Group #23752956 — subgroups

Subgroups — #23752956

Order 9664.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#18012731	C₂	1	1	yes	maximal, characteristic, complemented
2	#18012727	C₂	1	1	yes	maximal, characteristic, complemented
3	#18012723	C₂	1	2	yes	maximal, complemented
4	#18012721	C₂	1	2	yes	maximal, complemented
5	#18012716	C₂	1	2	yes	maximal, complemented
6	#18012715	C₂	1	2	yes	maximal, complemented
7	#18012712	C₂	1	2	yes	maximal, complemented
8	#18012711	C₂	1	2	yes	maximal, complemented
9	#124×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₀₂	C₂²	1	1	yes	characteristic
11	C₂×D₆₀₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₅₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₅₁	C₂²	1	2	yes	—
14	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
15	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
16	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
17	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
19	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
20	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
21	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₅₁	C₂²	1	2	yes	—
23	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₁₅₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₁₅₁	C₂²	1	2	yes	—
30	C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
31	C₂²×D₁₅₁	C₂³	1	2	yes	—
32	Dic₃₀₂	—	4	1	no	—
33	D₆₀₄	—	4	1	no	—
34	C₂²⋊D₁₅₁	—	4	2	no	—
35	C₂²⋊D₁₅₁	—	4	2	no	—
36	C₂×Dic₁₅₁	—	4	2	no	—
37	C₂×Dic₁₅₁	C₂³	1	2	yes	—
38	C₂×Dic₁₅₁	C₂³	1	2	yes	—
39	C₂×Dic₁₅₁	C₂³	1	2	yes	—
40	C₄×D₁₅₁	—	4	2	no	—
41	C₂³×C₁₅₁	C₂³	1	2	yes	—
42	Q₈×C₁₅₁	—	4	1	no	—
43	D₄×C₁₅₁	—	4	1	no	—
44	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
46	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
48	C₂²×C₁₅₁	—	2	2	no	—
49	C₂²×C₁₅₁	—	2	2	no	—
50	C₂²×C₁₅₁	—	2	2	no	—
51	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₅₁	—	2	2	no	—
53	C₂×D₁₅₁	—	2	2	no	—
54	C₂×D₁₅₁	—	2	2	no	—
55	C₄×C₁₅₁	—	2	1	no	—
56	C₄×C₁₅₁	—	2	2	no	—
57	C₄×C₁₅₁	—	2	1	no	—
58	C₄×C₁₅₁	—	2	1	no	—
59	Dic₁₅₁	—	2	2	no	—
60	Dic₁₅₁	—	2	2	no	—
61	Dic₁₅₁	—	2	2	no	—
62	C₂×C₁₅₁	—	4	2	no	—
63	C₂×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₅₁	—	4	2	no	—
67	C₁₅₁	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	151	1	no	maximal, Sylow
69	#125	—	151	1	no	—
70	#124	—	151	1	no	—
71	#124	D₁₅₁	1	1	yes	characteristic, complemented
72	#123	—	151	2	no	—
73	#123	—	151	2	no	—
74	C₂²⋊Q₈	—	151	2	no	—
75	C₂²⋊Q₈	—	151	2	no	—
76	#121	—	151	2	no	—
77	#121	—	151	2	no	—
78	C₂×Q₈	—	151	1	no	—
79	C₂×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
80	C₂×D₄	—	151	2	no	—
81	C₂×D₄	—	151	2	no	—
82	C₂×D₄	—	151	1	no	—
83	C₂×D₄	C₂×D₁₅₁	1	1	yes	characteristic
84	C₂²×C₄	—	151	2	no	—
85	C₂²×C₄	—	151	2	no	—
86	C₄⋊C₄	—	151	2	no	—
87	C₄⋊C₄	—	151	2	no	—
88	C₄⋊C₄	—	151	2	no	—
89	C₄⋊C₄	—	151	2	no	—
90	C₄⋊C₄	—	151	1	no	—
91	C₄⋊C₄	—	151	1	no	—
92	C₂²⋊C₄	—	151	2	no	—
93	C₂²⋊C₄	—	151	2	no	—
94	C₂²⋊C₄	—	151	2	no	—
95	C₂²⋊C₄	—	151	2	no	—
96	C₂²⋊C₄	C₂×D₁₅₁	1	2	yes	—
97	C₂²⋊C₄	C₂×D₁₅₁	1	2	yes	—
98	C₄²	C₂×D₁₅₁	1	1	yes	characteristic
99	C₂³	—	151	2	no	—
100	C₂³	C₂²×D₁₅₁	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	604	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	604	1	no	—
105	D₄	—	604	2	no	—
106	D₄	—	604	2	no	—
107	C₂×C₄	—	604	2	no	—
108	C₂×C₄	—	604	2	no	—
109	C₂×C₄	—	151	2	no	—
110	C₂×C₄	—	151	2	no	—
111	C₂×C₄	—	151	2	no	—
112	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	302	2	no	—
120	C₂²	—	302	2	no	—
121	C₂²	—	302	2	no	—
122	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	302	2	no	—
128	C₄	—	302	2	no	—
129	C₄	—	302	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	604	2	no	—
132	C₂	#18012784	1	1	yes	characteristic
133	C₂	#18012785	1	1	yes	characteristic
134	C₂	#18012786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#18012731

C₂

yes

maximal, characteristic, complemented

#18012727

C₂

yes

maximal, characteristic, complemented

#18012723

C₂

yes

maximal, complemented

#18012721

C₂

yes

maximal, complemented

#18012716

C₂

yes

maximal, complemented

#18012715

C₂

yes

maximal, complemented

#18012712

C₂

yes

maximal, complemented

#18012711

C₂

yes

maximal, complemented

#124×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₀₂

C₂²

yes

characteristic

C₂×D₆₀₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₁

C₂²

yes

—

C₂×C₂²⋊D₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₂×Q₈×C₁₅₁

C₂²

yes

characteristic

C₂×D₄×C₁₅₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

C₂³

yes

—

Dic₃₀₂

—

D₆₀₄

—

C₂²⋊D₁₅₁

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

2₋¹⁺⁴

yes

characteristic

D₁₅₁

—

C₁₅₁

#562

yes

characteristic, complemented, Sylow

#562

—

151

maximal, Sylow

#125

—

151

—

#124

—

151

—

#124

D₁₅₁

yes

characteristic, complemented

#123

—

151

—

#123

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

—

151

—

#121

—

151

—

#121

—

151

—

C₂×Q₈

—

151

—

C₂×Q₈

C₂×D₁₅₁

yes

characteristic

C₂×D₄

—

151

—

C₂×D₄

—

151

—

C₂×D₄

—

151

—

C₂×D₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

C₂×D₁₅₁

yes

—

C₂²⋊C₄

C₂×D₁₅₁

yes

—

C₄²

C₂×D₁₅₁

yes

characteristic

C₂³

—

151

—

100

C₂³

C₂²×D₁₅₁

yes

—

101

Q₈

—

102

Q₈

—

604

—

103

D₄

—

104

D₄

—

604

—

105

D₄

—

604

—

106

D₄

—

604

—

107

C₂×C₄

—

604

—

108

C₂×C₄

—

604

—

109

C₂×C₄

—

151

—

110

C₂×C₄

—

151

—

111

C₂×C₄

—

151

—

112

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

113

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

114

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

115

C₂×C₄

C₂²×D₁₅₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

302

—

120

C₂²

—

302

—

121

C₂²

—

302

—

122

C₂²

C₂³×D₁₅₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

302

—

128

C₄

—

302

—

129

C₄

—

302

—

130

C₂

—

131

C₂

—

604

—

132

C₂

#18012784

yes

characteristic

133

C₂

#18012785

yes

characteristic

134

C₂

#18012786

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁴
Fitting	#124×C₁₅₁	C₂
hypercenter	#124	D₁₅₁
hypocenter	C₁₅₁	#562
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁴

Fitting

#124×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #18012731	C₂	semidirect
2. #18012727	C₂	semidirect
3. #18012723	C₂	semidirect
4. #18012721	C₂	semidirect
5. #18012716	C₂	semidirect
6. #18012715	C₂	semidirect
7. #18012712	C₂	semidirect
8. #18012711	C₂	semidirect
9. #124×C₁₅₁	C₂	semidirect
67. C₁₅₁	#562	semidirect
71. #124	D₁₅₁	semidirect

H ≅ G/N

type

1. #18012731

C₂

semidirect

2. #18012727

C₂

semidirect

3. #18012723

C₂

semidirect

4. #18012721

C₂

semidirect

5. #18012716

C₂

semidirect

6. #18012715

C₂

semidirect

7. #18012712

C₂

semidirect

8. #18012711

C₂

semidirect

9. #124×C₁₅₁

C₂

semidirect

67. C₁₅₁

#562

semidirect

71. #124

D₁₅₁

semidirect

Subgroups — #23752956

Conjugacy classes of subgroups

Complements