Group #34070672 — subgroups

Subgroups — #34070672

Order 19328.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23752110	C₂	1	1	yes	maximal, characteristic
2	#23751974	C₂	1	2	yes	maximal
3	#23751974	C₂	1	2	yes	maximal
4	#23751967	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₅₁⋊C₄	—	2	1	no	—
10	C₁₅₁⋊#95	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₁	—	2	1	no	—
16	C₂²×Dic₁₅₁	—	2	1	no	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂×C₁₅₁⋊C₈	—	2	2	no	—
19	C₂×C₁₅₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₅₁	—	2	1	no	—
21	C₄⋊Dic₁₅₁	—	2	2	no	—
22	C₄⋊Dic₁₅₁	Q₈	1	2	yes	—
23	Dic₁₅₁⋊C₄	—	4	2	no	—
24	C₂²×C₄×C₁₅₁	—	2	2	no	—
25	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₁	—	2	4	no	—
29	C₄⋊C₄×C₁₅₁	—	4	2	no	—
30	C₄⋊C₄×C₁₅₁	—	4	2	no	—
31	C₄²×C₁₅₁	—	2	2	no	—
32	C₂×Dic₁₅₁	—	2	2	no	—
33	C₂×Dic₁₅₁	—	4	1	no	—
34	C₂×Dic₁₅₁	—	4	1	no	—
35	C₂×Dic₁₅₁	—	4	1	no	—
36	C₂×Dic₁₅₁	—	4	1	no	—
37	C₂×Dic₁₅₁	—	4	1	no	—
38	C₁₅₁⋊C₈	—	4	2	no	—
39	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	—	2	1	no	—
41	C₂×C₄×C₁₅₁	—	2	1	no	—
42	C₂×C₄×C₁₅₁	—	2	1	no	—
43	C₂×C₄×C₁₅₁	—	2	1	no	—
44	C₂×C₄×C₁₅₁	—	4	2	no	—
45	C₂×C₄×C₁₅₁	—	4	2	no	—
46	C₂×C₄×C₁₅₁	—	4	2	no	—
47	C₂×C₄×C₁₅₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₅₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₅₁	#41	1	2	yes	—
51	C₂×C₄×C₁₅₁	#41	1	2	yes	—
52	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
53	C₂²×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₅₁	#123	1	1	yes	characteristic
55	C₂²×C₁₅₁	#124	1	1	yes	characteristic
56	C₂²×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₁₅₁	—	2	2	no	—
61	C₄×C₁₅₁	—	2	2	no	—
62	C₄×C₁₅₁	—	8	2	no	—
63	C₄×C₁₅₁	#125	1	2	yes	—
64	C₄×C₁₅₁	#126	1	2	yes	—
65	Dic₁₅₁	—	4	2	no	—
66	Dic₁₅₁	—	8	1	no	—
67	C₂×C₁₅₁	#400	1	1	yes	characteristic
68	C₂×C₁₅₁	#476	1	1	yes	characteristic
69	C₂×C₁₅₁	#477	1	1	yes	characteristic
70	C₂×C₁₅₁	#481	1	1	yes	characteristic
71	C₂×C₁₅₁	#484	1	1	yes	characteristic
72	C₂×C₁₅₁	#487	1	1	yes	characteristic
73	C₂×C₁₅₁	#490	1	1	yes	characteristic
74	C₁₅₁	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	151	1	no	maximal, Sylow
76	#389	—	151	1	no	—
77	#384	D₁₅₁	1	1	yes	characteristic
78	#340	—	151	2	no	—
79	#340	—	151	2	no	—
80	#336	—	151	1	no	—
81	C₂²×C₈	—	151	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	302	1	no	—
85	C₂×C₄⋊C₄	—	151	1	no	—
86	C₂×C₄⋊C₄	—	151	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
88	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
89	#95	—	302	1	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	302	1	no	—
92	C₂²×C₄	—	302	1	no	—
93	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
96	C₂×C₈	—	302	2	no	—
97	C₂×C₈	—	302	2	no	—
98	C₂×C₈	—	302	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	302	1	no	—
103	C₄⋊C₄	—	302	2	no	—
104	C₄⋊C₄	—	604	2	no	—
105	C₄⋊C₄	—	151	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	302	2	no	—
116	C₂×C₄	—	604	1	no	—
117	C₂×C₄	—	604	1	no	—
118	C₂×C₄	—	604	1	no	—
119	C₂×C₄	—	604	1	no	—
120	C₂×C₄	—	604	1	no	—
121	C₂×C₄	#14558991	1	2	yes	—
122	C₂×C₄	#14558994	1	2	yes	—
123	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
124	C₂×C₄	#14559003	1	2	yes	—
125	C₂×C₄	#14559004	1	2	yes	—
126	C₂×C₄	#14559005	1	1	yes	characteristic
127	C₈	—	604	2	no	—
128	C₂²	#18012715	1	1	yes	characteristic
129	C₂²	#18012722	1	1	yes	characteristic
130	C₂²	#18012726	1	1	yes	characteristic
131	C₂²	C₂×#14558991	1	1	yes	characteristic
132	C₂²	C₂×#14558994	1	1	yes	characteristic
133	C₂²	#18012765	1	1	yes	characteristic
134	C₂²	#18012765	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	604	2	no	—
139	C₄	—	1208	1	no	—
140	C₄	#18012729	1	2	yes	—
141	C₄	#18012734	1	2	yes	—
142	C₂	#23752150	1	1	yes	characteristic
143	C₂	#23752499	1	1	yes	characteristic
144	C₂	#23752504	1	1	yes	characteristic
145	C₂	#23752530	1	1	yes	characteristic
146	C₂	#23752547	1	1	yes	characteristic
147	C₂	#23752562	1	1	yes	characteristic
148	C₂	#23752576	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752110

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

#23751974

C₂

yes

maximal

#23751967

C₂

yes

maximal, characteristic

#384×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₂×Dic₁₅₁⋊C₄

—

C₁₅₁⋊#95

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

Q₈

yes

—

Dic₁₅₁⋊C₄

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

SD₁₆

yes

—

C₂×C₄×C₁₅₁

Q₁₆

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₂²×C₁₅₁

C₂²⋊Q₈

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₁

C₂×Q₁₆

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#125

yes

—

C₄×C₁₅₁

#126

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#2100

—

151

maximal, Sylow

#389

—

151

—

#384

D₁₅₁

yes

characteristic

#340

—

151

—

#340

—

151

—

#336

—

151

—

C₂²×C₈

—

151

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#95

—

302

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

302

—

103

C₄⋊C₄

—

302

—

104

C₄⋊C₄

—

604

—

105

C₄⋊C₄

—

151

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

302

—

116

C₂×C₄

—

604

—

117

C₂×C₄

—

604

—

118

C₂×C₄

—

604

—

119

C₂×C₄

—

604

—

120

C₂×C₄

—

604

—

121

C₂×C₄

#14558991

yes

—

122

C₂×C₄

#14558994

yes

—

123

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

124

C₂×C₄

#14559003

yes

—

125

C₂×C₄

#14559004

yes

—

126

C₂×C₄

#14559005

yes

characteristic

127

C₈

—

604

—

128

C₂²

#18012715

yes

characteristic

129

C₂²

#18012722

yes

characteristic

130

C₂²

#18012726

yes

characteristic

131

C₂²

C₂×#14558991

yes

characteristic

132

C₂²

C₂×#14558994

yes

characteristic

133

C₂²

#18012765

yes

characteristic

134

C₂²

#18012765

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

604

—

139

C₄

—

1208

—

140

C₄

#18012729

yes

—

141

C₄

#18012734

yes

—

142

C₂

#23752150

yes

characteristic

143

C₂

#23752499

yes

characteristic

144

C₂

#23752504

yes

characteristic

145

C₂

#23752530

yes

characteristic

146

C₂

#23752547

yes

characteristic

147

C₂

#23752562

yes

characteristic

148

C₂

#23752576

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#384×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
hypercenter	#384	D₁₅₁
hypocenter	C₁₅₁	#2100
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#384×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
74. C₁₅₁	#2100	semidirect

H ≅ G/N

type

74. C₁₅₁

#2100

semidirect

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

subgroup	index
#23752110	2
#23751974	2
#23751967	2
#384×C₁₅₁	2
#2100	151

subgroup

index

#384×C₁₅₁

151

Subgroups — #34070672

Conjugacy classes of subgroups

Complements

Maximal subgroups