Group #34100037 — subgroups

Subgroups — #34100037

Order 20096.

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

normal subgroups only

This group has 228 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	C₂×#19097706	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₅₇	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757038	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₂²×#11929434	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	#19097706	C₄	1	8	yes	complemented
10	#19097681	—	2	2	no	—
11	C₈×Dic₁₅₇	—	2	2	no	—
12	C₈×Dic₁₅₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₇	Q₈	1	2	yes	—
16	#14578938	—	2	2	no	—
17	#14578938	C₂×C₄	1	4	yes	—
18	C₂×#11929434	—	2	2	no	—
19	C₂×#11929434	C₂×C₄	1	2	yes	—
20	C₂×#11929434	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
23	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₅₇	—	2	2	no	—
26	M₁₆×C₁₅₇	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₅₇	—	2	2	no	—
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₅₇	—	2	4	no	—
31	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
33	#11929434	—	2	4	no	—
34	#11929434	C₄²	1	4	yes	—
35	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	#41	1	2	yes	—
40	C₈×C₁₅₇	—	2	2	no	—
41	C₈×C₁₅₇	—	2	4	no	—
42	C₂²×C₁₅₇	—	2	2	no	—
43	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
45	C₄×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₅₇	#117	1	1	yes	characteristic
47	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
48	Dic₁₅₇	—	2	2	no	—
49	Dic₁₅₇	#108	1	4	yes	complemented
50	C₂×C₁₅₇	—	2	2	no	—
51	C₂×C₁₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₅₇	C₂×#108	1	2	yes	—
53	C₁₅₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
54	C₄×#108	—	157	1	no	maximal, Sylow
55	C₂×#108	—	157	2	no	—
56	C₄×M₁₆	—	157	2	no	—
57	C₂×C₄×C₈	—	157	1	no	—
58	#341	—	157	1	no	—
59	#341	D₁₅₇	1	1	yes	characteristic
60	C₂×M₁₆	—	157	2	no	—
61	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
62	C₂²×C₈	—	157	1	no	—
63	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
64	C₂×C₄²	—	157	1	no	—
65	#108	—	157	8	no	—
66	#97	—	314	2	no	—
67	C₄×C₈	—	314	2	no	—
68	C₄×C₈	—	157	4	no	—
69	C₂²×C₄	—	157	2	no	—
70	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	314	2	no	—
73	M₁₆	—	157	4	no	—
74	M₁₆	C₄×D₁₅₇	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	314	2	no	—
78	C₂×C₈	—	157	2	no	—
79	C₂×C₈	—	157	4	no	—
80	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
81	C₄²	—	157	2	no	—
82	C₄²	—	157	1	no	—
83	C₄²	—	157	1	no	—
84	C₂³	Q₈×D₁₅₇	1	1	yes	characteristic
85	C₂×C₄	—	314	4	no	—
86	C₂×C₄	—	157	2	no	—
87	C₂×C₄	—	157	2	no	—
88	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
90	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
91	C₂×C₄	#14578953	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	314	4	no	—
95	C₈	—	157	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
98	C₂²	#19097736	1	2	yes	—
99	C₄	—	314	2	no	—
100	C₄	—	157	4	no	—
101	C₄	C₄²×D₁₅₇	1	1	yes	characteristic
102	C₄	#19097727	1	1	yes	characteristic
103	C₄	#19097731	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#23757124	1	1	yes	characteristic
106	C₂	#108×D₁₅₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097706

C₂

yes

maximal

M₁₆×Dic₁₅₇

C₂

yes

maximal

C₂×C₈×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757038

C₂

yes

maximal, characteristic

#341×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097706

C₄

yes

complemented

#19097681

—

C₈×Dic₁₅₇

—

C₈×Dic₁₅₇

C₄

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

Q₈

yes

—

#14578938

—

#14578938

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

C₂×C₄

yes

—

C₂×#11929434

C₂×C₄

yes

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₄×Dic₁₅₇

D₄

yes

characteristic

C₄×Dic₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

#11929434

—

#11929434

C₄²

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

C₂×C₄²

yes

characteristic

C₄×C₁₅₇

#117

yes

characteristic

C₄×C₁₅₇

C₄×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

#108

yes

complemented

C₂×C₁₅₇

—

C₂×C₁₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

C₂×#108

yes

—

C₁₅₇

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

157

maximal, Sylow

C₂×#108

—

157

—

C₄×M₁₆

—

157

—

C₂×C₄×C₈

—

157

—

#341

—

157

—

#341

D₁₅₇

yes

characteristic

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₂×C₄²

—

157

—

#108

—

157

—

#97

—

314

—

C₄×C₈

—

314

—

C₄×C₈

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

314

—

M₁₆

—

157

—

M₁₆

C₄×D₁₅₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

Q₈×D₁₅₇

yes

characteristic

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

C₂×C₄

D₄×D₁₅₇

yes

characteristic

C₂×C₄

#14578953

yes

—

C₈

—

C₈

—

C₈

—

314

—

C₈

—

157

—

C₂²

—

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

C₂²

#19097736

yes

—

C₄

—

314

—

100

C₄

—

157

—

101

C₄

C₄²×D₁₅₇

yes

characteristic

102

C₄

#19097727

yes

characteristic

103

C₄

#19097731

yes

—

104

C₂

—

105

C₂

#23757124

yes

characteristic

106

C₂

#108×D₁₅₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₄×C₁₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#341×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#341	D₁₅₇
hypocenter	C₁₅₇	C₄×#108
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₄×C₁₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#341×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₅₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #19097706	C₄	semidirect
49. Dic₁₅₇	#108	semidirect
53. C₁₅₇	C₄×#108	semidirect

H ≅ G/N

type

9. #19097706

C₄

semidirect

49. Dic₁₅₇

#108

semidirect

53. C₁₅₇

C₄×#108

semidirect

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

subgroup	index
C₂×#19097706	2
M₁₆×Dic₁₅₇	2
C₂×C₈×Dic₁₅₇	2
#23757038	2
#341×C₁₅₇	2
C₄×#108	157

subgroup

index

C₂×#19097706

M₁₆×Dic₁₅₇

C₂×C₈×Dic₁₅₇

#23757038

#341×C₁₅₇

C₄×#108

157

Subgroups — #34100037

Conjugacy classes of subgroups

Complements

Maximal subgroups