Subgroups — #31884779

Order 18176.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	C₄	1	2	yes	—
5	C₄×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₇₁	C₄	1	2	yes	—
8	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₇₁⋊C₁₆	C₈	1	2	yes	—
10	C₄×C₇₁⋊C₈	C₈	1	2	yes	—
11	C₇₁⋊C₃₂	C₈	1	4	yes	complemented
12	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₇₁	C₈	1	4	yes	complemented
16	C₂×C₇₁⋊C₈	C₁₆	1	4	yes	—
17	C₇₁⋊C₁₆	C₁₆	1	4	yes	—
18	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
19	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
20	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
21	C₁₆×C₇₁	C₂×C₈	1	2	yes	—
22	C₁₆×C₇₁	C₂×C₈	1	2	yes	—
23	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
25	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
26	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
27	C₈×C₇₁	C₂×C₁₆	1	2	yes	—
28	C₇₁⋊C₈	—	2	4	no	—
29	C₂²×C₇₁	C₄×C₁₆	1	1	yes	characteristic
30	C₄×C₇₁	C₈²	1	1	yes	characteristic
31	C₄×C₇₁	C₄×C₁₆	1	1	yes	characteristic
32	C₄×C₇₁	M₆₄	1	4	yes	—
33	#7329	—	71	1	no	maximal, Sylow
34	C₂×C₇₁	C₈×C₁₆	1	1	yes	characteristic
35	C₂×C₇₁	#1397	1	2	yes	—
36	C₄×C₃₂	—	71	1	no	—
37	C₄×C₃₂	D₇₁	1	1	yes	characteristic
38	C₈×C₁₆	—	71	1	no	—
39	C₇₁	#7329	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃₂	—	71	2	no	—
41	C₂×C₃₂	Dic₇₁	1	2	yes	—
42	C₄×C₁₆	—	71	1	no	—
43	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
44	C₈²	—	71	1	no	—
45	C₂×C₁₆	—	71	2	no	—
46	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
48	C₄×C₈	—	71	2	no	—
49	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
50	C₃₂	—	71	4	no	—
51	C₃₂	C₇₁⋊C₈	1	4	yes	complemented
52	C₂×C₈	—	71	4	no	—
53	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
54	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
55	C₄²	C₈×D₇₁	1	1	yes	characteristic
56	C₁₆	—	71	4	no	—
57	C₁₆	C₈×D₇₁	1	2	yes	—
58	C₁₆	C₂×C₇₁⋊C₈	1	2	yes	—
59	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
60	C₂×C₄	C₁₆×D₇₁	1	2	yes	—
61	C₈	—	142	4	no	—
62	C₈	C₈×Dic₇₁	1	1	yes	characteristic
63	C₈	C₄×C₇₁⋊C₈	1	1	yes	characteristic
64	C₈	C₁₆×D₇₁	1	2	yes	—
65	C₂²	C₁₆×Dic₇₁	1	1	yes	characteristic
66	C₄	C₈×C₇₁⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×Dic₇₁	1	1	yes	characteristic
68	C₄	#17985832	1	4	yes	—
69	C₂	C₁₆×C₇₁⋊C₈	1	1	yes	characteristic
70	C₂	#22586473	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₈×C₁₆
Fitting	C₄×C₃₂×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₄×C₁₆
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	C₄×C₃₂	D₇₁
hypocenter	C₇₁	#7329
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
11. C₇₁⋊C₃₂	C₈	semidirect
15. C₃₂×C₇₁	C₈	semidirect
39. C₇₁	#7329	semidirect
51. C₃₂	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇₁⋊C₃₂	2
C₁₆×C₇₁⋊C₈	2
C₄×C₃₂×C₇₁	2
#7329	71