Group #34098539 — subgroups

Subgroups — #34098539

Order 20096.

216 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 97 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	#23757326	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11929434	C₂	1	4	yes	maximal, complemented
4	#533×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929434	—	2	1	no	—
6	C₄⋊#11929434	—	2	1	no	—
7	C₄⋊#11929434	—	2	1	no	—
8	C₄⋊#11929434	C₂²	1	2	yes	—
9	C₂²⋊#11929434	—	2	1	no	—
10	C₄×#11929434	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₅₇	C₄	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₅₇	C₄	1	1	yes	characteristic
13	#125×C₁₅₇	—	2	1	no	—
14	C₂²⋊Q₈×C₁₅₇	—	2	1	no	—
15	C₂²⋊Q₈×C₁₅₇	C₄	1	2	yes	—
16	C₄×Q₈×C₁₅₇	—	2	1	no	—
17	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
18	#117×C₁₅₇	—	2	1	no	—
19	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	—	2	2	no	—
22	C₂×#11929434	—	4	2	no	—
23	C₂×Q₈×C₁₅₇	—	2	1	no	—
24	C₂×Q₈×C₁₅₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₇	—	2	1	no	—
29	C₄⋊C₄×C₁₅₇	—	2	1	no	—
30	C₄⋊C₄×C₁₅₇	—	2	1	no	—
31	C₄⋊C₄×C₁₅₇	—	2	2	no	—
32	C₄⋊C₄×C₁₅₇	—	2	2	no	—
33	C₄⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
35	C₄²×C₁₅₇	—	2	2	no	—
36	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
39	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
40	#11929434	—	4	2	no	—
41	#11929434	—	4	2	no	—
42	C₂³×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₁₅₇	—	2	4	no	—
44	Q₈×C₁₅₇	—	4	2	no	—
45	C₂×C₄×C₁₅₇	—	2	2	no	—
46	C₂×C₄×C₁₅₇	—	2	2	no	—
47	C₂×C₄×C₁₅₇	—	2	2	no	—
48	C₂×C₄×C₁₅₇	—	4	1	no	—
49	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₅₇	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₁₅₇	—	2	1	no	—
58	C₂²×C₁₅₇	—	2	1	no	—
59	C₂²×C₁₅₇	#103	1	2	yes	—
60	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₅₇	—	2	1	no	—
62	C₄×C₁₅₇	—	2	1	no	—
63	C₄×C₁₅₇	—	2	1	no	—
64	C₄×C₁₅₇	—	4	2	no	—
65	C₄×C₁₅₇	—	4	2	no	—
66	C₄×C₁₅₇	#103	1	2	yes	—
67	C₄×C₁₅₇	#131	1	2	yes	—
68	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₅₇	—	2	2	no	—
71	C₂×C₁₅₇	#408	1	1	yes	characteristic
72	C₂×C₁₅₇	C₂×#103	1	1	yes	characteristic
73	C₂×C₁₅₇	#421	1	1	yes	characteristic
74	C₁₅₇	#1481	1	1	yes	characteristic, complemented, Sylow
75	#1481	—	157	1	no	maximal, Sylow
76	#533	D₁₅₇	1	1	yes	characteristic
77	#431	—	157	1	no	—
78	C₂×C₄⋊C₈	—	157	1	no	—
79	Q₈⋊C₈	—	157	4	no	—
80	C₂²×C₈	—	314	1	no	—
81	C₄⋊Q₈	Dic₁₅₇	1	1	yes	characteristic
82	C₄⋊Q₈	Dic₁₅₇	1	1	yes	characteristic
83	#125	—	2	1	no	—
84	C₂²⋊Q₈	—	2	1	no	—
85	C₂²⋊Q₈	Dic₁₅₇	1	2	yes	—
86	C₄×Q₈	—	2	1	no	—
87	C₄×Q₈	C₂×D₁₅₇	1	2	yes	—
88	#117	—	2	1	no	—
89	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
90	C₄⋊C₈	—	314	1	no	—
91	C₄⋊C₈	—	314	1	no	—
92	C₄⋊C₈	—	157	2	no	—
93	C₂²⋊C₈	—	314	1	no	—
94	C₄×C₈	—	157	2	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	C₂×Dic₁₅₇	1	2	yes	—
97	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
100	C₂×C₈	—	314	2	no	—
101	C₂×C₈	—	314	2	no	—
102	C₂×C₈	—	628	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
112	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
113	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
114	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
115	C₂³	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
125	C₂×C₄	#14578941	1	2	yes	—
126	C₂×C₄	#14578944	1	2	yes	—
127	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
130	C₈	—	628	2	no	—
131	C₈	—	628	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#19097695	1	2	yes	—
135	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	4	2	no	—
141	C₄	#19097695	1	2	yes	—
142	C₄	#19097778	1	2	yes	—
143	C₄	C₂×#14578941	1	1	yes	characteristic
144	C₄	C₂×#14578944	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	#23757240	1	1	yes	characteristic
147	C₂	C₂×#19097695	1	1	yes	characteristic
148	C₂	#23757290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757326

C₂

yes

maximal, characteristic

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

Q₈⋊#11929434

C₂

yes

maximal, complemented

#533×C₁₅₇

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂²⋊#11929434

—

C₄×#11929434

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₄

yes

characteristic

C₄⋊Q₈×C₁₅₇

C₄

yes

characteristic

#125×C₁₅₇

—

C₂²⋊Q₈×C₁₅₇

—

C₂²⋊Q₈×C₁₅₇

C₄

yes

—

C₄×Q₈×C₁₅₇

—

C₄×Q₈×C₁₅₇

C₂²

yes

—

#117×C₁₅₇

—

C₂×C₄²×C₁₅₇

C₂²

yes

—

—

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

C₂³

yes

characteristic

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#103

yes

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#103

yes

—

C₄×C₁₅₇

#131

yes

—

C₄×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₅₇

C₂×Q₁₆

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1481

—

157

maximal, Sylow

#533

D₁₅₇

yes

characteristic

#431

—

157

—

C₂×C₄⋊C₈

—

157

—

Q₈⋊C₈

—

157

—

C₂²×C₈

—

314

—

C₄⋊Q₈

Dic₁₅₇

yes

characteristic

C₄⋊Q₈

Dic₁₅₇

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₁₅₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₅₇

yes

—

#117

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

C₂²⋊C₈

—

314

—

C₄×C₈

—

157

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂×Dic₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

100

C₂×C₈

—

314

—

101

C₂×C₈

—

314

—

102

C₂×C₈

—

628

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₂×Dic₁₅₇

yes

characteristic

112

C₄²

C₂²⋊D₁₅₇

yes

characteristic

113

C₄²

C₂²⋊D₁₅₇

yes

characteristic

114

C₄²

C₂²×D₁₅₇

yes

characteristic

115

C₂³

C₂²⋊Dic₁₅₇

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²⋊Dic₁₅₇

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₁₅₇

yes

characteristic

124

C₂×C₄

C₂²⋊Dic₁₅₇

yes

characteristic

125

C₂×C₄

#14578941

yes

—

126

C₂×C₄

#14578944

yes

—

127

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

130

C₈

—

628

—

131

C₈

—

628

—

132

C₂²

—

133

C₂²

—

134

C₂²

#19097695

yes

—

135

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#19097695

yes

—

142

C₄

#19097778

yes

—

143

C₄

C₂×#14578941

yes

characteristic

144

C₄

C₂×#14578944

yes

characteristic

145

C₂

—

146

C₂

#23757240

yes

characteristic

147

C₂

C₂×#19097695

yes

characteristic

148

C₂

#23757290

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#533×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₅₇	C₂²⋊C₄
cosocle	C₄²×C₁₅₇	C₂³
hypercenter	#533	D₁₅₇
hypocenter	C₁₅₇	#1481
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#533×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₅₇

C₂²⋊C₄

cosocle

C₄²×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. Q₈⋊#11929434	C₂	semidirect
74. C₁₅₇	#1481	semidirect

H ≅ G/N

type

3. Q₈⋊#11929434

C₂

semidirect

74. C₁₅₇

#1481

semidirect

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

subgroup	index
#23757326	2
C₂×C₄⋊#11929434	2
Q₈⋊#11929434	2
#533×C₁₅₇	2
#1481	157

subgroup

index

#533×C₁₅₇

157

Subgroups — #34098539

Conjugacy classes of subgroups

Complements

Maximal subgroups