Group #34101105 — subgroups

Subgroups — #34101105

Order 20096.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#19097763	C₂	1	1	yes	maximal, characteristic
2	#23757143	C₂	1	1	yes	maximal, characteristic
3	#23757036	C₂	1	4	yes	maximal
4	C₂×#97×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#19097763	—	2	2	no	—
6	#19097763	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
10	C₂×#14578933	—	2	2	no	—
11	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	#97×C₁₅₇	—	2	2	no	—
14	C₂²×Dic₁₅₇	—	2	2	no	—
15	C₂²×Dic₁₅₇	—	2	2	no	—
16	C₄⋊Dic₁₅₇	—	2	2	no	—
17	C₄⋊Dic₁₅₇	—	2	4	no	—
18	C₄⋊Dic₁₅₇	—	4	4	no	—
19	C₄⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
20	#14578933	—	2	8	no	—
21	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₅₇	—	2	2	no	—
24	C₂×C₈×C₁₅₇	—	2	2	no	—
25	C₂×C₈×C₁₅₇	Q₈	1	4	yes	—
26	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
27	C₄²×C₁₅₇	D₄	1	2	yes	—
28	C₂×Dic₁₅₇	—	2	4	no	—
29	C₂×Dic₁₅₇	—	4	2	no	—
30	C₂×Dic₁₅₇	—	4	2	no	—
31	C₂×Dic₁₅₇	—	4	2	no	—
32	C₂×Dic₁₅₇	—	4	4	no	—
33	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	—	2	2	no	—
35	C₂×C₄×C₁₅₇	—	2	2	no	—
36	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₅₇	#41	1	2	yes	—
41	C₈×C₁₅₇	—	2	4	no	—
42	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₅₇	#124	1	1	yes	characteristic
44	C₂²×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₅₇	#137	1	2	yes	—
47	C₄×C₁₅₇	—	2	4	no	—
48	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
49	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
50	Dic₁₅₇	—	4	4	no	—
51	Dic₁₅₇	—	8	2	no	—
52	C₂×C₁₅₇	#391	1	1	yes	characteristic
53	C₂×C₁₅₇	#417	1	2	yes	—
54	C₂×C₁₅₇	#490	1	1	yes	characteristic
55	C₂×C₁₅₇	#491	1	1	yes	characteristic
56	C₂×C₁₅₇	#501	1	2	yes	—
57	C₁₅₇	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	157	1	no	maximal, Sylow
59	C₂×#125	—	157	1	no	—
60	C₂×#97	D₁₅₇	1	1	yes	characteristic
61	#384	—	157	1	no	—
62	#340	—	157	4	no	—
63	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
64	#125	—	314	2	no	—
65	#125	—	157	4	no	—
66	C₂×C₄⋊C₄	—	314	2	no	—
67	C₂×C₄⋊C₄	—	314	2	no	—
68	C₂×C₄⋊C₄	—	157	2	no	—
69	C₂×C₄⋊C₄	—	157	2	no	—
70	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	314	2	no	—
73	C₂²×C₄	—	314	2	no	—
74	C₂²×C₄	D₆₂₈	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₃₁₄	1	4	yes	—
79	C₄⋊C₄	—	314	2	no	—
80	C₄⋊C₄	—	314	8	no	—
81	C₄⋊C₄	—	314	4	no	—
82	C₄⋊C₄	—	628	4	no	—
83	C₄⋊C₄	—	157	4	no	—
84	C₄²	C₄×D₁₅₇	1	2	yes	—
85	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
86	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	314	4	no	—
90	C₂×C₄	—	628	2	no	—
91	C₂×C₄	—	628	2	no	—
92	C₂×C₄	—	628	2	no	—
93	C₂×C₄	—	628	4	no	—
94	C₂×C₄	#14578931	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
98	C₂×C₄	#14578954	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×#14578931	1	1	yes	characteristic
101	C₂²	#19097759	1	1	yes	characteristic
102	C₂²	#19097771	1	1	yes	characteristic
103	C₂²	#19097799	1	2	yes	—
104	C₂²	#19097804	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	628	4	no	—
107	C₄	—	1256	2	no	—
108	C₄	C₄×Dic₃₁₄	1	2	yes	—
109	C₄	#19097750	1	2	yes	—
110	C₂	#23757179	1	1	yes	characteristic
111	C₂	#23757275	1	2	yes	—
112	C₂	#23757638	1	1	yes	characteristic
113	C₂	#23757642	1	1	yes	characteristic
114	C₂	#23757682	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097763

C₂

yes

maximal, characteristic

#23757143

C₂

yes

maximal, characteristic

#23757036

C₂

yes

maximal

C₂×#97×C₁₅₇

C₂

yes

maximal, characteristic

#19097763

—

#19097763

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×#14578933

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#97×C₁₅₇

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

#14578933

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

Q₈

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

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

#124

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₅₇

#137

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#391

yes

characteristic

C₂×C₁₅₇

#417

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₁₅₇

#1992

yes

characteristic, complemented, Sylow

#1992

—

157

maximal, Sylow

C₂×#125

—

157

—

C₂×#97

D₁₅₇

yes

characteristic

#384

—

157

—

#340

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

#125

—

314

—

#125

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#97

—

C₂²×C₄

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

D₆₂₈

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₁₄

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

628

—

C₄⋊C₄

—

157

—

C₄²

C₄×D₁₅₇

yes

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₂³

C₂×D₆₂₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

#14578931

yes

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

#14578954

yes

—

C₈

—

100

C₂²

C₂×#14578931

yes

characteristic

101

C₂²

#19097759

yes

characteristic

102

C₂²

#19097771

yes

characteristic

103

C₂²

#19097799

yes

—

104

C₂²

#19097804

yes

—

105

C₄

—

106

C₄

—

628

—

107

C₄

—

1256

—

108

C₄

C₄×Dic₃₁₄

yes

—

109

C₄

#19097750

yes

—

110

C₂

#23757179

yes

characteristic

111

C₂

#23757275

yes

—

112

C₂

#23757638

yes

characteristic

113

C₂

#23757642

yes

characteristic

114

C₂

#23757682

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#97×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#97	D₁₅₇
hypocenter	C₁₅₇	#1992
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₆₂₈

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#97×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×#97

D₁₅₇

hypocenter

C₁₅₇

#1992

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #19097763	C₄	semidirect
57. C₁₅₇	#1992	semidirect

H ≅ G/N

type

6. #19097763

C₄

semidirect

57. C₁₅₇

#1992

semidirect

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

subgroup	index
C₂×#19097763	2
#23757143	2
#23757036	2
C₂×#97×C₁₅₇	2
#1992	157

subgroup

index

C₂×#97×C₁₅₇

157

Subgroups — #34101105

Conjugacy classes of subgroups

Complements

Maximal subgroups