Group #34100689 — subgroups

Subgroups — #34100689

Order 20096.

308 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23758040	C₂	1	1	yes	maximal, characteristic
2	#23757257	C₂	1	1	yes	maximal, characteristic
3	C₂×#19097690	C₂	1	2	yes	maximal, complemented
4	#23757043	C₂	1	2	yes	maximal, complemented
5	#536×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	#19097778	—	2	4	no	—
7	C₂×#14578938	—	2	2	no	—
8	C₂×#14578938	C₂²	1	2	yes	—
9	C₂×#14578938	C₂²	1	2	yes	—
10	#19097690	—	2	4	no	—
11	#19097690	—	2	1	no	—
12	#19097688	—	2	1	no	—
13	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₅₇	—	2	1	no	—
16	#124×C₁₅₇	—	2	1	no	—
17	#123×C₁₅₇	C₄	1	4	yes	—
18	C₂²⋊Q₈×C₁₅₇	—	2	2	no	—
19	#117×C₁₅₇	C₂²	1	1	yes	characteristic
20	#14578938	—	2	4	no	—
21	#14578938	—	2	2	no	—
22	#14578938	—	2	2	no	—
23	#14578938	—	2	2	no	—
24	#14578938	D₄	1	4	yes	—
25	C₂×#11929434	—	2	4	no	—
26	C₂×#11929434	—	2	2	no	—
27	C₂×#11929434	—	2	2	no	—
28	#41×C₁₅₇	—	2	2	no	—
29	#41×C₁₅₇	—	2	2	no	—
30	C₂×Q₈×C₁₅₇	—	2	1	no	—
31	C₂×Q₈×C₁₅₇	—	2	1	no	—
32	C₂×Q₈×C₁₅₇	—	2	2	no	—
33	C₂×Q₈×C₁₅₇	—	2	2	no	—
34	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₅₇	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₁₅₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
39	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₅₇	—	2	4	no	—
41	C₄⋊C₄×C₁₅₇	—	2	1	no	—
42	C₂²⋊C₄×C₁₅₇	—	2	4	no	—
43	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
44	C₄²×C₁₅₇	—	2	1	no	—
45	#11929434	—	2	4	no	—
46	#11929434	—	4	2	no	—
47	C₂³×C₁₅₇	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₅₇	—	2	2	no	—
50	Q₈×C₁₅₇	—	4	1	no	—
51	Q₈×C₁₅₇	—	4	2	no	—
52	Q₈×C₁₅₇	—	4	1	no	—
53	D₄×C₁₅₇	—	2	4	no	—
54	D₄×C₁₅₇	—	2	2	no	—
55	C₂×C₄×C₁₅₇	—	2	2	no	—
56	C₂×C₄×C₁₅₇	—	2	2	no	—
57	C₂×C₄×C₁₅₇	—	2	2	no	—
58	C₂×C₄×C₁₅₇	—	2	2	no	—
59	C₂×C₄×C₁₅₇	—	2	4	no	—
60	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₅₇	#41	1	2	yes	—
65	C₂²×C₁₅₇	—	2	2	no	—
66	C₂²×C₁₅₇	—	4	1	no	—
67	C₂²×C₁₅₇	—	4	2	no	—
68	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
70	C₄×C₁₅₇	—	2	2	no	—
71	C₄×C₁₅₇	—	4	2	no	—
72	C₄×C₁₅₇	—	4	2	no	—
73	C₄×C₁₅₇	#120	1	1	yes	characteristic
74	C₄×C₁₅₇	#121	1	2	yes	—
75	C₄×C₁₅₇	#123	1	1	yes	characteristic
76	C₂×C₁₅₇	—	2	1	no	—
77	C₂×C₁₅₇	—	2	2	no	—
78	C₂×C₁₅₇	—	4	2	no	—
79	C₂×C₁₅₇	#386	1	1	yes	characteristic
80	C₁₅₇	#1915	1	1	yes	characteristic, complemented, Sylow
81	#1915	—	157	1	no	maximal, Sylow
82	#568	—	157	1	no	—
83	#536	D₁₅₇	1	1	yes	characteristic
84	#413	—	157	1	no	—
85	C₂×#101	—	157	2	no	—
86	#344	—	157	2	no	—
87	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
89	#131	—	314	4	no	—
90	C₂×M₁₆	—	314	2	no	—
91	C₂×M₁₆	—	157	2	no	—
92	C₂×M₁₆	—	157	2	no	—
93	C₄⋊Q₈	—	2	1	no	—
94	#124	—	2	1	no	—
95	#123	Dic₁₅₇	1	4	yes	—
96	C₂²⋊Q₈	—	2	2	no	—
97	#117	C₂×D₁₅₇	1	1	yes	characteristic
98	#101	—	314	1	no	—
99	#101	—	314	4	no	—
100	#100	—	314	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
108	C₂×D₄	C₂×Dic₁₅₇	1	2	yes	—
109	C₂×D₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
111	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
113	M₁₆	—	314	4	no	—
114	M₁₆	—	314	2	no	—
115	M₁₆	—	314	2	no	—
116	M₁₆	—	314	2	no	—
117	M₁₆	—	157	4	no	—
118	C₂×C₈	—	314	4	no	—
119	C₂×C₈	—	314	2	no	—
120	C₂×C₈	—	314	2	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊Dic₁₅₇	1	2	yes	—
127	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
140	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
143	C₂×C₄	#14578953	1	2	yes	—
144	C₈	—	314	4	no	—
145	C₈	—	628	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
150	C₂²	D₄×Dic₁₅₇	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#19097739	1	1	yes	characteristic
155	C₄	#19097743	1	2	yes	—
156	C₄	#19097752	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#23757153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758040

C₂

yes

maximal, characteristic

#23757257

C₂

yes

maximal, characteristic

C₂×#19097690

C₂

yes

maximal, complemented

#23757043

C₂

yes

maximal, complemented

#536×C₁₅₇

C₂

yes

maximal, characteristic

#19097778

—

C₂×#14578938

—

C₂×#14578938

C₂²

yes

—

C₂×#14578938

C₂²

yes

—

#19097690

—

#19097690

—

#19097688

—

C₂×#41×C₁₅₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₇

—

#124×C₁₅₇

—

#123×C₁₅₇

C₄

yes

—

C₂²⋊Q₈×C₁₅₇

—

#117×C₁₅₇

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

—

#41×C₁₅₇

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

C₂×D₄×C₁₅₇

C₂×C₄

yes

—

C₂×D₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

—

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

#121

yes

—

C₄×C₁₅₇

#123

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#386

yes

characteristic

C₁₅₇

#1915

yes

characteristic, complemented, Sylow

#1915

—

157

maximal, Sylow

#568

—

157

—

#536

D₁₅₇

yes

characteristic

#413

—

157

—

C₂×#101

—

157

—

#344

—

157

—

C₂×#41

C₂×D₁₅₇

yes

characteristic

C₂²×Q₈

C₂×D₁₅₇

yes

characteristic

#131

—

314

—

C₂×M₁₆

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₄⋊Q₈

—

#124

—

#123

Dic₁₅₇

yes

—

C₂²⋊Q₈

—

#117

C₂×D₁₅₇

yes

characteristic

#101

—

314

—

#101

—

314

—

100

#100

—

314

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₁₅₇

yes

characteristic

108

C₂×D₄

C₂×Dic₁₅₇

yes

—

109

C₂×D₄

C₂²⋊D₁₅₇

yes

characteristic

110

C₂²×C₄

C₂×Dic₁₅₇

yes

—

111

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

112

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

113

M₁₆

—

314

—

114

M₁₆

—

314

—

115

M₁₆

—

314

—

116

M₁₆

—

314

—

117

M₁₆

—

157

—

118

C₂×C₈

—

314

—

119

C₂×C₈

—

314

—

120

C₂×C₈

—

314

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×Dic₁₅₇

yes

—

125

C₄²

—

126

C₂³

C₂²⋊Dic₁₅₇

yes

—

127

C₂³

C₂²×Dic₁₅₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

140

C₂×C₄

D₄×D₁₅₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

143

C₂×C₄

#14578953

yes

—

144

C₈

—

314

—

145

C₈

—

628

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

150

C₂²

D₄×Dic₁₅₇

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#19097739

yes

characteristic

155

C₄

#19097743

yes

—

156

C₄

#19097752

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#23757153

yes

characteristic

	subgroup	G/H
center	C₂	#23757153
commutator	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#536×C₁₅₇	C₂
socle	C₂×C₁₅₇	#386
prime radical	C₂×#41×C₁₅₇	C₂²
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#536	D₁₅₇
hypocenter	C₁₅₇	#1915
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#23757153

commutator

C₂³×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#536×C₁₅₇

C₂

socle

C₂×C₁₅₇

#386

prime radical

C₂×#41×C₁₅₇

C₂²

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
3. C₂×#19097690	C₂	semidirect
4. #23757043	C₂	semidirect
80. C₁₅₇	#1915	semidirect

H ≅ G/N

type

3. C₂×#19097690

C₂

semidirect

4. #23757043

C₂

semidirect

80. C₁₅₇

#1915

semidirect

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

subgroup	index
#23758040	2
#23757257	2
C₂×#19097690	2
#23757043	2
#536×C₁₅₇	2
#1915	157

subgroup

index

#536×C₁₅₇

157

Subgroups — #34100689

Conjugacy classes of subgroups

Complements

Maximal subgroups