Group #34111940 — subgroups

Subgroups — #34111940

Order 20096.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#23758086	C₂	1	4	yes	maximal, complemented
2	#23757545	C₂	1	4	yes	maximal, complemented
3	#23757455	C₂	1	4	yes	maximal, complemented
4	#23757391	C₂	1	2	yes	maximal, complemented
5	#494×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097819	C₂²	1	4	yes	—
7	#19097803	—	2	8	no	—
8	#19097802	C₂²	1	8	yes	—
9	#19097794	—	2	4	no	—
10	#19097793	C₂²	1	2	yes	—
11	Q₁₆×D₁₅₇	—	2	4	no	—
12	#19097777	C₂²	1	4	yes	—
13	#19097707	C₂²	1	2	yes	—
14	#19097699	C₂²	1	4	yes	—
15	#19097690	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₅₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	#14578955	—	2	8	no	—
20	#14578954	—	4	4	no	—
21	#14578954	C₂³	1	4	yes	—
22	Q₈×D₁₅₇	—	2	8	no	—
23	Dic₆₂₈	D₄	1	2	yes	—
24	#14578944	—	2	8	no	—
25	#14578943	D₄	1	4	yes	—
26	#14578941	—	2	8	no	—
27	D₁₂₅₆	D₄	1	2	yes	—
28	#14578938	C₂³	1	4	yes	—
29	#14578937	—	2	4	no	—
30	C₈×D₁₅₇	—	2	4	no	—
31	C₂×Q₈×C₁₅₇	C₂³	1	4	yes	—
32	Q₁₆×C₁₅₇	—	2	4	no	—
33	Q₁₆×C₁₅₇	—	2	4	no	—
34	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₅₇	—	4	2	no	—
36	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
37	Dic₃₁₄	—	4	4	no	—
38	Dic₃₁₄	C₂×D₄	1	4	yes	—
39	D₆₂₈	—	4	4	no	—
40	D₆₂₈	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₅₇	—	2	4	no	—
42	C₄×D₁₅₇	—	2	4	no	—
43	C₄×D₁₅₇	—	4	8	no	—
44	#11929434	—	2	4	no	—
45	Q₈×C₁₅₇	—	2	8	no	—
46	C₂×C₄×C₁₅₇	—	2	1	no	—
47	C₂×C₄×C₁₅₇	—	2	4	no	—
48	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
50	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
51	C₂²×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₅₇	—	2	4	no	—
53	C₄×C₁₅₇	—	2	2	no	—
54	C₄×C₁₅₇	—	4	4	no	—
55	C₄×C₁₅₇	C₂²×D₄	1	2	yes	—
56	Dic₁₅₇	—	2	4	no	—
57	C₂×C₁₅₇	—	2	1	no	—
58	C₂×C₁₅₇	D₄²	1	1	yes	characteristic
59	D₁₅₇	—	4	4	no	—
60	C₁₅₇	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	157	1	no	maximal, Sylow
62	#578	—	157	4	no	—
63	#494	D₁₅₇	1	1	yes	characteristic, complemented
64	#473	—	157	4	no	—
65	#456	—	157	4	no	—
66	#443	—	157	2	no	—
67	2₋¹⁺⁴	—	157	4	no	—
68	#137	—	314	8	no	—
69	#137	—	157	8	no	—
70	#135	—	314	4	no	—
71	#135	—	157	2	no	—
72	C₂×Q₁₆	—	314	4	no	—
73	C₂×Q₁₆	C₂×D₁₅₇	1	4	yes	—
74	#131	—	157	4	no	—
75	C₄⋊Q₈	C₂×D₁₅₇	1	2	yes	—
76	#108	—	157	2	no	—
77	#104	—	157	4	no	—
78	#101	—	157	4	no	—
79	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
80	#41	—	314	8	no	—
81	#41	—	628	4	no	—
82	#41	—	157	4	no	—
83	C₂×Q₈	—	314	8	no	—
84	C₂×Q₈	C₂²×D₁₅₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	314	8	no	—
88	Q₁₆	—	157	2	no	—
89	SD₁₆	—	314	8	no	—
90	SD₁₆	—	157	4	no	—
91	D₈	—	157	2	no	—
92	M₁₆	—	314	4	no	—
93	M₁₆	—	157	4	no	—
94	C₂×C₈	—	314	4	no	—
95	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	628	4	no	—
100	Q₈	—	157	4	no	—
101	D₄	—	314	4	no	—
102	D₄	—	628	4	no	—
103	D₄	—	157	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	314	4	no	—
107	C₂×C₄	—	628	8	no	—
108	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
109	C₈	—	314	4	no	—
110	C₈	D₄×D₁₅₇	1	2	yes	—
111	C₈	D₄×D₁₅₇	1	2	yes	—
112	C₂²	—	314	4	no	—
113	C₂²	#19097816	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	314	4	no	—
117	C₄	C₂×D₄×D₁₅₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	628	4	no	—
120	C₂	#23757908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758086

C₂

yes

maximal, complemented

#23757545

C₂

yes

maximal, complemented

#23757455

C₂

yes

maximal, complemented

#23757391

C₂

yes

maximal, complemented

#494×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#19097819

C₂²

yes

—

#19097803

—

#19097802

C₂²

yes

—

#19097794

—

#19097793

C₂²

yes

—

Q₁₆×D₁₅₇

—

#19097777

C₂²

yes

—

#19097707

C₂²

yes

—

#19097699

C₂²

yes

—

#19097690

C₂²

yes

—

C₂×Q₁₆×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₅₇

—

Dic₆₂₈

D₄

yes

—

#14578944

—

#14578943

D₄

yes

—

#14578941

—

D₁₂₅₆

D₄

yes

—

#14578938

C₂³

yes

—

#14578937

—

C₈×D₁₅₇

—

C₂×Q₈×C₁₅₇

C₂³

yes

—

Q₁₆×C₁₅₇

—

Q₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

C₂×D₄

yes

—

D₆₂₈

—

D₆₂₈

C₂×D₄

yes

—

C₂²⋊D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×D₄

yes

—

C₈×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²×D₄

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

D₄²

yes

characteristic

D₁₅₇

—

C₁₅₇

#3293

yes

characteristic, complemented, Sylow

#3293

—

157

maximal, Sylow

#578

—

157

—

#494

D₁₅₇

yes

characteristic, complemented

#473

—

157

—

#456

—

157

—

#443

—

157

—

2₋¹⁺⁴

—

157

—

#137

—

314

—

#137

—

157

—

#135

—

314

—

#135

—

157

—

C₂×Q₁₆

—

314

—

C₂×Q₁₆

C₂×D₁₅₇

yes

—

#131

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

—

#108

—

157

—

#104

—

157

—

#101

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

#41

—

314

—

#41

—

628

—

#41

—

157

—

C₂×Q₈

—

314

—

C₂×Q₈

C₂²×D₁₅₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

314

—

Q₁₆

—

157

—

SD₁₆

—

314

—

SD₁₆

—

157

—

D₈

—

157

—

M₁₆

—

314

—

M₁₆

—

157

—

C₂×C₈

—

314

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

628

—

100

Q₈

—

157

—

101

D₄

—

314

—

102

D₄

—

628

—

103

D₄

—

157

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

314

—

107

C₂×C₄

—

628

—

108

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

109

C₈

—

314

—

110

C₈

D₄×D₁₅₇

yes

—

111

C₈

D₄×D₁₅₇

yes

—

112

C₂²

—

314

—

113

C₂²

#19097816

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

314

—

117

C₄

C₂×D₄×D₁₅₇

yes

—

118

C₂

—

119

C₂

—

628

—

120

C₂

#23757908

yes

characteristic

	subgroup	G/H
center	C₂	#23757908
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#494×C₁₅₇	C₂
socle	C₂×C₁₅₇	D₄²
hypercenter	#494	D₁₅₇
hypocenter	C₁₅₇	#3293
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#23757908

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

#494×C₁₅₇

C₂

socle

C₂×C₁₅₇

D₄²

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #23758086	C₂	semidirect
2. #23757545	C₂	semidirect
3. #23757455	C₂	semidirect
4. #23757391	C₂	semidirect
5. #494×C₁₅₇	C₂	semidirect
60. C₁₅₇	#3293	semidirect
63. #494	D₁₅₇	semidirect

H ≅ G/N

type

1. #23758086

C₂

semidirect

2. #23757545

C₂

semidirect

3. #23757455

C₂

semidirect

4. #23757391

C₂

semidirect

5. #494×C₁₅₇

C₂

semidirect

60. C₁₅₇

#3293

semidirect

63. #494

D₁₅₇

semidirect

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

subgroup	index
#23758086	2
#23757545	2
#23757455	2
#23757391	2
#494×C₁₅₇	2
#3293	157

subgroup

index

#494×C₁₅₇

157

Subgroups — #34111940

Conjugacy classes of subgroups

Complements

Maximal subgroups