Group #34101076 — subgroups

Subgroups — #34101076

Order 20096.

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

normal subgroups only

This group has 280 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×C₄×Dic₃₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
3	#23757027	C₂	1	4	yes	maximal
4	C₂×#97×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₄	C₄	1	1	yes	characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₄×Dic₃₁₄	—	2	2	no	—
8	C₄×Dic₃₁₄	—	2	2	no	—
9	C₂×C₄⋊Dic₁₅₇	C₄	1	1	yes	characteristic
10	C₂×#14578933	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
12	C₄⋊#11929434	—	2	2	no	—
13	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	#97×C₁₅₇	—	2	2	no	—
16	C₂×Dic₃₁₄	—	2	4	no	—
17	C₂×Dic₃₁₄	—	2	2	no	—
18	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	—	2	4	no	—
22	C₄⋊Dic₁₅₇	—	2	2	no	—
23	#14578933	—	2	4	no	—
24	C₄×Dic₁₅₇	—	2	2	no	—
25	C₄×Dic₁₅₇	—	2	2	no	—
26	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	—	2	2	no	—
31	C₂×C₈×C₁₅₇	Q₈	1	4	yes	complemented
32	C₄²×C₁₅₇	D₄	1	2	yes	—
33	C₄²×C₁₅₇	D₄	1	2	yes	—
34	Dic₃₁₄	—	2	8	no	—
35	C₂×Dic₁₅₇	—	2	2	no	—
36	C₂×Dic₁₅₇	—	2	2	no	—
37	C₂×Dic₁₅₇	—	2	2	no	—
38	C₂×Dic₁₅₇	—	2	4	no	—
39	C₂×Dic₁₅₇	M₁₆	1	4	yes	—
40	#11929434	—	4	2	no	—
41	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	—	2	2	no	—
44	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₅₇	#41	1	2	yes	—
50	C₈×C₁₅₇	—	2	4	no	—
51	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₁₅₇	#131	1	2	yes	—
55	C₄×C₁₅₇	—	2	2	no	—
56	C₄×C₁₅₇	—	2	2	no	—
57	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₅₇	#124	1	1	yes	characteristic
59	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₅₇	—	2	4	no	—
61	Dic₁₅₇	—	4	2	no	—
62	C₂×C₁₅₇	#391	1	1	yes	characteristic
63	C₂×C₁₅₇	#407	1	1	yes	characteristic
64	C₂×C₁₅₇	#408	1	1	yes	characteristic
65	C₂×C₁₅₇	#446	1	4	yes	—
66	C₁₅₇	#1985	1	1	yes	characteristic, complemented, Sylow
67	#1985	—	157	1	no	maximal, Sylow
68	C₂×C₄×Q₈	—	157	1	no	—
69	C₂×C₄⋊C₈	—	157	1	no	—
70	C₂×#97	D₁₅₇	1	1	yes	characteristic
71	#336	—	157	4	no	—
72	C₂²×Q₈	—	157	1	no	—
73	C₂²×C₈	—	157	2	no	—
74	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
75	C₄×Q₈	—	314	2	no	—
76	C₄×Q₈	—	314	2	no	—
77	C₂×C₄⋊C₄	—	157	2	no	—
78	C₂×C₄⋊C₄	—	157	1	no	—
79	C₂×C₄²	—	157	2	no	—
80	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
81	C₄⋊C₈	—	314	2	no	—
82	#97	—	2	2	no	—
83	C₂×Q₈	—	314	4	no	—
84	C₂×Q₈	—	314	2	no	—
85	C₂²×C₄	—	157	2	no	—
86	C₂²×C₄	—	157	2	no	—
87	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	314	2	no	—
93	C₂×C₈	—	314	4	no	—
94	C₂×C₈	Dic₃₁₄	1	4	yes	complemented
95	C₄⋊C₄	—	314	4	no	—
96	C₄⋊C₄	—	314	2	no	—
97	C₄²	—	314	2	no	—
98	C₄²	—	314	2	no	—
99	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
100	C₄²	D₆₂₈	1	2	yes	—
101	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
102	Q₈	—	314	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	314	2	no	—
106	C₂×C₄	—	314	2	no	—
107	C₂×C₄	—	314	2	no	—
108	C₂×C₄	—	314	4	no	—
109	C₂×C₄	—	157	4	no	—
110	C₂×C₄	#14578931	1	2	yes	—
111	C₂×C₄	#14578931	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
115	C₂×C₄	#14578954	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	628	2	no	—
118	C₂²	C₂×#14578931	1	1	yes	characteristic
119	C₂²	C₄×Dic₃₁₄	1	2	yes	—
120	C₂²	M₁₆×D₁₅₇	1	2	yes	—
121	C₂²	#19097779	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	314	4	no	—
125	C₄	—	628	2	no	—
126	C₄	#19097750	1	2	yes	—
127	C₄	#19097759	1	1	yes	characteristic
128	C₄	#19097771	1	1	yes	characteristic
129	C₂	#23757179	1	1	yes	characteristic
130	C₂	#23757236	1	1	yes	characteristic
131	C₂	#23757241	1	1	yes	characteristic
132	C₂	#23757401	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₃₁₄

C₂

yes

maximal, characteristic

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

C₂×#97×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₄

yes

characteristic

C₂²×#11929434

C₂²

yes

—

C₄×Dic₃₁₄

—

C₄×Dic₃₁₄

—

C₂×C₄⋊Dic₁₅₇

C₄

yes

characteristic

C₂×#14578933

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₄⋊#11929434

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#97×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

#14578933

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

Q₈

yes

complemented

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

M₁₆

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₂²×C₁₅₇

C₂×M₁₆

yes

—

C₂²×C₁₅₇

#131

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

#124

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₁₅₇

#1985

yes

characteristic, complemented, Sylow

#1985

—

157

maximal, Sylow

C₂×C₄×Q₈

—

157

—

C₂×C₄⋊C₈

—

157

—

C₂×#97

D₁₅₇

yes

characteristic

#336

—

157

—

C₂²×Q₈

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₄×Q₈

—

314

—

C₄×Q₈

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

#97

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

Dic₃₁₄

yes

complemented

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

C₂²⋊D₁₅₇

yes

—

100

C₄²

D₆₂₈

yes

—

101

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

102

Q₈

—

314

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

314

—

106

C₂×C₄

—

314

—

107

C₂×C₄

—

314

—

108

C₂×C₄

—

314

—

109

C₂×C₄

—

157

—

110

C₂×C₄

#14578931

yes

—

111

C₂×C₄

#14578931

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

113

C₂×C₄

C₂×D₆₂₈

yes

characteristic

114

C₂×C₄

C₂×Dic₃₁₄

yes

—

115

C₂×C₄

#14578954

yes

—

116

C₈

—

117

C₈

—

628

—

118

C₂²

C₂×#14578931

yes

characteristic

119

C₂²

C₄×Dic₃₁₄

yes

—

120

C₂²

M₁₆×D₁₅₇

yes

—

121

C₂²

#19097779

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

314

—

125

C₄

—

628

—

126

C₄

#19097750

yes

—

127

C₄

#19097759

yes

characteristic

128

C₄

#19097771

yes

characteristic

129

C₂

#23757179

yes

characteristic

130

C₂

#23757236

yes

characteristic

131

C₂

#23757241

yes

characteristic

132

C₂

#23757401

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator, socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
Fitting	C₂×#97×C₁₅₇	C₂
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#97	D₁₅₇
hypocenter	C₁₅₇	#1985
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator, socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

Fitting

C₂×#97×C₁₅₇

C₂

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×#97

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
31. C₂×C₈×C₁₅₇	Q₈	semidirect
66. C₁₅₇	#1985	semidirect
94. C₂×C₈	Dic₃₁₄	semidirect

H ≅ G/N

type

31. C₂×C₈×C₁₅₇

Q₈

semidirect

66. C₁₅₇

#1985

semidirect

94. C₂×C₈

Dic₃₁₄

semidirect

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

subgroup	index
C₂×C₄×Dic₃₁₄	2
C₂×C₄⋊#11929434	2
#23757027	2
C₂×#97×C₁₅₇	2
#1985	157

subgroup

index

C₂×C₄×Dic₃₁₄

C₂×C₄⋊#11929434

#23757027

C₂×#97×C₁₅₇

#1985

157

Subgroups — #34101076

Conjugacy classes of subgroups

Complements

Maximal subgroups