characteristic
232	C₂²×C₄	#11765848	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
234	M₁₆	—	142	4	no	—
235	M₁₆	—	142	4	no	—
236	C₂×C₈	—	71	4	no	—
237	C₂×C₈	—	142	2	no	—
238	C₂×C₈	—	142	2	no	—
239	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
240	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
241	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
242	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
243	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
245	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
246	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
247	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
248	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
249	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
250	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
253	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
254	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
255	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
256	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
257	C₄⋊C₄	—	142	2	no	—
258	C₄⋊C₄	—	142	2	no	—
259	C₂²⋊C₄	—	142	2	no	—
260	C₂²⋊C₄	—	142	2	no	—
261	C₄²	—	142	2	no	—
262	C₄²	—	142	2	no	—
263	C₁₆	—	2	4	no	—
264	C₁₆	—	142	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	C₈×Dic₇₁	1	2	yes	—
267	C₂³	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
268	C₂³	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
269	C₂³	Dic₇₁⋊C₈	1	2	yes	—
270	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
271	C₂×C₄	—	284	4	no	—
272	C₂×C₄	—	142	4	no	—
273	C₂×C₄	C₇₁⋊#95	1	2	yes	—
274	C₂×C₄	C₇₁⋊#95	1	2	yes	—
275	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
276	C₂×C₄	#14556038	1	2	yes	—
277	C₂×C₄	#14556038	1	2	yes	—
278	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
279	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
280	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
281	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
282	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
283	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
284	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
286	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
290	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
291	C₈	—	142	4	no	—
292	C₈	C₇₁⋊#95	1	2	yes	—
293	C₈	C₇₁⋊#95	1	2	yes	—
294	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
295	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
296	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
297	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#17985753	1	2	yes	—
304	C₂²	#17985753	1	2	yes	—
305	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
306	C₂²	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
307	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
308	C₄	—	284	4	no	—
309	C₄	#17985753	1	2	yes	—
310	C₄	#17985753	1	2	yes	—
311	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
312	C₄	C₂×#14556038	1	1	yes	characteristic
313	C₄	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
314	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	C₂×#17985753	1	1	yes	characteristic
320	C₂	#22589960	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₁)⋊C₄
commutator	C₂²×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂²×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂²×M₃₂	D₇₁
hypocenter	C₇₁	#12551
LCS	G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22586423	C₂	semidirect
9. C₂×#14556068	C₄	semidirect
21. C₂×M₃₂×C₇₁	C₄	semidirect
165. C₇₁	#12551	semidirect
175. C₂×M₃₂	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556068	2
C₂×#17985954	2
#22586423	2
C₂²×M₃₂×C₇₁	2
#12551	71