Group #34099654 — subgroups

Subgroups — #34099654

Order 20096.

152 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 57 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	#23757678	C₂	1	1	yes	maximal, characteristic
2	#23757483	C₂	1	2	yes	maximal
3	#23757024	C₂	1	1	yes	maximal, characteristic
4	#23757018	C₂	1	2	yes	maximal
5	#446×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578944	—	2	2	no	—
7	#19097771	C₂²	1	2	yes	—
8	#19097705	—	2	1	no	—
9	#19097705	—	2	1	no	—
10	C₄⋊#11929434	C₂²	1	2	yes	—
11	#19097694	—	2	2	no	—
12	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	—	2	1	no	—
14	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	#97×C₁₅₇	—	2	1	no	—
16	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁₄	—	2	2	no	—
18	#14578944	—	4	4	no	—
19	C₂×#11929434	—	2	2	no	—
20	C₄⋊Dic₁₅₇	—	2	2	no	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₅₇	—	2	1	no	—
24	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₁₅₇	—	2	1	no	—
27	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
28	C₄²×C₁₅₇	—	2	1	no	—
29	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
30	Dic₃₁₄	—	4	2	no	—
31	Dic₃₁₄	—	4	2	no	—
32	C₂×Dic₁₅₇	—	4	2	no	—
33	#11929434	—	4	2	no	—
34	Q₈×C₁₅₇	—	2	2	no	—
35	C₂×C₄×C₁₅₇	—	2	1	no	—
36	C₂×C₄×C₁₅₇	—	2	1	no	—
37	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
40	C₈×C₁₅₇	—	2	1	no	—
41	C₈×C₁₅₇	—	4	1	no	—
42	C₈×C₁₅₇	Q₁₆	1	2	yes	complemented
43	C₂²×C₁₅₇	#121	1	1	yes	characteristic
44	C₄×C₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	4	1	no	—
46	C₄×C₁₅₇	—	4	1	no	—
47	C₄×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
48	C₄×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
49	C₄×C₁₅₇	#137	1	1	yes	characteristic
50	Dic₁₅₇	—	8	2	no	—
51	C₂×C₁₅₇	#462	1	1	yes	characteristic
52	C₂×C₁₅₇	#469	1	1	yes	characteristic
53	C₂×C₁₅₇	#473	1	1	yes	characteristic
54	C₁₅₇	#1694	1	1	yes	characteristic, complemented, Sylow
55	#1694	—	157	1	no	maximal, Sylow
56	#500	—	157	1	no	—
57	#462	—	157	2	no	—
58	#446	D₁₅₇	1	1	yes	characteristic
59	#334	—	157	1	no	—
60	#332	—	157	2	no	—
61	C₂×Q₁₆	—	314	2	no	—
62	C₄⋊Q₈	—	157	2	no	—
63	C₄×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
64	#107	—	314	1	no	—
65	#107	—	314	1	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	157	2	no	—
68	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
69	#103	—	314	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
72	C₂×Q₈	—	314	2	no	—
73	C₂×Q₈	D₆₂₈	1	1	yes	characteristic
74	Q₁₆	—	628	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	314	2	no	—
77	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	314	2	no	—
81	C₄⋊C₄	—	314	2	no	—
82	C₄⋊C₄	D₆₂₈	1	1	yes	characteristic
83	C₄²	—	2	1	no	—
84	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
85	Q₈	—	2	2	no	—
86	Q₈	—	628	2	no	—
87	Q₈	—	628	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	628	2	no	—
91	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
92	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
93	C₂×C₄	#14578954	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	1	no	—
96	C₈	—	628	2	no	—
97	C₈	#14578944	1	2	yes	complemented
98	C₂²	#19097745	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	1256	2	no	—
103	C₄	C₂×#14578944	1	1	yes	characteristic
104	C₄	#19097799	1	2	yes	—
105	C₄	#19097801	1	1	yes	characteristic
106	C₂	#23757483	1	1	yes	characteristic
107	C₂	#23757526	1	1	yes	characteristic
108	C₂	#23757548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757678

C₂

yes

maximal, characteristic

#23757483

C₂

yes

maximal

#23757024

C₂

yes

maximal, characteristic

#23757018

C₂

yes

maximal

#446×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578944

—

#19097771

C₂²

yes

—

#19097705

—

#19097705

—

C₄⋊#11929434

C₂²

yes

—

#19097694

—

C₄×Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₄⋊C₈×C₁₅₇

C₂²

yes

characteristic

#97×C₁₅₇

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

#14578944

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

#11929434

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

Q₁₆

yes

complemented

C₂²×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₇

#137

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1694

—

157

maximal, Sylow

#500

—

157

—

#462

—

157

—

#446

D₁₅₇

yes

characteristic

#334

—

157

—

#332

—

157

—

C₂×Q₁₆

—

314

—

C₄⋊Q₈

—

157

—

C₄×Q₈

C₂×D₁₅₇

yes

characteristic

#107

—

314

—

#107

—

314

—

C₄⋊C₈

—

C₄⋊C₈

—

157

—

C₄⋊C₈

C₂×D₁₅₇

yes

characteristic

#103

—

314

—

#97

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×Q₈

D₆₂₈

yes

characteristic

Q₁₆

—

628

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

C₂²⋊D₁₅₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

D₆₂₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

628

—

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

628

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂×D₆₂₈

yes

characteristic

C₂×C₄

#14578954

yes

characteristic

C₈

—

C₈

—

C₈

—

628

—

C₈

yes

C₂²

yes

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

1256

—

103

C₄

C₂×#14578944

yes

characteristic

104

C₄

#19097799

yes

—

105

C₄

#19097801

yes

characteristic

106

C₂

#23757483

yes

characteristic

107

C₂

#23757526

yes

characteristic

108

C₂

#23757548

yes

characteristic

	subgroup	G/H
center	C₂²	#19097745
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#446×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#121
hypercenter	#446	D₁₅₇
hypocenter	C₁₅₇	#1694
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#19097745

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#446×C₁₅₇

C₂

C₂²×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
42. C₈×C₁₅₇	Q₁₆	semidirect
54. C₁₅₇	#1694	semidirect
97. C₈	#14578944	semidirect

H ≅ G/N

type

42. C₈×C₁₅₇

Q₁₆

semidirect

54. C₁₅₇

#1694

semidirect

97. C₈

#14578944

semidirect

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

subgroup	index
#23757678	2
#23757483	2
#23757024	2
#23757018	2
#446×C₁₅₇	2
#1694	157

subgroup

index

#446×C₁₅₇

157

Subgroups — #34099654

Conjugacy classes of subgroups

Complements

Maximal subgroups