Subgroups — #31884780

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	2	yes	maximal
2	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇₁⋊C₃₂	C₄	1	4	yes	—
4	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
6	C₈²×C₇₁	C₄	1	1	yes	characteristic
7	C₇₁⋊C₃₂	C₈	1	8	yes	complemented
8	C₂×C₁₆×C₇₁	C₈	1	2	yes	—
9	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
10	C₄×C₈×C₇₁	C₈	1	2	yes	—
11	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₇₁	C₁₆	1	4	yes	—
13	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
14	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
16	C₁₆×C₇₁	C₁₆	1	4	yes	—
17	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
18	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
20	C₈×C₇₁	—	2	4	no	—
21	C₈×C₇₁	C₄×C₈	1	2	yes	—
22	C₈×C₇₁	C₂×C₁₆	1	2	yes	—
23	C₂²×C₇₁	C₄×C₁₆	1	1	yes	characteristic
24	C₄×C₇₁	C₈²	1	1	yes	characteristic
25	C₄×C₇₁	C₄×C₁₆	1	1	yes	characteristic
26	C₄×C₇₁	M₆₄	1	4	yes	—
27	#7329	—	71	1	no	maximal, Sylow
28	C₂×C₇₁	C₈×C₁₆	1	1	yes	characteristic
29	C₂×C₇₁	#1397	1	2	yes	—
30	C₄×C₃₂	—	71	2	no	—
31	C₈×C₁₆	D₇₁	1	1	yes	characteristic
32	C₇₁	#7329	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃₂	—	71	4	no	—
34	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
35	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
36	C₈²	Dic₇₁	1	1	yes	characteristic
37	C₂×C₁₆	C₇₁⋊C₈	1	2	yes	—
38	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
39	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
40	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
41	C₃₂	—	71	8	no	—
42	C₂×C₈	C₇₁⋊C₁₆	1	4	yes	—
43	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
44	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
45	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
46	C₁₆	C₇₁⋊C₁₆	1	4	yes	—
47	C₁₆	C₈×D₇₁	1	4	yes	—
48	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
49	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
50	C₈	—	2	4	no	—
51	C₈	C₈×Dic₇₁	1	2	yes	—
52	C₈	C₂×C₇₁⋊C₁₆	1	2	yes	—
53	C₂²	C₄×C₇₁⋊C₁₆	1	1	yes	characteristic
54	C₄	C₈×C₇₁⋊C₈	1	1	yes	characteristic
55	C₄	C₄×C₇₁⋊C₁₆	1	1	yes	characteristic
56	C₄	#17985833	1	4	yes	—
57	C₂	C₈×C₇₁⋊C₁₆	1	1	yes	characteristic
58	C₂	#22586474	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
7. C₇₁⋊C₃₂	C₈	semidirect
32. C₇₁	#7329	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
C₈×C₁₆×C₇₁	2
#7329	71