Group #34100254 — subgroups

Subgroups — #34100254

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×#19097700	C₂	1	2	yes	maximal
2	#23757119	C₂	1	1	yes	maximal, characteristic
3	#23757028	C₂	1	1	yes	maximal, characteristic
4	#23757027	C₂	1	2	yes	maximal
5	#336×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	#19097700	—	2	4	no	—
10	#19097678	C₄	1	2	yes	—
11	#19097678	C₄	1	2	yes	—
12	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	—	2	2	no	—
19	C₂×#11929434	—	2	2	no	—
20	C₄×Dic₁₅₇	—	2	2	no	—
21	C₄×Dic₁₅₇	D₄	1	2	yes	—
22	C₄×Dic₁₅₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₅₇	—	2	2	no	—
26	C₂×C₈×C₁₅₇	—	2	2	no	—
27	C₂×C₈×C₁₅₇	—	2	2	no	—
28	C₄²×C₁₅₇	—	2	2	no	—
29	C₂×Dic₁₅₇	—	2	2	no	—
30	C₂×Dic₁₅₇	—	2	2	no	—
31	C₂×Dic₁₅₇	—	2	1	no	—
32	C₂×Dic₁₅₇	—	2	1	no	—
33	C₂×Dic₁₅₇	—	2	2	no	—
34	C₂×Dic₁₅₇	—	2	2	no	—
35	C₂×Dic₁₅₇	C₄⋊C₄	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	—	2	2	no	—
40	C₂×C₄×C₁₅₇	—	2	2	no	—
41	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	#41	1	2	yes	—
44	C₂×C₄×C₁₅₇	#41	1	2	yes	—
45	C₈×C₁₅₇	—	4	2	no	—
46	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₇	#117	1	1	yes	characteristic
48	C₂²×C₁₅₇	#117	1	1	yes	characteristic
49	C₂²×C₁₅₇	#131	1	2	yes	—
50	C₂²×C₁₅₇	#131	1	2	yes	—
51	C₄×C₁₅₇	—	4	2	no	—
52	C₄×C₁₅₇	#124	1	2	yes	—
53	C₄×C₁₅₇	#125	1	2	yes	—
54	Dic₁₅₇	—	2	4	no	—
55	Dic₁₅₇	—	4	1	no	—
56	Dic₁₅₇	—	4	1	no	—
57	C₂×C₁₅₇	#382	1	1	yes	characteristic
58	C₂×C₁₅₇	#424	1	2	yes	—
59	C₂×C₁₅₇	#433	1	2	yes	—
60	C₂×C₁₅₇	#433	1	1	yes	characteristic
61	C₂×C₁₅₇	#433	1	1	yes	characteristic
62	C₁₅₇	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	157	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	157	2	no	—
65	#376	—	157	1	no	—
66	#336	—	157	2	no	—
67	#336	—	157	1	no	—
68	#336	D₁₅₇	1	1	yes	characteristic
69	C₂²×C₈	—	157	2	no	—
70	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
71	C₂×C₄²	—	157	1	no	—
72	C₂×C₄²	—	157	1	no	—
73	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
74	C₄⋊C₈	—	314	4	no	—
75	#95	—	157	2	no	—
76	#95	—	157	2	no	—
77	C₂²×C₄	—	157	1	no	—
78	C₂²×C₄	—	157	1	no	—
79	C₂²×C₄	—	157	2	no	—
80	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	314	2	no	—
86	C₂×C₈	—	314	2	no	—
87	C₂×C₈	—	314	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	314	2	no	—
90	C₄²	—	157	2	no	—
91	C₄²	—	157	2	no	—
92	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	314	2	no	—
97	C₂×C₄	—	314	2	no	—
98	C₂×C₄	—	314	1	no	—
99	C₂×C₄	—	314	1	no	—
100	C₂×C₄	—	314	2	no	—
101	C₂×C₄	—	314	2	no	—
102	C₂×C₄	—	157	4	no	—
103	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
105	C₂×C₄	#14578953	1	2	yes	—
106	C₂×C₄	#14578954	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	628	2	no	—
109	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
110	C₂²	#19097727	1	1	yes	characteristic
111	C₂²	#19097728	1	1	yes	characteristic
112	C₂²	#19097777	1	2	yes	—
113	C₂²	#19097779	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	314	4	no	—
116	C₄	—	628	1	no	—
117	C₄	—	628	1	no	—
118	C₄	#19097757	1	2	yes	—
119	C₄	#19097761	1	2	yes	—
120	C₂	#23757137	1	1	yes	characteristic
121	C₂	#23757298	1	2	yes	—
122	C₂	#23757331	1	2	yes	—
123	C₂	#23757332	1	1	yes	characteristic
124	C₂	#23757335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097700

C₂

yes

maximal

#23757119

C₂

yes

maximal, characteristic

#23757028

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

#336×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097700

—

#19097678

C₄

yes

—

#19097678

C₄

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

—

C₄×Dic₁₅₇

—

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

—

C₄²×C₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

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

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₂²×C₁₅₇

#131

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#124

yes

—

C₄×C₁₅₇

#125

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#382

yes

characteristic

C₂×C₁₅₇

#424

yes

—

C₂×C₁₅₇

#433

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1832

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

#376

—

157

—

#336

—

157

—

#336

—

157

—

#336

D₁₅₇

yes

characteristic

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

#95

—

157

—

#95

—

157

—

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

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₄²

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

100

C₂×C₄

—

314

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

157

—

103

C₂×C₄

D₄×D₁₅₇

yes

characteristic

104

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

105

C₂×C₄

#14578953

yes

—

106

C₂×C₄

#14578954

yes

—

107

C₈

—

108

C₈

—

628

—

109

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

110

C₂²

#19097727

yes

characteristic

111

C₂²

#19097728

yes

characteristic

112

C₂²

#19097777

yes

—

113

C₂²

#19097779

yes

—

114

C₄

—

115

C₄

—

314

—

116

C₄

—

628

—

117

C₄

—

628

—

118

C₄

#19097757

yes

—

119

C₄

#19097761

yes

—

120

C₂

#23757137

yes

characteristic

121

C₂

#23757298

yes

—

122

C₂

#23757331

yes

—

123

C₂

#23757332

yes

characteristic

124

C₂

#23757335

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator, socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
Fitting	#336×C₁₅₇	C₂
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#336	D₁₅₇
hypocenter	C₁₅₇	#1832
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator, socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

Fitting

#336×C₁₅₇

C₂

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₅₇	#1832	semidirect

H ≅ G/N

type

62. C₁₅₇

#1832

semidirect

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

subgroup	index
C₂×#19097700	2
#23757119	2
#23757028	2
#23757027	2
#336×C₁₅₇	2
#1832	157

subgroup

index

#336×C₁₅₇

157

Subgroups — #34100254

Conjugacy classes of subgroups

Complements

Maximal subgroups