Group #34100270 — subgroups

Subgroups — #34100270

Order 20096.

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

normal subgroups only

This group has 224 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	#23757143	C₂	1	1	yes	maximal, characteristic
2	#23757139	C₂	1	1	yes	maximal, characteristic
3	#23757036	C₂	1	4	yes	maximal
4	C₂×#97×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
6	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
8	#19097678	—	2	2	no	—
9	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
11	#97×C₁₅₇	C₄	1	4	yes	complemented
12	C₂²×Dic₁₅₇	—	2	2	no	—
13	C₂²×Dic₁₅₇	—	2	2	no	—
14	C₄⋊Dic₁₅₇	—	2	4	no	—
15	C₄⋊Dic₁₅₇	—	2	4	no	—
16	C₄⋊Dic₁₅₇	—	4	4	no	—
17	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
18	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₁₅₇	—	2	2	no	—
20	C₂×C₈×C₁₅₇	—	2	2	no	—
21	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
22	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₅₇	Q₈	1	2	yes	—
24	C₂×Dic₁₅₇	—	4	2	no	—
25	C₂×Dic₁₅₇	—	4	2	no	—
26	C₂×Dic₁₅₇	—	4	2	no	—
27	C₂×Dic₁₅₇	—	4	4	no	—
28	C₂×Dic₁₅₇	—	4	2	no	—
29	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	—	2	2	no	—
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	#41	1	2	yes	—
36	C₂×C₄×C₁₅₇	#41	1	2	yes	—
37	C₈×C₁₅₇	—	2	4	no	—
38	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₅₇	#124	1	1	yes	characteristic
40	C₂²×C₁₅₇	#124	1	1	yes	characteristic
41	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₅₇	#137	1	2	yes	—
43	C₄×C₁₅₇	—	2	4	no	—
44	C₄×C₁₅₇	#117	1	2	yes	—
45	C₄×C₁₅₇	#125	1	2	yes	—
46	Dic₁₅₇	—	8	2	no	—
47	Dic₁₅₇	—	8	2	no	—
48	C₂×C₁₅₇	#382	1	1	yes	characteristic
49	C₂×C₁₅₇	#428	1	2	yes	—
50	C₂×C₁₅₇	#489	1	2	yes	—
51	C₂×C₁₅₇	#490	1	1	yes	characteristic
52	C₂×C₁₅₇	#491	1	1	yes	characteristic
53	C₁₅₇	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	157	1	no	maximal, Sylow
55	C₂×#97	D₁₅₇	1	1	yes	characteristic
56	#384	—	157	1	no	—
57	#382	—	157	1	no	—
58	#340	—	157	4	no	—
59	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
60	C₂×C₄⋊C₄	—	314	2	no	—
61	C₂×C₄⋊C₄	—	157	2	no	—
62	C₂×C₄⋊C₄	—	157	2	no	—
63	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
64	#97	Dic₁₅₇	1	4	yes	complemented
65	#95	—	314	2	no	—
66	C₂²×C₄	—	314	2	no	—
67	C₂²×C₄	—	314	2	no	—
68	C₂²×C₄	D₆₂₈	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	—
73	C₄⋊C₄	—	314	4	no	—
74	C₄⋊C₄	—	314	4	no	—
75	C₄⋊C₄	—	628	4	no	—
76	C₄²	C₂×Dic₁₅₇	1	2	yes	—
77	C₄²	Dic₃₁₄	1	2	yes	—
78	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	628	2	no	—
82	C₂×C₄	—	628	2	no	—
83	C₂×C₄	—	628	4	no	—
84	C₂×C₄	—	628	2	no	—
85	C₂×C₄	—	628	2	no	—
86	C₂×C₄	C₄⋊Dic₁₅₇	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
89	C₂×C₄	#14578954	1	2	yes	—
90	C₂×C₄	#14578954	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
93	C₂²	#19097759	1	1	yes	characteristic
94	C₂²	#19097759	1	1	yes	characteristic
95	C₂²	#19097799	1	2	yes	—
96	C₂²	#19097804	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	1256	2	no	—
99	C₄	—	1256	2	no	—
100	C₄	#19097730	1	2	yes	—
101	C₄	#19097763	1	2	yes	—
102	C₂	#23757139	1	1	yes	characteristic
103	C₂	#23757315	1	2	yes	—
104	C₂	#23757633	1	2	yes	—
105	C₂	#23757638	1	1	yes	characteristic
106	C₂	#23757642	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757143

C₂

yes

maximal, characteristic

#23757139

C₂

yes

maximal, characteristic

#23757036

C₂

yes

maximal

C₂×#97×C₁₅₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

#19097678

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#97×C₁₅₇

C₄

yes

complemented

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

Q₈

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

C₄⋊C₄

yes

—

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

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₂²×C₁₅₇

#137

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#117

yes

—

C₄×C₁₅₇

#125

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#382

yes

characteristic

C₂×C₁₅₇

#428

yes

—

C₂×C₁₅₇

#489

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1836

—

157

maximal, Sylow

C₂×#97

D₁₅₇

yes

characteristic

#384

—

157

—

#382

—

157

—

#340

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#97

Dic₁₅₇

yes

complemented

#95

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

D₆₂₈

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

628

—

C₄²

C₂×Dic₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂×D₆₂₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

C₄⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

C₂×C₄

#14578954

yes

—

C₂×C₄

#14578954

yes

—

C₈

—

C₂²

C₂×C₄⋊Dic₁₅₇

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#19097804

yes

—

C₄

—

C₄

—

1256

—

C₄

—

1256

—

100

C₄

#19097730

yes

—

101

C₄

#19097763

yes

—

102

C₂

#23757139

yes

characteristic

103

C₂

#23757315

yes

—

104

C₂

#23757633

yes

—

105

C₂

#23757638

yes

characteristic

106

C₂

#23757642

yes

characteristic

	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	#1836
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

#1836

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₁₅₇	C₄	semidirect
53. C₁₅₇	#1836	semidirect
64. #97	Dic₁₅₇	semidirect

H ≅ G/N

type

11. #97×C₁₅₇

C₄

semidirect

53. C₁₅₇

#1836

semidirect

64. #97

Dic₁₅₇

semidirect

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

subgroup	index
#23757143	2
#23757139	2
#23757036	2
C₂×#97×C₁₅₇	2
#1836	157

subgroup

index

C₂×#97×C₁₅₇

157

Subgroups — #34100270

Conjugacy classes of subgroups

Complements

Maximal subgroups