Subgroups — #31885256

Order 18176.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	C₄	1	1	yes	characteristic
5	C₁₆×Dic₇₁	C₄	1	1	yes	characteristic
6	C₇₁⋊C₆₄	C₄	1	2	yes	complemented
7	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
8	C₆₄×C₇₁	C₄	1	2	yes	complemented
9	C₂×C₇₁⋊C₁₆	C₈	1	1	yes	characteristic
10	C₈×Dic₇₁	C₈	1	1	yes	characteristic
11	C₇₁⋊C₃₂	C₈	1	2	yes	—
12	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₇₁⋊C₈	C₁₆	1	1	yes	characteristic
16	C₄×Dic₇₁	C₁₆	1	1	yes	characteristic
17	C₇₁⋊C₁₆	C₁₆	1	2	yes	—
18	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₇₁	C₄²	1	1	yes	characteristic
20	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
21	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₇₁	C₃₂	1	2	yes	—
23	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
24	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
25	C₇₁⋊C₈	C₃₂	1	2	yes	—
26	C₂²×C₇₁	C₂×C₃₂	1	1	yes	characteristic
27	C₄×C₇₁	C₄×C₁₆	1	1	yes	characteristic
28	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
29	Dic₇₁	—	2	2	no	—
30	#7510	—	71	1	no	maximal, Sylow
31	C₂×C₇₁	C₄×C₃₂	1	1	yes	characteristic
32	C₂×C₇₁	M₁₂₈	1	2	yes	—
33	C₂×C₆₄	—	71	1	no	—
34	C₂×C₆₄	D₇₁	1	1	yes	characteristic
35	C₄×C₃₂	—	71	1	no	—
36	C₇₁	#7510	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃₂	—	71	1	no	—
38	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
39	C₄×C₁₆	—	71	1	no	—
40	C₆₄	—	71	2	no	—
41	C₆₄	Dic₇₁	1	2	yes	complemented
42	C₂×C₁₆	—	71	1	no	—
43	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
44	C₄×C₈	—	71	1	no	—
45	C₃₂	—	71	2	no	—
46	C₃₂	C₄×D₇₁	1	1	yes	characteristic
47	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
48	C₂×C₈	—	71	1	no	—
49	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
50	C₄²	—	71	1	no	—
51	C₁₆	—	71	2	no	—
52	C₁₆	C₄×Dic₇₁	1	1	yes	characteristic
53	C₁₆	C₈×D₇₁	1	1	yes	characteristic
54	C₂×C₄	—	71	2	no	—
55	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
56	C₈	—	71	2	no	—
57	C₈	C₈×Dic₇₁	1	1	yes	characteristic
58	C₈	C₁₆×D₇₁	1	1	yes	characteristic
59	C₂²	C₃₂×D₇₁	1	1	yes	characteristic
60	C₄	—	142	2	no	—
61	C₄	C₁₆×Dic₇₁	1	1	yes	characteristic
62	C₄	C₃₂×D₇₁	1	1	yes	characteristic
63	C₂	C₃₂×Dic₇₁	1	1	yes	characteristic
64	C₂	#22586550	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₇₁
commutator	C₂×C₇₁	C₄×C₃₂
Fitting	C₂×C₆₄×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×C₃₂
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	C₂×C₆₄	D₇₁
hypocenter	C₇₁	#7510
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
6. C₇₁⋊C₆₄	C₄	semidirect
8. C₆₄×C₇₁	C₄	semidirect
36. C₇₁	#7510	semidirect
41. C₆₄	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇₁⋊C₆₄	2
C₃₂×Dic₇₁	2
C₂×C₆₄×C₇₁	2
#7510	71