Group #34100272 — subgroups

Subgroups — #34100272

Order 20096.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#23757296	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
3	#23757038	C₂	1	1	yes	maximal, characteristic
4	#23757037	C₂	1	2	yes	maximal
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	#19097700	—	2	4	no	—
10	C₈×Dic₁₅₇	C₄	1	2	yes	—
11	C₈×Dic₁₅₇	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₇	Q₈	1	2	yes	—
15	#14578938	—	2	2	no	—
16	#14578938	—	2	2	no	—
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	—	2	2	no	—
19	C₂×#11929434	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₅₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₅₇	—	2	2	no	—
25	M₁₆×C₁₅₇	—	2	2	no	—
26	C₂×C₈×C₁₅₇	—	2	2	no	—
27	C₂×C₈×C₁₅₇	—	2	2	no	—
28	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₅₇	—	2	4	no	—
30	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
32	#11929434	—	2	2	no	—
33	#11929434	—	4	2	no	—
34	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	#41	1	2	yes	—
38	C₂×C₄×C₁₅₇	#41	1	2	yes	—
39	C₈×C₁₅₇	—	2	2	no	—
40	C₈×C₁₅₇	—	4	2	no	—
41	C₂²×C₁₅₇	—	2	2	no	—
42	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₅₇	#125	1	2	yes	—
44	C₄×C₁₅₇	#117	1	1	yes	characteristic
45	C₄×C₁₅₇	#117	1	1	yes	characteristic
46	C₄×C₁₅₇	#124	1	2	yes	—
47	Dic₁₅₇	—	2	2	no	—
48	Dic₁₅₇	#108	1	4	yes	—
49	C₂×C₁₅₇	—	2	2	no	—
50	C₂×C₁₅₇	#382	1	1	yes	characteristic
51	C₂×C₁₅₇	C₂×#108	1	2	yes	—
52	C₁₅₇	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	157	1	no	maximal, Sylow
54	#423	—	157	2	no	—
55	C₂×C₄×C₈	—	157	1	no	—
56	#341	—	157	2	no	—
57	#341	—	157	1	no	—
58	#341	D₁₅₇	1	1	yes	characteristic
59	C₂×M₁₆	—	157	2	no	—
60	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
61	C₂²×C₈	—	157	1	no	—
62	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
63	C₂×C₄²	—	157	1	no	—
64	C₄⋊C₈	—	314	4	no	—
65	C₄×C₈	—	157	2	no	—
66	C₄×C₈	—	157	2	no	—
67	C₂²×C₄	—	157	2	no	—
68	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	314	2	no	—
72	M₁₆	—	314	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	314	2	no	—
76	C₂×C₈	—	314	2	no	—
77	C₂×C₈	—	157	2	no	—
78	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
79	C₄²	—	157	1	no	—
80	C₄²	—	157	1	no	—
81	C₄²	—	157	2	no	—
82	C₂³	Q₈×D₁₅₇	1	1	yes	characteristic
83	C₂×C₄	—	314	4	no	—
84	C₂×C₄	—	157	2	no	—
85	C₂×C₄	—	157	2	no	—
86	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
87	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
88	C₂×C₄	#14578953	1	2	yes	—
89	C₂×C₄	#14578954	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	314	2	no	—
93	C₈	—	628	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
96	C₂²	#19097761	1	2	yes	—
97	C₄	—	314	2	no	—
98	C₄	—	157	4	no	—
99	C₄	#19097727	1	1	yes	characteristic
100	C₄	#19097728	1	1	yes	characteristic
101	C₄	#19097757	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#23757137	1	1	yes	characteristic
104	C₂	#108×D₁₅₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757296

C₂

yes

maximal

C₂×C₈×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757038

C₂

yes

maximal, characteristic

#23757037

C₂

yes

maximal

#341×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097700

—

C₈×Dic₁₅₇

C₄

yes

—

C₈×Dic₁₅₇

C₄

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₁₅₇

C₂×C₄

yes

characteristic

C₄×Dic₁₅₇

C₂×C₄

yes

characteristic

C₄×Dic₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

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

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#125

yes

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

Dic₁₅₇

#108

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#382

yes

characteristic

C₂×C₁₅₇

C₂×#108

yes

—

C₁₅₇

#1837

yes

characteristic, complemented, Sylow

#1837

—

157

maximal, Sylow

#423

—

157

—

C₂×C₄×C₈

—

157

—

#341

—

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

—

C₄⋊C₈

—

314

—

C₄×C₈

—

157

—

C₄×C₈

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

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

D₄×D₁₅₇

yes

characteristic

C₂×C₄

#14578953

yes

—

C₂×C₄

#14578954

yes

—

C₈

—

C₈

—

C₈

—

314

—

C₈

—

628

—

C₂²

—

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

C₂²

#19097761

yes

—

C₄

—

314

—

C₄

—

157

—

C₄

#19097727

yes

characteristic

100

C₄

#19097728

yes

characteristic

101

C₄

#19097757

yes

—

102

C₂

—

103

C₂

#23757137

yes

characteristic

104

C₂

#108×D₁₅₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
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	#1837
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

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 ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₁₅₇	#1837	semidirect

H ≅ G/N

type

52. C₁₅₇

#1837

semidirect

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

subgroup	index
#23757296	2
C₂×C₈×Dic₁₅₇	2
#23757038	2
#23757037	2
#341×C₁₅₇	2
#1837	157

subgroup

index

C₂×C₈×Dic₁₅₇

#341×C₁₅₇

157

Subgroups — #34100272

Conjugacy classes of subgroups

Complements

Maximal subgroups