Group #23752920 — subgroups

Subgroups — #23752920

Order 9664.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#18012738	C₂	1	1	yes	maximal, characteristic, complemented
2	#18012729	C₂	1	2	yes	maximal, complemented
3	#18012719	C₂	1	2	yes	maximal, complemented
4	#18012716	C₂	1	2	yes	maximal, complemented
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	(C₂²⋊D₁₅₁)⋊C₄	C₂	1	2	yes	maximal
9	C₂×C₄⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, 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×Dic₃₀₂	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₅₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
15	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
17	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
20	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
21	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
22	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
23	Dic₁₅₁⋊C₄	C₂²	1	1	yes	characteristic
24	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
26	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₅₁)⋊C₄	C₂²	1	1	yes	characteristic
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	Dic₃₀₂	—	4	2	no	—
37	C₂²⋊D₁₅₁	Q₈	1	4	yes	complemented
38	C₂×Dic₁₅₁	—	4	2	no	—
39	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₅₁	C₂³	1	2	yes	—
41	C₂×Dic₁₅₁	C₂³	1	2	yes	—
42	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
44	C₄×D₁₅₁	—	2	2	no	—
45	C₄×D₁₅₁	—	4	2	no	—
46	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
47	Q₈×C₁₅₁	—	2	2	no	—
48	C₂×C₄×C₁₅₁	—	2	2	no	—
49	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
50	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
52	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
53	C₂²×C₁₅₁	—	2	1	no	—
54	C₂²×C₁₅₁	—	2	1	no	—
55	C₂²×C₁₅₁	C₂×Q₈	1	2	yes	—
56	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
57	C₂×D₁₅₁	—	2	1	no	—
58	C₂×D₁₅₁	—	2	1	no	—
59	C₂×D₁₅₁	C₂×Q₈	1	2	yes	—
60	C₄×C₁₅₁	—	2	1	no	—
61	C₄×C₁₅₁	—	2	2	no	—
62	C₄×C₁₅₁	—	2	2	no	—
63	C₄×C₁₅₁	#41	1	2	yes	—
64	Dic₁₅₁	—	2	1	no	—
65	Dic₁₅₁	—	2	2	no	—
66	Dic₁₅₁	—	2	2	no	—
67	Dic₁₅₁	—	2	1	no	—
68	Dic₁₅₁	C₂×Q₈	1	2	yes	—
69	C₂×C₁₅₁	—	2	2	no	—
70	C₂×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
72	C₂×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
73	D₁₅₁	—	2	2	no	—
74	C₁₅₁	#554	1	1	yes	characteristic, complemented, Sylow
75	#554	—	151	1	no	maximal, Sylow
76	C₄⋊Q₈	—	151	1	no	—
77	#125	—	151	2	no	—
78	C₂²⋊Q₈	—	151	1	no	—
79	C₂²⋊Q₈	—	151	2	no	—
80	C₂²⋊Q₈	—	151	2	no	—
81	C₂²⋊Q₈	D₁₅₁	1	1	yes	characteristic, complemented
82	C₄×Q₈	—	151	1	no	—
83	C₄×D₄	—	151	1	no	—
84	C₄×D₄	—	151	2	no	—
85	C₂×C₄⋊C₄	—	151	1	no	—
86	C₂×C₄⋊C₄	—	151	1	no	—
87	C₂×Q₈	—	151	2	no	—
88	C₂×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
89	C₂×D₄	—	151	1	no	—
90	C₂²×C₄	—	151	1	no	—
91	C₂²×C₄	—	151	2	no	—
92	C₂²×C₄	—	151	2	no	—
93	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
94	C₄⋊C₄	—	151	2	no	—
95	C₄⋊C₄	—	151	1	no	—
96	C₄⋊C₄	—	151	2	no	—
97	C₄⋊C₄	—	151	2	no	—
98	C₄⋊C₄	—	151	1	no	—
99	C₄⋊C₄	—	151	2	no	—
100	C₄⋊C₄	—	151	1	no	—
101	C₄⋊C₄	—	151	2	no	—
102	C₄⋊C₄	C₂×D₁₅₁	1	1	yes	characteristic
103	C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
104	C₂²⋊C₄	—	151	1	no	—
105	C₂²⋊C₄	—	151	1	no	—
106	C₂²⋊C₄	—	151	2	no	—
107	C₂²⋊C₄	C₂×D₁₅₁	1	2	yes	—
108	C₄²	—	151	2	no	—
109	C₄²	—	151	1	no	—
110	C₂³	—	151	1	no	—
111	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	Q₈	—	604	2	no	—
114	D₄	—	151	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	302	2	no	—
117	C₂×C₄	—	604	2	no	—
118	C₂×C₄	—	604	2	no	—
119	C₂×C₄	—	151	1	no	—
120	C₂×C₄	—	151	2	no	—
121	C₂×C₄	—	151	1	no	—
122	C₂×C₄	—	151	2	no	—
123	C₂×C₄	—	151	1	no	—
124	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
125	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
127	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	302	1	no	—
131	C₂²	—	302	1	no	—
132	C₂²	—	151	2	no	—
133	C₂²	Q₈×D₁₅₁	1	2	yes	—
134	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	302	1	no	—
139	C₄	—	302	2	no	—
140	C₄	—	302	2	no	—
141	C₄	—	302	1	no	—
142	C₄	—	151	2	no	—
143	C₄	#14559003	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	302	2	no	—
146	C₂	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
147	C₂	C₂×#14559003	1	1	yes	characteristic
148	C₂	#18012785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#18012738

C₂

yes

maximal, characteristic, complemented

#18012729

C₂

yes

maximal, complemented

#18012719

C₂

yes

maximal, complemented

#18012716

C₂

yes

maximal, complemented

#18012715

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₅₁

C₂

yes

maximal, characteristic

(C₂²⋊D₁₅₁)⋊C₄

C₂

yes

maximal

C₂×C₄⋊Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₀₂

C₂²

yes

—

C₂×C₂²⋊D₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

C₂×C₄×D₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

characteristic

Dic₁₅₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

characteristic

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

characteristic

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

Dic₃₀₂

—

C₂²⋊D₁₅₁

Q₈

yes

complemented

C₂×Dic₁₅₁

—

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

C₂×Q₈

yes

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

C₂×Q₈

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#41

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₂×Q₈

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂²×Q₈

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

H(5,𝔽₂)

yes

characteristic

D₁₅₁

—

C₁₅₁

#554

yes

characteristic, complemented, Sylow

#554

—

151

maximal, Sylow

C₄⋊Q₈

—

151

—

#125

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

D₁₅₁

yes

characteristic, complemented

C₄×Q₈

—

151

—

C₄×D₄

—

151

—

C₄×D₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×Q₈

—

151

—

C₂×Q₈

C₂×D₁₅₁

yes

characteristic

C₂×D₄

—

151

—

C₂²×C₄

—

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

—

100

C₄⋊C₄

—

151

—

101

C₄⋊C₄

—

151

—

102

C₄⋊C₄

C₂×D₁₅₁

yes

characteristic

103

C₄⋊C₄

C₂×D₁₅₁

yes

—

104

C₂²⋊C₄

—

151

—

105

C₂²⋊C₄

—

151

—

106

C₂²⋊C₄

—

151

—

107

C₂²⋊C₄

C₂×D₁₅₁

yes

—

108

C₄²

—

151

—

109

C₄²

—

151

—

110

C₂³

—

151

—

111

C₂³

C₂²×D₁₅₁

yes

characteristic

112

Q₈

—

113

Q₈

—

604

—

114

D₄

—

151

—

115

C₂×C₄

—

116

C₂×C₄

—

302

—

117

C₂×C₄

—

604

—

118

C₂×C₄

—

604

—

119

C₂×C₄

—

151

—

120

C₂×C₄

—

151

—

121

C₂×C₄

—

151

—

122

C₂×C₄

—

151

—

123

C₂×C₄

—

151

—

124

C₂×C₄

C₂²×D₁₅₁

yes

—

125

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

126

C₂×C₄

C₂²×D₁₅₁

yes

—

127

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

302

—

131

C₂²

—

302

—

132

C₂²

—

151

—

133

C₂²

Q₈×D₁₅₁

yes

—

134

C₂²

C₂³×D₁₅₁

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

302

—

139

C₄

—

302

—

140

C₄

—

302

—

141

C₄

—

302

—

142

C₄

—

151

—

143

C₄

#14559003

yes

—

144

C₂

—

145

C₂

—

302

—

146

C₂

C₂×Q₈×D₁₅₁

yes

characteristic

147

C₂

C₂×#14559003

yes

characteristic

148

C₂

#18012785

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	#554
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. #18012738	C₂	semidirect
2. #18012729	C₂	semidirect
3. #18012719	C₂	semidirect
4. #18012716	C₂	semidirect
5. #18012715	C₂	semidirect
6. Q₈×Dic₁₅₁	C₂	semidirect
9. C₂×C₄⋊Dic₁₅₁	C₂	semidirect
10. C₄⋊C₄×D₁₅₁	C₂	semidirect
11. C₂²⋊Q₈×C₁₅₁	C₂	semidirect
37. C₂²⋊D₁₅₁	Q₈	semidirect
74. C₁₅₁	#554	semidirect
81. C₂²⋊Q₈	D₁₅₁	semidirect

H ≅ G/N

type

1. #18012738

C₂

semidirect

2. #18012729

C₂

semidirect

3. #18012719

C₂

semidirect

4. #18012716

C₂

semidirect

5. #18012715

C₂

semidirect

6. Q₈×Dic₁₅₁

C₂

semidirect

9. C₂×C₄⋊Dic₁₅₁

C₂

semidirect

10. C₄⋊C₄×D₁₅₁

C₂

semidirect

11. C₂²⋊Q₈×C₁₅₁

C₂

semidirect

37. C₂²⋊D₁₅₁

Q₈

semidirect

74. C₁₅₁

#554

semidirect

81. C₂²⋊Q₈

D₁₅₁

semidirect

Subgroups — #23752920

Conjugacy classes of subgroups

Complements