Subgroups — #31928658

Order 18176.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985869	C₂	1	14	yes	maximal
2	C₂×C₄³×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Dic₇₁	C₂²	1	28	yes	—
4	#17985869	—	2	56	no	—
5	C₂²×C₄²×C₇₁	C₂²	1	7	yes	—
6	C₄³×C₇₁	C₄	1	8	yes	complemented
7	C₂³×Dic₇₁	C₂³	1	8	yes	—
8	C₂×C₄⋊Dic₇₁	—	2	168	no	—
9	C₂³×C₄×C₇₁	C₂³	1	7	yes	—
10	C₂×C₄²×C₇₁	C₂×C₄	1	28	yes	—
11	C₂×C₄²×C₇₁	D₄	1	28	yes	—
12	C₂×C₄²×C₇₁	Q₈	1	28	yes	—
13	C₂²×Dic₇₁	—	2	56	no	—
14	C₄⋊Dic₇₁	—	4	112	no	—
15	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
16	C₂²×C₄×C₇₁	C₂²×C₄	1	14	yes	—
17	C₂²×C₄×C₇₁	C₂×D₄	1	42	yes	—
18	C₂²×C₄×C₇₁	C₂×Q₈	1	42	yes	—
19	C₄²×C₇₁	C₄⋊C₄	1	112	yes	complemented
20	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
21	C₂³×C₇₁	C₂²×D₄	1	7	yes	—
22	C₂³×C₇₁	C₂²×Q₈	1	7	yes	—
23	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	84	yes	—
24	C₂×C₄×C₇₁	#127	1	28	yes	—
25	C₂×C₄×C₇₁	C₄⋊Q₈	1	84	yes	—
26	C₂×Dic₇₁	—	4	56	no	—
27	C₂²×C₇₁	C₂²×C₄⋊C₄	1	7	yes	—
28	C₂²×C₇₁	C₂×#127	1	7	yes	—
29	C₂²×C₇₁	C₂×C₄⋊Q₈	1	21	yes	—
30	C₄×C₇₁	#384	1	56	yes	complemented
31	Dic₇₁	—	8	8	no	—
32	#14692	—	71	1	no	maximal, Sylow
33	C₂×C₇₁	C₂×#384	1	7	yes	—
34	C₂×C₇₁	#2867	1	1	yes	characteristic
35	C₂×C₇₁	#2868	1	7	yes	—
36	C₂×#384	—	71	14	no	—
37	C₂×C₄³	D₇₁	1	1	yes	characteristic
38	C₇₁	#14692	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₄⋊C₄	—	71	28	no	—
40	C₂²×C₄²	C₂×D₇₁	1	7	yes	—
41	#384	—	142	56	no	—
42	C₄³	Dic₇₁	1	8	yes	complemented
43	C₂³×C₄	—	71	8	no	—
44	C₂³×C₄	C₂²×D₇₁	1	7	yes	—
45	C₂×C₄⋊C₄	—	142	168	no	—
46	C₂×C₄²	Dic₁₄₂	1	28	yes	—
47	C₂×C₄²	D₂₈₄	1	28	yes	—
48	C₂×C₄²	C₂×Dic₇₁	1	28	yes	—
49	C₂⁴	C₂³×D₇₁	1	1	yes	characteristic
50	C₂²×C₄	—	142	56	no	—
51	C₂²×C₄	C₂²×Dic₇₁	1	14	yes	—
52	C₂²×C₄	C₂×D₂₈₄	1	42	yes	—
53	C₂²×C₄	C₂×Dic₁₄₂	1	42	yes	—
54	C₄⋊C₄	—	284	112	no	—
55	C₄²	C₄⋊Dic₇₁	1	112	yes	complemented
56	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
57	C₂³	C₂²×D₂₈₄	1	7	yes	—
58	C₂³	C₂²×Dic₁₄₂	1	7	yes	—
59	C₂×C₄	—	284	56	no	—
60	C₂×C₄	C₂×C₄⋊Dic₇₁	1	84	yes	—
61	C₂×C₄	#14556125	1	28	yes	—
62	C₂×C₄	#14556128	1	84	yes	—
63	C₂²	C₂²×C₄⋊Dic₇₁	1	7	yes	—
64	C₂²	C₂×#14556125	1	7	yes	—
65	C₂²	C₂×#14556128	1	21	yes	—
66	C₄	—	568	8	no	—
67	C₄	#17985869	1	56	yes	complemented
68	C₂	C₂×#17985869	1	7	yes	—
69	C₂	#22596735	1	1	yes	characteristic
70	C₂	#22596738	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₁
commutator	C₂³×C₇₁	C₂³×C₄
Fitting	C₂×C₄³×C₇₁	C₂
socle, prime radical, cosocle	C₂⁴×C₇₁	C₂⁴
hypercenter	C₂×C₄³	D₇₁
hypocenter	C₇₁	#14692
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
6. C₄³×C₇₁	C₄	semidirect
19. C₄²×C₇₁	C₄⋊C₄	semidirect
30. C₄×C₇₁	#384	semidirect
38. C₇₁	#14692	semidirect
42. C₄³	Dic₇₁	semidirect
55. C₄²	C₄⋊Dic₇₁	semidirect
67. C₄	#17985869	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985869	2
C₂×C₄³×C₇₁	2
#14692	71