Group #34068912 — subgroups

Subgroups — #34068912

Order 19328.

284 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 113 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×#18012746	C₂	1	1	yes	maximal, characteristic
2	#23752196	C₂	1	1	yes	maximal, characteristic
3	#23752168	C₂	1	2	yes	maximal, complemented
4	#23751982	C₂	1	2	yes	maximal, complemented
5	#410×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#18012746	—	2	2	no	—
7	#18012746	—	2	2	no	—
8	C₂×#14558988	C₄	1	2	yes	complemented
9	C₂×#14558988	C₂²	1	2	yes	—
10	C₂×#14558988	C₂²	1	1	yes	characteristic
11	C₂²×C₁₅₁⋊C₈	C₄	1	2	yes	complemented
12	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
13	#18012658	C₄	1	2	yes	complemented
14	#18012656	C₄	1	2	yes	complemented
15	#18012650	—	2	2	no	—
16	C₄×C₁₅₁⋊C₈	—	2	2	no	—
17	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic
18	#117×C₁₅₁	C₂²	1	2	yes	—
19	#99×C₁₅₁	C₄	1	4	yes	—
20	#14558988	—	2	4	no	—
21	#14558988	—	2	2	no	—
22	#14558988	—	2	2	no	—
23	#14558988	C₂×C₄	1	4	yes	—
24	C₂×C₁₅₁⋊C₈	—	2	2	no	—
25	C₂×C₁₅₁⋊C₈	—	2	2	no	—
26	C₂×C₁₅₁⋊C₈	—	2	2	no	—
27	C₂×C₁₅₁⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₁₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
31	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
32	#41×C₁₅₁	—	2	2	no	—
33	#41×C₁₅₁	—	2	2	no	—
34	C₂×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₁₅₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₅₁	—	2	2	no	—
40	C₂²⋊C₄×C₁₅₁	C₂×C₄	1	4	yes	—
41	C₄²×C₁₅₁	—	2	2	no	—
42	C₁₅₁⋊C₈	—	2	2	no	—
43	C₁₅₁⋊C₈	—	2	4	no	—
44	C₁₅₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₁₅₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₅₁⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₅₁	C₄²	1	2	yes	—
48	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₅₁	—	2	2	no	—
50	D₄×C₁₅₁	—	2	4	no	—
51	D₄×C₁₅₁	—	2	2	no	—
52	C₂×C₄×C₁₅₁	—	2	2	no	—
53	C₂×C₄×C₁₅₁	—	2	2	no	—
54	C₂×C₄×C₁₅₁	—	2	4	no	—
55	C₂×C₄×C₁₅₁	C₄²	1	2	yes	—
56	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₅₁	#41	1	2	yes	—
60	C₂²×C₁₅₁	—	2	2	no	—
61	C₂²×C₁₅₁	—	4	1	no	—
62	C₂²×C₁₅₁	—	4	2	no	—
63	C₂²×C₁₅₁	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₁₅₁	C₄×D₄	1	2	yes	—
65	C₄×C₁₅₁	—	2	2	no	—
66	C₄×C₁₅₁	—	4	4	no	—
67	C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₁₅₁	#117	1	1	yes	characteristic
69	C₄×C₁₅₁	C₄×D₄	1	2	yes	—
70	C₂×C₁₅₁	—	2	2	no	—
71	C₂×C₁₅₁	—	2	1	no	—
72	C₂×C₁₅₁	—	4	2	no	—
73	C₂×C₁₅₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	C₁₅₁	#1757	1	1	yes	characteristic, complemented, Sylow
75	#1757	—	151	1	no	maximal, Sylow
76	C₂×#131	—	151	1	no	—
77	#413	—	151	1	no	—
78	#410	D₁₅₁	1	1	yes	characteristic
79	#405	—	151	2	no	—
80	#344	—	151	2	no	—
81	C₂×#41	C₂×D₁₅₁	1	1	yes	characteristic
82	#131	—	302	2	no	—
83	#131	—	302	2	no	—
84	C₂×M₁₆	—	151	2	no	—
85	C₂×M₁₆	—	151	2	no	—
86	C₂×M₁₆	—	151	1	no	—
87	C₂²×C₈	—	151	2	no	—
88	C₂²×C₈	—	151	1	no	—
89	#117	C₂×D₁₅₁	1	2	yes	—
90	#101	—	151	2	no	—
91	#100	—	151	2	no	—
92	#99	Dic₁₅₁	1	4	yes	—
93	#97	—	302	2	no	—
94	C₄×C₈	—	302	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₄×D₁₅₁	1	1	yes	characteristic
98	C₂×D₄	C₄×D₁₅₁	1	1	yes	characteristic
99	C₂×D₄	C₂×Dic₁₅₁	1	2	yes	—
100	C₂²×C₄	C₄×D₁₅₁	1	2	yes	—
101	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
102	M₁₆	—	302	2	no	—
103	M₁₆	—	302	2	no	—
104	M₁₆	—	302	4	no	—
105	M₁₆	—	151	4	no	—
106	C₂×C₈	—	302	2	no	—
107	C₂×C₈	—	302	2	no	—
108	C₂×C₈	—	302	2	no	—
109	C₂×C₈	—	151	2	no	—
110	C₂×C₈	—	151	1	no	—
111	C₂×C₈	—	151	1	no	—
112	C₂×C₈	—	151	2	no	—
113	C₂×C₈	—	151	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	C₂×Dic₁₅₁	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₂³	C₄×Dic₁₅₁	1	2	yes	—
118	C₂³	C₂²×Dic₁₅₁	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	C₄×Dic₁₅₁	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
128	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
129	C₂×C₄	#14559003	1	2	yes	—
130	C₈	—	302	2	no	—
131	C₈	—	302	4	no	—
132	C₈	—	151	1	no	—
133	C₈	—	151	1	no	—
134	C₈	—	151	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×Dic₁₅₁	1	1	yes	characteristic
139	C₂²	D₄×Dic₁₅₁	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂²⋊C₄×D₁₅₁	1	1	yes	characteristic
143	C₄	#18012695	1	1	yes	characteristic
144	C₄	(C₂²⋊D₁₅₁)⋊C₄	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₂²⋊C₄×Dic₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012746

C₂

yes

maximal, characteristic

#23752196

C₂

yes

maximal, characteristic

#23752168

C₂

yes

maximal, complemented

#23751982

C₂

yes

maximal, complemented

#410×C₁₅₁

C₂

yes

maximal, characteristic

#18012746

—

#18012746

—

C₂×#14558988

C₄

yes

complemented

C₂×#14558988

C₂²

yes

—

C₂×#14558988

C₂²

yes

characteristic

C₂²×C₁₅₁⋊C₈

C₄

yes

complemented

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

#18012658

C₄

yes

complemented

#18012656

C₄

yes

complemented

#18012650

—

C₄×C₁₅₁⋊C₈

—

C₂×#41×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

—

#99×C₁₅₁

C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

#41×C₁₅₁

—

#41×C₁₅₁

—

C₂×Q₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×D₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂×D₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₁₅₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₁₅₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₅₁

C₄²

yes

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄²

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₄²

yes

characteristic

C₂²×C₁₅₁

C₄×D₄

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₁

#117

yes

characteristic

C₄×C₁₅₁

C₄×D₄

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₄×C₂²⋊C₄

yes

characteristic

C₁₅₁

#1757

yes

characteristic, complemented, Sylow

#1757

—

151

maximal, Sylow

C₂×#131

—

151

—

#413

—

151

—

#410

D₁₅₁

yes

characteristic

#405

—

151

—

#344

—

151

—

C₂×#41

C₂×D₁₅₁

yes

characteristic

#131

—

302

—

#131

—

302

—

C₂×M₁₆

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

—

151

—

#117

C₂×D₁₅₁

yes

—

#101

—

151

—

#100

—

151

—

#99

Dic₁₅₁

yes

—

#97

—

302

—

C₄×C₈

—

302

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₅₁

yes

characteristic

C₂×D₄

C₄×D₁₅₁

yes

characteristic

C₂×D₄

C₂×Dic₁₅₁

yes

—

100

C₂²×C₄

C₄×D₁₅₁

yes

—

101

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

102

M₁₆

—

302

—

103

M₁₆

—

302

—

104

M₁₆

—

302

—

105

M₁₆

—

151

—

106

C₂×C₈

—

302

—

107

C₂×C₈

—

302

—

108

C₂×C₈

—

302

—

109

C₂×C₈

—

151

—

110

C₂×C₈

—

151

—

111

C₂×C₈

—

151

—

112

C₂×C₈

—

151

—

113

C₂×C₈

—

151

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

C₂×Dic₁₅₁

yes

—

116

C₄²

—

117

C₂³

C₄×Dic₁₅₁

yes

—

118

C₂³

C₂²×Dic₁₅₁

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₄×Dic₁₅₁

yes

—

126

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

127

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

128

C₂×C₄

D₄×D₁₅₁

yes

—

129

C₂×C₄

#14559003

yes

—

130

C₈

—

302

—

131

C₈

—

302

—

132

C₈

—

151

—

133

C₈

—

151

—

134

C₈

—

151

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₁₅₁

yes

characteristic

139

C₂²

D₄×Dic₁₅₁

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂²⋊C₄×D₁₅₁

yes

characteristic

143

C₄

#18012695

yes

characteristic

144

C₄

(C₂²⋊D₁₅₁)⋊C₄

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂²⋊C₄×Dic₁₅₁

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₁₅₁
commutator	C₂²×C₁₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#410×C₁₅₁	C₂
socle	C₂×C₁₅₁	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₅₁	C₂²
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#410	D₁₅₁
hypocenter	C₁₅₁	#1757
LCS	G ▷ C₂²×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

C₂²⋊C₄×D₁₅₁

commutator

C₂²×C₁₅₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#410×C₁₅₁

C₂

socle

C₂×C₁₅₁

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₁₅₁

C₂²

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
3. #23752168	C₂	semidirect
4. #23751982	C₂	semidirect
8. C₂×#14558988	C₄	semidirect
11. C₂²×C₁₅₁⋊C₈	C₄	semidirect
13. #18012658	C₄	semidirect
14. #18012656	C₄	semidirect
74. C₁₅₁	#1757	semidirect

H ≅ G/N

type

3. #23752168

C₂

semidirect

4. #23751982

C₂

semidirect

8. C₂×#14558988

C₄

semidirect

11. C₂²×C₁₅₁⋊C₈

C₄

semidirect

13. #18012658

C₄

semidirect

14. #18012656

C₄

semidirect

74. C₁₅₁

#1757

semidirect

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

subgroup	index
C₂×#18012746	2
#23752196	2
#23752168	2
#23751982	2
#410×C₁₅₁	2
#1757	151

subgroup

index

#410×C₁₅₁

151

Subgroups — #34068912

Conjugacy classes of subgroups

Complements

Maximal subgroups