Subgroups — #31914342

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601443	C₂	1	1	yes	maximal, characteristic, complemented
2	#22589968	C₂	1	2	yes	maximal, complemented
3	#22589961	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586422	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
6	#17986416	—	2	4	no	—
7	C₂×#14556068	—	2	2	no	—
8	C₂×#14556068	C₂²	1	2	yes	—
9	C₂×#14556068	C₂²	1	2	yes	—
10	#17985783	—	2	4	no	—
11	#17985781	—	2	2	no	—
12	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
14	#532×C₇₁	C₄	1	1	yes	characteristic
15	#436×C₇₁	—	2	2	no	—
16	C₈×D₄×C₇₁	—	2	2	no	—
17	#423×C₇₁	C₄	1	1	yes	characteristic
18	#408×C₇₁	C₄	1	1	yes	characteristic
19	#408×C₇₁	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	#14556068	—	2	4	no	—
22	#14556068	—	2	2	no	—
23	#14556068	—	2	2	no	—
24	#14556068	—	2	2	no	—
25	#14556068	D₄	1	4	yes	complemented
26	C₂×C₇₁⋊C₁₆	—	2	4	no	—
27	C₂×C₇₁⋊C₁₆	—	2	2	no	—
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
31	#131×C₇₁	—	2	2	no	—
32	#131×C₇₁	—	2	2	no	—
33	#131×C₇₁	—	2	2	no	—
34	#131×C₇₁	—	2	2	no	—
35	C₂×M₁₆×C₇₁	—	2	1	no	—
36	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
39	C₂²×C₈×C₇₁	—	2	1	no	—
40	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	D₄	1	2	yes	—
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₇₁	C₈	1	1	yes	characteristic
44	#127×C₇₁	C₈	1	1	yes	characteristic
45	#124×C₇₁	C₈	1	1	yes	characteristic
46	#124×C₇₁	C₈	1	1	yes	characteristic
47	#121×C₇₁	—	2	2	no	—
48	C₄×D₄×C₇₁	—	2	2	no	—
49	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₇₁	—	2	2	no	—
51	C₂²⋊C₈×C₇₁	—	2	2	no	—
52	C₂²⋊C₈×C₇₁	—	2	2	no	—
53	C₄×C₈×C₇₁	—	2	1	no	—
54	C₄×C₈×C₇₁	—	2	1	no	—
55	C₇₁⋊C₁₆	—	2	4	no	—
56	C₇₁⋊C₁₆	—	4	2	no	—
57	#41×C₇₁	—	2	2	no	—
58	#41×C₇₁	—	2	2	no	—
59	#41×C₇₁	—	2	2	no	—
60	#41×C₇₁	—	2	2	no	—
61	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
62	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
63	C₂×D₄×C₇₁	—	2	2	no	—
64	C₂×D₄×C₇₁	—	2	1	no	—
65	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₇₁	M₁₆	1	2	yes	—
68	C₂²×C₄×C₇₁	—	2	1	no	—
69	C₂²×C₄×C₇₁	—	2	1	no	—
70	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
72	M₁₆×C₇₁	—	2	2	no	—
73	M₁₆×C₇₁	—	2	4	no	—
74	M₁₆×C₇₁	—	2	2	no	—
75	M₁₆×C₇₁	—	4	2	no	—
76	C₂×C₈×C₇₁	—	2	2	no	—
77	C₂×C₈×C₇₁	—	2	2	no	—
78	C₂×C₈×C₇₁	—	2	1	no	—
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	2	no	—
81	C₂×C₈×C₇₁	—	4	1	no	—
82	C₂×C₈×C₇₁	—	4	1	no	—
83	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	#41	1	2	yes	—
88	C₄⋊C₄×C₇₁	—	2	2	no	—
89	C₂²⋊C₄×C₇₁	—	2	2	no	—
90	C₂²⋊C₄×C₇₁	—	2	2	no	—
91	C₄²×C₇₁	—	2	1	no	—
92	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
94	C₂³×C₇₁	—	2	1	no	—
95	C₂³×C₇₁	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
97	Q₈×C₇₁	—	2	2	no	—
98	Q₈×C₇₁	—	2	2	no	—
99	D₄×C₇₁	—	2	4	no	—
100	D₄×C₇₁	—	2	2	no	—
101	D₄×C₇₁	—	2	2	no	—
102	D₄×C₇₁	—	4	4	no	—
103	D₄×C₇₁	—	4	2	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	4	2	no	—
110	C₂×C₄×C₇₁	—	4	1	no	—
111	C₂×C₄×C₇₁	—	4	1	no	—
112	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#131	1	2	yes	—
117	C₈×C₇₁	—	2	2	no	—
118	C₈×C₇₁	—	4	1	no	—
119	C₈×C₇₁	—	4	1	no	—
120	C₈×C₇₁	#120	1	1	yes	characteristic
121	C₈×C₇₁	#121	1	2	yes	—
122	C₈×C₇₁	#123	1	1	yes	characteristic
123	C₂²×C₇₁	—	2	2	no	—
124	C₂²×C₇₁	—	4	1	no	—
125	C₂²×C₇₁	—	4	2	no	—
126	C₂²×C₇₁	—	4	2	no	—
127	C₂²×C₇₁	—	4	1	no	—
128	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₇₁	#435	1	2	yes	—
130	C₄×C₇₁	—	2	2	no	—
131	C₄×C₇₁	—	2	2	no	—
132	C₄×C₇₁	—	4	1	no	—
133	C₄×C₇₁	—	4	1	no	—
134	C₄×C₇₁	#386	1	1	yes	characteristic
135	C₄×C₇₁	#408	1	1	yes	characteristic
136	C₄×C₇₁	C₈×D₄	1	2	yes	—
137	#12982	—	71	1	no	maximal, Sylow
138	C₂×C₇₁	—	2	1	no	—
139	C₂×C₇₁	—	2	2	no	—
140	C₂×C₇₁	—	4	2	no	—
141	C₂×C₇₁	—	8	1	no	—
142	C₂×C₇₁	#1869	1	1	yes	characteristic
143	#3407	—	71	1	no	—
144	#2931	D₇₁	1	1	yes	characteristic
145	#2117	—	71	2	no	—
146	#2115	—	71	1	no	—
147	#1376	—	71	2	no	—
148	C₇₁	#12982	1	1	yes	characteristic, complemented, Sylow
149	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
150	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
151	#532	Dic₇₁	1	1	yes	characteristic
152	#504	—	142	4	no	—
153	C₂×M₃₂	—	71	2	no	—
154	C₂×M₃₂	—	71	2	no	—
155	C₂×M₃₂	—	142	2	no	—
156	#436	—	2	2	no	—
157	C₈×D₄	—	2	2	no	—
158	#423	Dic₇₁	1	1	yes	characteristic
159	#408	Dic₇₁	1	1	yes	characteristic
160	#408	Dic₇₁	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
162	#350	—	142	4	no	—
163	#349	—	142	2	no	—
164	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
165	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
166	#131	—	2	2	no	—
167	#131	—	2	2	no	—
168	#131	—	2	2	no	—
169	#131	—	2	2	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
172	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
173	C₂×M₁₆	C₂²⋊D₇₁	1	2	yes	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
176	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
177	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
178	C₄⋊Q₈	C₇₁⋊C₈	1	1	yes	characteristic
179	#127	C₇₁⋊C₈	1	1	yes	characteristic
180	#124	C₇₁⋊C₈	1	1	yes	characteristic
181	#124	C₇₁⋊C₈	1	1	yes	characteristic
182	#121	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
185	M₃₂	—	71	4	no	—
186	M₃₂	—	142	2	no	—
187	M₃₂	—	142	2	no	—
188	M₃₂	—	142	4	no	—
189	M₃₂	—	142	2	no	—
190	C₂×C₁₆	—	142	2	no	—
191	C₂×C₁₆	—	142	4	no	—
192	C₂×C₁₆	—	142	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
203	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
207	C₂×D₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
208	C₂×D₄	#11765849	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
225	C₂×C₈	D₄×D₇₁	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
227	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
228	C₂×C₈	#11765864	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
234	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
235	C₁₆	—	284	2	no	—
236	C₁₆	—	142	4	no	—
237	C₂³	—	2	1	no	—
238	C₂³	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
239	C₂³	C₂×#11765849	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
258	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
259	C₂×C₄	#14556135	1	2	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	#14556096	1	1	yes	characteristic
264	C₈	#14556100	1	2	yes	—
265	C₈	#14556109	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
272	C₂²	#17986071	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	#17985879	1	1	yes	characteristic
278	C₄	#17985966	1	1	yes	characteristic
279	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	#22588633	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556096
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2931	C₂
socle	C₂×C₇₁	#1869
prime radical	#532×C₇₁	C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2931	D₇₁
hypocenter	C₇₁	#12982
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #22601443	C₂	semidirect
2. #22589968	C₂	semidirect
3. #22589961	C₂	semidirect
4. #22586422	C₂	semidirect
25. #14556068	D₄	semidirect
148. C₇₁	#12982	semidirect

Maximal subgroups

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

subgroup	index
#22601443	2
#22589968	2
#22589961	2
#22586422	2
C₇₁×#2931	2
#12982	71