Group #34082512 — subgroups

Subgroups — #34082512

Order 19328.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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	#23752841	C₂	1	12	yes	maximal, complemented
2	#23752838	C₂	1	3	yes	maximal
3	#23752764	C₂	1	12	yes	maximal, complemented
4	C₂×#18012695	C₂	1	3	yes	maximal
5	#543×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₅₁	C₂²	1	1	yes	characteristic
7	#18012738	C₂²	1	12	yes	complemented
8	#18012737	C₂²	1	12	yes	complemented
9	#18012729	C₂²	1	24	yes	complemented
10	#18012718	C₂²	1	12	yes	—
11	#18012716	C₂²	1	24	yes	—
12	Q₈×Dic₁₅₁	C₂²	1	4	yes	complemented
13	#18012704	C₂²	1	12	yes	complemented
14	#18012695	C₂²	1	12	yes	—
15	#18012695	C₂²	1	12	yes	—
16	C₂×Dic₁₅₁⋊C₄	C₂²	1	6	yes	—
17	C₂×C₂²⋊Dic₁₅₁	C₂²	1	3	yes	—
18	C₂×C₄×Dic₁₅₁	C₂²	1	6	yes	—
19	C₂²⋊Q₈×C₁₅₁	C₂²	1	12	yes	—
20	C₂×C₂²⋊C₄×C₁₅₁	C₂²	1	3	yes	—
21	C₂×Dic₃₀₂	C₂³	1	12	yes	—
22	C₂²×Dic₁₅₁	Q₈	1	8	yes	complemented
23	C₂²×Dic₁₅₁	C₂³	1	6	yes	—
24	C₂²×Dic₁₅₁	C₂³	1	3	yes	—
25	C₂²×Dic₁₅₁	C₂³	1	3	yes	—
26	C₄⋊Dic₁₅₁	C₂³	1	12	yes	—
27	Dic₁₅₁⋊C₄	C₂³	1	24	yes	—
28	Dic₁₅₁⋊C₄	C₂³	1	24	yes	—
29	C₂²⋊Dic₁₅₁	C₂³	1	12	yes	—
30	C₄×Dic₁₅₁	C₂³	1	12	yes	—
31	C₄×Dic₁₅₁	C₂³	1	12	yes	—
32	C₂⁴×C₁₅₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₅₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₅₁	C₂³	1	6	yes	—
35	C₄⋊C₄×C₁₅₁	C₂³	1	12	yes	—
36	C₂²⋊C₄×C₁₅₁	C₂³	1	12	yes	—
37	Dic₃₀₂	—	4	12	no	—
38	C₂×Dic₁₅₁	—	4	6	no	—
39	C₂×Dic₁₅₁	C₂×Q₈	1	12	yes	—
40	C₂×Dic₁₅₁	C₂×Q₈	1	12	yes	—
41	C₂×Dic₁₅₁	C₂⁴	1	12	yes	—
42	C₂×Dic₁₅₁	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₁₅₁	C₂⁴	1	3	yes	—
44	C₂³×C₁₅₁	—	4	2	no	—
45	C₂³×C₁₅₁	C₂×Q₈	1	4	yes	—
46	C₂³×C₁₅₁	C₂⁴	1	3	yes	—
47	Q₈×C₁₅₁	—	4	4	no	—
48	C₂×C₄×C₁₅₁	—	4	6	no	—
49	C₂×C₄×C₁₅₁	C₂⁴	1	12	yes	—
50	C₂²×C₁₅₁	—	2	6	no	—
51	C₂²×C₁₅₁	—	4	4	no	—
52	C₂²×C₁₅₁	C₂²×Q₈	1	6	yes	—
53	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
54	C₄×C₁₅₁	—	2	12	no	—
55	Dic₁₅₁	—	2	12	no	—
56	Dic₁₅₁	C₂²×Q₈	1	2	yes	—
57	Dic₁₅₁	C₂²×Q₈	1	6	yes	—
58	C₂×C₁₅₁	—	2	6	no	—
59	C₂×C₁₅₁	C₂³×Q₈	1	1	yes	characteristic
60	C₂×C₁₅₁	#585	1	2	yes	—
61	C₁₅₁	#3458	1	1	yes	characteristic, complemented, Sylow
62	#3458	—	151	1	no	maximal, Sylow
63	#544	—	151	12	no	—
64	#543	—	151	3	no	—
65	#543	D₁₅₁	1	1	yes	characteristic
66	#533	—	151	12	no	—
67	C₂×#117	—	151	3	no	—
68	C₂³×C₄	—	151	1	no	—
69	C₄⋊Q₈	—	151	12	no	—
70	C₄⋊Q₈	—	151	12	no	—
71	#125	—	151	24	no	—
72	C₂²⋊Q₈	—	151	12	no	—
73	C₂²⋊Q₈	—	151	24	no	—
74	C₂²⋊Q₈	C₂×D₁₅₁	1	12	yes	—
75	C₄×Q₈	—	151	4	no	—
76	C₄×Q₈	—	151	12	no	—
77	#117	—	151	12	no	—
78	#117	—	151	12	no	—
79	C₂×C₄⋊C₄	—	151	6	no	—
80	C₂×C₂²⋊C₄	—	151	3	no	—
81	C₂×C₂²⋊C₄	C₂×D₁₅₁	1	3	yes	—
82	C₂×C₄²	—	151	6	no	—
83	C₂⁴	C₂²×D₁₅₁	1	1	yes	characteristic
84	C₂×Q₈	—	151	12	no	—
85	C₂×Q₈	C₂²×D₁₅₁	1	4	yes	—
86	C₂²×C₄	—	151	6	no	—
87	C₂²×C₄	—	151	8	no	—
88	C₂²×C₄	—	151	3	no	—
89	C₂²×C₄	—	151	3	no	—
90	C₂²×C₄	C₂²×D₁₅₁	1	6	yes	—
91	C₄⋊C₄	—	151	24	no	—
92	C₄⋊C₄	—	151	24	no	—
93	C₄⋊C₄	—	151	12	no	—
94	C₄⋊C₄	C₂²×D₁₅₁	1	12	yes	—
95	C₂²⋊C₄	—	151	12	no	—
96	C₂²⋊C₄	C₂²×D₁₅₁	1	12	yes	—
97	C₄²	—	151	12	no	—
98	C₄²	—	151	12	no	—
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₁₅₁	1	4	yes	—
101	C₂³	C₂³×D₁₅₁	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	604	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	604	6	no	—
106	C₂×C₄	—	151	12	no	—
107	C₂×C₄	—	151	12	no	—
108	C₂×C₄	—	151	12	no	—
109	C₂×C₄	—	151	1	no	—
110	C₂×C₄	—	151	3	no	—
111	C₂×C₄	C₂³×D₁₅₁	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₁₅₁	1	6	yes	—
115	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	302	12	no	—
118	C₄	—	151	2	no	—
119	C₄	—	151	6	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₁₅₁	1	1	yes	characteristic
122	C₂	#23753047	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752841

C₂

yes

maximal, complemented

#23752838

C₂

yes

maximal

#23752764

C₂

yes

maximal, complemented

C₂×#18012695

C₂

yes

maximal

#543×C₁₅₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₅₁

C₂²

yes

characteristic

#18012738

C₂²

yes

complemented

#18012737

C₂²

yes

complemented

#18012729

C₂²

yes

complemented

#18012718

C₂²

yes

—

#18012716

C₂²

yes

—

Q₈×Dic₁₅₁

C₂²

yes

complemented

#18012704

C₂²

yes

complemented

#18012695

C₂²

yes

—

#18012695

C₂²

yes

—

C₂×Dic₁₅₁⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₂²⋊Q₈×C₁₅₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

C₂³

yes

—

C₂²×Dic₁₅₁

Q₈

yes

complemented

C₂²×Dic₁₅₁

C₂³

yes

—

C₂²×Dic₁₅₁

C₂³

yes

—

C₂²×Dic₁₅₁

C₂³

yes

—

C₄⋊Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

C₂³

yes

—

Dic₁₅₁⋊C₄

C₂³

yes

—

C₂²⋊Dic₁₅₁

C₂³

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₂⁴×C₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

C₂³

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₂²⋊C₄×C₁₅₁

C₂³

yes

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×Q₈

yes

—

C₂×Dic₁₅₁

C₂×Q₈

yes

—

C₂×Dic₁₅₁

C₂⁴

yes

—

C₂×Dic₁₅₁

C₂⁴

yes

characteristic

C₂×Dic₁₅₁

C₂⁴

yes

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

C₂×Q₈

yes

—

C₂³×C₁₅₁

C₂⁴

yes

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂⁴

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²×Q₈

yes

—

C₂²×C₁₅₁

C₂⁵

yes

characteristic

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₂²×Q₈

yes

—

Dic₁₅₁

C₂²×Q₈

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂³×Q₈

yes

characteristic

C₂×C₁₅₁

#585

yes

—

C₁₅₁

#3458

yes

characteristic, complemented, Sylow

#3458

—

151

maximal, Sylow

#544

—

151

—

#543

—

151

—

#543

D₁₅₁

yes

characteristic

#533

—

151

—

C₂×#117

—

151

—

C₂³×C₄

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

—

151

—

#125

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

C₂×D₁₅₁

yes

—

C₄×Q₈

—

151

—

C₄×Q₈

—

151

—

#117

—

151

—

#117

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂×C₂²⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₂⁴

C₂²×D₁₅₁

yes

characteristic

C₂×Q₈

—

151

—

C₂×Q₈

C₂²×D₁₅₁

yes

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂²×D₁₅₁

yes

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

C₂²×D₁₅₁

yes

—

C₄²

—

151

—

C₄²

—

151

—

C₂³

—

100

C₂³

Q₈×D₁₅₁

yes

—

101

C₂³

C₂³×D₁₅₁

yes

—

102

Q₈

—

103

Q₈

—

604

—

104

C₂×C₄

—

105

C₂×C₄

—

604

—

106

C₂×C₄

—

151

—

107

C₂×C₄

—

151

—

108

C₂×C₄

—

151

—

109

C₂×C₄

—

151

—

110

C₂×C₄

—

151

—

111

C₂×C₄

C₂³×D₁₅₁

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₁₅₁

yes

—

115

C₂²

C₂⁴×D₁₅₁

yes

characteristic

116

C₄

—

117

C₄

—

302

—

118

C₄

—

151

—

119

C₄

—

151

—

120

C₂

—

121

C₂

C₂²×Q₈×D₁₅₁

yes

characteristic

122

C₂

#23753047

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Fitting	#543×C₁₅₁	C₂
prime radical	C₂⁴×C₁₅₁	C₂³
hypercenter	#543	D₁₅₁
hypocenter	C₁₅₁	#3458
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁵

Fitting

#543×C₁₅₁

C₂

prime radical

C₂⁴×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
1. #23752841	C₂	semidirect
3. #23752764	C₂	semidirect
7. #18012738	C₂²	semidirect
8. #18012737	C₂²	semidirect
9. #18012729	C₂²	semidirect
12. Q₈×Dic₁₅₁	C₂²	semidirect
13. #18012704	C₂²	semidirect
22. C₂²×Dic₁₅₁	Q₈	semidirect
61. C₁₅₁	#3458	semidirect

H ≅ G/N

type

1. #23752841

C₂

semidirect

3. #23752764

C₂

semidirect

7. #18012738

C₂²

semidirect

8. #18012737

C₂²

semidirect

9. #18012729

C₂²

semidirect

12. Q₈×Dic₁₅₁

C₂²

semidirect

13. #18012704

C₂²

semidirect

22. C₂²×Dic₁₅₁

Q₈

semidirect

61. C₁₅₁

#3458

semidirect

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

subgroup	index
#23752841	2
#23752838	2
#23752764	2
C₂×#18012695	2
#543×C₁₅₁	2
#3458	151

subgroup

index

#543×C₁₅₁

151

Subgroups — #34082512

Conjugacy classes of subgroups

Complements

Maximal subgroups