Subgroups — #22596732

Order 9088.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17985890	C₂	1	7	yes	maximal
2	#17985883	C₂	1	7	yes	maximal, complemented
3	C₄³×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₇₁⋊C₄	C₂²	1	7	yes	—
5	C₂×(C₂×D₇₁)⋊C₄	C₂²	1	7	yes	—
6	C₇₁⋊#95	C₂²	1	14	yes	—
7	C₂×C₄²×C₇₁	C₂²	1	7	yes	—
8	C₂³×D₇₁	C₂³	1	1	yes	characteristic
9	C₂²×Dic₇₁	C₂³	1	7	yes	—
10	Dic₇₁⋊C₄	—	2	14	no	—
11	(C₂×D₇₁)⋊C₄	—	2	14	no	—
12	C₂²×C₄×C₇₁	C₂³	1	7	yes	—
13	C₄²×C₇₁	—	2	14	no	—
14	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
15	C₂²×D₇₁	—	2	7	no	—
16	C₂×C₄×C₇₁	—	2	14	no	—
17	C₂×C₄×C₇₁	#41	1	14	yes	—
18	C₂×Dic₇₁	—	2	7	no	—
19	C₂×Dic₇₁	—	2	14	no	—
20	C₂²×C₇₁	C₂×#41	1	7	yes	—
21	C₂×D₇₁	—	4	7	no	—
22	C₄×C₇₁	—	2	14	no	—
23	Dic₇₁	—	4	7	no	—
24	C₂×C₇₁	#529	1	7	yes	—
25	D₇₁	—	8	1	no	—
26	#2866	—	71	1	no	maximal, Sylow
27	C₇₁	#2866	1	1	yes	characteristic, complemented, Sylow
28	#388	—	71	7	no	—
29	#387	—	71	7	no	—
30	C₄³	D₇₁	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	71	7	no	—
32	C₂×C₂²⋊C₄	—	71	7	no	—
33	C₂×C₄²	C₂×D₇₁	1	7	yes	—
34	#95	—	71	14	no	—
35	C₂⁴	—	71	1	no	—
36	C₂²×C₄	—	71	7	no	—
37	C₂²×C₄	C₂²×D₇₁	1	7	yes	—
38	C₄⋊C₄	—	142	14	no	—
39	C₂²⋊C₄	—	142	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	142	7	no	—
42	C₂³	C₂³×D₇₁	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	142	7	no	—
45	C₂×C₄	—	142	14	no	—
46	C₂×C₄	#11765865	1	14	yes	—
47	C₂²	—	284	7	no	—
48	C₂²	C₂×#11765865	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	284	7	no	—
51	C₂	—	568	1	no	—
52	C₂	#17986533	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₁
commutator, socle	C₂³×C₇₁	C₂⁴
Fitting	C₄³×C₇₁	C₂
hypercenter	C₄³	D₇₁
hypocenter	C₇₁	#2866
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #17985883	C₂	semidirect
3. C₄³×C₇₁	C₂	semidirect
27. C₇₁	#2866	semidirect
30. C₄³	D₇₁	semidirect