Group #34077466 — subgroups

Subgroups — #34077466

Order 19328.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#18012739	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#18012726	C₂	1	4	yes	maximal, complemented
3	#23752115	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₀₂	C₂²	1	2	yes	—
7	#18012739	—	2	4	no	—
8	#18012726	—	2	16	no	—
9	C₂×C₄⋊Dic₁₅₁	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₁₅₁	C₂²	1	16	yes	—
11	C₂×C₄×Dic₁₅₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₅₁	C₂²	1	2	yes	—
14	C₂²×D₄×C₁₅₁	C₂²	1	4	yes	—
15	#127×C₁₅₁	—	2	4	no	—
16	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₀₂	—	2	8	no	—
18	C₂×Dic₃₀₂	—	2	4	no	—
19	C₂²×Dic₁₅₁	C₂³	1	4	yes	—
20	C₂²×Dic₁₅₁	C₂³	1	4	yes	—
21	C₄⋊Dic₁₅₁	—	2	8	no	—
22	C₂²⋊Dic₁₅₁	—	2	32	no	—
23	C₄×Dic₁₅₁	—	2	8	no	—
24	C₄×Dic₁₅₁	D₄	1	8	yes	complemented
25	C₂⁴×C₁₅₁	C₂³	1	4	yes	—
26	C₂×D₄×C₁₅₁	—	2	8	no	—
27	C₂×D₄×C₁₅₁	—	2	8	no	—
28	C₂×D₄×C₁₅₁	—	2	4	no	—
29	C₂×D₄×C₁₅₁	—	2	16	no	—
30	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
32	C₄²×C₁₅₁	D₄	1	4	yes	—
33	Dic₃₀₂	—	4	8	no	—
34	C₂×Dic₁₅₁	—	2	8	no	—
35	C₂×Dic₁₅₁	—	2	8	no	—
36	C₂×Dic₁₅₁	—	2	4	no	—
37	C₂×Dic₁₅₁	C₂×D₄	1	8	yes	—
38	C₂³×C₁₅₁	—	2	8	no	—
39	C₂³×C₁₅₁	—	2	4	no	—
40	C₂³×C₁₅₁	—	2	16	no	—
41	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
42	D₄×C₁₅₁	—	4	16	no	—
43	D₄×C₁₅₁	—	4	8	no	—
44	C₂×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₅₁	#41	1	8	yes	—
48	C₂²×C₁₅₁	—	4	8	no	—
49	C₂²×C₁₅₁	—	4	4	no	—
50	C₂²×C₁₅₁	—	4	16	no	—
51	C₂²×C₁₅₁	C₂²×D₄	1	2	yes	—
52	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
53	C₂²×C₁₅₁	C₂×#41	1	4	yes	—
54	C₄×C₁₅₁	#124	1	8	yes	complemented
55	C₄×C₁₅₁	#127	1	4	yes	—
56	Dic₁₅₁	—	2	8	no	—
57	Dic₁₅₁	—	4	4	no	—
58	C₂×C₁₅₁	—	8	4	no	—
59	C₂×C₁₅₁	C₂×#124	1	2	yes	—
60	C₂×C₁₅₁	C₂×#127	1	1	yes	characteristic
61	C₂×C₁₅₁	#532	1	4	yes	—
62	C₁₅₁	#2858	1	1	yes	characteristic, complemented, Sylow
63	#2858	—	151	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	151	1	no	—
65	C₂×#127	D₁₅₁	1	1	yes	characteristic
66	C₂×#124	—	151	4	no	—
67	#390	—	151	8	no	—
68	C₄³	—	151	1	no	—
69	C₂²×Q₈	—	151	2	no	—
70	C₂²×D₄	C₂×D₁₅₁	1	2	yes	—
71	C₂²×D₄	C₂×D₁₅₁	1	4	yes	—
72	C₄⋊Q₈	—	302	4	no	—
73	#127	—	2	4	no	—
74	#124	—	302	16	no	—
75	C₂×C₄⋊C₄	—	151	4	no	—
76	C₂×C₂²⋊C₄	—	151	16	no	—
77	C₂×C₄²	—	151	4	no	—
78	C₂×C₄²	—	151	2	no	—
79	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
80	C₂⁴	C₂²×D₁₅₁	1	4	yes	—
81	C₂×Q₈	—	302	8	no	—
82	C₂×Q₈	—	302	4	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	8	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	2	16	no	—
87	C₂²×C₄	—	151	4	no	—
88	C₂²×C₄	—	151	4	no	—
89	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
91	C₄⋊C₄	—	302	8	no	—
92	C₂²⋊C₄	—	302	32	no	—
93	C₄²	—	302	8	no	—
94	C₄²	—	151	8	no	—
95	C₄²	C₂²⋊D₁₅₁	1	4	yes	—
96	C₂³	—	2	8	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	16	no	—
99	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
100	Q₈	—	604	8	no	—
101	D₄	—	4	16	no	—
102	D₄	—	4	8	no	—
103	C₂×C₄	—	302	8	no	—
104	C₂×C₄	—	302	8	no	—
105	C₂×C₄	—	302	4	no	—
106	C₂×C₄	—	151	8	no	—
107	C₂×C₄	D₄×D₁₅₁	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
110	C₂×C₄	#14559003	1	8	yes	—
111	C₂²	—	4	8	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	4	16	no	—
114	C₂²	C₂×D₄×D₁₅₁	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
116	C₂²	C₂×#14559003	1	4	yes	—
117	C₄	—	302	8	no	—
118	C₄	—	604	4	no	—
119	C₄	#18012726	1	8	yes	complemented
120	C₄	#18012735	1	4	yes	—
121	C₂	—	8	4	no	—
122	C₂	C₂×#18012726	1	2	yes	—
123	C₂	C₂×#18012735	1	1	yes	characteristic
124	C₂	#23752760	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012739

C₂

yes

maximal, characteristic, complemented

C₂×#18012726

C₂

yes

maximal, complemented

#23752115

C₂

yes

maximal, complemented

C₄²×Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

—

#18012739

—

#18012726

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₂²×D₄×C₁₅₁

C₂²

yes

—

C₂²×D₄×C₁₅₁

C₂²

yes

—

#127×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂²×Dic₁₅₁

C₂³

yes

—

C₂²×Dic₁₅₁

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

D₄

yes

complemented

C₂⁴×C₁₅₁

C₂³

yes

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

D₄

yes

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

C₂⁴

yes

characteristic

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

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

—

C₄×C₁₅₁

#124

yes

complemented

C₄×C₁₅₁

#127

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#124

yes

—

C₂×C₁₅₁

C₂×#127

yes

characteristic

C₂×C₁₅₁

#532

yes

—

C₁₅₁

#2858

yes

characteristic, complemented, Sylow

#2858

—

151

maximal, Sylow

C₂×C₄⋊Q₈

—

151

—

C₂×#127

D₁₅₁

yes

characteristic

C₂×#124

—

151

—

#390

—

151

—

C₄³

—

151

—

C₂²×Q₈

—

151

—

C₂²×D₄

C₂×D₁₅₁

yes

—

C₂²×D₄

C₂×D₁₅₁

yes

—

C₄⋊Q₈

—

302

—

#127

—

#124

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₂⁴

C₂²×D₁₅₁

yes

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

C₄²

—

302

—

C₄²

—

151

—

C₄²

C₂²⋊D₁₅₁

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₅₁

yes

characteristic

100

Q₈

—

604

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

302

—

104

C₂×C₄

—

302

—

105

C₂×C₄

—

302

—

106

C₂×C₄

—

151

—

107

C₂×C₄

D₄×D₁₅₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

110

C₂×C₄

#14559003

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₁₅₁

yes

—

115

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

116

C₂²

C₂×#14559003

yes

—

117

C₄

—

302

—

118

C₄

—

604

—

119

C₄

#18012726

yes

complemented

120

C₄

#18012735

yes

—

121

C₂

—

122

C₂

C₂×#18012726

yes

—

123

C₂

C₂×#18012735

yes

characteristic

124

C₂

#23752760

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₁
commutator, socle, cosocle	C₂³×C₁₅₁	C₂⁴
Fitting, prime radical	C₂×#127×C₁₅₁	C₂
hypercenter	C₂×#127	D₁₅₁
hypocenter	C₁₅₁	#2858
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, prime radical

C₂×#127×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#18012739	C₂	semidirect
2. C₂×#18012726	C₂	semidirect
3. #23752115	C₂	semidirect
4. C₄²×Dic₁₅₁	C₂	semidirect
24. C₄×Dic₁₅₁	D₄	semidirect
54. C₄×C₁₅₁	#124	semidirect
62. C₁₅₁	#2858	semidirect
119. C₄	#18012726	semidirect

H ≅ G/N

type

1. C₂×#18012739

C₂

semidirect

2. C₂×#18012726

C₂

semidirect

3. #23752115

C₂

semidirect

4. C₄²×Dic₁₅₁

C₂

semidirect

24. C₄×Dic₁₅₁

D₄

semidirect

54. C₄×C₁₅₁

#124

semidirect

62. C₁₅₁

#2858

semidirect

119. C₄

#18012726

semidirect

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

subgroup	index
C₂×#18012739	2
C₂×#18012726	2
#23752115	2
C₄²×Dic₁₅₁	2
C₂×#127×C₁₅₁	2
#2858	151

subgroup

index

C₂×#18012739

C₂×#18012726

#23752115

C₄²×Dic₁₅₁

C₂×#127×C₁₅₁

#2858

151

Subgroups — #34077466

Conjugacy classes of subgroups

Complements

Maximal subgroups