Group #34100857 — subgroups

Subgroups — #34100857

Order 20096.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#19097700	C₂	1	2	yes	maximal
2	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757027	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	1	yes	characteristic
7	C₂²×#11929434	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
10	#19097700	—	2	4	no	—
11	#19097682	—	2	2	no	—
12	#19097678	C₄	1	1	yes	characteristic
13	#19097678	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₅₇	—	2	2	no	—
17	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
19	C₂×#11929434	—	2	1	no	—
20	C₂×#11929434	—	2	1	no	—
21	C₂×#11929434	—	2	1	no	—
22	C₂×#11929434	—	2	1	no	—
23	C₂×#11929434	D₄	1	2	yes	—
24	C₂×#11929434	Q₈	1	2	yes	—
25	C₄⋊Dic₁₅₇	—	2	4	no	—
26	C₄×Dic₁₅₇	—	2	2	no	—
27	C₄×Dic₁₅₇	—	2	2	no	—
28	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₅₇	—	2	2	no	—
32	C₂×C₈×C₁₅₇	—	2	2	no	—
33	C₂×C₈×C₁₅₇	—	2	2	no	—
34	C₄²×C₁₅₇	D₄	1	2	yes	—
35	C₄²×C₁₅₇	Q₈	1	2	yes	—
36	C₂×Dic₁₅₇	—	2	2	no	—
37	C₂×Dic₁₅₇	—	2	2	no	—
38	C₂×Dic₁₅₇	—	2	2	no	—
39	C₂×Dic₁₅₇	—	2	2	no	—
40	C₂×Dic₁₅₇	—	2	2	no	—
41	C₂×Dic₁₅₇	M₁₆	1	4	yes	—
42	#11929434	—	2	2	no	—
43	#11929434	—	2	2	no	—
44	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	—	2	2	no	—
46	C₂×C₄×C₁₅₇	—	2	2	no	—
47	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
55	C₈×C₁₅₇	—	4	2	no	—
56	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₅₇	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₅₇	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
60	C₂²×C₁₅₇	#131	1	2	yes	—
61	C₄×C₁₅₇	—	2	2	no	—
62	C₄×C₁₅₇	—	2	2	no	—
63	C₄×C₁₅₇	#121	1	1	yes	characteristic
64	C₄×C₁₅₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₁₅₇	#125	1	1	yes	characteristic
66	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₅₇	—	2	4	no	—
68	Dic₁₅₇	—	4	2	no	—
69	C₂×C₁₅₇	#389	1	1	yes	characteristic
70	C₂×C₁₅₇	#423	1	1	yes	characteristic
71	C₂×C₁₅₇	#424	1	1	yes	characteristic
72	C₂×C₁₅₇	#435	1	2	yes	—
73	C₂×C₁₅₇	#446	1	2	yes	—
74	C₁₅₇	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	157	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	157	2	no	—
77	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
78	C₂×#97	—	157	1	no	—
79	C₄×C₄⋊C₄	—	157	1	no	—
80	#336	—	157	2	no	—
81	C₂²×C₈	—	157	1	no	—
82	C₂²×C₈	—	157	1	no	—
83	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
84	C₂×C₄⋊C₄	—	157	2	no	—
85	C₂×C₄²	—	157	2	no	—
86	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	314	4	no	—
89	#97	—	314	2	no	—
90	#95	—	157	1	no	—
91	#95	—	157	1	no	—
92	C₂²×C₄	—	157	2	no	—
93	C₂²×C₄	—	157	2	no	—
94	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	314	1	no	—
101	C₂×C₈	—	314	1	no	—
102	C₂×C₈	—	314	1	no	—
103	C₂×C₈	—	314	1	no	—
104	C₂×C₈	—	157	2	no	—
105	C₂×C₈	—	157	2	no	—
106	C₄⋊C₄	—	314	4	no	—
107	C₄²	—	314	2	no	—
108	C₄²	—	314	2	no	—
109	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
110	C₄²	Dic₃₁₄	1	2	yes	—
111	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	314	2	no	—
115	C₂×C₄	—	314	2	no	—
116	C₂×C₄	—	314	2	no	—
117	C₂×C₄	—	314	2	no	—
118	C₂×C₄	—	314	2	no	—
119	C₂×C₄	—	157	4	no	—
120	C₂×C₄	#14578933	1	2	yes	—
121	C₂×C₄	#14578933	1	2	yes	—
122	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
126	C₂×C₄	#14578953	1	1	yes	characteristic
127	C₂×C₄	#14578955	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	314	2	no	—
130	C₈	—	314	2	no	—
131	C₂²	C₂×#14578933	1	1	yes	characteristic
132	C₂²	#19097733	1	1	yes	characteristic
133	C₂²	#19097736	1	1	yes	characteristic
134	C₂²	M₁₆×D₁₅₇	1	2	yes	—
135	C₂²	#19097777	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	314	4	no	—
139	C₄	—	628	2	no	—
140	C₄	#19097742	1	1	yes	characteristic
141	C₄	#19097747	1	1	yes	characteristic
142	C₄	#19097762	1	1	yes	characteristic
143	C₄	#19097770	1	1	yes	characteristic
144	C₂	#23757169	1	1	yes	characteristic
145	C₂	#23757296	1	1	yes	characteristic
146	C₂	#23757299	1	1	yes	characteristic
147	C₂	#23757343	1	2	yes	—
148	C₂	#23757402	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097700

C₂

yes

maximal

C₂×#19097682

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

characteristic

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₇

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

#19097700

—

#19097682

—

#19097678

C₄

yes

characteristic

#19097678

C₄

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#11929434

Q₈

yes

—

C₄⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

M₁₆

yes

—

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×D₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

characteristic

C₂²×C₁₅₇

C₂×M₁₆

yes

—

C₂²×C₁₅₇

#131

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₅₇

#125

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₂×C₁₅₇

#446

yes

—

C₁₅₇

#1940

yes

characteristic, complemented, Sylow

#1940

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×C₄⋊C₈

D₁₅₇

yes

characteristic

C₂×#97

—

157

—

C₄×C₄⋊C₄

—

157

—

#336

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

314

—

#97

—

314

—

#95

—

157

—

#95

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

314

—

101

C₂×C₈

—

314

—

102

C₂×C₈

—

314

—

103

C₂×C₈

—

314

—

104

C₂×C₈

—

157

—

105

C₂×C₈

—

157

—

106

C₄⋊C₄

—

314

—

107

C₄²

—

314

—

108

C₄²

—

314

—

109

C₄²

C₂²⋊D₁₅₇

yes

—

110

C₄²

Dic₃₁₄

yes

—

111

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

314

—

115

C₂×C₄

—

314

—

116

C₂×C₄

—

314

—

117

C₂×C₄

—

314

—

118

C₂×C₄

—

314

—

119

C₂×C₄

—

157

—

120

C₂×C₄

#14578933

yes

—

121

C₂×C₄

#14578933

yes

—

122

C₂×C₄

D₄×D₁₅₇

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

124

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

125

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

126

C₂×C₄

#14578953

yes

characteristic

127

C₂×C₄

#14578955

yes

characteristic

128

C₈

—

129

C₈

—

314

—

130

C₈

—

314

—

131

C₂²

C₂×#14578933

yes

characteristic

132

C₂²

#19097733

yes

characteristic

133

C₂²

#19097736

yes

characteristic

134

C₂²

M₁₆×D₁₅₇

yes

—

135

C₂²

#19097777

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

314

—

139

C₄

—

628

—

140

C₄

#19097742

yes

characteristic

141

C₄

#19097747

yes

characteristic

142

C₄

#19097762

yes

characteristic

143

C₄

#19097770

yes

characteristic

144

C₂

#23757169

yes

characteristic

145

C₂

#23757296

yes

characteristic

146

C₂

#23757299

yes

characteristic

147

C₂

#23757343

yes

—

148

C₂

#23757402

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator, socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₅₇	C₂
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#1940
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator, socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₅₇

C₂

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
74. C₁₅₇	#1940	semidirect

H ≅ G/N

type

74. C₁₅₇

#1940

semidirect

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

subgroup	index
C₂×#19097700	2
C₂×#19097682	2
C₄×C₄⋊Dic₁₅₇	2
#23757027	2
C₂×C₄⋊C₈×C₁₅₇	2
#1940	157

subgroup

index

C₄×C₄⋊Dic₁₅₇

C₂×C₄⋊C₈×C₁₅₇

157

Subgroups — #34100857

Conjugacy classes of subgroups

Complements

Maximal subgroups