Group #34101482 — subgroups

Subgroups — #34101482

Order 20096.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#23757904	C₂	1	1	yes	maximal, characteristic
2	#23757315	C₂	1	2	yes	maximal
3	#23757142	C₂	1	1	yes	maximal, characteristic
4	#23757034	C₂	1	2	yes	maximal
5	#343×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	#19097770	—	2	2	no	—
7	#19097762	—	2	2	no	—
8	#19097750	—	2	2	no	—
9	#19097730	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×#14578933	—	2	2	no	—
12	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
13	#19097705	—	2	2	no	—
14	#19097703	—	2	2	no	—
15	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁₄	—	2	2	no	—
18	C₂²×Dic₁₅₇	—	2	1	no	—
19	C₂²×Dic₁₅₇	—	2	1	no	—
20	C₄⋊Dic₁₅₇	—	2	2	no	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₄⋊Dic₁₅₇	D₄	1	2	yes	—
23	C₄⋊Dic₁₅₇	D₄	1	2	yes	—
24	#14578933	—	2	4	no	—
25	#14578933	—	4	4	no	—
26	C₂²⋊Dic₁₅₇	—	2	2	no	—
27	C₄×Dic₁₅₇	—	2	2	no	—
28	C₄×Dic₁₅₇	—	2	1	no	—
29	C₄×Dic₁₅₇	Q₈	1	2	yes	complemented
30	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₅₇	—	2	2	no	—
33	M₁₆×C₁₅₇	Q₈	1	4	yes	complemented
34	C₂×C₈×C₁₅₇	—	2	2	no	—
35	C₄⋊C₄×C₁₅₇	—	2	2	no	—
36	Dic₃₁₄	—	8	2	no	—
37	C₂×Dic₁₅₇	—	2	2	no	—
38	C₂×Dic₁₅₇	—	2	2	no	—
39	C₂×Dic₁₅₇	—	2	2	no	—
40	C₂×Dic₁₅₇	—	2	2	no	—
41	C₂×Dic₁₅₇	—	2	2	no	—
42	C₂×Dic₁₅₇	—	4	2	no	—
43	C₂×Dic₁₅₇	—	4	1	no	—
44	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	—	2	2	no	—
46	C₂×C₄×C₁₅₇	—	4	2	no	—
47	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
51	C₈×C₁₅₇	—	2	4	no	—
52	C₂²×C₁₅₇	—	2	1	no	—
53	C₂²×C₁₅₇	—	2	1	no	—
54	C₂²×C₁₅₇	#120	1	1	yes	characteristic
55	C₂²×C₁₅₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₅₇	—	4	2	no	—
58	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
60	Dic₁₅₇	—	2	4	no	—
61	Dic₁₅₇	—	4	2	no	—
62	Dic₁₅₇	—	4	2	no	—
63	Dic₁₅₇	—	8	1	no	—
64	C₂×C₁₅₇	—	2	1	no	—
65	C₂×C₁₅₇	—	2	1	no	—
66	C₂×C₁₅₇	#395	1	1	yes	characteristic
67	C₂×C₁₅₇	#453	1	1	yes	characteristic
68	C₂×C₁₅₇	#456	1	1	yes	characteristic
69	C₁₅₇	#2060	1	1	yes	characteristic, complemented, Sylow
70	#2060	—	157	1	no	maximal, Sylow
71	#544	—	157	1	no	—
72	#428	—	157	2	no	—
73	#384	—	157	1	no	—
74	#343	D₁₅₇	1	1	yes	characteristic
75	#339	—	157	2	no	—
76	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
77	C₄⋊Q₈	—	314	2	no	—
78	#125	—	314	2	no	—
79	C₂²⋊Q₈	—	314	2	no	—
80	#117	—	157	2	no	—
81	C₂×C₄⋊C₄	—	314	2	no	—
82	C₂×C₄⋊C₄	—	157	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
84	C₂×C₄²	—	157	1	no	—
85	#107	—	314	2	no	—
86	#106	—	314	2	no	—
87	C₂×Q₈	—	314	2	no	—
88	C₂²×C₄	—	314	1	no	—
89	C₂²×C₄	—	314	1	no	—
90	C₂²×C₄	D₆₂₈	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	Dic₃₁₄	1	4	yes	complemented
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	314	2	no	—
97	C₄⋊C₄	—	314	4	no	—
98	C₄⋊C₄	—	314	2	no	—
99	C₄⋊C₄	—	628	4	no	—
100	C₄⋊C₄	—	157	2	no	—
101	C₄⋊C₄	—	157	2	no	—
102	C₂²⋊C₄	—	314	2	no	—
103	C₄²	—	314	2	no	—
104	C₄²	—	314	1	no	—
105	C₄²	—	157	2	no	—
106	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
107	Q₈	—	1256	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	314	2	no	—
111	C₂×C₄	—	314	2	no	—
112	C₂×C₄	—	314	2	no	—
113	C₂×C₄	—	314	2	no	—
114	C₂×C₄	—	314	2	no	—
115	C₂×C₄	—	628	2	no	—
116	C₂×C₄	—	628	1	no	—
117	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
118	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
120	C₂×C₄	#14578953	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#19097741	1	1	yes	characteristic
125	C₂²	#19097751	1	1	yes	characteristic
126	C₂²	#19097771	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	314	4	no	—
129	C₄	—	628	2	no	—
130	C₄	—	628	2	no	—
131	C₄	—	1256	1	no	—
132	C₄	#19097748	1	2	yes	—
133	C₄	#19097770	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#23757194	1	1	yes	characteristic
137	C₂	#23757443	1	1	yes	characteristic
138	C₂	#23757458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757904

C₂

yes

maximal, characteristic

#23757315

C₂

yes

maximal

#23757142

C₂

yes

maximal, characteristic

#23757034

C₂

yes

maximal

#343×C₁₅₇

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₂×#14578933

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097705

—

#19097703

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

D₄

yes

—

C₄⋊Dic₁₅₇

D₄

yes

—

#14578933

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

Q₈

yes

complemented

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

Q₈

yes

complemented

C₂×C₈×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#120

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#2060

—

157

maximal, Sylow

#544

—

157

—

#428

—

157

—

#384

—

157

—

#343

D₁₅₇

yes

characteristic

#339

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

314

—

#125

—

314

—

C₂²⋊Q₈

—

314

—

#117

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂×C₄²

—

157

—

#107

—

314

—

#106

—

314

—

C₂×Q₈

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

D₆₂₈

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

Dic₃₁₄

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

628

—

100

C₄⋊C₄

—

157

—

101

C₄⋊C₄

—

157

—

102

C₂²⋊C₄

—

314

—

103

C₄²

—

314

—

104

C₄²

—

314

—

105

C₄²

—

157

—

106

C₂³

C₂×D₆₂₈

yes

characteristic

107

Q₈

—

1256

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

314

—

111

C₂×C₄

—

314

—

112

C₂×C₄

—

314

—

113

C₂×C₄

—

314

—

114

C₂×C₄

—

314

—

115

C₂×C₄

—

628

—

116

C₂×C₄

—

628

—

117

C₂×C₄

D₄×D₁₅₇

yes

—

118

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

119

C₂×C₄

C₂×Dic₃₁₄

yes

—

120

C₂×C₄

#14578953

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#19097741

yes

characteristic

125

C₂²

#19097751

yes

characteristic

126

C₂²

#19097771

yes

characteristic

127

C₄

—

128

C₄

—

314

—

129

C₄

—

628

—

130

C₄

—

628

—

131

C₄

—

1256

—

132

C₄

#19097748

yes

—

133

C₄

#19097770

yes

—

134

C₂

—

135

C₂

—

136

C₂

#23757194

yes

characteristic

137

C₂

#23757443

yes

characteristic

138

C₂

#23757458

yes

characteristic

	subgroup	G/H
center	C₂²	#19097741
commutator, cosocle	C₂²×C₄×C₁₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#343×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#120
prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	#343	D₁₅₇
hypocenter	C₁₅₇	#2060
LCS	G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#19097741

commutator, cosocle

C₂²×C₄×C₁₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#343×C₁₅₇

C₂

socle

C₂²×C₁₅₇

#120

prime radical

C₂³×C₁₅₇

C₂×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
29. C₄×Dic₁₅₇	Q₈	semidirect
33. M₁₆×C₁₅₇	Q₈	semidirect
69. C₁₅₇	#2060	semidirect
93. M₁₆	Dic₃₁₄	semidirect

H ≅ G/N

type

29. C₄×Dic₁₅₇

Q₈

semidirect

33. M₁₆×C₁₅₇

Q₈

semidirect

69. C₁₅₇

#2060

semidirect

93. M₁₆

Dic₃₁₄

semidirect

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

subgroup	index
#23757904	2
#23757315	2
#23757142	2
#23757034	2
#343×C₁₅₇	2
#2060	157

subgroup

index

#343×C₁₅₇

157

Subgroups — #34101482

Conjugacy classes of subgroups

Complements

Maximal subgroups