Group #34077452 — subgroups

Subgroups — #34077452

Order 19328.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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	C₂×#18012710	C₂	1	6	yes	maximal, complemented
2	C₂×C₄²×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752115	C₂	1	6	yes	maximal, complemented
4	#23752082	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₁₅₁	C₂²	1	4	yes	—
7	C₂²×C₄×D₁₅₁	C₂²	1	3	yes	—
8	#18012710	—	2	24	no	—
9	C₂×C₄⋊Dic₁₅₁	C₂²	1	6	yes	—
10	C₂×C₂²⋊Dic₁₅₁	C₂²	1	12	yes	—
11	C₂×C₄×Dic₁₅₁	C₂²	1	3	yes	—
12	C₄²×D₁₅₁	—	2	4	no	—
13	C₂²×D₄×C₁₅₁	C₂²	1	6	yes	—
14	#127×C₁₅₁	—	2	4	no	—
15	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂³×D₁₅₁	C₂³	1	1	yes	characteristic
17	C₂×C₂²⋊D₁₅₁	—	2	24	no	—
18	C₂²×Dic₁₅₁	C₂³	1	4	yes	—
19	C₂²×Dic₁₅₁	C₂³	1	3	yes	—
20	C₂×C₄×D₁₅₁	—	2	6	no	—
21	C₂×C₄×D₁₅₁	—	2	6	no	—
22	C₂×C₄×D₁₅₁	D₄	1	12	yes	complemented
23	C₄⋊Dic₁₅₁	—	2	12	no	—
24	C₂²⋊Dic₁₅₁	—	2	24	no	—
25	C₄×Dic₁₅₁	—	2	6	no	—
26	C₂⁴×C₁₅₁	C₂³	1	4	yes	—
27	C₂×D₄×C₁₅₁	—	2	12	no	—
28	C₂×D₄×C₁₅₁	—	2	12	no	—
29	C₂×D₄×C₁₅₁	—	2	12	no	—
30	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
31	C₄²×C₁₅₁	D₄	1	4	yes	complemented
32	C₂²×D₁₅₁	—	2	1	no	—
33	C₂²×D₁₅₁	—	2	3	no	—
34	C₂²×D₁₅₁	C₂×D₄	1	6	yes	—
35	C₂²⋊D₁₅₁	—	4	16	no	—
36	C₂×Dic₁₅₁	—	2	3	no	—
37	C₂×Dic₁₅₁	—	2	3	no	—
38	C₂×Dic₁₅₁	—	2	12	no	—
39	C₂×Dic₁₅₁	C₂×D₄	1	6	yes	—
40	C₄×D₁₅₁	—	2	24	no	—
41	C₂³×C₁₅₁	—	2	12	no	—
42	C₂³×C₁₅₁	—	2	4	no	—
43	C₂³×C₁₅₁	—	2	12	no	—
44	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
45	D₄×C₁₅₁	—	4	24	no	—
46	C₂×C₄×C₁₅₁	C₂×D₄	1	6	yes	—
47	C₂×C₄×C₁₅₁	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₁₅₁	#41	1	6	yes	—
49	C₂²×C₁₅₁	—	4	12	no	—
50	C₂²×C₁₅₁	—	4	4	no	—
51	C₂²×C₁₅₁	—	4	12	no	—
52	C₂²×C₁₅₁	C₂²×D₄	1	3	yes	—
53	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₁₅₁	C₂×#41	1	3	yes	—
55	C₂×D₁₅₁	—	2	6	no	—
56	C₂×D₁₅₁	—	2	6	no	—
57	C₂×D₁₅₁	#127	1	4	yes	complemented
58	C₄×C₁₅₁	#121	1	12	yes	complemented
59	Dic₁₅₁	—	2	6	no	—
60	Dic₁₅₁	—	4	4	no	—
61	C₂×C₁₅₁	—	8	4	no	—
62	C₂×C₁₅₁	C₂×#121	1	3	yes	—
63	C₂×C₁₅₁	C₂×#127	1	1	yes	characteristic
64	C₂×C₁₅₁	#532	1	3	yes	—
65	D₁₅₁	—	2	4	no	—
66	C₁₅₁	#2856	1	1	yes	characteristic, complemented, Sylow
67	#2856	—	151	1	no	maximal, Sylow
68	C₂×#127	D₁₅₁	1	1	yes	characteristic, complemented
69	C₂×#121	—	151	6	no	—
70	C₂²×C₄²	—	151	1	no	—
71	#390	—	151	6	no	—
72	#384	—	151	1	no	—
73	C₂²×D₄	—	151	4	no	—
74	C₂²×D₄	C₂×D₁₅₁	1	6	yes	—
75	C₂³×C₄	—	151	3	no	—
76	#127	—	2	4	no	—
77	#121	—	302	24	no	—
78	C₂×C₄⋊C₄	—	151	6	no	—
79	C₂×C₂²⋊C₄	—	151	12	no	—
80	C₂×C₄²	—	302	4	no	—
81	C₂×C₄²	—	151	3	no	—
82	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
83	C₂⁴	—	151	1	no	—
84	C₂⁴	C₂²×D₁₅₁	1	4	yes	—
85	C₂×D₄	—	2	12	no	—
86	C₂×D₄	—	2	12	no	—
87	C₂×D₄	—	2	12	no	—
88	C₂×D₄	—	302	24	no	—
89	C₂²×C₄	—	302	6	no	—
90	C₂²×C₄	—	302	6	no	—
91	C₂²×C₄	—	151	4	no	—
92	C₂²×C₄	—	151	3	no	—
93	C₂²×C₄	—	151	12	no	—
94	C₂²×C₄	C₂²×D₁₅₁	1	3	yes	—
95	C₄⋊C₄	—	302	12	no	—
96	C₂²⋊C₄	—	302	24	no	—
97	C₄²	—	302	6	no	—
98	C₄²	C₂²⋊D₁₅₁	1	4	yes	complemented
99	C₂³	—	2	12	no	—
100	C₂³	—	2	4	no	—
101	C₂³	—	2	12	no	—
102	C₂³	—	302	1	no	—
103	C₂³	—	302	3	no	—
104	C₂³	—	151	6	no	—
105	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
106	D₄	—	4	24	no	—
107	D₄	—	604	16	no	—
108	C₂×C₄	—	302	3	no	—
109	C₂×C₄	—	302	3	no	—
110	C₂×C₄	—	302	24	no	—
111	C₂×C₄	—	302	12	no	—
112	C₂×C₄	—	151	6	no	—
113	C₂×C₄	D₄×D₁₅₁	1	6	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	6	yes	—
115	C₂×C₄	#14559003	1	6	yes	—
116	C₂²	—	4	4	no	—
117	C₂²	—	4	12	no	—
118	C₂²	—	4	12	no	—
119	C₂²	—	302	6	no	—
120	C₂²	—	302	6	no	—
121	C₂²	—	151	4	no	—
122	C₂²	C₂×D₄×D₁₅₁	1	3	yes	—
123	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
124	C₂²	C₂×#14559003	1	3	yes	—
125	C₄	—	302	6	no	—
126	C₄	—	604	4	no	—
127	C₄	#18012710	1	12	yes	complemented
128	C₂	—	8	4	no	—
129	C₂	—	302	4	no	—
130	C₂	C₂×#18012710	1	3	yes	—
131	C₂	#127×D₁₅₁	1	1	yes	characteristic
132	C₂	#23752760	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012710

C₂

yes

maximal, complemented

C₂×C₄²×D₁₅₁

C₂

yes

maximal, characteristic, complemented

#23752115

C₂

yes

maximal, complemented

#23752082

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₁₅₁

C₂²

yes

—

C₂²×C₄×D₁₅₁

C₂²

yes

—

#18012710

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₄²×D₁₅₁

—

C₂²×D₄×C₁₅₁

C₂²

yes

—

#127×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂³×D₁₅₁

C₂³

yes

characteristic

C₂×C₂²⋊D₁₅₁

—

C₂²×Dic₁₅₁

C₂³

yes

—

C₂²×Dic₁₅₁

C₂³

yes

—

C₂×C₄×D₁₅₁

—

C₂×C₄×D₁₅₁

—

C₂×C₄×D₁₅₁

D₄

yes

complemented

C₄⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₂⁴×C₁₅₁

C₂³

yes

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

D₄

yes

complemented

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

C₂×D₄

yes

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

C₂⁴

yes

characteristic

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²×D₄

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₂²×C₁₅₁

C₂×#41

yes

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

yes

C₄×C₁₅₁

yes

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#121

yes

—

C₂×C₁₅₁

C₂×#127

yes

characteristic

C₂×C₁₅₁

#532

yes

—

D₁₅₁

—

C₁₅₁

#2856

yes

characteristic, complemented, Sylow

#2856

—

151

maximal, Sylow

C₂×#127

D₁₅₁

yes

characteristic, complemented

C₂×#121

—

151

—

C₂²×C₄²

—

151

—

#390

—

151

—

#384

—

151

—

C₂²×D₄

—

151

—

C₂²×D₄

C₂×D₁₅₁

yes

—

C₂³×C₄

—

151

—

#127

—

#121

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂×C₄²

—

302

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₂⁴

—

151

—

C₂⁴

C₂²×D₁₅₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

C₄²

—

302

—

C₄²

C₂²⋊D₁₅₁

yes

complemented

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

302

—

103

C₂³

—

302

—

104

C₂³

—

151

—

105

C₂³

C₂³×D₁₅₁

yes

characteristic

106

D₄

—

107

D₄

—

604

—

108

C₂×C₄

—

302

—

109

C₂×C₄

—

302

—

110

C₂×C₄

—

302

—

111

C₂×C₄

—

302

—

112

C₂×C₄

—

151

—

113

C₂×C₄

D₄×D₁₅₁

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

115

C₂×C₄

#14559003

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

302

—

120

C₂²

—

302

—

121

C₂²

—

151

—

122

C₂²

C₂×D₄×D₁₅₁

yes

—

123

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

124

C₂²

C₂×#14559003

yes

—

125

C₄

—

302

—

126

C₄

—

604

—

127

C₄

#18012710

yes

complemented

128

C₂

—

129

C₂

—

302

—

130

C₂

C₂×#18012710

yes

—

131

C₂

#127×D₁₅₁

yes

characteristic

132

C₂

#23752760

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₁
commutator, socle, cosocle	C₂³×C₁₅₁	C₂⁴
Fitting	C₂×#127×C₁₅₁	C₂
hypercenter	C₂×#127	D₁₅₁
hypocenter	C₁₅₁	#2856
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₅₁

commutator, socle, cosocle

C₂³×C₁₅₁

C₂⁴

Fitting

C₂×#127×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#18012710	C₂	semidirect
2. C₂×C₄²×D₁₅₁	C₂	semidirect
3. #23752115	C₂	semidirect
4. #23752082	C₂	semidirect
5. C₂×#127×C₁₅₁	C₂	semidirect
22. C₂×C₄×D₁₅₁	D₄	semidirect
31. C₄²×C₁₅₁	D₄	semidirect
57. C₂×D₁₅₁	#127	semidirect
58. C₄×C₁₅₁	#121	semidirect
66. C₁₅₁	#2856	semidirect
68. C₂×#127	D₁₅₁	semidirect
98. C₄²	C₂²⋊D₁₅₁	semidirect
127. C₄	#18012710	semidirect

H ≅ G/N

type

1. C₂×#18012710

C₂

semidirect

2. C₂×C₄²×D₁₅₁

C₂

semidirect

3. #23752115

C₂

semidirect

4. #23752082

C₂

semidirect

5. C₂×#127×C₁₅₁

C₂

semidirect

22. C₂×C₄×D₁₅₁

D₄

semidirect

31. C₄²×C₁₅₁

D₄

semidirect

57. C₂×D₁₅₁

#127

semidirect

58. C₄×C₁₅₁

#121

semidirect

66. C₁₅₁

#2856

semidirect

68. C₂×#127

D₁₅₁

semidirect

98. C₄²

C₂²⋊D₁₅₁

semidirect

127. C₄

#18012710

semidirect

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

subgroup	index
C₂×#18012710	2
C₂×C₄²×D₁₅₁	2
#23752115	2
#23752082	2
C₂×#127×C₁₅₁	2
#2856	151

subgroup

index

C₂×C₄²×D₁₅₁

C₂×#127×C₁₅₁

151

Subgroups — #34077452

Conjugacy classes of subgroups

Complements

Maximal subgroups