Group #34099803 — subgroups

Subgroups — #34099803

Order 20096.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23757649	C₂	1	2	yes	maximal, complemented
2	#23757646	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757619	C₂	1	2	yes	maximal, complemented
4	#23756994	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14578943	—	2	2	no	—
7	C₂×#14578939	—	2	4	no	—
8	#19097771	C₂²	1	1	yes	characteristic
9	#19097768	C₂²	1	1	yes	characteristic
10	#19097692	—	2	4	no	—
11	C₄×#11929434	C₂²	1	2	yes	—
12	C₂×D₈×C₁₅₇	—	2	2	no	—
13	#127×C₁₅₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₄	—	2	1	no	—
16	C₂×D₆₂₈	—	2	2	no	—
17	C₂×D₆₂₈	—	2	1	no	—
18	#14578943	—	4	4	no	—
19	#14578939	—	4	8	no	—
20	C₂×#11929434	D₄	1	4	yes	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₂×D₄×C₁₅₇	—	2	2	no	—
23	C₂×D₄×C₁₅₇	—	2	2	no	—
24	C₂×D₄×C₁₅₇	—	2	2	no	—
25	D₈×C₁₅₇	—	4	4	no	—
26	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
27	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
28	C₂²×D₁₅₇	—	4	1	no	—
29	Dic₃₁₄	—	4	2	no	—
30	D₆₂₈	—	4	2	no	—
31	D₆₂₈	—	4	4	no	—
32	C₂×Dic₁₅₇	—	4	1	no	—
33	#11929434	—	2	4	no	—
34	C₂³×C₁₅₇	—	4	2	no	—
35	D₄×C₁₅₇	—	4	4	no	—
36	D₄×C₁₅₇	—	4	4	no	—
37	D₄×C₁₅₇	—	4	4	no	—
38	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₅₇	D₈	1	4	yes	complemented
41	C₂²×C₁₅₇	—	8	2	no	—
42	C₂²×C₁₅₇	—	8	2	no	—
43	C₂²×C₁₅₇	—	8	2	no	—
44	C₂²×C₁₅₇	#127	1	1	yes	characteristic
45	C₂×D₁₅₇	—	8	1	no	—
46	C₂×D₁₅₇	—	8	2	no	—
47	C₄×C₁₅₇	C₂×D₈	1	2	yes	—
48	C₄×C₁₅₇	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₅₇	—	8	1	no	—
50	C₂×C₁₅₇	—	16	2	no	—
51	C₂×C₁₅₇	#493	1	1	yes	characteristic
52	C₂×C₁₅₇	#496	1	2	yes	—
53	D₁₅₇	—	16	1	no	—
54	C₁₅₇	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	157	1	no	maximal, Sylow
56	#493	—	157	2	no	—
57	#493	D₁₅₇	1	1	yes	characteristic, complemented
58	#492	—	157	1	no	—
59	#486	—	157	2	no	—
60	#322	—	157	1	no	—
61	C₂×SD₁₆	—	314	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	314	4	no	—
64	C₄⋊Q₈	—	157	1	no	—
65	#127	—	157	1	no	—
66	#127	C₂×D₁₅₇	1	2	yes	—
67	#102	—	314	4	no	—
68	C₄×C₈	—	157	2	no	—
69	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
70	C₂×Q₈	—	314	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	314	2	no	—
75	C₂×D₄	—	314	1	no	—
76	SD₁₆	—	628	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	628	8	no	—
79	C₂×C₈	—	157	4	no	—
80	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
81	C₄⋊C₄	—	314	2	no	—
82	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	628	1	no	—
85	Q₈	—	628	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	628	2	no	—
90	D₄	—	628	4	no	—
91	C₂×C₄	—	628	1	no	—
92	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
94	C₈	—	314	4	no	—
95	C₈	#14578939	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	1256	1	no	—
100	C₂²	—	1256	2	no	—
101	C₂²	#19097767	1	1	yes	characteristic
102	C₄	—	1256	1	no	—
103	C₄	C₂×#14578939	1	2	yes	—
104	C₄	#19097795	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	2512	1	no	—
107	C₂	#23757649	1	1	yes	characteristic
108	C₂	#23757661	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757649

C₂

yes

maximal, complemented

#23757646

C₂

yes

maximal, characteristic, complemented

#23757619

C₂

yes

maximal, complemented

#23756994

C₂

yes

maximal, characteristic, complemented

#493×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578943

—

C₂×#14578939

—

#19097771

C₂²

yes

characteristic

#19097768

C₂²

yes

characteristic

#19097692

—

C₄×#11929434

C₂²

yes

—

C₂×D₈×C₁₅₇

—

#127×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

#14578943

—

#14578939

—

C₂×#11929434

D₄

yes

—

C₄⋊Dic₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

—

D₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂²×D₁₅₇

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₈×C₁₅₇

D₈

yes

complemented

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#127

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂×D₈

yes

—

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#493

yes

characteristic

C₂×C₁₅₇

#496

yes

—

D₁₅₇

—

C₁₅₇

#1721

yes

characteristic, complemented, Sylow

#1721

—

157

maximal, Sylow

#493

—

157

—

#493

D₁₅₇

yes

characteristic, complemented

#492

—

157

—

#486

—

157

—

#322

—

157

—

C₂×SD₁₆

—

314

—

C₂×D₈

—

C₂×D₈

—

314

—

C₄⋊Q₈

—

157

—

#127

—

157

—

#127

C₂×D₁₅₇

yes

—

#102

—

314

—

C₄×C₈

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

SD₁₆

—

628

—

D₈

—

D₈

—

628

—

C₂×C₈

—

157

—

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

—

C₂³

—

628

—

Q₈

—

628

—

D₄

—

D₄

—

D₄

—

D₄

—

628

—

D₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

D₄×D₁₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₈

—

314

—

C₈

#14578939

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1256

—

100

C₂²

—

1256

—

101

C₂²

#19097767

yes

characteristic

102

C₄

—

1256

—

103

C₄

C₂×#14578939

yes

—

104

C₄

#19097795

yes

—

105

C₂

—

106

C₂

—

2512

—

107

C₂

#23757649

yes

characteristic

108

C₂

#23757661

yes

—

	subgroup	G/H
center	C₂²	#19097767
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#493×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#127
hypercenter	#493	D₁₅₇
hypocenter	C₁₅₇	#1721
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#19097767

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#493×C₁₅₇

C₂

C₂²×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #23757649	C₂	semidirect
2. #23757646	C₂	semidirect
3. #23757619	C₂	semidirect
4. #23756994	C₂	semidirect
5. #493×C₁₅₇	C₂	semidirect
40. C₈×C₁₅₇	D₈	semidirect
54. C₁₅₇	#1721	semidirect
57. #493	D₁₅₇	semidirect
95. C₈	#14578939	semidirect

H ≅ G/N

type

1. #23757649

C₂

semidirect

2. #23757646

C₂

semidirect

3. #23757619

C₂

semidirect

4. #23756994

C₂

semidirect

5. #493×C₁₅₇

C₂

semidirect

40. C₈×C₁₅₇

D₈

semidirect

54. C₁₅₇

#1721

semidirect

57. #493

D₁₅₇

semidirect

95. C₈

#14578939

semidirect

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

subgroup	index
#23757649	2
#23757646	2
#23757619	2
#23756994	2
#493×C₁₅₇	2
#1721	157

subgroup

index

#493×C₁₅₇

157

Subgroups — #34099803

Conjugacy classes of subgroups

Complements

Maximal subgroups