Group #34100043 — subgroups

Subgroups — #34100043

Order 20096.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄×#14578938	C₂	1	1	yes	maximal, characteristic
3	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
4	#23757037	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₂²×#11929434	C₂²	1	1	yes	characteristic
8	C₂²×#11929434	C₂²	1	1	yes	characteristic
9	#19097706	C₄	1	8	yes	complemented
10	#19097682	—	2	1	no	—
11	#19097682	C₄	1	4	yes	—
12	C₄×#11929434	—	2	1	no	—
13	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	#97×C₁₅₇	—	2	1	no	—
16	C₄×C₈×C₁₅₇	—	2	1	no	—
17	#14578938	—	2	2	no	—
18	#14578938	C₂×C₄	1	4	yes	—
19	C₂×#11929434	—	2	2	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	C₂×C₄	1	2	yes	—
22	C₂×#11929434	C₂×C₄	1	2	yes	—
23	C₂×#11929434	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₅₇	—	2	2	no	—
27	M₁₆×C₁₅₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
32	#11929434	—	2	2	no	—
33	#11929434	—	2	4	no	—
34	#11929434	C₄²	1	4	yes	—
35	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	—	4	2	no	—
37	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	#41	1	2	yes	—
43	C₈×C₁₅₇	—	2	4	no	—
44	C₂²×C₁₅₇	—	2	2	no	—
45	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
47	C₄×C₁₅₇	—	2	2	no	—
48	C₄×C₁₅₇	—	2	2	no	—
49	C₄×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₅₇	#117	1	1	yes	characteristic
51	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
52	C₂×C₁₅₇	—	2	2	no	—
53	C₂×C₁₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₅₇	#430	1	2	yes	—
55	C₁₅₇	#1778	1	1	yes	characteristic, complemented, Sylow
56	#1778	—	157	1	no	maximal, Sylow
57	C₂×#108	—	157	2	no	—
58	C₄×M₁₆	—	157	1	no	—
59	C₄×M₁₆	D₁₅₇	1	1	yes	characteristic
60	C₂×#97	—	157	1	no	—
61	#341	—	157	2	no	—
62	C₂×M₁₆	—	157	2	no	—
63	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
64	C₂²×C₈	—	157	1	no	—
65	C₂²×C₈	—	157	1	no	—
66	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
67	#108	—	157	8	no	—
68	#97	—	2	1	no	—
69	#97	—	314	1	no	—
70	#97	—	157	4	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	314	1	no	—
73	C₂²×C₄	Dic₃₁₄	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	314	2	no	—
77	M₁₆	—	157	4	no	—
78	M₁₆	C₄×D₁₅₇	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	314	2	no	—
81	C₂×C₈	—	314	2	no	—
82	C₂×C₈	—	157	2	no	—
83	C₂×C₈	—	157	2	no	—
84	C₂×C₈	—	157	4	no	—
85	C₄²	C₄×D₁₅₇	1	2	yes	—
86	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
88	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#14578933	1	2	yes	—
91	C₂×C₄	#14578933	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
95	C₂×C₄	#14578954	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	314	2	no	—
98	C₈	—	314	4	no	—
99	C₈	—	157	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#14578933	1	1	yes	characteristic
102	C₂²	C₄×Dic₃₁₄	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₁₅₇	1	1	yes	characteristic
106	C₄	#19097728	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₁₅₇	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#14578933	1	1	yes	characteristic
110	C₂	#23757321	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097706

C₂

yes

maximal

C₄×#14578938

C₂

yes

maximal, characteristic

C₂×#19097682

C₂

yes

maximal, characteristic

#23757037

C₂

yes

maximal

C₄×M₁₆×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂²×#11929434

C₂²

yes

characteristic

#19097706

C₄

yes

complemented

#19097682

—

#19097682

C₄

yes

—

C₄×#11929434

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#97×C₁₅₇

—

C₄×C₈×C₁₅₇

—

#14578938

—

#14578938

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

C₂×C₄

yes

—

C₂×#11929434

C₂×C₄

yes

—

C₂×#11929434

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

Q₈

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

—

—

C₄²

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

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

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₄²

yes

characteristic

C₄×C₁₅₇

#117

yes

characteristic

C₄×C₁₅₇

C₄×D₄

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

#430

yes

—

C₁₅₇

#1778

yes

characteristic, complemented, Sylow

#1778

—

157

maximal, Sylow

C₂×#108

—

157

—

C₄×M₁₆

—

157

—

C₄×M₁₆

D₁₅₇

yes

characteristic

C₂×#97

—

157

—

#341

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#108

—

157

—

#97

—

#97

—

314

—

#97

—

157

—

C₄×C₈

—

C₄×C₈

—

314

—

C₂²×C₄

Dic₃₁₄

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

314

—

M₁₆

—

157

—

M₁₆

C₄×D₁₅₇

yes

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₄²

C₄×D₁₅₇

yes

—

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₂³

C₂×Dic₃₁₄

yes

characteristic

C₂×C₄

—

C₂×C₄

#14578933

yes

—

C₂×C₄

#14578933

yes

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

#14578954

yes

—

C₈

—

C₈

—

314

—

C₈

—

314

—

C₈

—

157

—

100

C₂²

—

101

C₂²

C₂×#14578933

yes

characteristic

102

C₂²

C₄×Dic₃₁₄

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₁₅₇

yes

characteristic

106

C₄

#19097728

yes

characteristic

107

C₄

C₄×C₂²⋊D₁₅₇

yes

—

108

C₂

—

109

C₂

C₄×#14578933

yes

characteristic

110

C₂

#23757321

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₄×C₁₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₄×M₁₆×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₄×M₁₆	D₁₅₇
hypocenter	C₁₅₇	#1778
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₅₇

commutator

C₄×C₁₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₄×M₁₆×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₅₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₄×M₁₆

D₁₅₇

hypocenter

C₁₅₇

#1778

LCS

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #19097706	C₄	semidirect
55. C₁₅₇	#1778	semidirect

H ≅ G/N

type

9. #19097706

C₄

semidirect

55. C₁₅₇

#1778

semidirect

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

subgroup	index
C₂×#19097706	2
C₄×#14578938	2
C₂×#19097682	2
#23757037	2
C₄×M₁₆×C₁₅₇	2
#1778	157

subgroup

index

C₄×M₁₆×C₁₅₇

157

Subgroups — #34100043

Conjugacy classes of subgroups

Complements

Maximal subgroups