Group #34079486 — subgroups

Subgroups — #34079486

Order 19328.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#18012736	C₂	1	1	yes	maximal, characteristic
2	#23752578	C₂	1	4	yes	maximal, complemented
3	#23752560	C₂	1	4	yes	maximal, complemented
4	#23752216	C₂	1	4	yes	maximal, complemented
5	C₄×#14558988	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₀₄	—	2	2	no	—
8	C₂²×D₆₀₄	C₂²	1	2	yes	—
9	C₂×#14558988	C₂²	1	2	yes	—
10	#18012736	—	2	2	no	—
11	#18012736	C₂²	1	2	yes	—
12	#18012736	C₂²	1	2	yes	—
13	#18012659	C₂²	1	16	yes	complemented
14	#18012650	C₂²	1	2	yes	complemented
15	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	complemented
17	#125×C₁₅₁	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
19	C₄×Q₈×C₁₅₁	—	2	2	no	—
20	#117×C₁₅₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
22	C₂³×D₁₅₁	—	2	2	no	—
23	C₂×D₆₀₄	—	2	2	no	—
24	C₂×D₆₀₄	—	2	4	no	—
25	C₂×D₆₀₄	—	2	4	no	—
26	C₂×D₆₀₄	—	2	8	no	—
27	C₂×D₆₀₄	—	4	8	no	—
28	C₂×D₆₀₄	C₂³	1	4	yes	—
29	#14558988	—	2	4	no	—
30	C₂×C₁₅₁⋊C₈	C₂³	1	4	yes	—
31	C₂×Q₈×C₁₅₁	—	2	2	no	—
32	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
33	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₅₁	—	2	4	no	—
35	C₄⋊C₄×C₁₅₁	—	2	2	no	—
36	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
38	C₄²×C₁₅₁	—	2	2	no	—
39	C₄²×C₁₅₁	D₄	1	2	yes	—
40	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
42	C₂²×D₁₅₁	—	2	4	no	—
43	C₂²×D₁₅₁	—	4	8	no	—
44	C₂²×D₁₅₁	—	4	4	no	—
45	D₆₀₄	—	2	8	no	—
46	D₆₀₄	—	4	8	no	—
47	D₆₀₄	—	4	4	no	—
48	D₆₀₄	—	4	8	no	—
49	C₁₅₁⋊C₈	—	2	4	no	—
50	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₅₁	—	4	4	no	—
52	C₂×C₄×C₁₅₁	—	2	4	no	—
53	C₂×C₄×C₁₅₁	—	2	4	no	—
54	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₅₁	#41	1	4	yes	—
58	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₅₁	—	2	2	no	—
60	C₂²×C₁₅₁	#124	1	2	yes	—
61	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₁₅₁	—	4	4	no	—
63	C₂×D₁₅₁	—	4	8	no	—
64	C₂×D₁₅₁	—	8	8	no	—
65	C₄×C₁₅₁	—	2	2	no	—
66	C₄×C₁₅₁	—	4	4	no	—
67	C₄×C₁₅₁	#124	1	2	yes	—
68	C₄×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₁₅₁	C₂×#41	1	2	yes	—
70	C₂×C₁₅₁	—	2	2	no	—
71	C₂×C₁₅₁	C₂×#124	1	1	yes	characteristic
72	C₂×C₁₅₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
73	D₁₅₁	—	8	4	no	—
74	C₁₅₁	#3138	1	1	yes	characteristic, complemented, Sylow
75	#3138	—	151	1	no	maximal, Sylow
76	#533	D₁₅₁	1	1	yes	characteristic, complemented
77	C₂×#127	—	151	1	no	—
78	#490	—	151	4	no	—
79	#486	—	151	4	no	—
80	#418	—	151	4	no	—
81	C₄×M₁₆	—	151	1	no	—
82	C₂²×D₄	—	302	2	no	—
83	C₂²×D₄	—	151	2	no	—
84	C₂×M₁₆	—	151	2	no	—
85	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	complemented
86	#127	—	302	2	no	—
87	#127	—	151	2	no	—
88	#127	—	151	2	no	—
89	#125	C₂×D₁₅₁	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₁₅₁	1	2	yes	—
93	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
94	#102	—	151	16	no	—
95	#97	—	151	2	no	—
96	C₄×C₈	—	151	2	no	—
97	C₂⁴	—	302	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	302	2	no	—
100	C₂×D₄	—	302	4	no	—
101	C₂×D₄	—	302	4	no	—
102	C₂×D₄	—	302	8	no	—
103	C₂×D₄	—	604	8	no	—
104	C₂×D₄	—	151	4	no	—
105	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
107	M₁₆	—	302	4	no	—
108	C₂×C₈	—	151	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
115	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
116	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
117	C₂³	—	302	4	no	—
118	C₂³	—	604	8	no	—
119	C₂³	—	604	4	no	—
120	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	302	8	no	—
123	D₄	—	604	8	no	—
124	D₄	—	604	4	no	—
125	D₄	—	604	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
131	C₂×C₄	#14559005	1	4	yes	—
132	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
133	C₈	—	302	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	604	4	no	—
136	C₂²	—	604	8	no	—
137	C₂²	—	1208	8	no	—
138	C₂²	#18012728	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#18012728	1	2	yes	—
143	C₄	#18012764	1	4	yes	—
144	C₄	C₂×#14559005	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	1208	4	no	—
147	C₂	C₂×#18012728	1	1	yes	characteristic
148	C₂	C₂×#18012764	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012736

C₂

yes

maximal, characteristic

#23752578

C₂

yes

maximal, complemented

#23752560

C₂

yes

maximal, complemented

#23752216

C₂

yes

maximal, complemented

C₄×#14558988

C₂

yes

maximal, characteristic, complemented

#533×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₀₄

—

C₂²×D₆₀₄

C₂²

yes

—

C₂×#14558988

C₂²

yes

—

#18012736

—

#18012736

C₂²

yes

—

#18012736

C₂²

yes

—

#18012659

C₂²

yes

complemented

#18012650

C₂²

yes

complemented

C₄×C₁₅₁⋊C₈

C₂²

yes

complemented

C₄⋊Q₈×C₁₅₁

C₂²

yes

complemented

#125×C₁₅₁

C₂²

yes

complemented

C₂²⋊Q₈×C₁₅₁

—

C₄×Q₈×C₁₅₁

—

#117×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂³×D₁₅₁

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

C₂³

yes

—

#14558988

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₂×Q₈×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#124

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#124

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₁

C₂×#41

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#124

yes

characteristic

C₂×C₁₅₁

C₂×AGL(1,ℤ/8)

yes

—

D₁₅₁

—

C₁₅₁

#3138

yes

characteristic, complemented, Sylow

#3138

—

151

maximal, Sylow

#533

D₁₅₁

yes

characteristic, complemented

C₂×#127

—

151

—

#490

—

151

—

#486

—

151

—

#418

—

151

—

C₄×M₁₆

—

151

—

C₂²×D₄

—

302

—

C₂²×D₄

—

151

—

C₂×M₁₆

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

complemented

#127

—

302

—

#127

—

151

—

#127

—

151

—

#125

C₂×D₁₅₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#102

—

151

—

#97

—

151

—

C₄×C₈

—

151

—

C₂⁴

—

302

—

C₂×Q₈

—

C₂×D₄

—

302

—

100

C₂×D₄

—

302

—

101

C₂×D₄

—

302

—

102

C₂×D₄

—

302

—

103

C₂×D₄

—

604

—

104

C₂×D₄

—

151

—

105

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

106

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

107

M₁₆

—

302

—

108

C₂×C₈

—

151

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₁₅₁

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₁₅₁

yes

—

115

C₄²

C₂²×D₁₅₁

yes

characteristic

116

C₄²

C₂²×D₁₅₁

yes

characteristic

117

C₂³

—

302

—

118

C₂³

—

604

—

119

C₂³

—

604

—

120

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

121

Q₈

—

122

D₄

—

302

—

123

D₄

—

604

—

124

D₄

—

604

—

125

D₄

—

604

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

131

C₂×C₄

#14559005

yes

—

132

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

133

C₈

—

302

—

134

C₂²

—

135

C₂²

—

604

—

136

C₂²

—

604

—

137

C₂²

—

1208

—

138

C₂²

#18012728

yes

—

139

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#18012728

yes

—

143

C₄

#18012764

yes

—

144

C₄

C₂×#14559005

yes

—

145

C₂

—

146

C₂

—

1208

—

147

C₂

C₂×#18012728

yes

characteristic

148

C₂

C₂×#18012764

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#533×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×#18012736	C₂
hypercenter	#533	D₁₅₁
hypocenter	C₁₅₁	#3138
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

#533×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

C₂

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #23752578	C₂	semidirect
3. #23752560	C₂	semidirect
4. #23752216	C₂	semidirect
5. C₄×#14558988	C₂	semidirect
6. #533×C₁₅₁	C₂	semidirect
13. #18012659	C₂²	semidirect
14. #18012650	C₂²	semidirect
15. C₄×C₁₅₁⋊C₈	C₂²	semidirect
16. C₄⋊Q₈×C₁₅₁	C₂²	semidirect
17. #125×C₁₅₁	C₂²	semidirect
74. C₁₅₁	#3138	semidirect
76. #533	D₁₅₁	semidirect
85. C₄⋊Q₈	C₂×D₁₅₁	semidirect
89. #125	C₂×D₁₅₁	semidirect

H ≅ G/N

type

2. #23752578

C₂

semidirect

3. #23752560

C₂

semidirect

4. #23752216

C₂

semidirect

5. C₄×#14558988

C₂

semidirect

6. #533×C₁₅₁

C₂

semidirect

13. #18012659

C₂²

semidirect

14. #18012650

C₂²

semidirect

15. C₄×C₁₅₁⋊C₈

C₂²

semidirect

16. C₄⋊Q₈×C₁₅₁

C₂²

semidirect

17. #125×C₁₅₁

C₂²

semidirect

74. C₁₅₁

#3138

semidirect

76. #533

D₁₅₁

semidirect

85. C₄⋊Q₈

C₂×D₁₅₁

semidirect

89. #125

C₂×D₁₅₁

semidirect

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

subgroup	index
C₂×#18012736	2
#23752578	2
#23752560	2
#23752216	2
C₄×#14558988	2
#533×C₁₅₁	2
#3138	151

subgroup

index

#533×C₁₅₁

151

Subgroups — #34079486

Conjugacy classes of subgroups

Complements

Maximal subgroups