Group #34101106 — subgroups

Subgroups — #34101106

Order 20096.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19097682	C₂	1	1	yes	maximal, characteristic
2	#23757143	C₂	1	1	yes	maximal, characteristic
3	#23757035	C₂	1	4	yes	maximal
4	C₂×#125×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929434	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
7	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
8	#19097682	—	2	2	no	—
9	#125×C₁₅₇	—	2	2	no	—
10	#125×C₁₅₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₇	—	2	2	no	—
15	C₂×#11929434	—	2	2	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	Q₈	1	4	yes	—
18	C₄⋊Dic₁₅₇	—	2	4	no	—
19	C₄⋊Dic₁₅₇	—	4	4	no	—
20	C₂²×C₄×C₁₅₇	—	2	2	no	—
21	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₅₇	—	2	8	no	—
24	C₄⋊C₄×C₁₅₇	—	2	2	no	—
25	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
26	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
27	C₄²×C₁₅₇	D₄	1	2	yes	—
28	C₂×Dic₁₅₇	—	4	2	no	—
29	C₂×Dic₁₅₇	—	4	4	no	—
30	#11929434	—	2	4	no	—
31	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₅₇	—	2	2	no	—
33	C₂×C₄×C₁₅₇	—	2	2	no	—
34	C₂×C₄×C₁₅₇	—	2	4	no	—
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	—	4	2	no	—
37	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₅₇	#41	1	2	yes	—
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	—	4	4	no	—
49	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
50	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
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	D₁₅₇	1	1	yes	characteristic
60	C₂×#97	—	157	1	no	—
61	#384	—	157	1	no	—
62	#340	—	157	4	no	—
63	C₂²×C₈	—	157	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₁₅₇	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	314	2	no	—
68	C₂×C₄⋊C₄	—	157	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
70	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
71	#97	—	314	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	314	2	no	—
74	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
76	C₂×C₈	—	314	2	no	—
77	C₂×C₈	—	314	2	no	—
78	C₂×C₈	—	157	4	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	314	4	no	—
82	C₄⋊C₄	—	628	4	no	—
83	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
84	C₄²	C₂×Dic₁₅₇	1	2	yes	—
85	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	628	2	no	—
93	C₂×C₄	—	628	4	no	—
94	C₂×C₄	C₂²⋊Dic₁₅₇	1	4	yes	—
95	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
98	C₂×C₄	#14578955	1	2	yes	—
99	C₈	—	314	4	no	—
100	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
101	C₂²	#19097760	1	1	yes	characteristic
102	C₂²	#19097769	1	1	yes	characteristic
103	C₂²	#19097796	1	2	yes	—
104	C₂²	#19097801	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	1256	2	no	—
108	C₄	Q₈×Dic₁₅₇	1	2	yes	—
109	C₄	#19097747	1	2	yes	—
110	C₂	#23757180	1	1	yes	characteristic
111	C₂	#23757274	1	2	yes	—
112	C₂	#23757639	1	1	yes	characteristic
113	C₂	#23757643	1	1	yes	characteristic
114	C₂	#23757683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097682

C₂

yes

maximal, characteristic

#23757143

C₂

yes

maximal, characteristic

#23757035

C₂

yes

maximal

C₂×#125×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

#19097682

—

#125×C₁₅₇

—

#125×C₁₅₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

Q₈

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

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

—

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×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

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

D₁₅₇

yes

characteristic

C₂×#97

—

157

—

#384

—

157

—

#340

—

157

—

C₂²×C₈

—

157

—

#125

—

#125

Dic₁₅₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#97

—

314

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

628

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₄²

C₂×Dic₁₅₇

yes

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

Q₈×D₁₅₇

yes

—

C₂×C₄

#14578955

yes

—

C₈

—

314

—

100

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

101

C₂²

#19097760

yes

characteristic

102

C₂²

#19097769

yes

characteristic

103

C₂²

#19097796

yes

—

104

C₂²

#19097801

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1256

—

108

C₄

Q₈×Dic₁₅₇

yes

—

109

C₄

#19097747

yes

—

110

C₂

#23757180

yes

characteristic

111

C₂

#23757274

yes

—

112

C₂

#23757639

yes

characteristic

113

C₂

#23757643

yes

characteristic

114

C₂

#23757683

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#125×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#125	D₁₅₇
hypocenter	C₁₅₇	#1992
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#125×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 ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₅₇	C₄	semidirect
57. C₁₅₇	#1992	semidirect
65. #125	Dic₁₅₇	semidirect

H ≅ G/N

type

10. #125×C₁₅₇

C₄

semidirect

57. C₁₅₇

#1992

semidirect

65. #125

Dic₁₅₇

semidirect

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

subgroup	index
C₂×#19097682	2
#23757143	2
#23757035	2
C₂×#125×C₁₅₇	2
#1992	157

subgroup

index

C₂×#125×C₁₅₇

157

Subgroups — #34101106

Conjugacy classes of subgroups

Complements

Maximal subgroups