Group #34102213 — subgroups

Subgroups — #34102213

Order 20096.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#23757711	C₂	1	2	yes	maximal, complemented
2	#23757650	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757391	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757087	C₂	1	2	yes	maximal, complemented
5	#364×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097793	C₂²	1	1	yes	characteristic
7	C₂×D₁₂₅₆	—	2	1	no	—
8	C₂×D₁₂₅₆	C₂²	1	2	yes	—
9	#19097777	—	2	1	no	—
10	#19097768	—	2	1	no	—
11	#19097716	—	2	2	no	—
12	D₂₅₁₂	—	2	2	no	—
13	#19097707	C₂²	1	1	yes	characteristic
14	#19097699	—	2	1	no	—
15	M₃₂×C₁₅₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
17	#14578954	—	2	1	no	—
18	C₂×D₆₂₈	—	2	2	no	—
19	C₂×D₆₂₈	—	4	2	no	—
20	Dic₆₂₈	D₄	1	1	yes	characteristic, complemented
21	#14578943	—	2	1	no	—
22	D₁₂₅₆	—	2	2	no	—
23	D₁₂₅₆	—	2	2	no	—
24	D₁₂₅₆	—	4	1	no	—
25	D₁₂₅₆	—	4	1	no	—
26	D₁₂₅₆	D₄	1	1	yes	characteristic, complemented
27	#14578938	—	2	1	no	—
28	#14578937	—	4	1	no	—
29	C₈×D₁₅₇	—	4	1	no	—
30	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
33	C₁₆×C₁₅₇	—	2	2	no	—
34	C₂²×D₁₅₇	—	4	2	no	—
35	Dic₃₁₄	—	2	1	no	—
36	D₆₂₈	—	2	1	no	—
37	D₆₂₈	—	4	2	no	—
38	D₆₂₈	—	4	2	no	—
39	D₆₂₈	—	8	4	no	—
40	C₂²⋊D₁₅₇	—	4	1	no	—
41	C₄×D₁₅₇	—	4	1	no	—
42	#11929434	—	4	1	no	—
43	C₂×C₄×C₁₅₇	—	2	1	no	—
44	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₅₇	D₈	1	1	yes	characteristic
46	C₈×C₁₅₇	D₈	1	1	yes	characteristic
47	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₅₇	#41	1	1	yes	characteristic
49	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₅₇	—	4	1	no	—
51	C₂×D₁₅₇	—	8	4	no	—
52	C₂×D₁₅₇	—	8	2	no	—
53	C₄×C₁₅₇	—	2	2	no	—
54	C₄×C₁₅₇	#121	1	1	yes	characteristic
55	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
56	Dic₁₅₇	—	4	1	no	—
57	C₂×C₁₅₇	—	2	1	no	—
58	C₂×C₁₅₇	#459	1	1	yes	characteristic
59	D₁₅₇	—	8	1	no	—
60	D₁₅₇	—	16	2	no	—
61	C₁₅₇	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	157	1	no	maximal, Sylow
63	#509	—	157	2	no	—
64	#493	—	157	1	no	—
65	#443	—	157	1	no	—
66	#364	D₁₅₇	1	1	yes	characteristic, complemented
67	#361	—	157	2	no	—
68	#135	—	157	1	no	—
69	C₂×D₈	—	314	1	no	—
70	C₂×D₈	—	157	2	no	—
71	#131	—	314	1	no	—
72	#127	—	314	1	no	—
73	SD₃₂	—	314	2	no	—
74	D₁₆	—	314	2	no	—
75	M₃₂	C₂×D₁₅₇	1	2	yes	—
76	#108	—	157	1	no	—
77	#104	—	314	1	no	—
78	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
79	#41	—	314	1	no	—
80	C₂×D₄	—	314	2	no	—
81	C₂×D₄	—	628	2	no	—
82	Q₁₆	—	157	1	no	—
83	SD₁₆	—	314	1	no	—
84	D₈	—	314	2	no	—
85	D₈	—	314	2	no	—
86	D₈	—	628	1	no	—
87	D₈	—	628	1	no	—
88	D₈	—	157	1	no	—
89	M₁₆	—	314	1	no	—
90	M₁₆	—	628	1	no	—
91	C₂×C₈	—	628	1	no	—
92	C₂×C₈	D₆₂₈	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
94	C₄²	D₆₂₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	628	2	no	—
97	Q₈	—	314	1	no	—
98	D₄	—	314	1	no	—
99	D₄	—	628	1	no	—
100	D₄	—	628	2	no	—
101	D₄	—	628	2	no	—
102	D₄	—	1256	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	628	1	no	—
105	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
106	C₈	—	628	1	no	—
107	C₈	D₁₂₅₆	1	1	yes	characteristic
108	C₈	D₁₂₅₆	1	1	yes	characteristic
109	C₈	D₄×D₁₅₇	1	1	yes	characteristic
110	C₈	#14578955	1	1	yes	characteristic
111	C₂²	—	628	1	no	—
112	C₂²	—	1256	4	no	—
113	C₂²	—	1256	2	no	—
114	C₂²	#19097799	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	628	1	no	—
117	C₄	#19097746	1	1	yes	characteristic
118	C₄	C₂×D₁₂₅₆	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2512	2	no	—
121	C₂	—	1256	1	no	—
122	C₂	#23757469	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757711

C₂

yes

maximal, complemented

#23757650

C₂

yes

maximal, characteristic, complemented

#23757391

C₂

yes

maximal, characteristic, complemented

#23757087

C₂

yes

maximal, complemented

#364×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#19097793

C₂²

yes

characteristic

C₂×D₁₂₅₆

—

C₂×D₁₂₅₆

C₂²

yes

—

#19097777

—

#19097768

—

#19097716

—

D₂₅₁₂

—

#19097707

C₂²

yes

characteristic

#19097699

—

M₃₂×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

#14578954

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

Dic₆₂₈

D₄

yes

characteristic, complemented

#14578943

—

D₁₂₅₆

—

D₁₂₅₆

—

D₁₂₅₆

—

D₁₂₅₆

—

D₁₂₅₆

D₄

yes

characteristic, complemented

#14578938

—

#14578937

—

C₈×D₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂³

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₁₆×C₁₅₇

—

C₂²×D₁₅₇

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₈×C₁₅₇

D₈

yes

characteristic

C₈×C₁₅₇

D₈

yes

characteristic

C₈×C₁₅₇

C₂×D₄

yes

C₈×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

C₂×D₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#459

yes

characteristic

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#2213

yes

characteristic, complemented, Sylow

#2213

—

157

maximal, Sylow

#509

—

157

—

#493

—

157

—

#443

—

157

—

#364

D₁₅₇

yes

characteristic, complemented

#361

—

157

—

#135

—

157

—

C₂×D₈

—

314

—

C₂×D₈

—

157

—

#131

—

314

—

#127

—

314

—

SD₃₂

—

314

—

D₁₆

—

314

—

M₃₂

C₂×D₁₅₇

yes

—

#108

—

157

—

#104

—

314

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

#41

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

628

—

Q₁₆

—

157

—

SD₁₆

—

314

—

D₈

—

314

—

D₈

—

314

—

D₈

—

628

—

D₈

—

628

—

D₈

—

157

—

M₁₆

—

314

—

M₁₆

—

628

—

C₂×C₈

—

628

—

C₂×C₈

D₆₂₈

yes

characteristic

C₂×C₈

C₂²×D₁₅₇

yes

characteristic

C₄²

D₆₂₈

yes

characteristic

C₁₆

—

C₂³

—

628

—

Q₈

—

314

—

D₄

—

314

—

D₄

—

628

—

100

D₄

—

628

—

101

D₄

—

628

—

102

D₄

—

1256

—

103

C₂×C₄

—

104

C₂×C₄

—

628

—

105

C₂×C₄

C₂×D₆₂₈

yes

characteristic

106

C₈

—

628

—

107

C₈

D₁₂₅₆

yes

characteristic

108

C₈

D₁₂₅₆

yes

characteristic

109

C₈

D₄×D₁₅₇

yes

characteristic

110

C₈

#14578955

yes

characteristic

111

C₂²

—

628

—

112

C₂²

—

1256

—

113

C₂²

—

1256

—

114

C₂²

#19097799

yes

characteristic

115

C₄

—

116

C₄

—

628

—

117

C₄

#19097746

yes

characteristic

118

C₄

C₂×D₁₂₅₆

yes

characteristic

119

C₂

—

120

C₂

—

2512

—

121

C₂

—

1256

—

122

C₂

#23757469

yes

characteristic

	subgroup	G/H
center	C₂	#23757469
commutator, cosocle	C₂×C₈×C₁₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	#364×C₁₅₇	C₂
socle	C₂×C₁₅₇	#459
hypercenter	#364	D₁₅₇
hypocenter	C₁₅₇	#2213
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#23757469

commutator, cosocle

C₂×C₈×C₁₅₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₅₇

Fitting

#364×C₁₅₇

C₂

C₂×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₂×C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #23757711	C₂	semidirect
2. #23757650	C₂	semidirect
3. #23757391	C₂	semidirect
4. #23757087	C₂	semidirect
5. #364×C₁₅₇	C₂	semidirect
20. Dic₆₂₈	D₄	semidirect
26. D₁₂₅₆	D₄	semidirect
61. C₁₅₇	#2213	semidirect
66. #364	D₁₅₇	semidirect

H ≅ G/N

type

1. #23757711

C₂

semidirect

2. #23757650

C₂

semidirect

3. #23757391

C₂

semidirect

4. #23757087

C₂

semidirect

5. #364×C₁₅₇

C₂

semidirect

20. Dic₆₂₈

D₄

semidirect

26. D₁₂₅₆

D₄

semidirect

61. C₁₅₇

#2213

semidirect

66. #364

D₁₅₇

semidirect

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

subgroup	index
#23757711	2
#23757650	2
#23757391	2
#23757087	2
#364×C₁₅₇	2
#2213	157

subgroup

index

#364×C₁₅₇

157

Subgroups — #34102213

Conjugacy classes of subgroups

Complements

Maximal subgroups