Group #34077102 — subgroups

Subgroups — #34077102

Order 19328.

432 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752132	C₂	1	3	yes	maximal
2	#23752131	C₂	1	2	yes	maximal
3	#23752117	C₂	1	3	yes	maximal
4	#23752116	C₂	1	6	yes	maximal
5	#395×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₀₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃₀₂	C₂²	1	3	yes	—
8	C₂×Dic₁₅₁⋊C₄	C₂²	1	6	yes	—
9	C₂×C₄×Dic₁₅₁	C₂²	1	3	yes	—
10	C₁₅₁⋊#95	C₂²	1	6	yes	—
11	C₁₅₁⋊#95	C₂²	1	6	yes	—
12	C₁₅₁⋊#95	C₂²	1	3	yes	—
13	C₂²×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	3	yes	—
15	#95×C₁₅₁	C₂²	1	3	yes	—
16	C₂×Dic₃₀₂	—	4	3	no	—
17	C₂×Dic₃₀₂	—	4	3	no	—
18	C₂×Dic₃₀₂	—	4	3	no	—
19	C₂×Dic₃₀₂	Q₈	1	12	yes	complemented
20	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
21	C₂²×Dic₁₅₁	C₂³	1	6	yes	—
22	Dic₁₅₁⋊C₄	—	4	6	no	—
23	C₄×Dic₁₅₁	—	4	3	no	—
24	C₂×Q₈×C₁₅₁	—	4	3	no	—
25	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
28	C₄⋊C₄×C₁₅₁	—	4	3	no	—
29	Dic₃₀₂	—	4	12	no	—
30	Dic₃₀₂	—	4	4	no	—
31	C₂×Dic₁₅₁	—	2	6	no	—
32	C₂×Dic₁₅₁	—	2	6	no	—
33	C₂×Dic₁₅₁	—	2	6	no	—
34	C₂×Dic₁₅₁	C₂×Q₈	1	12	yes	—
35	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
36	Q₈×C₁₅₁	—	4	4	no	—
37	C₂×C₄×C₁₅₁	—	2	3	no	—
38	C₂×C₄×C₁₅₁	—	2	3	no	—
39	C₂×C₄×C₁₅₁	—	2	6	no	—
40	C₂×C₄×C₁₅₁	—	2	3	no	—
41	C₂×C₄×C₁₅₁	—	2	3	no	—
42	C₂×C₄×C₁₅₁	C₂×Q₈	1	6	yes	—
43	C₂²×C₁₅₁	C₂²×Q₈	1	3	yes	—
44	C₂²×C₁₅₁	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
46	C₄×C₁₅₁	—	2	6	no	—
47	C₄×C₁₅₁	—	4	3	no	—
48	C₄×C₁₅₁	—	4	1	no	—
49	Dic₁₅₁	—	2	12	no	—
50	Dic₁₅₁	—	4	2	no	—
51	C₂×C₁₅₁	#543	1	3	yes	—
52	C₂×C₁₅₁	Q₈²	1	3	yes	—
53	C₂×C₁₅₁	H(3,𝔽₄)	1	1	yes	characteristic
54	C₁₅₁	#2811	1	1	yes	characteristic, complemented, Sylow
55	#2811	—	151	1	no	maximal, Sylow
56	#395	—	151	2	no	—
57	#395	—	151	3	no	—
58	#395	D₁₅₁	1	1	yes	characteristic
59	#391	—	151	3	no	—
60	#391	—	151	6	no	—
61	C₂²×Q₈	—	151	1	no	—
62	C₂²×Q₈	—	151	3	no	—
63	C₂²×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	151	6	no	—
65	C₂×C₄⋊C₄	C₂×D₁₅₁	1	3	yes	—
66	C₂×C₄²	—	151	3	no	—
67	#95	—	151	6	no	—
68	#95	—	151	3	no	—
69	#95	—	151	6	no	—
70	#95	C₂×D₁₅₁	1	3	yes	—
71	C₂×Q₈	—	4	3	no	—
72	C₂×Q₈	—	604	3	no	—
73	C₂×Q₈	—	604	3	no	—
74	C₂×Q₈	—	604	3	no	—
75	C₂×Q₈	—	151	12	no	—
76	C₂²×C₄	—	151	2	no	—
77	C₂²×C₄	—	151	6	no	—
78	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₁₅₁	1	3	yes	—
80	C₂²×C₄	C₂²×D₁₅₁	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	604	6	no	—
83	C₄²	—	604	3	no	—
84	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
85	Q₈	—	4	4	no	—
86	Q₈	—	604	12	no	—
87	Q₈	—	604	4	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	302	6	no	—
94	C₂×C₄	—	302	6	no	—
95	C₂×C₄	—	302	6	no	—
96	C₂×C₄	—	151	12	no	—
97	C₂×C₄	Q₈×D₁₅₁	1	6	yes	—
98	C₂²	C₂×Q₈×D₁₅₁	1	3	yes	—
99	C₂²	#18012784	1	3	yes	—
100	C₂²	#18012785	1	1	yes	characteristic
101	C₄	—	2	6	no	—
102	C₄	—	4	3	no	—
103	C₄	—	4	1	no	—
104	C₄	—	302	12	no	—
105	C₄	—	604	2	no	—
106	C₂	#23752837	1	3	yes	—
107	C₂	#23752942	1	3	yes	—
108	C₂	#23752955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752132

C₂

yes

maximal

#23752131

C₂

yes

maximal

#23752117

C₂

yes

maximal

#23752116

C₂

yes

maximal

#395×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

characteristic

C₂²×Dic₃₀₂

C₂²

yes

—

C₂×Dic₁₅₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₁₅₁⋊#95

C₂²

yes

—

C₁₅₁⋊#95

C₂²

yes

—

C₁₅₁⋊#95

C₂²

yes

—

C₂²×Q₈×C₁₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

#95×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

Q₈

yes

complemented

C₂²×Dic₁₅₁

C₂³

yes

—

C₂²×Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

—

C₄×Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×Q₈

yes

—

C₂³×C₁₅₁

C₂⁴

yes

characteristic

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂²×C₁₅₁

C₂²×Q₈

yes

—

C₂²×C₁₅₁

H(5,𝔽₂)

yes

—

C₂²×C₁₅₁

H(5,𝔽₂)

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#543

yes

—

C₂×C₁₅₁

Q₈²

yes

—

C₂×C₁₅₁

H(3,𝔽₄)

yes

characteristic

C₁₅₁

#2811

yes

characteristic, complemented, Sylow

#2811

—

151

maximal, Sylow

#395

—

151

—

#395

—

151

—

#395

D₁₅₁

yes

characteristic

#391

—

151

—

#391

—

151

—

C₂²×Q₈

—

151

—

C₂²×Q₈

—

151

—

C₂²×Q₈

C₂×D₁₅₁

yes

characteristic

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

#95

—

151

—

#95

—

151

—

#95

—

151

—

#95

C₂×D₁₅₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

604

—

C₂×Q₈

—

604

—

C₂×Q₈

—

604

—

C₂×Q₈

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

604

—

C₄²

—

604

—

C₂³

C₂³×D₁₅₁

yes

characteristic

Q₈

—

Q₈

—

604

—

Q₈

—

604

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

Q₈×D₁₅₁

yes

—

C₂²

C₂×Q₈×D₁₅₁

yes

—

C₂²

#18012784

yes

—

100

C₂²

#18012785

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

302

—

105

C₄

—

604

—

106

C₂

#23752837

yes

—

107

C₂

#23752942

yes

—

108

C₂

#23752955

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₁
commutator, socle, prime radical, cosocle	C₂³×C₁₅₁	C₂⁴
Fitting	#395×C₁₅₁	C₂
hypercenter	#395	D₁₅₁
hypocenter	C₁₅₁	#2811
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₅₁

commutator, socle, prime radical, cosocle

C₂³×C₁₅₁

C₂⁴

Fitting

#395×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
19. C₂×Dic₃₀₂	Q₈	semidirect
54. C₁₅₁	#2811	semidirect

H ≅ G/N

type

19. C₂×Dic₃₀₂

Q₈

semidirect

54. C₁₅₁

#2811

semidirect

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

subgroup	index
#23752132	2
#23752131	2
#23752117	2
#23752116	2
#395×C₁₅₁	2
#2811	151

subgroup

index

#395×C₁₅₁

151

Subgroups — #34077102

Conjugacy classes of subgroups

Complements

Maximal subgroups