Subgroups — #31884066

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	2	yes	maximal
2	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	—
4	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
5	C₈²×C₇₁	C₂²	1	1	yes	characteristic
6	C₂×C₇₁⋊C₁₆	C₈	1	8	yes	—
7	C₂×C₁₆×C₇₁	C₈	1	4	yes	—
8	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
9	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
10	C₇₁⋊C₁₆	—	2	8	no	—
11	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
12	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
13	C₄²×C₇₁	C₄²	1	1	yes	characteristic
14	C₁₆×C₇₁	C₁₆	1	8	yes	complemented
15	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
16	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
17	C₈×C₇₁	C₂×C₁₆	1	4	yes	—
18	C₈×C₇₁	M₃₂	1	8	yes	—
19	C₂²×C₇₁	C₈²	1	1	yes	characteristic
20	C₄×C₇₁	C₄×C₁₆	1	2	yes	—
21	C₄×C₇₁	#346	1	4	yes	—
22	#7052	—	71	1	no	maximal, Sylow
23	C₂×C₇₁	C₈×C₁₆	1	1	yes	characteristic
24	C₂×C₇₁	#1311	1	2	yes	—
25	C₈×C₁₆	—	71	2	no	—
26	C₈×C₁₆	D₇₁	1	1	yes	characteristic
27	C₇₁	#7052	1	1	yes	characteristic, complemented, Sylow
28	C₄×C₁₆	—	71	4	no	—
29	C₄×C₁₆	Dic₇₁	1	2	yes	—
30	C₈²	C₂×D₇₁	1	1	yes	characteristic
31	C₂×C₁₆	—	71	8	no	—
32	C₂×C₁₆	C₇₁⋊C₈	1	4	yes	—
33	C₄×C₈	C₄×D₇₁	1	2	yes	—
34	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
35	C₂×C₈	C₈×D₇₁	1	4	yes	—
36	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
37	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
38	C₁₆	—	142	8	no	—
39	C₁₆	C₇₁⋊C₁₆	1	8	yes	complemented
40	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
41	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
42	C₈	C₂×C₇₁⋊C₁₆	1	4	yes	—
43	C₈	#14556067	1	8	yes	—
44	C₂²	C₈×C₇₁⋊C₈	1	1	yes	characteristic
45	C₄	C₄×C₇₁⋊C₁₆	1	2	yes	—
46	C₄	#17985774	1	4	yes	—
47	C₂	C₈×C₇₁⋊C₁₆	1	1	yes	characteristic
48	C₂	#22586256	1	2	yes	—

	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
14. C₁₆×C₇₁	C₁₆	semidirect
27. C₇₁	#7052	semidirect
39. C₁₆	C₇₁⋊C₁₆	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