Subgroups — #32150539

Order 18176.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590147	C₂	1	8	yes	maximal, complemented
2	#22590096	C₂	1	2	yes	maximal, complemented
3	#22589951	C₂	1	2	yes	maximal
4	C₄×#14556068	C₂	1	2	yes	maximal, complemented
5	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	4	yes	—
7	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	8	yes	complemented
8	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	8	yes	—
9	#17985775	C₂²	1	4	yes	complemented
10	C₄×C₇₁⋊C₁₆	C₂²	1	4	yes	complemented
11	C₂²×M₁₆×C₇₁	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#117×C₇₁	C₄	1	1	yes	characteristic
14	#405×C₇₁	C₄	1	4	yes	—
15	#405×C₇₁	C₂²	1	4	yes	—
16	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
17	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	#14556068	—	2	8	no	—
21	C₂×C₇₁⋊C₁₆	C₂³	1	8	yes	—
22	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₇₁	—	2	4	no	—
24	C₂×M₁₆×C₇₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₇₁	—	2	4	no	—
26	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
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	C₂³	1	2	yes	—
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	#117×C₇₁	C₈	1	4	yes	—
33	#117×C₇₁	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₇₁	C₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₇₁	C₈	1	2	yes	—
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	#97×C₇₁	C₂×C₄	1	4	yes	—
39	#97×C₇₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₁	C₂³	1	2	yes	—
43	C₄×C₈×C₇₁	C₂³	1	2	yes	—
44	C₇₁⋊C₁₆	—	2	8	no	—
45	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	—	2	2	no	—
47	C₂²×C₄×C₇₁	—	2	2	no	—
48	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₇₁	—	2	8	no	—
55	C₂×C₈×C₇₁	—	2	2	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	4	no	—
58	C₂×C₈×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₇₁	#41	1	4	yes	—
65	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
67	C₂²⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
68	C₄²×C₇₁	C₂×C₈	1	2	yes	—
69	C₄²×C₇₁	C₂×C₈	1	2	yes	—
70	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
72	C₂³×C₇₁	—	2	2	no	—
73	C₂³×C₇₁	—	4	2	no	—
74	C₂³×C₇₁	C₂²×C₈	1	2	yes	—
75	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	—	2	4	no	—
80	C₂×C₄×C₇₁	—	4	2	no	—
81	C₂×C₄×C₇₁	—	4	2	no	—
82	C₂×C₄×C₇₁	#117	1	2	yes	—
83	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₇₁	—	2	2	no	—
90	C₈×C₇₁	—	2	2	no	—
91	C₈×C₇₁	#117	1	2	yes	—
92	C₈×C₇₁	C₂×M₁₆	1	4	yes	—
93	C₈×C₇₁	C₂×#41	1	2	yes	—
94	C₂²×C₇₁	—	2	2	no	—
95	C₂²×C₇₁	—	2	4	no	—
96	C₂²×C₇₁	—	2	2	no	—
97	C₂²×C₇₁	—	4	4	no	—
98	C₂²×C₇₁	#431	1	2	yes	—
99	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
100	C₄×C₇₁	—	2	4	no	—
101	C₄×C₇₁	—	2	4	no	—
102	C₄×C₇₁	—	2	2	no	—
103	C₄×C₇₁	#431	1	2	yes	—
104	C₄×C₇₁	C₂×#117	1	1	yes	characteristic
105	C₄×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
106	#33419	—	71	1	no	maximal, Sylow
107	C₂×C₇₁	—	2	2	no	—
108	C₂×C₇₁	—	4	2	no	—
109	C₂×C₇₁	C₂×#431	1	1	yes	characteristic
110	C₂×C₇₁	#3407	1	2	yes	—
111	C₂×#405	D₇₁	1	1	yes	characteristic
112	#2164	—	71	8	no	—
113	#2150	—	71	2	no	—
114	#2112	—	71	2	no	—
115	C₄×M₃₂	—	71	2	no	—
116	C₇₁	#33419	1	1	yes	characteristic, complemented, Sylow
117	C₂²×M₁₆	Dic₇₁	1	1	yes	characteristic
118	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
119	C₂×#117	Dic₇₁	1	1	yes	characteristic
120	C₂×M₃₂	—	71	4	no	—
121	#405	Dic₇₁	1	4	yes	—
122	#405	C₂×D₇₁	1	4	yes	—
123	C₂×#97	Dic₇₁	1	1	yes	characteristic
124	C₂×#97	Dic₇₁	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	71	8	no	—
128	C₂²⋊C₁₆	—	71	8	no	—
129	#346	—	71	4	no	—
130	C₄×C₁₆	—	71	4	no	—
131	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	C₂×Dic₇₁	1	4	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
136	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
137	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
140	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
141	#117	C₇₁⋊C₈	1	4	yes	—
142	#117	C₂×Dic₇₁	1	4	yes	—
143	C₂×C₄⋊C₄	C₇₁⋊C₈	1	2	yes	—
144	C₂×C₂²⋊C₄	C₇₁⋊C₈	1	2	yes	—
145	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
147	M₃₂	—	142	8	no	—
148	C₂×C₁₆	—	71	8	no	—
149	#97	C₂×Dic₇₁	1	4	yes	—
150	#97	C₂×Dic₇₁	1	4	yes	—
151	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
152	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
153	C₄×C₈	C₂²×D₇₁	1	2	yes	—
154	C₄×C₈	C₂²×D₇₁	1	2	yes	—
155	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
160	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
161	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	#11765849	1	4	yes	—
170	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
171	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
172	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
174	C₂×C₈	#11765865	1	4	yes	—
175	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
177	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
178	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
179	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
180	C₄²	C₂²×Dic₇₁	1	2	yes	—
181	C₄²	C₂²×Dic₇₁	1	2	yes	—
182	C₁₆	—	142	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×C₇₁⋊C₈	1	2	yes	—
186	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#14556087	1	2	yes	—
194	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
195	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
196	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
197	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#11765849	1	2	yes	—
199	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	#14556087	1	2	yes	—
203	C₈	C₂×#11765849	1	4	yes	—
204	C₈	C₂×#11765865	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#17986052	1	2	yes	—
210	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	#17986052	1	2	yes	—
215	C₄	C₂×#14556087	1	1	yes	characteristic
216	C₄	C₂²×#11765849	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#17986052	1	1	yes	characteristic
220	C₂	#22601443	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₂×#405×C₇₁	C₂
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	#33419
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #22590147	C₂	semidirect
2. #22590096	C₂	semidirect
4. C₄×#14556068	C₂	semidirect
7. C₄⋊(C₇₁⋊C₁₆)	C₂²	semidirect
9. #17985775	C₂²	semidirect
10. C₄×C₇₁⋊C₁₆	C₂²	semidirect
116. C₇₁	#33419	semidirect

Maximal subgroups

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

subgroup	index
#22590147	2
#22590096	2
#22589951	2
C₄×#14556068	2
C₂×#405×C₇₁	2
#33419	71