Group #34100252 — subgroups

Subgroups — #34100252

Order 20096.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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⋊#11929434	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
3	#23757027	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929434	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₅₇	C₄	1	2	yes	—
7	C₂×C₄×Dic₁₅₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
10	C₄⋊#11929434	—	2	2	no	—
11	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	—	2	2	no	—
14	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	—	2	4	no	—
18	C₄×Dic₁₅₇	—	2	4	no	—
19	C₄×Dic₁₅₇	—	2	2	no	—
20	C₄×Dic₁₅₇	—	2	2	no	—
21	C₄×Dic₁₅₇	—	2	2	no	—
22	C₄×Dic₁₅₇	C₈	1	4	yes	complemented
23	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₅₇	—	2	2	no	—
27	C₂×C₈×C₁₅₇	—	2	4	no	—
28	C₄²×C₁₅₇	D₄	1	2	yes	—
29	C₄²×C₁₅₇	Q₈	1	2	yes	—
30	C₂×Dic₁₅₇	—	2	2	no	—
31	C₂×Dic₁₅₇	—	2	2	no	—
32	C₂×Dic₁₅₇	—	2	2	no	—
33	C₂×Dic₁₅₇	—	2	2	no	—
34	C₂×Dic₁₅₇	C₂×C₈	1	4	yes	—
35	C₂×Dic₁₅₇	M₁₆	1	4	yes	—
36	#11929434	—	4	2	no	—
37	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	—	2	2	no	—
39	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₅₇	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₁₅₇	M₁₆	1	2	yes	—
43	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	#41	1	2	yes	—
46	C₂×C₄×C₁₅₇	#41	1	2	yes	—
47	C₈×C₁₅₇	—	4	2	no	—
48	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₅₇	#117	1	2	yes	—
50	C₂²×C₁₅₇	C₂²×C₈	1	1	yes	characteristic
51	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
52	C₂²×C₁₅₇	C₂×M₁₆	1	1	yes	characteristic
53	C₄×C₁₅₇	—	2	2	no	—
54	C₄×C₁₅₇	C₄⋊C₈	1	4	yes	—
55	C₄×C₁₅₇	#124	1	1	yes	characteristic
56	C₄×C₁₅₇	#124	1	1	yes	characteristic
57	C₄×C₁₅₇	#125	1	2	yes	—
58	Dic₁₅₇	—	2	4	no	—
59	Dic₁₅₇	—	2	4	no	—
60	C₂×C₁₅₇	#382	1	1	yes	characteristic
61	C₂×C₁₅₇	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₁₅₇	#423	1	1	yes	characteristic
63	C₂×C₁₅₇	#431	1	2	yes	—
64	C₂×C₁₅₇	#432	1	2	yes	—
65	C₁₅₇	#1831	1	1	yes	characteristic, complemented, Sylow
66	#1831	—	157	1	no	maximal, Sylow
67	C₂×C₄⋊C₈	—	157	1	no	—
68	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
69	C₄³	—	157	1	no	—
70	#336	—	157	4	no	—
71	C₂²×C₈	—	157	2	no	—
72	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
73	C₂×C₄²	—	157	2	no	—
74	C₂×C₄²	—	157	1	no	—
75	C₂×C₄²	—	157	2	no	—
76	C₂×C₄²	—	157	1	no	—
77	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	314	2	no	—
80	C₂²×C₄	—	157	2	no	—
81	C₂²×C₄	—	157	2	no	—
82	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	314	2	no	—
88	C₂×C₈	—	314	4	no	—
89	C₄²	—	314	4	no	—
90	C₄²	—	314	2	no	—
91	C₄²	—	314	2	no	—
92	C₄²	—	314	2	no	—
93	C₄²	—	157	4	no	—
94	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
95	C₄²	Dic₃₁₄	1	2	yes	—
96	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	314	2	no	—
99	C₂×C₄	—	314	2	no	—
100	C₂×C₄	—	314	2	no	—
101	C₂×C₄	—	314	2	no	—
102	C₂×C₄	—	157	4	no	—
103	C₂×C₄	—	157	4	no	—
104	C₂×C₄	#14578933	1	2	yes	—
105	C₂×C₄	#14578933	1	2	yes	—
106	C₂×C₄	C₈×D₁₅₇	1	2	yes	—
107	C₂×C₄	#14578937	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
110	C₂×C₄	#14578953	1	2	yes	—
111	C₂×C₄	#14578955	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	628	2	no	—
114	C₂²	C₂×#14578933	1	1	yes	characteristic
115	C₂²	#19097729	1	2	yes	—
116	C₂²	C₂×C₈×D₁₅₇	1	1	yes	characteristic
117	C₂²	M₁₆×D₁₅₇	1	2	yes	—
118	C₂²	C₂×#14578937	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	314	4	no	—
121	C₄	—	314	4	no	—
122	C₄	#19097700	1	4	yes	—
123	C₄	#19097758	1	1	yes	characteristic
124	C₄	#19097760	1	1	yes	characteristic
125	C₄	#19097762	1	2	yes	—
126	C₂	#23757138	1	1	yes	characteristic
127	C₂	C₂×#19097700	1	1	yes	characteristic
128	C₂	#23757296	1	1	yes	characteristic
129	C₂	#23757325	1	2	yes	—
130	C₂	#23757329	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

C₄²×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₄

yes

characteristic

C₂×C₄×Dic₁₅₇

C₄

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₄⋊#11929434

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₈

yes

complemented

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

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×C₈

yes

—

C₂×Dic₁₅₇

M₁₆

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×C₈

yes

—

C₂×C₄×C₁₅₇

M₁₆

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#117

yes

—

C₂²×C₁₅₇

C₂²×C₈

yes

characteristic

C₂²×C₁₅₇

C₂×M₁₆

yes

—

C₂²×C₁₅₇

C₂×M₁₆

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄⋊C₈

yes

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#382

yes

characteristic

C₂×C₁₅₇

C₂×C₄⋊C₈

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₂×C₁₅₇

#432

yes

—

C₁₅₇

#1831

yes

characteristic, complemented, Sylow

#1831

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×C₄⋊C₈

D₁₅₇

yes

characteristic

C₄³

—

157

—

#336

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

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

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

100

C₂×C₄

—

314

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

157

—

103

C₂×C₄

—

157

—

104

C₂×C₄

#14578933

yes

—

105

C₂×C₄

#14578933

yes

—

106

C₂×C₄

C₈×D₁₅₇

yes

—

107

C₂×C₄

#14578937

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

109

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

110

C₂×C₄

#14578953

yes

—

111

C₂×C₄

#14578955

yes

—

112

C₈

—

113

C₈

—

628

—

114

C₂²

C₂×#14578933

yes

characteristic

115

C₂²

#19097729

yes

—

116

C₂²

C₂×C₈×D₁₅₇

yes

characteristic

117

C₂²

M₁₆×D₁₅₇

yes

—

118

C₂²

C₂×#14578937

yes

characteristic

119

C₄

—

120

C₄

—

314

—

121

C₄

—

314

—

122

C₄

#19097700

yes

—

123

C₄

#19097758

yes

characteristic

124

C₄

#19097760

yes

characteristic

125

C₄

#19097762

yes

—

126

C₂

#23757138

yes

characteristic

127

C₂

C₂×#19097700

yes

characteristic

128

C₂

#23757296

yes

characteristic

129

C₂

#23757325

yes

—

130

C₂

#23757329

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#1831
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator

C₂²×C₁₅₇

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₄×Dic₁₅₇	C₈	semidirect
65. C₁₅₇	#1831	semidirect

H ≅ G/N

type

22. C₄×Dic₁₅₇

C₈

semidirect

65. C₁₅₇

#1831

semidirect

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

subgroup	index
C₂×C₄⋊#11929434	2
C₄²×Dic₁₅₇	2
#23757027	2
C₂×C₄⋊C₈×C₁₅₇	2
#1831	157

subgroup

index

C₂×C₄⋊#11929434

C₄²×Dic₁₅₇

#23757027

C₂×C₄⋊C₈×C₁₅₇

#1831

157

Subgroups — #34100252

Conjugacy classes of subgroups

Complements

Maximal subgroups