Group #34110659 — subgroups

Subgroups — #34110659

Order 20096.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#23757826	C₂	1	2	yes	maximal
2	#23757681	C₂	1	4	yes	maximal, complemented
3	#23757680	C₂	1	4	yes	maximal, complemented
4	#23757311	C₂	1	1	yes	maximal, characteristic
5	#23757309	C₂	1	2	yes	maximal
6	C₂×#19097681	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11929434	C₂²	1	1	yes	characteristic
9	#19097770	C₂²	1	4	yes	—
10	#19097762	C₂²	1	4	yes	—
11	#19097748	—	2	4	no	—
12	C₄×Dic₃₁₄	—	2	4	no	—
13	#19097730	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
15	#19097705	C₂²	1	2	yes	—
16	#19097704	C₂²	1	4	yes	—
17	#19097703	C₂²	1	2	yes	—
18	#19097702	C₂²	1	4	yes	—
19	#19097681	C₂²	1	2	yes	—
20	#19097681	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
22	#117×C₁₅₇	C₂²	1	2	yes	—
23	#107×C₁₅₇	C₂²	1	2	yes	—
24	#106×C₁₅₇	C₂²	1	2	yes	—
25	C₂×Dic₃₁₄	—	2	4	no	—
26	C₂²×Dic₁₅₇	D₄	1	2	yes	—
27	C₂×#11929434	Q₈	1	4	yes	complemented
28	C₂×#11929434	C₂³	1	2	yes	—
29	C₄⋊Dic₁₅₇	—	2	4	no	—
30	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
31	#14578933	—	2	4	no	—
32	#14578933	—	2	4	no	—
33	C₂²⋊Dic₁₅₇	—	2	2	no	—
34	C₄×Dic₁₅₇	—	2	2	no	—
35	C₄×Dic₁₅₇	D₄	1	2	yes	—
36	C₄×Dic₁₅₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₅₇	Q₈	1	4	yes	complemented
39	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₅₇	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
43	C₄²×C₁₅₇	—	2	2	no	—
44	Dic₃₁₄	—	4	8	no	—
45	C₂×Dic₁₅₇	—	2	4	no	—
46	C₂×Dic₁₅₇	—	4	2	no	—
47	C₂×Dic₁₅₇	C₂×D₄	1	4	yes	—
48	#11929434	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	—	2	4	no	—
51	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
56	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₁₅₇	—	2	2	no	—
58	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₁₅₇	—	4	4	no	—
61	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
62	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
64	Dic₁₅₇	—	2	4	no	—
65	Dic₁₅₇	—	4	4	no	—
66	C₂×C₁₅₇	—	2	2	no	—
67	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₅₇	#575	1	2	yes	—
69	C₁₅₇	#3162	1	1	yes	characteristic, complemented, Sylow
70	#3162	—	157	1	no	maximal, Sylow
71	#533	—	157	2	no	—
72	#501	—	157	4	no	—
73	#501	—	157	4	no	—
74	#427	—	157	1	no	—
75	#427	—	157	2	no	—
76	#427	D₁₅₇	1	1	yes	characteristic
77	C₂×#97	—	157	1	no	—
78	C₂²×C₈	—	157	1	no	—
79	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
80	C₄⋊Q₈	—	157	4	no	—
81	#125	—	157	4	no	—
82	C₂²⋊Q₈	—	314	4	no	—
83	C₄×Q₈	—	314	4	no	—
84	#117	—	157	2	no	—
85	#117	C₂×D₁₅₇	1	2	yes	—
86	C₂×C₄²	—	157	1	no	—
87	#107	—	157	2	no	—
88	#107	—	157	4	no	—
89	#107	C₂×D₁₅₇	1	2	yes	—
90	#106	—	157	2	no	—
91	#106	—	157	4	no	—
92	#106	C₂×D₁₅₇	1	2	yes	—
93	#97	—	157	2	no	—
94	#97	—	157	2	no	—
95	C₂×Q₈	—	314	4	no	—
96	C₂²×C₄	—	157	2	no	—
97	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂×C₈	—	157	4	no	—
99	C₂×C₈	—	157	2	no	—
100	C₂×C₈	Dic₃₁₄	1	4	yes	complemented
101	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
103	C₄⋊C₄	—	314	4	no	—
104	C₄⋊C₄	—	314	4	no	—
105	C₄⋊C₄	—	314	4	no	—
106	C₄⋊C₄	—	157	4	no	—
107	C₄⋊C₄	C₂²×D₁₅₇	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	314	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	314	2	no	—
112	C₄²	—	157	2	no	—
113	C₄²	—	157	2	no	—
114	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
115	Q₈	—	628	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	314	4	no	—
118	C₂×C₄	—	628	2	no	—
119	C₂×C₄	—	157	4	no	—
120	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
122	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
123	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
124	C₈	—	157	4	no	—
125	C₈	C₂×Dic₃₁₄	1	2	yes	—
126	C₈	C₂×Dic₃₁₄	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#19097770	1	2	yes	—
129	C₂²	C₂×D₄×D₁₅₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	314	4	no	—
132	C₄	—	628	4	no	—
133	C₄	#19097770	1	2	yes	—
134	C₄	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
135	C₄	C₂²×Dic₃₁₄	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#19097770	1	1	yes	characteristic
138	C₂	#23758070	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757826

C₂

yes

maximal

#23757681

C₂

yes

maximal, complemented

#23757680

C₂

yes

maximal, complemented

#23757311

C₂

yes

maximal, characteristic

#23757309

C₂

yes

maximal

C₂×#19097681

C₂

yes

maximal, characteristic

#427×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

characteristic

#19097770

C₂²

yes

—

#19097762

C₂²

yes

—

#19097748

—

C₄×Dic₃₁₄

—

#19097730

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097705

C₂²

yes

—

#19097704

C₂²

yes

—

#19097703

C₂²

yes

—

#19097702

C₂²

yes

—

#19097681

C₂²

yes

—

#19097681

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

—

#107×C₁₅₇

C₂²

yes

—

#106×C₁₅₇

C₂²

yes

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

D₄

yes

—

C₂×#11929434

Q₈

yes

complemented

C₂×#11929434

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

#14578933

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

Q₈

yes

complemented

C₂×C₈×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂³

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×D₄

yes

—

#11929434

C₂×Q₈

yes

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×Q₈

yes

—

C₈×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₄⋊Q₈

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

#575

yes

—

C₁₅₇

#3162

yes

characteristic, complemented, Sylow

#3162

—

157

maximal, Sylow

#533

—

157

—

#501

—

157

—

#501

—

157

—

#427

—

157

—

#427

—

157

—

#427

D₁₅₇

yes

characteristic

C₂×#97

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊Q₈

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

314

—

C₄×Q₈

—

314

—

#117

—

157

—

#117

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

#107

—

157

—

#107

—

157

—

#107

C₂×D₁₅₇

yes

—

#106

—

157

—

#106

—

157

—

#106

C₂×D₁₅₇

yes

—

#97

—

157

—

#97

—

157

—

C₂×Q₈

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

157

—

C₂×C₈

—

157

—

100

C₂×C₈

Dic₃₁₄

yes

complemented

101

C₂×C₈

C₂²×D₁₅₇

yes

characteristic

102

C₂×C₈

C₂²×D₁₅₇

yes

characteristic

103

C₄⋊C₄

—

314

—

104

C₄⋊C₄

—

314

—

105

C₄⋊C₄

—

314

—

106

C₄⋊C₄

—

157

—

107

C₄⋊C₄

C₂²×D₁₅₇

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

314

—

110

C₄²

—

111

C₄²

—

314

—

112

C₄²

—

157

—

113

C₄²

—

157

—

114

C₂³

D₄×D₁₅₇

yes

characteristic

115

Q₈

—

628

—

116

C₂×C₄

—

117

C₂×C₄

—

314

—

118

C₂×C₄

—

628

—

119

C₂×C₄

—

157

—

120

C₂×C₄

D₄×D₁₅₇

yes

characteristic

121

C₂×C₄

Q₈×D₁₅₇

yes

—

122

C₂×C₄

C₂×Dic₃₁₄

yes

—

123

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

124

C₈

—

157

—

125

C₈

C₂×Dic₃₁₄

yes

—

126

C₈

C₂×Dic₃₁₄

yes

—

127

C₂²

—

128

C₂²

#19097770

yes

—

129

C₂²

C₂×D₄×D₁₅₇

yes

characteristic

130

C₄

—

131

C₄

—

314

—

132

C₄

—

628

—

133

C₄

#19097770

yes

—

134

C₄

C₂×Q₈×D₁₅₇

yes

characteristic

135

C₄

C₂²×Dic₃₁₄

yes

characteristic

136

C₂

—

137

C₂

C₂×#19097770

yes

characteristic

138

C₂

#23758070

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#427×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	#427	D₁₅₇
hypocenter	C₁₅₇	#3162
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

#427×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂²×D₄

prime radical

C₂³×C₁₅₇

C₂×D₄

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #23757681	C₂	semidirect
3. #23757680	C₂	semidirect
27. C₂×#11929434	Q₈	semidirect
38. C₂×C₈×C₁₅₇	Q₈	semidirect
69. C₁₅₇	#3162	semidirect
100. C₂×C₈	Dic₃₁₄	semidirect

H ≅ G/N

type

2. #23757681

C₂

semidirect

3. #23757680

C₂

semidirect

27. C₂×#11929434

Q₈

semidirect

38. C₂×C₈×C₁₅₇

Q₈

semidirect

69. C₁₅₇

#3162

semidirect

100. C₂×C₈

Dic₃₁₄

semidirect

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

subgroup	index
#23757826	2
#23757681	2
#23757680	2
#23757311	2
#23757309	2
C₂×#19097681	2
#427×C₁₅₇	2
#3162	157

subgroup

index

#427×C₁₅₇

157

Subgroups — #34110659

Conjugacy classes of subgroups

Complements

Maximal subgroups