Group #34068733 — subgroups

Subgroups — #34068733

Order 19328.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23752608	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752586	C₂	1	1	yes	maximal, characteristic
3	#23752585	C₂	1	1	yes	maximal, characteristic
4	#23752562	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752560	C₂	1	1	yes	maximal, characteristic
6	#23751933	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14558993	—	2	2	no	—
9	C₂×#14558991	—	2	2	no	—
10	#18012739	C₂²	1	1	yes	characteristic
11	#18012736	C₂²	1	1	yes	characteristic
12	#18012670	—	2	2	no	—
13	#18012663	—	2	2	no	—
14	#18012659	—	2	2	no	—
15	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	#107×C₁₅₁	—	2	2	no	—
20	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₀₂	—	2	1	no	—
22	C₂×D₆₀₄	—	2	1	no	—
23	C₂×D₆₀₄	—	2	1	no	—
24	C₂×D₆₀₄	—	2	1	no	—
25	#14558993	—	4	4	no	—
26	#14558991	—	4	4	no	—
27	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
28	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₁₅₁	—	2	1	no	—
30	C₄⋊Dic₁₅₁	—	2	1	no	—
31	C₂×Q₈×C₁₅₁	—	2	1	no	—
32	C₂×Q₈×C₁₅₁	—	2	1	no	—
33	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₅₁	—	2	1	no	—
35	C₄⋊C₄×C₁₅₁	—	2	1	no	—
36	C₄⋊C₄×C₁₅₁	—	2	1	no	—
37	C₄⋊C₄×C₁₅₁	—	2	1	no	—
38	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
39	C₂²×D₁₅₁	—	4	1	no	—
40	Dic₃₀₂	—	4	2	no	—
41	D₆₀₄	—	4	2	no	—
42	D₆₀₄	—	4	2	no	—
43	D₆₀₄	—	4	2	no	—
44	C₂×Dic₁₅₁	—	4	1	no	—
45	C₁₅₁⋊C₈	—	2	2	no	—
46	C₁₅₁⋊C₈	—	2	2	no	—
47	Q₈×C₁₅₁	—	4	2	no	—
48	Q₈×C₁₅₁	—	4	2	no	—
49	C₂×C₄×C₁₅₁	—	4	1	no	—
50	C₂×C₄×C₁₅₁	—	4	1	no	—
51	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₅₁	SD₁₆	1	4	yes	complemented
55	C₂²×C₁₅₁	#127	1	1	yes	characteristic
56	C₂×D₁₅₁	—	8	1	no	—
57	C₂×D₁₅₁	—	8	1	no	—
58	C₂×D₁₅₁	—	8	1	no	—
59	C₄×C₁₅₁	—	8	1	no	—
60	C₄×C₁₅₁	—	8	1	no	—
61	C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
62	C₄×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₅₁	#137	1	2	yes	—
64	Dic₁₅₁	—	8	1	no	—
65	C₂×C₁₅₁	#492	1	1	yes	characteristic
66	C₂×C₁₅₁	#496	1	1	yes	characteristic
67	C₂×C₁₅₁	#497	1	1	yes	characteristic
68	D₁₅₁	—	16	1	no	—
69	C₁₅₁	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	151	1	no	maximal, Sylow
71	#500	D₁₅₁	1	1	yes	characteristic, complemented
72	#498	—	151	1	no	—
73	#492	—	151	1	no	—
74	#492	—	151	1	no	—
75	#487	—	151	1	no	—
76	#486	—	151	1	no	—
77	#322	—	151	1	no	—
78	C₂×SD₁₆	—	302	2	no	—
79	C₂×SD₁₆	—	302	2	no	—
80	C₄⋊Q₈	—	151	1	no	—
81	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
83	#127	—	151	1	no	—
84	#107	—	2	2	no	—
85	#106	—	302	2	no	—
86	#103	—	302	2	no	—
87	#102	—	302	2	no	—
88	C₄×C₈	—	151	1	no	—
89	C₄×C₈	—	151	1	no	—
90	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	302	1	no	—
94	C₂×D₄	—	302	1	no	—
95	C₂×D₄	—	302	1	no	—
96	C₂×D₄	—	302	1	no	—
97	SD₁₆	—	604	4	no	—
98	SD₁₆	—	604	4	no	—
99	C₂×C₈	—	151	2	no	—
100	C₂×C₈	—	151	2	no	—
101	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	302	1	no	—
107	C₄⋊C₄	—	302	1	no	—
108	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
109	C₂³	—	604	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	604	2	no	—
113	D₄	—	604	2	no	—
114	D₄	—	604	2	no	—
115	D₄	—	604	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	604	1	no	—
119	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
122	C₈	—	302	2	no	—
123	C₈	—	302	2	no	—
124	C₈	#14558991	1	4	yes	complemented
125	C₂²	—	1208	1	no	—
126	C₂²	—	1208	1	no	—
127	C₂²	—	1208	1	no	—
128	C₂²	#18012735	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	1208	1	no	—
132	C₄	C₂×#14558991	1	2	yes	—
133	C₄	#18012763	1	2	yes	—
134	C₄	#18012771	1	2	yes	—
135	C₂	—	2416	1	no	—
136	C₂	#23752586	1	1	yes	characteristic
137	C₂	#23752600	1	1	yes	characteristic
138	C₂	#23752605	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752608

C₂

yes

maximal, characteristic, complemented

#23752586

C₂

yes

maximal, characteristic

#23752585

C₂

yes

maximal, characteristic

#23752562

C₂

yes

maximal, characteristic, complemented

#23752560

C₂

yes

maximal, characteristic

#23751933

C₂

yes

maximal, characteristic, complemented

#500×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14558993

—

C₂×#14558991

—

#18012739

C₂²

yes

characteristic

#18012736

C₂²

yes

—

—

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#107×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

#14558993

—

#14558991

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

Dic₃₀₂

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

SD₁₆

yes

complemented

C₂²×C₁₅₁

#127

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₁

#137

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

D₁₅₁

—

C₁₅₁

#1714

yes

characteristic, complemented, Sylow

#1714

—

151

maximal, Sylow

#500

D₁₅₁

yes

characteristic, complemented

#498

—

151

—

#492

—

151

—

#492

—

151

—

#487

—

151

—

#486

—

151

—

#322

—

151

—

C₂×SD₁₆

—

302

—

C₂×SD₁₆

—

302

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#127

—

151

—

#107

—

#106

—

302

—

#103

—

302

—

#102

—

302

—

C₄×C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

SD₁₆

—

604

—

SD₁₆

—

604

—

C₂×C₈

—

151

—

100

C₂×C₈

—

151

—

101

C₂×C₈

C₂²⋊D₁₅₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

302

—

107

C₄⋊C₄

—

302

—

108

C₄²

C₂²×D₁₅₁

yes

characteristic

109

C₂³

—

604

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

604

—

113

D₄

—

604

—

114

D₄

—

604

—

115

D₄

—

604

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

604

—

119

C₂×C₄

D₄×D₁₅₁

yes

characteristic

120

C₂×C₄

D₄×D₁₅₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

122

C₈

—

302

—

123

C₈

—

302

—

124

C₈

#14558991

yes

complemented

125

C₂²

—

1208

—

126

C₂²

—

1208

—

127

C₂²

—

1208

—

128

C₂²

#18012735

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

1208

—

132

C₄

C₂×#14558991

yes

—

133

C₄

#18012763

yes

—

134

C₄

#18012771

yes

—

135

C₂

—

2416

—

136

C₂

#23752586

yes

characteristic

137

C₂

#23752600

yes

characteristic

138

C₂

#23752605

yes

characteristic

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#500×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
prime radical	#18012736	C₂²
hypercenter	#500	D₁₅₁
hypocenter	C₁₅₁	#1714
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#18012735

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#500×C₁₅₁

C₂

C₂²×C₁₅₁

C₂²

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #23752608	C₂	semidirect
4. #23752562	C₂	semidirect
6. #23751933	C₂	semidirect
7. #500×C₁₅₁	C₂	semidirect
54. C₈×C₁₅₁	SD₁₆	semidirect
69. C₁₅₁	#1714	semidirect
71. #500	D₁₅₁	semidirect
124. C₈	#14558991	semidirect

H ≅ G/N

type

1. #23752608

C₂

semidirect

4. #23752562

C₂

semidirect

6. #23751933

C₂

semidirect

7. #500×C₁₅₁

C₂

semidirect

54. C₈×C₁₅₁

SD₁₆

semidirect

69. C₁₅₁

#1714

semidirect

71. #500

D₁₅₁

semidirect

124. C₈

#14558991

semidirect

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

subgroup	index
#23752608	2
#23752586	2
#23752585	2
#23752562	2
#23752560	2
#23751933	2
#500×C₁₅₁	2
#1714	151

subgroup

index

#500×C₁₅₁

151

Subgroups — #34068733

Conjugacy classes of subgroups

Complements

Maximal subgroups