Group #34099813 — subgroups

Subgroups — #34099813

Order 20096.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23757678	C₂	1	1	yes	maximal, characteristic
2	#23757678	C₂	1	1	yes	maximal, characteristic
3	#23757671	C₂	1	1	yes	maximal, characteristic
4	#23757653	C₂	1	1	yes	maximal, characteristic
5	#23757624	C₂	1	1	yes	maximal, characteristic
6	#23757624	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₂₈	—	2	2	no	—
9	#19097771	C₂²	1	1	yes	characteristic
10	#19097771	C₂²	1	1	yes	characteristic
11	#19097771	C₂²	1	1	yes	characteristic
12	#19097771	C₂²	1	1	yes	characteristic
13	#19097705	—	2	2	no	—
14	#19097705	—	2	2	no	—
15	#19097703	—	2	2	no	—
16	#19097696	—	2	2	no	—
17	#19097696	—	2	2	no	—
18	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₁₄	—	2	1	no	—
22	C₂×Dic₃₁₄	—	2	1	no	—
23	C₂×Dic₃₁₄	—	2	1	no	—
24	C₂×Dic₃₁₄	—	2	1	no	—
25	Dic₆₂₈	—	4	4	no	—
26	C₄⋊Dic₁₅₇	—	2	1	no	—
27	C₄⋊Dic₁₅₇	—	2	1	no	—
28	C₄⋊Dic₁₅₇	—	2	1	no	—
29	C₄⋊Dic₁₅₇	—	2	1	no	—
30	C₄⋊Dic₁₅₇	—	2	1	no	—
31	C₄⋊Dic₁₅₇	—	2	1	no	—
32	C₄⋊Dic₁₅₇	—	2	1	no	—
33	C₄⋊Dic₁₅₇	—	2	1	no	—
34	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
35	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
36	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
37	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
38	Dic₃₁₄	—	4	2	no	—
39	Dic₃₁₄	—	4	2	no	—
40	Dic₃₁₄	—	4	2	no	—
41	Dic₃₁₄	—	4	2	no	—
42	C₂×Dic₁₅₇	—	4	1	no	—
43	C₂×Dic₁₅₇	—	4	1	no	—
44	C₂×Dic₁₅₇	—	4	1	no	—
45	C₂×Dic₁₅₇	—	4	1	no	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₅₇	—	2	2	no	—
50	C₈×C₁₅₇	—	2	2	no	—
51	C₈×C₁₅₇	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₅₇	#127	1	1	yes	characteristic
53	C₄×C₁₅₇	C₂×Q₁₆	1	2	yes	—
54	C₄×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₅₇	#137	1	2	yes	—
56	Dic₁₅₇	—	8	1	no	—
57	Dic₁₅₇	—	8	1	no	—
58	Dic₁₅₇	—	8	1	no	—
59	Dic₁₅₇	—	8	1	no	—
60	C₂×C₁₅₇	#494	1	1	yes	characteristic
61	C₂×C₁₅₇	#496	1	1	yes	characteristic
62	C₂×C₁₅₇	#497	1	1	yes	characteristic
63	C₁₅₇	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	157	1	no	maximal, Sylow
65	#500	—	157	1	no	—
66	#500	—	157	1	no	—
67	#498	—	157	1	no	—
68	#494	—	157	1	no	—
69	#487	—	157	1	no	—
70	#487	—	157	1	no	—
71	#322	D₁₅₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	314	2	no	—
73	C₄⋊Q₈	—	157	1	no	—
74	C₄⋊Q₈	—	157	1	no	—
75	C₄⋊Q₈	—	157	1	no	—
76	C₄⋊Q₈	—	157	1	no	—
77	#107	—	314	2	no	—
78	#107	—	314	2	no	—
79	#106	—	314	2	no	—
80	#103	—	314	2	no	—
81	#103	—	314	2	no	—
82	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
85	C₂×Q₈	—	314	1	no	—
86	C₂×Q₈	—	314	1	no	—
87	C₂×Q₈	—	314	1	no	—
88	C₂×Q₈	—	314	1	no	—
89	Q₁₆	—	628	4	no	—
90	C₂×C₈	D₆₂₈	1	2	yes	—
91	C₂×C₈	D₆₂₈	1	2	yes	—
92	C₂×C₈	D₆₂₈	1	2	yes	—
93	C₄⋊C₄	—	314	1	no	—
94	C₄⋊C₄	—	314	1	no	—
95	C₄⋊C₄	—	314	1	no	—
96	C₄⋊C₄	—	314	1	no	—
97	C₄⋊C₄	—	314	1	no	—
98	C₄⋊C₄	—	314	1	no	—
99	C₄⋊C₄	—	314	1	no	—
100	C₄⋊C₄	—	314	1	no	—
101	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
102	Q₈	—	628	2	no	—
103	Q₈	—	628	2	no	—
104	Q₈	—	628	2	no	—
105	Q₈	—	628	2	no	—
106	C₂×C₄	—	628	1	no	—
107	C₂×C₄	—	628	1	no	—
108	C₂×C₄	—	628	1	no	—
109	C₂×C₄	—	628	1	no	—
110	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
111	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
112	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₆₂₈	1	4	yes	complemented
116	C₂²	#19097768	1	1	yes	characteristic
117	C₄	—	1256	1	no	—
118	C₄	—	1256	1	no	—
119	C₄	—	1256	1	no	—
120	C₄	—	1256	1	no	—
121	C₄	C₂×Dic₆₂₈	1	2	yes	—
122	C₄	#19097799	1	2	yes	—
123	C₄	#19097804	1	2	yes	—
124	C₂	#23757653	1	1	yes	characteristic
125	C₂	#23757662	1	1	yes	characteristic
126	C₂	#23757667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757678

C₂

yes

maximal, characteristic

#23757678

C₂

yes

maximal, characteristic

#23757671

C₂

yes

maximal, characteristic

#23757653

C₂

yes

maximal, characteristic

#23757624

C₂

yes

maximal, characteristic

#23757624

C₂

yes

maximal, characteristic

#322×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₆₂₈

—

#19097771

C₂²

yes

characteristic

#19097771

C₂²

yes

characteristic

#19097771

C₂²

yes

characteristic

#19097771

C₂²

yes

—

—

—

—

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

Dic₆₂₈

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

Q₁₆

yes

complemented

C₂²×C₁₅₇

#127

yes

characteristic

C₄×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₇

#137

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1723

—

157

maximal, Sylow

#500

—

157

—

#500

—

157

—

#498

—

157

—

#494

—

157

—

#487

—

157

—

#487

—

157

—

#322

D₁₅₇

yes

characteristic

C₂×Q₁₆

—

314

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

#107

—

314

—

#107

—

314

—

#106

—

314

—

#103

—

314

—

#103

—

314

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

Q₁₆

—

628

—

C₂×C₈

D₆₂₈

yes

—

C₂×C₈

D₆₂₈

yes

—

C₂×C₈

D₆₂₈

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

100

C₄⋊C₄

—

314

—

101

C₄²

C₂²×D₁₅₇

yes

characteristic

102

Q₈

—

628

—

103

Q₈

—

628

—

104

Q₈

—

628

—

105

Q₈

—

628

—

106

C₂×C₄

—

628

—

107

C₂×C₄

—

628

—

108

C₂×C₄

—

628

—

109

C₂×C₄

—

628

—

110

C₂×C₄

C₂×D₆₂₈

yes

characteristic

111

C₂×C₄

C₂×D₆₂₈

yes

characteristic

112

C₂×C₄

C₂×D₆₂₈

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₆₂₈

yes

complemented

116

C₂²

#19097768

yes

characteristic

117

C₄

—

1256

—

118

C₄

—

1256

—

119

C₄

—

1256

—

120

C₄

—

1256

—

121

C₄

C₂×Dic₆₂₈

yes

—

122

C₄

#19097799

yes

—

123

C₄

#19097804

yes

—

124

C₂

#23757653

yes

characteristic

125

C₂

#23757662

yes

characteristic

126

C₂

#23757667

yes

characteristic

	subgroup	G/H
center	C₂²	#19097768
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#322×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#127
hypercenter	#322	D₁₅₇
hypocenter	C₁₅₇	#1723
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#19097768

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#322×C₁₅₇

C₂

C₂²×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₁₅₇	Q₁₆	semidirect
63. C₁₅₇	#1723	semidirect
115. C₈	Dic₆₂₈	semidirect

H ≅ G/N

type

51. C₈×C₁₅₇

Q₁₆

semidirect

63. C₁₅₇

#1723

semidirect

115. C₈

Dic₆₂₈

semidirect

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

subgroup	index
#23757678	2
#23757671	2
#23757653	2
#23757624	2
#322×C₁₅₇	2
#1723	157

subgroup

index

#322×C₁₅₇

157

Subgroups — #34099813

Conjugacy classes of subgroups

Complements

Maximal subgroups