Group #34110650 — subgroups

Subgroups — #34110650

Order 20096.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#23757830	C₂	1	2	yes	maximal
2	#23757678	C₂	1	2	yes	maximal, complemented
3	#23757675	C₂	1	4	yes	maximal, complemented
4	#23757671	C₂	1	2	yes	maximal, complemented
5	#23757311	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	#19097771	C₂²	1	4	yes	—
8	#19097763	C₂²	1	4	yes	—
9	#19097750	—	2	4	no	—
10	C₄×Dic₃₁₄	—	2	4	no	—
11	#19097730	C₂²	1	4	yes	—
12	#19097705	C₂²	1	8	yes	—
13	#19097703	C₂²	1	8	yes	—
14	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₁₄	—	2	4	no	—
20	C₄⋊Dic₁₅₇	—	2	4	no	—
21	C₄⋊Dic₁₅₇	C₂³	1	8	yes	—
22	#14578933	—	2	8	no	—
23	C₂²⋊Dic₁₅₇	—	2	4	no	—
24	C₄×Dic₁₅₇	—	2	4	no	—
25	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₅₇	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
29	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
30	C₄²×C₁₅₇	D₄	1	2	yes	—
31	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
33	Dic₃₁₄	—	4	8	no	—
34	C₂×Dic₁₅₇	—	2	8	no	—
35	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	—	2	4	no	—
37	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
42	C₈×C₁₅₇	C₂×Q₈	1	4	yes	—
43	C₈×C₁₅₇	C₂×Q₈	1	4	yes	—
44	C₂²×C₁₅₇	—	2	2	no	—
45	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₅₇	—	2	2	no	—
48	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
49	C₄×C₁₅₇	#135	1	4	yes	—
50	C₄×C₁₅₇	C₂²×Q₈	1	2	yes	—
51	Dic₁₅₇	—	4	8	no	—
52	C₂×C₁₅₇	—	2	2	no	—
53	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₁₅₇	C₂×#135	1	2	yes	—
55	C₁₅₇	#3160	1	1	yes	characteristic, complemented, Sylow
56	#3160	—	157	1	no	maximal, Sylow
57	#533	—	157	2	no	—
58	#500	—	157	2	no	—
59	#499	—	157	4	no	—
60	#498	—	157	2	no	—
61	#427	—	157	4	no	—
62	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
64	C₄⋊Q₈	—	157	4	no	—
65	#125	—	157	4	no	—
66	C₂²⋊Q₈	—	314	4	no	—
67	C₄×Q₈	—	314	4	no	—
68	#117	—	157	4	no	—
69	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
70	#107	—	157	8	no	—
71	#106	—	157	8	no	—
72	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
74	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
75	C₂×Q₈	—	314	4	no	—
76	C₂²×C₄	D₆₂₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
78	C₂×C₈	Dic₃₁₄	1	8	yes	complemented
79	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
80	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
81	C₄⋊C₄	—	314	8	no	—
82	C₄⋊C₄	—	314	4	no	—
83	C₄⋊C₄	—	157	8	no	—
84	C₂²⋊C₄	—	314	4	no	—
85	C₄²	—	314	4	no	—
86	C₄²	D₆₂₈	1	2	yes	—
87	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
88	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
89	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
90	Q₈	—	628	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	314	8	no	—
93	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
94	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
95	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₃₁₄	1	4	yes	—
97	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
98	C₈	C₂×Dic₃₁₄	1	4	yes	—
99	C₈	C₂×Dic₃₁₄	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#19097771	1	2	yes	—
102	C₂²	C₂²×D₆₂₈	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	628	8	no	—
105	C₄	#19097771	1	2	yes	—
106	C₄	#19097793	1	4	yes	—
107	C₄	C₂²×Dic₃₁₄	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#19097771	1	1	yes	characteristic
110	C₂	C₂×#19097793	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757830

C₂

yes

maximal

#23757678

C₂

yes

maximal, complemented

#23757675

C₂

yes

maximal, complemented

#23757671

C₂

yes

maximal, complemented

#23757311

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

#19097771

C₂²

yes

—

#19097763

C₂²

yes

—

#19097750

—

C₄×Dic₃₁₄

—

#19097730

C₂²

yes

—

#19097705

C₂²

yes

—

#19097703

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

Q₈

yes

complemented

C₂×C₈×C₁₅₇

C₂³

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×Q₈

yes

—

C₈×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₄⋊Q₈

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

C₄×C₁₅₇

#135

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#135

yes

—

C₁₅₇

#3160

yes

characteristic, complemented, Sylow

#3160

—

157

maximal, Sylow

#533

—

157

—

#500

—

157

—

#499

—

157

—

#498

—

157

—

#427

—

157

—

C₂×C₄×C₈

D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

314

—

C₄×Q₈

—

314

—

#117

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#107

—

157

—

#106

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₄×C₈

C₂×D₁₅₇

yes

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂²×C₄

D₆₂₈

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

Dic₃₁₄

yes

complemented

C₂×C₈

C₂²×D₁₅₇

yes

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

314

—

C₄²

—

314

—

C₄²

D₆₂₈

yes

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

C₂×D₆₂₈

yes

characteristic

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

C₂×D₆₂₈

yes

—

C₂×C₄

C₂×D₆₂₈

yes

—

C₂×C₄

C₂×D₆₂₈

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

C₈

C₂×Dic₃₁₄

yes

—

C₈

C₂×Dic₃₁₄

yes

—

100

C₂²

—

101

C₂²

#19097771

yes

—

102

C₂²

C₂²×D₆₂₈

yes

characteristic

103

C₄

—

104

C₄

—

628

—

105

C₄

#19097771

yes

—

106

C₄

#19097793

yes

—

107

C₄

C₂²×Dic₃₁₄

yes

—

108

C₂

—

109

C₂

C₂×#19097771

yes

characteristic

110

C₂

C₂×#19097793

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₂₈
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	#3160
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₆₂₈

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×C₄×C₈×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂²×D₄

prime radical

C₂³×C₁₅₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₅₇

hypocenter

C₁₅₇

#3160

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #23757678	C₂	semidirect
3. #23757675	C₂	semidirect
4. #23757671	C₂	semidirect
27. C₂×C₈×C₁₅₇	Q₈	semidirect
55. C₁₅₇	#3160	semidirect
78. C₂×C₈	Dic₃₁₄	semidirect

H ≅ G/N

type

2. #23757678

C₂

semidirect

3. #23757675

C₂

semidirect

4. #23757671

C₂

semidirect

27. C₂×C₈×C₁₅₇

Q₈

semidirect

55. C₁₅₇

#3160

semidirect

78. C₂×C₈

Dic₃₁₄

semidirect

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

subgroup	index
#23757830	2
#23757678	2
#23757675	2
#23757671	2
#23757311	2
C₂×C₄×C₈×C₁₅₇	2
#3160	157

subgroup

index

C₂×C₄×C₈×C₁₅₇

157

Subgroups — #34110650

Conjugacy classes of subgroups

Complements

Maximal subgroups