Subgroups — #31912557

Order 18176.

426 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985784	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22588179	C₂	1	1	yes	maximal, characteristic
5	#22586429	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586421	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986415	—	2	2	no	—
10	#17986415	C₄	1	4	yes	complemented
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556067	—	2	1	no	—
13	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	—	2	1	no	—
15	C₂×#14556054	C₄	1	1	yes	characteristic
16	#17985993	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	#17985784	C₄	1	4	yes	complemented
19	C₁₆×Dic₇₁	—	2	2	no	—
20	#17985763	C₄	1	1	yes	characteristic
21	#17985758	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
24	#405×C₇₁	C₂²	1	1	yes	characteristic
25	#346×C₇₁	—	2	1	no	—
26	C₄×C₁₆×C₇₁	—	2	1	no	—
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556134	—	2	1	no	—
29	#14556134	—	2	1	no	—
30	#14556134	C₂×C₄	1	2	yes	—
31	#14556134	C₂×C₄	1	2	yes	—
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765848	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₁	—	2	1	no	—
36	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
37	#14556068	—	2	1	no	—
38	#14556068	C₂×C₄	1	2	yes	—
39	#14556067	—	2	4	no	—
40	C₂×C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₇₁⋊C₁₆	—	2	1	no	—
42	C₂×C₇₁⋊C₁₆	—	2	1	no	—
43	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
44	C₁₆×D₇₁	—	2	4	no	—
45	#14556054	C₈	1	4	yes	—
46	#14556051	C₈	1	2	yes	—
47	#14556048	C₈	1	2	yes	—
48	C₈×Dic₇₁	—	2	1	no	—
49	C₈×Dic₇₁	—	2	1	no	—
50	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
52	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₇₁	—	2	1	no	—
54	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
55	C₂×C₁₆×C₇₁	—	2	1	no	—
56	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
58	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
60	#97×C₇₁	—	2	1	no	—
61	C₄×C₈×C₇₁	—	2	1	no	—
62	#11765865	—	2	1	no	—
63	#11765865	—	2	1	no	—
64	#11765865	C₄²	1	2	yes	—
65	#11765865	C₂×C₈	1	2	yes	—
66	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
67	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₇₁	—	2	1	no	—
69	C₂²×Dic₇₁	—	2	1	no	—
70	C₂×C₄×D₇₁	—	2	1	no	—
71	#11765849	—	2	1	no	—
72	#11765849	C₄²	1	2	yes	—
73	#11765848	—	2	2	no	—
74	#11765848	—	2	2	no	—
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₂×C₇₁⋊C₈	—	2	1	no	—
77	C₂×C₇₁⋊C₈	—	2	1	no	—
78	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
79	C₈×D₇₁	—	2	2	no	—
80	C₈×D₇₁	—	2	2	no	—
81	C₄×Dic₇₁	—	2	1	no	—
82	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
83	C₇₁⋊C₁₆	—	2	2	no	—
84	C₇₁⋊C₁₆	—	2	2	no	—
85	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₇₁	—	2	1	no	—
87	M₁₆×C₇₁	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₇₁	—	2	1	no	—
89	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
96	C₂²⋊C₄×C₇₁	—	2	1	no	—
97	C₄²×C₇₁	—	2	1	no	—
98	C₁₆×C₇₁	—	2	2	no	—
99	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
100	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₇₁	#131	1	1	yes	characteristic
102	C₂²×D₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
110	C₂²⋊D₇₁	—	2	2	no	—
111	C₂²⋊D₇₁	—	2	2	no	—
112	C₂×Dic₇₁	—	2	4	no	—
113	C₂×Dic₇₁	—	2	2	no	—
114	C₂×Dic₇₁	—	2	1	no	—
115	D₂₈₄	—	2	1	no	—
116	D₂₈₄	C₄×C₈	1	2	yes	—
117	C₄×D₇₁	—	2	2	no	—
118	C₄×D₇₁	—	2	2	no	—
119	Dic₁₄₂	—	2	1	no	—
120	Dic₁₄₂	C₄×C₈	1	2	yes	—
121	C₈×C₇₁	—	2	2	no	—
122	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₇₁	#117	1	1	yes	characteristic
124	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
125	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
126	C₇₁⋊C₈	—	2	2	no	—
127	C₇₁⋊C₈	—	2	2	no	—
128	C₂²×C₇₁	—	2	2	no	—
129	C₂²×C₇₁	#405	1	1	yes	characteristic
130	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
132	C₂×D₇₁	—	2	2	no	—
133	C₂×D₇₁	—	4	2	no	—
134	C₄×C₇₁	—	4	2	no	—
135	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
138	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
139	Dic₇₁	—	2	4	no	—
140	Dic₇₁	—	2	2	no	—
141	#12614	—	71	1	no	maximal, Sylow
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	—	2	1	no	—
144	C₂×C₇₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₇₁	#2170	1	2	yes	—
146	D₇₁	—	4	2	no	—
147	C₂×#504	—	71	1	no	—
148	C₂×#350	—	71	1	no	—
149	#2108	D₇₁	1	1	yes	characteristic, complemented
150	C₂×C₄×C₁₆	—	71	1	no	—
151	#1764	—	71	1	no	—
152	#1379	—	71	1	no	—
153	#1376	—	71	1	no	—
154	C₇₁	#12614	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	71	1	no	—
156	#504	—	71	4	no	—
157	#504	—	142	2	no	—
158	C₂×M₃₂	—	71	1	no	—
159	C₂×M₃₂	—	142	1	no	—
160	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
161	C₂²×C₁₆	—	71	1	no	—
162	C₂²×C₁₆	—	142	1	no	—
163	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
164	C₂×#104	—	71	1	no	—
165	#416	—	71	1	no	—
166	#405	C₂×D₇₁	1	1	yes	characteristic
167	C₂×C₄×C₈	—	71	1	no	—
168	#350	—	71	4	no	—
169	#346	—	2	1	no	—
170	C₄×C₁₆	—	2	1	no	—
171	C₄×C₁₆	—	142	2	no	—
172	#341	—	71	1	no	—
173	#339	—	71	1	no	—
174	C₂×#41	—	71	1	no	—
175	#131	—	71	2	no	—
176	#131	—	71	2	no	—
177	#131	—	142	1	no	—
178	#131	—	142	1	no	—
179	C₂×M₁₆	—	71	1	no	—
180	C₂×M₁₆	—	142	1	no	—
181	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
182	C₂²×C₈	—	71	1	no	—
183	C₂²×C₈	—	142	1	no	—
184	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
185	#117	C₄×D₇₁	1	1	yes	characteristic
186	C₂×C₄²	—	71	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	71	2	no	—
189	M₃₂	—	142	1	no	—
190	M₃₂	—	142	4	no	—
191	M₃₂	C₄×D₇₁	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	71	2	no	—
194	C₂×C₁₆	—	142	1	no	—
195	C₂×C₁₆	—	142	1	no	—
196	C₂×C₁₆	—	142	1	no	—
197	C₂×C₁₆	—	142	4	no	—
198	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
199	C₂×C₁₆	D₂₈₄	1	2	yes	—
200	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
202	#104	—	71	4	no	—
203	#103	—	71	2	no	—
204	#102	—	71	2	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	142	1	no	—
208	C₄×C₈	—	142	1	no	—
209	#41	—	71	2	no	—
210	#41	—	71	2	no	—
211	#41	—	142	1	no	—
212	#41	—	142	1	no	—
213	C₂×Q₈	—	71	1	no	—
214	C₂×D₄	—	71	1	no	—
215	C₂×D₄	—	142	1	no	—
216	C₂²×C₄	—	142	1	no	—
217	C₂²×C₄	—	142	1	no	—
218	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	71	2	no	—
221	M₁₆	—	142	1	no	—
222	M₁₆	—	142	2	no	—
223	M₁₆	—	142	2	no	—
224	M₁₆	C₈×D₇₁	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	71	2	no	—
227	C₂×C₈	—	142	1	no	—
228	C₂×C₈	—	142	1	no	—
229	C₂×C₈	—	142	1	no	—
230	C₂×C₈	—	142	2	no	—
231	C₂×C₈	—	142	2	no	—
232	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
234	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
236	C₂×C₈	#11765865	1	1	yes	characteristic
237	C₂×C₈	#11765865	1	1	yes	characteristic
238	C₄⋊C₄	C₈×D₇₁	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	71	2	no	—
242	C₄²	—	142	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	142	2	no	—
245	C₁₆	—	142	2	no	—
246	C₁₆	(C₂×D₇₁)⋊C₄	1	2	yes	—
247	C₁₆	(C₂×D₇₁)⋊C₄	1	2	yes	—
248	C₂³	—	142	1	no	—
249	C₂³	#14556134	1	1	yes	characteristic
250	Q₈	—	71	2	no	—
251	Q₈	—	142	1	no	—
252	D₄	—	71	2	no	—
253	D₄	—	142	2	no	—
254	D₄	—	142	2	no	—
255	D₄	—	142	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	142	4	no	—
258	C₂×C₄	—	142	1	no	—
259	C₂×C₄	—	142	2	no	—
260	C₂×C₄	—	142	2	no	—
261	C₂×C₄	—	142	2	no	—
262	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
264	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
267	C₂×C₄	#14556134	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	142	2	no	—
270	C₈	—	142	2	no	—
271	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
272	C₈	#14556085	1	1	yes	characteristic
273	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
274	C₈	C₄×D₂₈₄	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	284	2	no	—
277	C₂²	—	142	2	no	—
278	C₂²	#17985953	1	1	yes	characteristic
279	C₂²	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
280	C₂²	C₈×D₂₈₄	1	1	yes	characteristic
281	C₄	—	4	2	no	—
282	C₄	—	142	4	no	—
283	C₄	—	142	2	no	—
284	C₄	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
285	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
286	C₄	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
287	C₄	C₈×D₂₈₄	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	284	2	no	—
291	C₂	C₈×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
292	C₂	#22590175	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×#11765865	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#12614
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×Dic₇₁	C₂	semidirect
5. #22586429	C₂	semidirect
6. #22586421	C₂	semidirect
7. C₇₁×#2108	C₂	semidirect
10. #17986415	C₄	semidirect
18. #17985784	C₄	semidirect
149. #2108	D₇₁	semidirect
154. C₇₁	#12614	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986415	2
C₂×#17985784	2
C₂×C₁₆×Dic₇₁	2
#22588179	2
#22586429	2
#22586421	2
C₇₁×#2108	2
#12614	71