Group #34111590 — subgroups

Subgroups — #34111590

Order 20096.

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

normal subgroups only

This group has 322 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	#23757885	C₂	1	2	yes	maximal
2	#23757581	C₂	1	4	yes	maximal, complemented
3	#23757531	C₂	1	2	yes	maximal
4	#23757514	C₂	1	2	yes	maximal
5	#23757484	C₂	1	4	yes	maximal, complemented
6	#424×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₆₂₈	C₂²	1	2	yes	—
8	C₂×Dic₆₂₈	C₂²	1	2	yes	—
9	#19097771	C₂²	1	2	yes	—
10	#19097764	—	2	4	no	—
11	#19097763	C₂²	1	2	yes	—
12	#19097761	—	2	4	no	—
13	#19097750	C₂²	1	4	yes	—
14	C₄×Dic₃₁₄	C₂²	1	4	yes	—
15	#19097705	C₂²	1	2	yes	—
16	#19097703	C₂²	1	2	yes	—
17	#19097696	C₂²	1	4	yes	—
18	#19097696	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
21	#117×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₅₇	C₂²	1	4	yes	—
23	C₂×Dic₃₁₄	C₂³	1	4	yes	—
24	Dic₆₂₈	—	4	2	no	—
25	Dic₆₂₈	—	4	2	no	—
26	C₄⋊Dic₁₅₇	—	2	4	no	—
27	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
28	#14578933	—	2	4	no	—
29	#14578933	—	2	4	no	—
30	#14578933	—	2	4	no	—
31	C₂²⋊Dic₁₅₇	—	2	4	no	—
32	C₄×Dic₁₅₇	—	2	4	no	—
33	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₅₇	—	4	1	no	—
35	C₂×C₈×C₁₅₇	—	4	1	no	—
36	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₅₇	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₅₇	D₄	1	2	yes	—
40	C₄²×C₁₅₇	C₂³	1	2	yes	—
41	Dic₃₁₄	—	2	4	no	—
42	Dic₃₁₄	—	2	4	no	—
43	C₂×Dic₁₅₇	—	2	4	no	—
44	C₂×Dic₁₅₇	—	2	4	no	—
45	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	—	2	1	no	—
47	C₂×C₄×C₁₅₇	—	2	1	no	—
48	C₂×C₄×C₁₅₇	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₅₇	—	2	2	no	—
52	C₈×C₁₅₇	—	2	2	no	—
53	C₂²×C₁₅₇	—	2	1	no	—
54	C₂²×C₁₅₇	—	2	2	no	—
55	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₅₇	—	2	1	no	—
57	C₄×C₁₅₇	—	2	4	no	—
58	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₅₇	—	4	4	no	—
61	Dic₁₅₇	—	4	4	no	—
62	C₂×C₁₅₇	—	4	1	no	—
63	C₂×C₁₅₇	#537	1	1	yes	characteristic
64	C₂×C₁₅₇	#578	1	2	yes	—
65	C₁₅₇	#3261	1	1	yes	characteristic, complemented, Sylow
66	#3261	—	157	1	no	maximal, Sylow
67	#541	—	157	2	no	—
68	#479	—	157	4	no	—
69	#470	—	157	2	no	—
70	#467	—	157	2	no	—
71	#462	—	157	4	no	—
72	#424	D₁₅₇	1	1	yes	characteristic
73	C₂×Q₁₆	—	157	2	no	—
74	C₂×Q₁₆	—	157	2	no	—
75	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
77	C₄⋊Q₈	—	157	2	no	—
78	#126	—	314	4	no	—
79	#125	—	314	4	no	—
80	#125	—	157	2	no	—
81	C₂²⋊Q₈	—	157	4	no	—
82	C₄×Q₈	—	157	4	no	—
83	#117	C₂×D₁₅₇	1	1	yes	characteristic
84	#107	—	157	2	no	—
85	#106	—	157	2	no	—
86	C₄⋊C₈	C₂×D₁₅₇	1	4	yes	—
87	#103	—	157	4	no	—
88	#103	—	157	4	no	—
89	C₂×Q₈	—	157	4	no	—
90	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
91	Q₁₆	—	628	2	no	—
92	Q₁₆	—	628	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
96	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
97	C₄⋊C₄	—	314	4	no	—
98	C₄⋊C₄	—	314	4	no	—
99	C₄⋊C₄	—	314	4	no	—
100	C₄⋊C₄	—	314	4	no	—
101	C₄⋊C₄	—	157	4	no	—
102	C₄⋊C₄	D₆₂₈	1	2	yes	—
103	C₂²⋊C₄	—	314	4	no	—
104	C₂²⋊C₄	D₆₂₈	1	2	yes	—
105	C₄²	—	314	4	no	—
106	C₄²	C₂²×D₁₅₇	1	2	yes	—
107	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
108	Q₈	—	314	4	no	—
109	Q₈	—	314	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	314	4	no	—
113	C₂×C₄	—	314	4	no	—
114	C₂×C₄	C₂×D₆₂₈	1	4	yes	—
115	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₆₂₈	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	628	4	no	—
125	C₄	—	628	4	no	—
126	C₄	#19097816	1	1	yes	characteristic
127	C₄	#19097819	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#23757854	1	1	yes	characteristic
130	C₂	#23758090	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757885

C₂

yes

maximal

#23757581

C₂

yes

maximal, complemented

#23757531

C₂

yes

maximal

#23757514

C₂

yes

maximal

#23757484

C₂

yes

maximal, complemented

#424×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₆₂₈

C₂²

yes

—

C₂×Dic₆₂₈

C₂²

yes

—

#19097771

C₂²

yes

—

#19097764

—

#19097763

C₂²

yes

—

#19097761

—

#19097750

C₂²

yes

—

C₄×Dic₃₁₄

C₂²

yes

—

#19097705

C₂²

yes

—

#19097703

C₂²

yes

—

#19097696

C₂²

yes

—

#19097696

C₂²

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₂²

yes

—

C₂×Dic₃₁₄

C₂³

yes

—

Dic₆₂₈

—

Dic₆₂₈

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

#14578933

—

#14578933

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

D₄

yes

—

C₂²⋊C₄×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₅₇

2₋¹⁺⁴

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#537

yes

characteristic

C₂×C₁₅₇

#578

yes

—

C₁₅₇

#3261

yes

characteristic, complemented, Sylow

#3261

—

157

maximal, Sylow

#541

—

157

—

#479

—

157

—

#470

—

157

—

#467

—

157

—

#462

—

157

—

#424

D₁₅₇

yes

characteristic

C₂×Q₁₆

—

157

—

C₂×Q₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊Q₈

—

157

—

#126

—

314

—

#125

—

314

—

#125

—

157

—

C₂²⋊Q₈

—

157

—

C₄×Q₈

—

157

—

#117

C₂×D₁₅₇

yes

characteristic

#107

—

157

—

#106

—

157

—

C₄⋊C₈

C₂×D₁₅₇

yes

—

#103

—

157

—

#103

—

157

—

C₂×Q₈

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

Q₁₆

—

628

—

Q₁₆

—

628

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

100

C₄⋊C₄

—

314

—

101

C₄⋊C₄

—

157

—

102

C₄⋊C₄

D₆₂₈

yes

—

103

C₂²⋊C₄

—

314

—

104

C₂²⋊C₄

D₆₂₈

yes

—

105

C₄²

—

314

—

106

C₄²

C₂²×D₁₅₇

yes

—

107

C₂³

C₂×D₆₂₈

yes

characteristic

108

Q₈

—

314

—

109

Q₈

—

314

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

314

—

113

C₂×C₄

—

314

—

114

C₂×C₄

C₂×D₆₂₈

yes

—

115

C₂×C₄

C₂×D₆₂₈

yes

characteristic

116

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₆₂₈

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

628

—

125

C₄

—

628

—

126

C₄

#19097816

yes

characteristic

127

C₄

#19097819

yes

characteristic

128

C₂

—

129

C₂

#23757854

yes

characteristic

130

C₂

#23758090

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₆₂₈
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#424×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	#424	D₁₅₇
hypocenter	C₁₅₇	#3261
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₆₂₈

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

#424×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂²×D₄

prime radical

C₂³×C₁₅₇

C₂×D₄

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #23757581	C₂	semidirect
5. #23757484	C₂	semidirect
65. C₁₅₇	#3261	semidirect

H ≅ G/N

type

2. #23757581

C₂

semidirect

5. #23757484

C₂

semidirect

65. C₁₅₇

#3261

semidirect

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

subgroup	index
#23757885	2
#23757581	2
#23757531	2
#23757514	2
#23757484	2
#424×C₁₅₇	2
#3261	157

subgroup

index

#424×C₁₅₇

157

Subgroups — #34111590

Conjugacy classes of subgroups

Complements

Maximal subgroups