Group #34099516 — subgroups

Subgroups — #34099516

Order 20096.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#23757669	C₂	1	1	yes	maximal, characteristic
2	#23757568	C₂	1	1	yes	maximal, characteristic
3	Q₈×#11929434	C₂	1	1	yes	maximal, characteristic
4	#23757023	C₂	1	1	yes	maximal, characteristic
5	#23757021	C₂	1	1	yes	maximal, characteristic
6	#23757019	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	#19097771	C₂²	1	1	yes	characteristic
9	#19097704	—	2	1	no	—
10	#19097702	—	2	1	no	—
11	#19097702	—	2	1	no	—
12	C₄⋊#11929434	—	2	1	no	—
13	C₄⋊#11929434	C₂²	1	1	yes	characteristic
14	C₄⋊#11929434	C₂²	1	1	yes	characteristic
15	#19097695	—	2	1	no	—
16	C₄×#11929434	—	2	1	no	—
17	C₄×#11929434	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₅₇	—	2	1	no	—
19	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
22	#103×C₁₅₇	—	2	1	no	—
23	C₂×Dic₃₁₄	—	2	1	no	—
24	C₂×#11929434	—	2	1	no	—
25	C₂×#11929434	—	2	1	no	—
26	C₂×#11929434	D₄	1	1	yes	characteristic
27	C₂×#11929434	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₅₇	—	2	1	no	—
29	C₄⋊Dic₁₅₇	—	2	1	no	—
30	C₂×Q₈×C₁₅₇	—	2	1	no	—
31	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
32	Q₁₆×C₁₅₇	—	4	2	no	—
33	C₂×C₈×C₁₅₇	—	2	1	no	—
34	C₄⋊C₄×C₁₅₇	—	2	1	no	—
35	C₄⋊C₄×C₁₅₇	—	2	1	no	—
36	C₄⋊C₄×C₁₅₇	—	2	1	no	—
37	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₅₇	—	2	1	no	—
39	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
40	Dic₃₁₄	—	8	1	no	—
41	C₂×Dic₁₅₇	—	4	1	no	—
42	#11929434	—	2	1	no	—
43	#11929434	—	4	1	no	—
44	#11929434	—	4	1	no	—
45	#11929434	Q₁₆	1	2	yes	complemented
46	Q₈×C₁₅₇	—	2	1	no	—
47	Q₈×C₁₅₇	—	4	1	no	—
48	Q₈×C₁₅₇	—	4	1	no	—
49	Q₈×C₁₅₇	SD₁₆	1	2	yes	—
50	C₂×C₄×C₁₅₇	—	2	1	no	—
51	C₂×C₄×C₁₅₇	—	2	1	no	—
52	C₂×C₄×C₁₅₇	—	4	1	no	—
53	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
56	C₈×C₁₅₇	—	4	1	no	—
57	C₂²×C₁₅₇	#121	1	1	yes	characteristic
58	C₄×C₁₅₇	—	2	2	no	—
59	C₄×C₁₅₇	—	2	1	no	—
60	C₄×C₁₅₇	—	4	1	no	—
61	C₄×C₁₅₇	—	8	1	no	—
62	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₅₇	#135	1	1	yes	characteristic
65	C₄×C₁₅₇	#137	1	1	yes	characteristic
66	Dic₁₅₇	—	8	1	no	—
67	C₂×C₁₅₇	#462	1	1	yes	characteristic
68	C₂×C₁₅₇	#465	1	1	yes	characteristic
69	C₂×C₁₅₇	#471	1	1	yes	characteristic
70	C₁₅₇	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	157	1	no	maximal, Sylow
72	#498	—	157	1	no	—
73	#477	—	157	1	no	—
74	#462	D₁₅₇	1	1	yes	characteristic
75	C₈×Q₈	—	157	1	no	—
76	#334	—	157	1	no	—
77	#333	—	157	1	no	—
78	#332	—	157	1	no	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	157	1	no	—
81	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
83	#107	—	314	1	no	—
84	#106	—	314	1	no	—
85	#106	—	314	1	no	—
86	C₄⋊C₈	—	314	1	no	—
87	C₄⋊C₈	—	157	1	no	—
88	C₄⋊C₈	—	157	1	no	—
89	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	314	1	no	—
92	C₄×C₈	—	314	1	no	—
93	C₄×C₈	—	157	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	314	1	no	—
96	C₂×Q₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	314	1	no	—
100	C₂×C₈	—	314	1	no	—
101	C₂×C₈	—	157	1	no	—
102	C₂×C₈	—	157	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	314	1	no	—
107	C₄⋊C₄	—	314	1	no	—
108	C₄⋊C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	1256	1	no	—
115	Q₈	#14578942	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	628	1	no	—
120	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
122	C₂×C₄	#14578953	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	314	1	no	—
125	C₈	—	628	1	no	—
126	C₈	—	628	1	no	—
127	C₈	—	157	2	no	—
128	C₂²	#19097743	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	1256	1	no	—
134	C₄	C₂×#14578942	1	1	yes	characteristic
135	C₄	Q₁₆×D₁₅₇	1	1	yes	characteristic
136	C₄	#19097791	1	1	yes	characteristic
137	C₄	#19097803	1	1	yes	characteristic
138	C₂	#23757481	1	1	yes	characteristic
139	C₂	#23757500	1	1	yes	characteristic
140	C₂	#23757538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757669

C₂

yes

maximal, characteristic

#23757568

C₂

yes

maximal, characteristic

Q₈×#11929434

C₂

yes

maximal, characteristic

#23757023

C₂

yes

maximal, characteristic

#23757021

C₂

yes

maximal, characteristic

#23757019

C₂

yes

maximal, characteristic

#462×C₁₅₇

C₂

yes

maximal, characteristic

#19097771

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊#11929434

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×Q₁₆×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

C₄×Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₂²

yes

characteristic

#103×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

D₄

yes

characteristic

C₂×#11929434

D₄

yes

characteristic

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

Q₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

C₂×Dic₁₅₇

—

—

—

—

Q₁₆

yes

complemented

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₅₇

C₂×Q₁₆

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1672

—

157

maximal, Sylow

#498

—

157

—

#477

—

157

—

#462

D₁₅₇

yes

characteristic

C₈×Q₈

—

157

—

#334

—

157

—

#333

—

157

—

#332

—

157

—

C₂×Q₁₆

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

C₄×Q₈

C₂×D₁₅₇

yes

characteristic

#107

—

314

—

#106

—

314

—

#106

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

C₄⋊C₈

—

157

—

C₄⋊C₈

C₂×D₁₅₇

yes

characteristic

#103

—

#103

—

314

—

C₄×C₈

—

314

—

C₄×C₈

—

157

—

C₂×Q₈

—

C₂×Q₈

—

314

—

C₂×Q₈

C₂²⋊D₁₅₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

314

—

100

C₂×C₈

—

314

—

101

C₂×C₈

—

157

—

102

C₂×C₈

—

157

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

314

—

107

C₄⋊C₄

—

314

—

108

C₄⋊C₄

C₂²⋊D₁₅₇

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₁₅₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

1256

—

115

Q₈

#14578942

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

628

—

120

C₂×C₄

D₄×D₁₅₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

122

C₂×C₄

#14578953

yes

characteristic

123

C₈

—

124

C₈

—

314

—

125

C₈

—

628

—

126

C₈

—

628

—

127

C₈

—

157

—

128

C₂²

#19097743

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

1256

—

134

C₄

C₂×#14578942

yes

characteristic

135

C₄

Q₁₆×D₁₅₇

yes

characteristic

136

C₄

#19097791

yes

characteristic

137

C₄

#19097803

yes

characteristic

138

C₂

#23757481

yes

characteristic

139

C₂

#23757500

yes

characteristic

140

C₂

#23757538

yes

characteristic

	subgroup	G/H
center	C₂²	#19097743
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#462×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#121
hypercenter	#462	D₁₅₇
hypocenter	C₁₅₇	#1672
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#19097743

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#462×C₁₅₇

C₂

C₂²×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
45. #11929434	Q₁₆	semidirect
70. C₁₅₇	#1672	semidirect

H ≅ G/N

type

45. #11929434

Q₁₆

semidirect

70. C₁₅₇

#1672

semidirect

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

subgroup	index
#23757669	2
#23757568	2
Q₈×#11929434	2
#23757023	2
#23757021	2
#23757019	2
#462×C₁₅₇	2
#1672	157

subgroup

index

#462×C₁₅₇

157

Subgroups — #34099516

Conjugacy classes of subgroups

Complements

Maximal subgroups