Group #34099994 — subgroups

Subgroups — #34099994

Order 20096.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×#19097700	C₂	1	2	yes	maximal
2	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
3	#23757119	C₂	1	1	yes	maximal, characteristic
4	#23757027	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	#19097700	C₄	1	8	yes	complemented
10	#19097682	—	2	2	no	—
11	#19097678	C₄	1	4	yes	—
12	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₅₇	—	2	2	no	—
15	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
16	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	—	2	2	no	—
19	C₂×#11929434	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₅₇	—	2	2	no	—
21	C₄×Dic₁₅₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₅₇	—	2	2	no	—
25	C₂×C₈×C₁₅₇	—	2	2	no	—
26	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
27	C₄²×C₁₅₇	D₄	1	2	yes	—
28	C₄²×C₁₅₇	Q₈	1	2	yes	—
29	C₂×Dic₁₅₇	—	2	4	no	—
30	C₂×Dic₁₅₇	—	2	2	no	—
31	C₂×Dic₁₅₇	—	2	4	no	—
32	C₂×Dic₁₅₇	C₄²	1	4	yes	—
33	#11929434	—	2	4	no	—
34	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	—	2	4	no	—
36	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	#41	1	2	yes	—
41	C₈×C₁₅₇	—	2	4	no	—
42	C₂²×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₅₇	#117	1	1	yes	characteristic
45	C₂²×C₁₅₇	#131	1	4	yes	—
46	C₄×C₁₅₇	—	2	4	no	—
47	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
48	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
49	Dic₁₅₇	—	2	4	no	—
50	Dic₁₅₇	—	4	2	no	—
51	C₂×C₁₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₅₇	#405	1	2	yes	—
53	C₂×C₁₅₇	#424	1	2	yes	—
54	C₂×C₁₅₇	#433	1	2	yes	—
55	C₁₅₇	#1768	1	1	yes	characteristic, complemented, Sylow
56	#1768	—	157	1	no	maximal, Sylow
57	C₂×C₄⋊C₈	—	157	2	no	—
58	C₂×#97	—	157	1	no	—
59	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
60	#376	—	157	1	no	—
61	#336	—	157	2	no	—
62	C₂²×C₈	—	157	2	no	—
63	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
64	C₂×C₄²	—	157	1	no	—
65	C₂×C₄²	—	157	1	no	—
66	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
67	C₄⋊C₈	—	157	8	no	—
68	#97	—	314	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	—	157	4	no	—
71	C₂²×C₄	—	157	2	no	—
72	C₂²×C₄	—	157	2	no	—
73	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	314	2	no	—
78	C₂×C₈	—	314	2	no	—
79	C₂×C₈	—	157	4	no	—
80	C₂×C₈	C₄×D₁₅₇	1	4	yes	—
81	C₄²	—	314	2	no	—
82	C₄²	—	157	4	no	—
83	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
84	C₄²	Dic₃₁₄	1	2	yes	—
85	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	314	4	no	—
88	C₂×C₄	—	314	2	no	—
89	C₂×C₄	—	314	4	no	—
90	C₂×C₄	—	157	4	no	—
91	C₂×C₄	#14578933	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
95	C₂×C₄	#14578954	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	314	4	no	—
98	C₂²	C₄²×D₁₅₇	1	1	yes	characteristic
99	C₂²	C₂×#14578933	1	1	yes	characteristic
100	C₂²	#19097728	1	1	yes	characteristic
101	C₂²	#19097777	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	314	4	no	—
104	C₄	—	628	2	no	—
105	C₄	C₄×C₂²⋊D₁₅₇	1	2	yes	—
106	C₄	C₄×Dic₃₁₄	1	2	yes	—
107	C₂	C₄×#14578933	1	1	yes	characteristic
108	C₂	#23757227	1	2	yes	—
109	C₂	#23757299	1	2	yes	—
110	C₂	#23757332	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097700

C₂

yes

maximal

C₂×#19097682

C₂

yes

maximal, characteristic

#23757119

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097700

C₄

yes

complemented

#19097682

—

#19097678

C₄

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

C₂×C₄

yes

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄²

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄²

yes

characteristic

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

C₄×Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

#405

yes

—

C₂×C₁₅₇

#424

yes

—

C₂×C₁₅₇

#433

yes

—

C₁₅₇

#1768

yes

characteristic, complemented, Sylow

#1768

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×#97

—

157

—

C₂×C₄×C₈

D₁₅₇

yes

characteristic

#376

—

157

—

#336

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

157

—

#97

—

314

—

C₄×C₈

—

#95

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

#14578933

yes

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

C₂×C₄

#14578954

yes

—

C₈

—

C₈

—

314

—

C₂²

C₄²×D₁₅₇

yes

characteristic

C₂²

C₂×#14578933

yes

characteristic

100

C₂²

#19097728

yes

characteristic

101

C₂²

#19097777

yes

—

102

C₄

—

103

C₄

—

314

—

104

C₄

—

628

—

105

C₄

C₄×C₂²⋊D₁₅₇

yes

—

106

C₄

C₄×Dic₃₁₄

yes

—

107

C₂

C₄×#14578933

yes

characteristic

108

C₂

#23757227

yes

—

109

C₂

#23757299

yes

—

110

C₂

#23757332

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂×C₄²
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	#1768
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator

C₂²×C₁₅₇

C₂×C₄²

Fitting

C₂×C₄×C₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. #19097700	C₄	semidirect
55. C₁₅₇	#1768	semidirect

H ≅ G/N

type

9. #19097700

C₄

semidirect

55. C₁₅₇

#1768

semidirect

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

subgroup	index
C₂×#19097700	2
C₂×#19097682	2
#23757119	2
#23757027	2
C₂×C₄×C₈×C₁₅₇	2
#1768	157

subgroup

index

C₂×C₄×C₈×C₁₅₇

157

Subgroups — #34099994

Conjugacy classes of subgroups

Complements

Maximal subgroups