Group #23752896 — subgroups

Subgroups — #23752896

Order 9664.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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	#18012732	C₂	1	2	yes	maximal, complemented
2	#18012730	C₂	1	1	yes	maximal, characteristic, complemented
3	#18012729	C₂	1	2	yes	maximal, complemented
4	#18012721	C₂	1	2	yes	maximal
5	#18012715	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₁₅₁	C₂	1	1	yes	maximal, characteristic
8	#18012698	C₂	1	1	yes	maximal, characteristic, complemented
9	#18012697	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₄×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊D₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
17	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
19	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
20	Dic₁₅₁⋊C₄	C₂²	1	1	yes	characteristic
21	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
22	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₅₁	C₂²	1	2	yes	—
24	C₂²⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
25	(C₂×D₁₅₁)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
27	C₄×Dic₁₅₁	C₂²	1	2	yes	—
28	C₄×Dic₁₅₁	C₂²	1	2	yes	—
29	C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₅₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₁₅₁	C₂²	1	2	yes	—
35	C₂²×D₁₅₁	C₂³	1	1	yes	characteristic
36	C₂²⋊D₁₅₁	—	2	2	no	—
37	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
38	C₂×Dic₁₅₁	C₂³	1	2	yes	—
39	C₂×Dic₁₅₁	C₂³	1	2	yes	—
40	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
42	C₄×D₁₅₁	—	2	4	no	—
43	C₄×D₁₅₁	—	2	2	no	—
44	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
45	Q₈×C₁₅₁	—	2	2	no	—
46	C₂×C₄×C₁₅₁	—	2	2	no	—
47	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
48	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
50	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
51	C₂²×C₁₅₁	—	2	1	no	—
52	C₂²×C₁₅₁	—	2	1	no	—
53	C₂²×C₁₅₁	—	2	1	no	—
54	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₅₁	—	2	1	no	—
56	C₂×D₁₅₁	—	2	1	no	—
57	C₂×D₁₅₁	#41	1	2	yes	—
58	C₄×C₁₅₁	—	2	1	no	—
59	C₄×C₁₅₁	—	2	2	no	—
60	C₄×C₁₅₁	—	2	2	no	—
61	C₄×C₁₅₁	#41	1	2	yes	—
62	Dic₁₅₁	—	2	2	no	—
63	Dic₁₅₁	—	2	1	no	—
64	Dic₁₅₁	—	2	1	no	—
65	Dic₁₅₁	—	2	1	no	—
66	Dic₁₅₁	—	2	2	no	—
67	C₂×C₁₅₁	—	4	1	no	—
68	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
71	D₁₅₁	—	2	2	no	—
72	C₁₅₁	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	151	1	no	maximal, Sylow
74	#126	—	151	2	no	—
75	#125	—	151	2	no	—
76	#125	—	151	1	no	—
77	#123	—	151	2	no	—
78	C₂²⋊Q₈	—	151	1	no	—
79	C₂²⋊Q₈	D₁₅₁	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	151	1	no	—
81	C₄×D₄	—	151	1	no	—
82	#117	—	151	1	no	—
83	#117	—	151	2	no	—
84	C₂×C₄⋊C₄	—	151	1	no	—
85	C₂×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
86	C₂×D₄	—	151	1	no	—
87	C₂²×C₄	—	151	2	no	—
88	C₂²×C₄	—	151	1	no	—
89	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
90	C₄⋊C₄	—	151	2	no	—
91	C₄⋊C₄	—	151	1	no	—
92	C₄⋊C₄	—	151	2	no	—
93	C₄⋊C₄	—	151	2	no	—
94	C₄⋊C₄	—	151	1	no	—
95	C₄⋊C₄	—	151	1	no	—
96	C₄⋊C₄	—	151	2	no	—
97	C₄⋊C₄	—	151	2	no	—
98	C₄⋊C₄	C₂×D₁₅₁	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
100	C₂²⋊C₄	—	151	2	no	—
101	C₂²⋊C₄	—	151	1	no	—
102	C₂²⋊C₄	—	151	1	no	—
103	C₂²⋊C₄	—	151	2	no	—
104	C₂²⋊C₄	C₂×D₁₅₁	1	2	yes	—
105	C₄²	—	151	2	no	—
106	C₄²	—	151	2	no	—
107	C₄²	—	151	1	no	—
108	C₂³	—	151	1	no	—
109	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	D₄	—	302	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	302	4	no	—
114	C₂×C₄	—	302	2	no	—
115	C₂×C₄	—	151	1	no	—
116	C₂×C₄	—	151	2	no	—
117	C₂×C₄	—	151	2	no	—
118	C₂×C₄	—	151	1	no	—
119	C₂×C₄	—	151	1	no	—
120	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
122	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
123	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	302	1	no	—
128	C₂²	—	302	1	no	—
129	C₂²	—	151	2	no	—
130	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	302	2	no	—
135	C₄	—	302	1	no	—
136	C₄	—	302	1	no	—
137	C₄	—	302	1	no	—
138	C₄	—	302	2	no	—
139	C₄	#14559003	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	302	2	no	—
142	C₂	#41×D₁₅₁	1	1	yes	characteristic
143	C₂	C₂×#14559003	1	1	yes	characteristic
144	C₂	#18012787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#18012732

C₂

yes

maximal, complemented

#18012730

C₂

yes

maximal, characteristic, complemented

#18012729

C₂

yes

maximal, complemented

#18012721

C₂

yes

maximal

#18012715

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₅₁

C₂

yes

maximal, characteristic

#18012698

C₂

yes

maximal, characteristic, complemented

#18012697

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₅₁

C₂²

yes

characteristic

C₂×C₄×D₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

characteristic

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

characteristic

(C₂×D₁₅₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

characteristic

C₂×Q₈×C₁₅₁

C₂²

yes

characteristic

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₂²×D₁₅₁

C₂³

yes

characteristic

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

characteristic

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

characteristic

C₂×Dic₁₅₁

C₂³

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

characteristic

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

#41

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#41

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

2₋¹⁺⁴

yes

characteristic

D₁₅₁

—

C₁₅₁

#552

yes

characteristic, complemented, Sylow

#552

—

151

maximal, Sylow

#126

—

151

—

#125

—

151

—

#125

—

151

—

#123

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

D₁₅₁

yes

characteristic, complemented

C₄×Q₈

—

151

—

C₄×D₄

—

151

—

#117

—

151

—

#117

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×Q₈

C₂×D₁₅₁

yes

characteristic

C₂×D₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

C₂×D₁₅₁

yes

—

100

C₂²⋊C₄

—

151

—

101

C₂²⋊C₄

—

151

—

102

C₂²⋊C₄

—

151

—

103

C₂²⋊C₄

—

151

—

104

C₂²⋊C₄

C₂×D₁₅₁

yes

—

105

C₄²

—

151

—

106

C₄²

—

151

—

107

C₄²

—

151

—

108

C₂³

—

151

—

109

C₂³

C₂²×D₁₅₁

yes

characteristic

110

Q₈

—

111

D₄

—

302

—

112

C₂×C₄

—

113

C₂×C₄

—

302

—

114

C₂×C₄

—

302

—

115

C₂×C₄

—

151

—

116

C₂×C₄

—

151

—

117

C₂×C₄

—

151

—

118

C₂×C₄

—

151

—

119

C₂×C₄

—

151

—

120

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

121

C₂×C₄

C₂²×D₁₅₁

yes

—

122

C₂×C₄

C₂²×D₁₅₁

yes

—

123

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

302

—

128

C₂²

—

302

—

129

C₂²

—

151

—

130

C₂²

C₂³×D₁₅₁

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

302

—

135

C₄

—

302

—

136

C₄

—

302

—

137

C₄

—

302

—

138

C₄

—

302

—

139

C₄

#14559003

yes

—

140

C₂

—

141

C₂

—

302

—

142

C₂

#41×D₁₅₁

yes

characteristic

143

C₂

C₂×#14559003

yes

characteristic

144

C₂

#18012787

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁴
Fitting	C₂²⋊Q₈×C₁₅₁	C₂
prime radical	C₂×C₂²⋊D₁₅₁	C₂²
hypercenter	C₂²⋊Q₈	D₁₅₁
hypocenter	C₁₅₁	#552
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁴

Fitting

C₂²⋊Q₈×C₁₅₁

C₂

prime radical

C₂×C₂²⋊D₁₅₁

C₂²

hypercenter

C₂²⋊Q₈

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #18012732	C₂	semidirect
2. #18012730	C₂	semidirect
3. #18012729	C₂	semidirect
5. #18012715	C₂	semidirect
6. Q₈×Dic₁₅₁	C₂	semidirect
8. #18012698	C₂	semidirect
9. #18012697	C₂	semidirect
10. C₄⋊C₄×D₁₅₁	C₂	semidirect
11. C₂²⋊Q₈×C₁₅₁	C₂	semidirect
72. C₁₅₁	#552	semidirect
79. C₂²⋊Q₈	D₁₅₁	semidirect

H ≅ G/N

type

1. #18012732

C₂

semidirect

2. #18012730

C₂

semidirect

3. #18012729

C₂

semidirect

5. #18012715

C₂

semidirect

6. Q₈×Dic₁₅₁

C₂

semidirect

8. #18012698

C₂

semidirect

9. #18012697

C₂

semidirect

10. C₄⋊C₄×D₁₅₁

C₂

semidirect

11. C₂²⋊Q₈×C₁₅₁

C₂

semidirect

72. C₁₅₁

#552

semidirect

79. C₂²⋊Q₈

D₁₅₁

semidirect

Subgroups — #23752896

Conjugacy classes of subgroups

Complements