Subgroups — #31884067

Order 18176.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
6	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
7	C₈²×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	—
9	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	—
10	C₂×C₁₆×C₇₁	C₈	1	4	yes	—
11	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₇₁⋊C₁₆	—	2	4	no	—
15	C₇₁⋊C₁₆	C₁₆	1	8	yes	complemented
16	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
17	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
18	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
19	C₄²×C₇₁	C₄²	1	1	yes	characteristic
20	C₁₆×C₇₁	—	2	4	no	—
21	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
22	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
23	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
24	C₈×C₇₁	C₂×C₁₆	1	4	yes	—
25	C₈×C₇₁	M₃₂	1	4	yes	—
26	C₈×C₇₁	M₃₂	1	4	yes	—
27	C₂²×C₇₁	C₈²	1	1	yes	characteristic
28	C₄×C₇₁	C₄×C₁₆	1	2	yes	—
29	C₄×C₇₁	#346	1	2	yes	—
30	C₄×C₇₁	#346	1	2	yes	—
31	#7052	—	71	1	no	maximal, Sylow
32	C₂×C₇₁	C₈×C₁₆	1	1	yes	characteristic
33	C₂×C₇₁	#1311	1	1	yes	characteristic
34	C₂×C₇₁	#1311	1	1	yes	characteristic
35	C₈×C₁₆	—	71	1	no	—
36	C₈×C₁₆	—	71	1	no	—
37	C₈×C₁₆	D₇₁	1	1	yes	characteristic
38	C₇₁	#7052	1	1	yes	characteristic, complemented, Sylow
39	C₄×C₁₆	—	71	2	no	—
40	C₄×C₁₆	—	71	2	no	—
41	C₄×C₁₆	Dic₇₁	1	2	yes	—
42	C₈²	C₂×D₇₁	1	1	yes	characteristic
43	C₂×C₁₆	—	71	4	no	—
44	C₂×C₁₆	—	71	4	no	—
45	C₂×C₁₆	C₇₁⋊C₈	1	4	yes	—
46	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
47	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
48	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
49	C₂×C₈	C₈×D₇₁	1	2	yes	—
50	C₂×C₈	C₈×D₇₁	1	2	yes	—
51	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
52	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
53	C₁₆	—	2	4	no	—
54	C₁₆	—	71	8	no	—
55	C₁₆	—	142	4	no	—
56	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
57	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
58	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
59	C₈	C₁₆×D₇₁	1	4	yes	—
60	C₈	#14556067	1	4	yes	—
61	C₈	#14556068	1	4	yes	—
62	C₂²	C₈×C₇₁⋊C₈	1	1	yes	characteristic
63	C₄	C₁₆×Dic₇₁	1	2	yes	—
64	C₄	#17985774	1	2	yes	—
65	C₄	#17985775	1	2	yes	—
66	C₂	C₁₆×C₇₁⋊C₈	1	1	yes	characteristic
67	C₂	#22586256	1	1	yes	characteristic
68	C₂	#22586257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈²	C₂×D₇₁
commutator	C₂×C₇₁	C₈×C₁₆
Fitting	C₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₈²
cosocle	C₈²×C₇₁	C₂²
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#7052
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₇₁⋊C₁₆	C₁₆	semidirect
38. C₇₁	#7052	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₇₁⋊C₁₆	2
C₈×C₁₆×C₇₁	2
#7052	71