Group #38742799 — subgroups

Subgroups — #38742799

Order 25216.

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

normal subgroups only

This group has 344 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	C₂²×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#27160939	C₂	1	1	yes	maximal, characteristic
3	#27160842	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₉₇	—	2	1	no	—
6	C₂×C₄⋊Dic₁₉₇	—	2	2	no	—
7	C₂×C₄⋊Dic₁₉₇	—	2	1	no	—
8	C₂×C₄⋊Dic₁₉₇	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₉₇	—	2	1	no	—
12	#20263233	—	2	1	no	—
13	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₉₇	—	2	2	no	—
16	C₂²×Dic₁₉₇	—	2	4	no	—
17	C₂²×Dic₁₉₇	—	2	2	no	—
18	C₂²×Dic₁₉₇	—	2	2	no	—
19	C₂²×Dic₁₉₇	—	4	2	no	—
20	C₂²×Dic₁₉₇	—	4	2	no	—
21	C₄⋊Dic₁₉₇	—	2	4	no	—
22	C₄⋊Dic₁₉₇	—	2	2	no	—
23	C₄⋊Dic₁₉₇	—	2	4	no	—
24	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
25	C₂²⋊Dic₁₉₇	—	4	2	no	—
26	C₂⁴×C₁₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₇	—	2	4	no	—
28	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
30	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₉₇	—	2	2	no	—
33	C₂×C₈×C₁₉₇	—	2	4	no	—
34	C₂×Dic₁₉₇	—	2	4	no	—
35	C₂×Dic₁₉₇	—	4	8	no	—
36	C₂×Dic₁₉₇	—	4	2	no	—
37	C₂×Dic₁₉₇	—	4	2	no	—
38	C₂×Dic₁₉₇	—	4	2	no	—
39	C₂×Dic₁₉₇	—	4	2	no	—
40	C₂×Dic₁₉₇	—	4	2	no	—
41	C₂³×C₁₉₇	—	2	2	no	—
42	C₂³×C₁₉₇	—	2	2	no	—
43	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₁₉₇	D₈	1	2	yes	—
45	C₂³×C₁₉₇	SD₁₆	1	2	yes	—
46	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	—	2	2	no	—
48	C₂×C₄×C₁₉₇	—	2	4	no	—
49	C₂×C₄×C₁₉₇	—	2	4	no	—
50	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	#41	1	4	yes	—
54	C₈×C₁₉₇	—	4	2	no	—
55	C₂²×C₁₉₇	—	2	2	no	—
56	C₂²×C₁₉₇	—	2	2	no	—
57	C₂²×C₁₉₇	—	2	4	no	—
58	C₂²×C₁₉₇	—	2	4	no	—
59	C₂²×C₁₉₇	#102	1	4	yes	—
60	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₉₇	#123	1	2	yes	—
62	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₁₉₇	#137	1	2	yes	—
65	C₄×C₁₉₇	—	2	2	no	—
66	C₄×C₁₉₇	#117	1	2	yes	—
67	C₄×C₁₉₇	#123	1	2	yes	—
68	Dic₁₉₇	—	4	4	no	—
69	Dic₁₉₇	—	8	2	no	—
70	C₂×C₁₉₇	—	2	4	no	—
71	C₂×C₁₉₇	#381	1	1	yes	characteristic
72	C₂×C₁₉₇	C₂×#102	1	1	yes	characteristic
73	C₂×C₁₉₇	#419	1	1	yes	characteristic
74	C₂×C₁₉₇	#480	1	2	yes	—
75	C₂×C₁₉₇	#483	1	2	yes	—
76	C₁₉₇	#1823	1	1	yes	characteristic, complemented, Sylow
77	#1823	—	197	1	no	maximal, Sylow
78	C₂²×C₄⋊C₄	—	197	1	no	—
79	C₂×C₂²⋊C₈	D₁₉₇	1	1	yes	characteristic
80	#380	—	197	1	no	—
81	#340	—	197	4	no	—
82	C₂³×C₄	—	394	1	no	—
83	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
85	C₂×C₄⋊C₄	—	394	2	no	—
86	C₂×C₄⋊C₄	—	394	1	no	—
87	C₂×C₄⋊C₄	—	197	2	no	—
88	C₂×C₄⋊C₄	—	197	4	no	—
89	C₂×C₄⋊C₄	—	197	2	no	—
90	C₂×C₂²⋊C₄	—	394	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	#95	—	394	1	no	—
93	C₂⁴	D₇₈₈	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	788	2	no	—
96	C₂²×C₄	—	788	2	no	—
97	C₂²×C₄	—	394	2	no	—
98	C₂²×C₄	—	394	4	no	—
99	C₂²×C₄	—	394	2	no	—
100	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
102	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	4	no	—
106	C₄⋊C₄	—	394	4	no	—
107	C₄⋊C₄	—	394	4	no	—
108	C₄⋊C₄	—	394	2	no	—
109	C₄⋊C₄	—	197	4	no	—
110	C₂²⋊C₄	—	788	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	#15691335	1	2	yes	—
114	C₂³	D₁₅₇₆	1	2	yes	—
115	C₂³	#15691347	1	2	yes	—
116	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	788	8	no	—
121	C₂×C₄	—	788	2	no	—
122	C₂×C₄	—	788	2	no	—
123	C₂×C₄	—	788	2	no	—
124	C₂×C₄	—	788	2	no	—
125	C₂×C₄	—	788	2	no	—
126	C₂×C₄	—	394	4	no	—
127	C₂×C₄	#15691335	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
130	C₂×C₄	#15691357	1	4	yes	—
131	C₈	—	4	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	4	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#20263248	1	4	yes	—
137	C₂²	C₂×#15691335	1	1	yes	characteristic
138	C₂²	#20263310	1	2	yes	—
139	C₂²	C₂×D₁₅₇₆	1	1	yes	characteristic
140	C₂²	C₂×#15691347	1	1	yes	characteristic
141	C₂²	#20263359	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	788	4	no	—
144	C₄	—	1576	2	no	—
145	C₄	#20263282	1	2	yes	—
146	C₄	#20263307	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#27160940	1	1	yes	characteristic
149	C₂	C₂×#20263248	1	1	yes	characteristic
150	C₂	#27161089	1	1	yes	characteristic
151	C₂	#27161393	1	2	yes	—
152	C₂	#27161410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₉₇

C₂

yes

maximal, characteristic

#27160939

C₂

yes

maximal, characteristic

#27160842

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂³×Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

C₄

yes

complemented

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₇

—

#20263233

—

C₂³×C₄×C₁₉₇

C₂²

yes

characteristic

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂²⋊C₈×C₁₉₇

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂×C₄

yes

—

C₂²⋊Dic₁₉₇

—

C₂⁴×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂²⋊C₄

yes

—

C₂³×C₁₉₇

D₈

yes

—

C₂³×C₁₉₇

SD₁₆

yes

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#102

yes

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#123

yes

—

C₂²×C₁₉₇

C₂×D₈

yes

characteristic

C₂²×C₁₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₇

#137

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#117

yes

—

C₄×C₁₉₇

#123

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

—

C₂×C₁₉₇

#483

yes

—

C₁₉₇

#1823

yes

characteristic, complemented, Sylow

#1823

—

197

maximal, Sylow

C₂²×C₄⋊C₄

—

197

—

C₂×C₂²⋊C₈

D₁₉₇

yes

characteristic

#380

—

197

—

#340

—

197

—

C₂³×C₄

—

394

—

C₂³×C₄

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

—

197

—

C₂×C₂²⋊C₄

—

394

—

C₂²⋊C₈

—

#95

—

394

—

C₂⁴

D₇₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

788

—

C₂²×C₄

—

788

—

C₂²×C₄

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

—

394

—

100

C₂²×C₄

C₄×D₁₉₇

yes

—

101

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

102

C₂²×C₄

D₇₈₈

yes

characteristic

103

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

394

—

107

C₄⋊C₄

—

394

—

108

C₄⋊C₄

—

394

—

109

C₄⋊C₄

—

197

—

110

C₂²⋊C₄

—

788

—

111

C₂³

—

112

C₂³

—

113

C₂³

#15691335

yes

—

114

C₂³

D₁₅₇₆

yes

—

115

C₂³

#15691347

yes

—

116

C₂³

C₂×D₇₈₈

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

788

—

121

C₂×C₄

—

788

—

122

C₂×C₄

—

788

—

123

C₂×C₄

—

788

—

124

C₂×C₄

—

788

—

125

C₂×C₄

—

788

—

126

C₂×C₄

—

394

—

127

C₂×C₄

#15691335

yes

—

128

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

130

C₂×C₄

#15691357

yes

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#20263248

yes

—

137

C₂²

C₂×#15691335

yes

characteristic

138

C₂²

#20263310

yes

—

139

C₂²

C₂×D₁₅₇₆

yes

characteristic

140

C₂²

C₂×#15691347

yes

characteristic

141

C₂²

#20263359

yes

—

142

C₄

—

143

C₄

—

788

—

144

C₄

—

1576

—

145

C₄

#20263282

yes

—

146

C₄

#20263307

yes

—

147

C₂

—

148

C₂

#27160940

yes

characteristic

149

C₂

C₂×#20263248

yes

characteristic

150

C₂

#27161089

yes

characteristic

151

C₂

#27161393

yes

—

152

C₂

#27161410

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₈₈
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×C₂²⋊C₈×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂×D₄
prime radical	C₂⁴×C₁₉₇	D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₉₇
hypocenter	C₁₉₇	#1823
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₇₈₈

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×C₂²⋊C₈×C₁₉₇

C₂

socle

C₂³×C₁₉₇

C₂×D₄

prime radical

C₂⁴×C₁₉₇

D₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₉₇

hypocenter

C₁₉₇

#1823

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #27160842	C₂	semidirect
8. C₂×C₄⋊Dic₁₉₇	C₄	semidirect
76. C₁₉₇	#1823	semidirect

H ≅ G/N

type

3. #27160842

C₂

semidirect

8. C₂×C₄⋊Dic₁₉₇

C₄

semidirect

76. C₁₉₇

#1823

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₉₇	2
#27160939	2
#27160842	2
C₂×C₂²⋊C₈×C₁₉₇	2
#1823	197

subgroup

index

C₂²×C₄⋊Dic₁₉₇

#27160939

#27160842

C₂×C₂²⋊C₈×C₁₉₇

#1823

197

Subgroups — #38742799

Conjugacy classes of subgroups

Complements

Maximal subgroups