Group #34099988 — subgroups

Subgroups — #34099988

Order 20096.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19097700	C₂	1	2	yes	maximal
2	C₂×#19097681	C₂	1	2	yes	maximal
3	C₄²×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757028	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₅₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
11	#19097700	C₄	1	8	yes	complemented
12	#19097681	—	2	4	no	—
13	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×#11929434	—	2	2	no	—
19	C₂×#11929434	—	2	2	no	—
20	C₂×#11929434	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₅₇	—	2	4	no	—
22	C₄×Dic₁₅₇	—	2	4	no	—
23	C₄×Dic₁₅₇	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₅₇	D₄	1	2	yes	—
25	C₄×Dic₁₅₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₅₇	—	2	2	no	—
32	C₂×Dic₁₅₇	—	2	2	no	—
33	C₂×Dic₁₅₇	—	2	2	no	—
34	C₂×Dic₁₅₇	—	2	2	no	—
35	C₂×Dic₁₅₇	C₄²	1	2	yes	—
36	C₂×Dic₁₅₇	C₄²	1	2	yes	—
37	C₂×Dic₁₅₇	C₄⋊C₄	1	4	yes	—
38	C₂×Dic₁₅₇	M₁₆	1	4	yes	—
39	#11929434	—	2	4	no	—
40	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	M₁₆	1	4	yes	—
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	#41	1	2	yes	—
48	C₈×C₁₅₇	—	2	4	no	—
49	C₂²×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₅₇	#117	1	1	yes	characteristic
52	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₅₇	—	2	4	no	—
55	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
56	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
57	Dic₁₅₇	—	2	2	no	—
58	Dic₁₅₇	—	2	4	no	—
59	Dic₁₅₇	#97	1	4	yes	complemented
60	C₂×C₁₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₅₇	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₅₇	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₅₇	#423	1	2	yes	—
64	C₂×C₁₅₇	#432	1	2	yes	—
65	C₁₅₇	#1767	1	1	yes	characteristic, complemented, Sylow
66	#1767	—	157	1	no	maximal, Sylow
67	C₂×C₄⋊C₈	—	157	2	no	—
68	C₂×#97	—	157	2	no	—
69	C₄³	—	157	1	no	—
70	#336	—	157	1	no	—
71	#336	D₁₅₇	1	1	yes	characteristic
72	C₂²×C₈	—	157	2	no	—
73	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
74	C₂×C₄²	—	157	2	no	—
75	C₂×C₄²	—	157	2	no	—
76	C₂×C₄²	—	157	1	no	—
77	C₂×C₄²	—	157	1	no	—
78	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
79	C₄⋊C₈	—	157	8	no	—
80	#97	—	314	4	no	—
81	C₂²×C₄	—	157	2	no	—
82	C₂²×C₄	—	157	1	no	—
83	C₂²×C₄	—	157	1	no	—
84	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	314	2	no	—
89	C₂×C₈	—	314	2	no	—
90	C₂×C₈	—	157	4	no	—
91	C₂×C₈	C₄×D₁₅₇	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	314	4	no	—
94	C₄²	—	314	4	no	—
95	C₄²	—	157	4	no	—
96	C₄²	—	157	2	no	—
97	C₄²	—	157	2	no	—
98	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	314	2	no	—
102	C₂×C₄	—	314	2	no	—
103	C₂×C₄	—	314	2	no	—
104	C₂×C₄	—	157	2	no	—
105	C₂×C₄	—	157	4	no	—
106	C₂×C₄	—	157	2	no	—
107	C₂×C₄	—	157	4	no	—
108	C₂×C₄	#14578937	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
110	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
112	C₂×C₄	#14578953	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	314	4	no	—
115	C₂²	C₄²×D₁₅₇	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
117	C₂²	#19097727	1	1	yes	characteristic
118	C₂²	M₁₆×D₁₅₇	1	2	yes	—
119	C₂²	C₂×#14578937	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	314	2	no	—
122	C₄	—	314	4	no	—
123	C₄	—	157	4	no	—
124	C₄	#19097731	1	2	yes	—
125	C₄	#19097736	1	2	yes	—
126	C₂	#23757124	1	1	yes	characteristic
127	C₂	#97×D₁₅₇	1	1	yes	characteristic
128	C₂	C₄×#14578937	1	1	yes	characteristic
129	C₂	#23757295	1	2	yes	—
130	C₂	#23757327	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097700

C₂

yes

maximal

C₂×#19097681

C₂

yes

maximal

C₄²×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757028

C₂

yes

maximal, characteristic

#336×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097700

C₄

yes

complemented

#19097681

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₄×Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

Q₈

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄²

yes

—

C₂×Dic₁₅₇

C₄²

yes

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

C₂×Dic₁₅₇

M₁₆

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

M₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄²

yes

characteristic

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#117

yes

characteristic

C₂²×C₁₅₇

C₂×M₁₆

yes

—

C₂²×C₁₅₇

C₂×M₁₆

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

C₄×Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

#97

yes

complemented

C₂×C₁₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

C₂×#97

yes

characteristic

C₂×C₁₅₇

C₄×M₁₆

yes

characteristic

C₂×C₁₅₇

#423

yes

—

C₂×C₁₅₇

#432

yes

—

C₁₅₇

#1767

yes

characteristic, complemented, Sylow

#1767

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×#97

—

157

—

C₄³

—

157

—

#336

—

157

—

#336

D₁₅₇

yes

characteristic

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

—

157

—

#97

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₄²

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

314

—

104

C₂×C₄

—

157

—

105

C₂×C₄

—

157

—

106

C₂×C₄

—

157

—

107

C₂×C₄

—

157

—

108

C₂×C₄

#14578937

yes

—

109

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

110

C₂×C₄

D₄×D₁₅₇

yes

characteristic

111

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

112

C₂×C₄

#14578953

yes

—

113

C₈

—

114

C₈

—

314

—

115

C₂²

C₄²×D₁₅₇

yes

characteristic

116

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

117

C₂²

#19097727

yes

characteristic

118

C₂²

M₁₆×D₁₅₇

yes

—

119

C₂²

C₂×#14578937

yes

—

120

C₄

—

121

C₄

—

314

—

122

C₄

—

314

—

123

C₄

—

157

—

124

C₄

#19097731

yes

—

125

C₄

#19097736

yes

—

126

C₂

#23757124

yes

characteristic

127

C₂

#97×D₁₅₇

yes

characteristic

128

C₂

C₄×#14578937

yes

characteristic

129

C₂

#23757295

yes

—

130

C₂

#23757327

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂×C₄²
Fitting	#336×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#336	D₁₅₇
hypocenter	C₁₅₇	#1767
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator

C₂²×C₁₅₇

C₂×C₄²

Fitting

#336×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #19097700	C₄	semidirect
59. Dic₁₅₇	#97	semidirect
65. C₁₅₇	#1767	semidirect

H ≅ G/N

type

11. #19097700

C₄

semidirect

59. Dic₁₅₇

#97

semidirect

65. C₁₅₇

#1767

semidirect

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

subgroup	index
C₂×#19097700	2
C₂×#19097681	2
C₄²×Dic₁₅₇	2
#23757028	2
#336×C₁₅₇	2
#1767	157

subgroup

index

C₄²×Dic₁₅₇

#336×C₁₅₇

157

Subgroups — #34099988

Conjugacy classes of subgroups

Complements

Maximal subgroups