Group #34100265 — subgroups

Subgroups — #34100265

Order 20096.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#23757138	C₂	1	2	yes	maximal
3	#23757036	C₂	1	1	yes	maximal, characteristic
4	#23757035	C₂	1	2	yes	maximal
5	#340×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₅₇	C₄	1	4	yes	complemented
10	#19097678	—	2	2	no	—
11	#19097678	—	2	2	no	—
12	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₅₇	—	2	2	no	—
15	C₂²×Dic₁₅₇	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₅₇	D₄	1	1	yes	characteristic
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₁₅₇	—	2	2	no	—
20	C₄⋊Dic₁₅₇	—	2	2	no	—
21	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₅₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₅₇	—	2	2	no	—
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₅₇	—	2	2	no	—
26	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₁₅₇	—	2	2	no	—
28	C₄⋊C₄×C₁₅₇	—	2	2	no	—
29	C₂×Dic₁₅₇	—	2	1	no	—
30	C₂×Dic₁₅₇	—	2	2	no	—
31	C₂×Dic₁₅₇	—	2	1	no	—
32	C₂×Dic₁₅₇	—	4	2	no	—
33	C₂×Dic₁₅₇	—	4	2	no	—
34	C₂×Dic₁₅₇	—	4	2	no	—
35	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
37	#11929434	—	2	2	no	—
38	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	—	4	2	no	—
40	C₂×C₄×C₁₅₇	—	4	2	no	—
41	C₂×C₄×C₁₅₇	—	4	2	no	—
42	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	#41	1	2	yes	—
45	C₂×C₄×C₁₅₇	#41	1	2	yes	—
46	C₈×C₁₅₇	—	2	2	no	—
47	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₅₇	#124	1	2	yes	—
49	C₂²×C₁₅₇	#135	1	1	yes	characteristic
50	C₂²×C₁₅₇	#135	1	2	yes	—
51	C₂²×C₁₅₇	#135	1	1	yes	characteristic
52	C₄×C₁₅₇	—	8	2	no	—
53	C₄×C₁₅₇	#117	1	1	yes	characteristic
54	C₄×C₁₅₇	#117	1	1	yes	characteristic
55	C₄×C₁₅₇	#125	1	2	yes	—
56	Dic₁₅₇	—	2	2	no	—
57	Dic₁₅₇	—	2	2	no	—
58	Dic₁₅₇	—	8	2	no	—
59	C₂×C₁₅₇	#382	1	1	yes	characteristic
60	C₂×C₁₅₇	#427	1	1	yes	characteristic
61	C₂×C₁₅₇	#427	1	1	yes	characteristic
62	C₂×C₁₅₇	#488	1	2	yes	—
63	C₂×C₁₅₇	#488	1	2	yes	—
64	C₁₅₇	#1835	1	1	yes	characteristic, complemented, Sylow
65	#1835	—	157	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	157	1	no	—
67	#382	—	157	2	no	—
68	#340	—	157	1	no	—
69	#340	—	157	2	no	—
70	#340	D₁₅₇	1	1	yes	characteristic
71	C₂²×C₈	—	157	1	no	—
72	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	157	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
75	C₂×C₄²	—	157	1	no	—
76	C₄×C₈	—	157	4	no	—
77	#95	—	314	2	no	—
78	#95	—	314	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	314	2	no	—
81	C₂²×C₄	—	157	1	no	—
82	C₂²×C₄	—	157	1	no	—
83	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	314	2	no	—
86	C₂×C₈	—	157	2	no	—
87	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	314	2	no	—
91	C₄⋊C₄	—	314	2	no	—
92	C₄²	—	157	2	no	—
93	C₄²	—	157	2	no	—
94	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	314	1	no	—
99	C₂×C₄	—	314	2	no	—
100	C₂×C₄	—	314	1	no	—
101	C₂×C₄	—	628	2	no	—
102	C₂×C₄	—	628	2	no	—
103	C₂×C₄	—	628	2	no	—
104	C₂×C₄	—	157	2	no	—
105	C₂×C₄	—	157	2	no	—
106	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
108	C₂×C₄	#14578953	1	2	yes	—
109	C₂×C₄	#14578954	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	314	2	no	—
112	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
113	C₂²	#19097757	1	2	yes	—
114	C₂²	#19097790	1	1	yes	characteristic
115	C₂²	#19097792	1	2	yes	—
116	C₂²	#19097794	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	314	2	no	—
119	C₄	—	314	2	no	—
120	C₄	—	1256	2	no	—
121	C₄	#19097727	1	1	yes	characteristic
122	C₄	#19097728	1	1	yes	characteristic
123	C₄	#19097761	1	2	yes	—
124	C₂	#23757137	1	1	yes	characteristic
125	C₂	#23757308	1	1	yes	characteristic
126	C₂	#23757310	1	1	yes	characteristic
127	C₂	#23757626	1	2	yes	—
128	C₂	#23757629	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757138

C₂

yes

maximal

#23757036

C₂

yes

maximal, characteristic

#23757035

C₂

yes

maximal

#340×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

C₈×Dic₁₅₇

C₄

yes

complemented

#19097678

—

#19097678

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

D₄

yes

characteristic

C₂²×Dic₁₅₇

D₄

yes

characteristic

C₂×#11929434

—

C₂×#11929434

C₂×C₄

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₄×Dic₁₅₇

Q₈

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

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

#124

yes

—

C₂²×C₁₅₇

#135

yes

characteristic

C₂²×C₁₅₇

#135

yes

—

C₂²×C₁₅₇

#135

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₂×C₁₅₇

#488

yes

—

C₁₅₇

#1835

yes

characteristic, complemented, Sylow

#1835

—

157

maximal, Sylow

C₂×C₄×C₈

—

157

—

#382

—

157

—

#340

—

157

—

#340

—

157

—

#340

D₁₅₇

yes

characteristic

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₄×C₈

—

157

—

#95

—

314

—

#95

—

314

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

D₄×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

100

C₂×C₄

—

314

—

101

C₂×C₄

—

628

—

102

C₂×C₄

—

628

—

103

C₂×C₄

—

628

—

104

C₂×C₄

—

157

—

105

C₂×C₄

—

157

—

106

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

107

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

108

C₂×C₄

#14578953

yes

—

109

C₂×C₄

#14578954

yes

—

110

C₈

—

111

C₈

—

314

—

112

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

113

C₂²

#19097757

yes

—

114

C₂²

#19097790

yes

characteristic

115

C₂²

#19097792

yes

—

116

C₂²

#19097794

yes

characteristic

117

C₄

—

118

C₄

—

314

—

119

C₄

—

314

—

120

C₄

—

1256

—

121

C₄

#19097727

yes

characteristic

122

C₄

#19097728

yes

characteristic

123

C₄

#19097761

yes

—

124

C₂

#23757137

yes

characteristic

125

C₂

#23757308

yes

characteristic

126

C₂

#23757310

yes

characteristic

127

C₂

#23757626

yes

—

128

C₂

#23757629

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#340×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#340	D₁₅₇
hypocenter	C₁₅₇	#1835
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#340×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₁₅₇	C₄	semidirect
64. C₁₅₇	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₁₅₇

C₄

semidirect

64. C₁₅₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₁₅₇	2
#23757138	2
#23757036	2
#23757035	2
#340×C₁₅₇	2
#1835	157

subgroup

index

C₂×C₈×Dic₁₅₇

#340×C₁₅₇

157

Subgroups — #34100265

Conjugacy classes of subgroups

Complements

Maximal subgroups