Group #34110640 — subgroups

Subgroups — C₂×#23757675 #34110640

Order 20096.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#19097763	C₂	1	2	yes	maximal
2	#23757675	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19097705	C₂	1	2	yes	maximal
4	C₂×#19097703	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	#19097763	—	2	4	no	—
7	#19097763	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
9	C₂×#14578933	—	2	4	no	—
10	#19097705	C₂²	1	8	yes	—
11	#19097703	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₅₇	C₂²	1	4	yes	—
16	C₂²×Dic₁₅₇	—	2	4	no	—
17	C₄⋊Dic₁₅₇	—	2	4	no	—
18	C₄⋊Dic₁₅₇	C₂³	1	8	yes	—
19	#14578933	—	2	16	no	—
20	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₅₇	Q₈	1	4	yes	—
23	C₂×C₈×C₁₅₇	Q₈	1	4	yes	—
24	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
25	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
26	C₄²×C₁₅₇	D₄	1	2	yes	—
27	C₄²×C₁₅₇	C₂³	1	2	yes	—
28	C₂×Dic₁₅₇	—	2	8	no	—
29	C₂×Dic₁₅₇	—	4	8	no	—
30	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	—	2	4	no	—
32	C₂×C₄×C₁₅₇	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
37	C₈×C₁₅₇	C₂×Q₈	1	4	yes	—
38	C₈×C₁₅₇	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₅₇	#135	1	4	yes	—
41	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₅₇	—	2	4	no	—
43	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
46	Dic₁₅₇	—	4	8	no	—
47	C₂×C₁₅₇	#499	1	4	yes	complemented
48	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₅₇	C₂×#135	1	2	yes	—
50	C₁₅₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	157	1	no	maximal, Sylow
52	C₂×#125	—	157	2	no	—
53	#499	—	157	8	no	—
54	C₂×#107	—	157	2	no	—
55	C₂×#106	—	157	2	no	—
56	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
59	#125	—	314	4	no	—
60	#125	—	157	8	no	—
61	C₂×C₄⋊C₄	—	314	4	no	—
62	C₂×C₄⋊C₄	—	157	4	no	—
63	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
64	#107	—	157	8	no	—
65	#106	—	157	8	no	—
66	C₄×C₈	C₂×D₁₅₇	1	4	yes	—
67	C₂²×C₄	—	314	4	no	—
68	C₂²×C₄	D₆₂₈	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
70	C₂×C₈	Dic₃₁₄	1	4	yes	—
71	C₂×C₈	Dic₃₁₄	1	4	yes	—
72	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
73	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
74	C₄⋊C₄	—	314	16	no	—
75	C₄⋊C₄	—	314	4	no	—
76	C₄⋊C₄	—	157	8	no	—
77	C₄²	D₆₂₈	1	2	yes	—
78	C₄²	C₂²×D₁₅₇	1	2	yes	—
79	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	314	8	no	—
82	C₂×C₄	—	628	8	no	—
83	C₂×C₄	C₂×D₆₂₈	1	4	yes	—
84	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
86	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
87	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
88	C₈	C₂×Dic₃₁₄	1	4	yes	—
89	C₈	C₂×Dic₃₁₄	1	4	yes	—
90	C₂²	#19097771	1	2	yes	—
91	C₂²	#19097793	1	4	yes	—
92	C₂²	C₂²×D₆₂₈	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	628	8	no	—
95	C₄	#19097771	1	2	yes	—
96	C₄	C₂²×Dic₃₁₄	1	1	yes	characteristic
97	C₄	C₂²×Dic₃₁₄	1	1	yes	characteristic
98	C₂	#23757675	1	4	yes	direct product factor
99	C₂	C₂×#19097771	1	1	yes	characteristic
100	C₂	C₂×#19097793	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097763

C₂

yes

maximal

#23757675

C₂

yes

maximal, direct product factor

C₂×#19097705

C₂

yes

maximal

C₂×#19097703

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

#19097763

—

#19097763

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×#14578933

—

#19097705

C₂²

yes

—

#19097703

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

#14578933

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

Q₈

yes

—

C₂×C₈×C₁₅₇

Q₈

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×Q₈

yes

—

C₈×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

C₄⋊Q₈

yes

—

C₂²×C₁₅₇

#135

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

#499

yes

complemented

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#135

yes

—

C₁₅₇

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

157

maximal, Sylow

C₂×#125

—

157

—

#499

—

157

—

C₂×#107

—

157

—

C₂×#106

—

157

—

C₂×C₄×C₈

D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#125

—

314

—

#125

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#107

—

157

—

#106

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

—

C₂²×C₄

—

314

—

C₂²×C₄

D₆₂₈

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

Dic₃₁₄

yes

—

C₂×C₈

Dic₃₁₄

yes

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₄²

D₆₂₈

yes

—

C₄²

C₂²×D₁₅₇

yes

—

C₂³

C₂×D₆₂₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

C₂×D₆₂₈

yes

—

C₂×C₄

C₂×D₆₂₈

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

C₈

C₂×Dic₃₁₄

yes

—

C₈

C₂×Dic₃₁₄

yes

—

C₂²

#19097771

yes

—

C₂²

#19097793

yes

—

C₂²

C₂²×D₆₂₈

yes

characteristic

C₄

—

C₄

—

628

—

C₄

#19097771

yes

—

C₄

C₂²×Dic₃₁₄

yes

characteristic

C₄

C₂²×Dic₃₁₄

yes

characteristic

C₂

#23757675

yes

direct product factor

C₂

C₂×#19097771

yes

characteristic

100

C₂

C₂×#19097793

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₂₈
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	C₂×#499
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₆₂₈

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×C₄×C₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₅₇

hypocenter

C₁₅₇

C₂×#499

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #23757675	C₂	direct (N × H)
47. C₂×C₁₅₇	#499	semidirect
50. C₁₅₇	C₂×#499	semidirect
98. C₂	#23757675	direct (N × H)

H ≅ G/N

type

2. #23757675

C₂

direct (N × H)

47. C₂×C₁₅₇

#499

semidirect

50. C₁₅₇

C₂×#499

semidirect

98. C₂

#23757675

direct (N × H)

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

subgroup	index
C₂×#19097763	2
#23757675	2
C₂×#19097705	2
C₂×#19097703	2
C₂×C₄×C₈×C₁₅₇	2
C₂×#499	157

subgroup

index

C₂×C₄×C₈×C₁₅₇

157

Subgroups — C₂×#23757675 #34110640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23757675 #34110640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757675 #34110640