Group #38742922 — subgroups

Subgroups — #38742922

Order 25216.

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

normal subgroups only

This group has 324 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	#27161122	C₂	1	2	yes	maximal, complemented
2	C₂×#20263243	C₂	1	1	yes	maximal, characteristic
3	#27160982	C₂	1	1	yes	maximal, characteristic
4	#27160838	C₂	1	2	yes	maximal, complemented
5	#518×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Dic₁₉₇	—	2	2	no	—
9	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
10	#20263259	—	2	2	no	—
11	#20263257	—	2	2	no	—
12	#20263243	C₄	1	4	yes	complemented
13	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₉₇	—	2	2	no	—
15	C₄×D₄×C₁₉₇	—	2	2	no	—
16	#117×C₁₉₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₉₇	—	2	2	no	—
18	C₂²×Dic₁₉₇	—	2	1	no	—
19	C₂²×Dic₁₉₇	—	2	1	no	—
20	#15691342	—	2	2	no	—
21	#15691342	C₂×C₄	1	4	yes	complemented
22	C₂×#13096802	—	2	2	no	—
23	C₄⋊Dic₁₉₇	—	2	2	no	—
24	C₂²⋊Dic₁₉₇	—	4	4	no	—
25	C₄×Dic₁₉₇	—	2	2	no	—
26	C₂⁴×C₁₉₇	D₄	1	2	yes	—
27	C₂×D₄×C₁₉₇	—	2	2	no	—
28	C₂×D₄×C₁₉₇	—	2	2	no	—
29	C₂×D₄×C₁₉₇	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₉₇	Q₈	1	2	yes	—
31	C₂²×C₄×C₁₉₇	—	2	2	no	—
32	C₂²×C₄×C₁₉₇	—	2	2	no	—
33	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₉₇	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₉₇	D₄	1	2	yes	—
36	C₂²⋊C₄×C₁₉₇	—	2	4	no	—
37	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
38	C₄²×C₁₉₇	—	2	2	no	—
39	C₂×Dic₁₉₇	—	2	2	no	—
40	C₂×Dic₁₉₇	—	4	1	no	—
41	C₂×Dic₁₉₇	—	4	1	no	—
42	C₂×Dic₁₉₇	—	4	2	no	—
43	C₂×Dic₁₉₇	—	4	1	no	—
44	#13096802	—	2	4	no	—
45	C₂³×C₁₉₇	—	2	2	no	—
46	C₂³×C₁₉₇	—	2	2	no	—
47	C₂³×C₁₉₇	—	4	2	no	—
48	C₂³×C₁₉₇	—	4	2	no	—
49	C₂³×C₁₉₇	C₄⋊C₄	1	4	yes	—
50	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	D₄×C₁₉₇	—	2	4	no	—
52	D₄×C₁₉₇	—	2	4	no	—
53	C₂×C₄×C₁₉₇	—	2	2	no	—
54	C₂×C₄×C₁₉₇	—	2	2	no	—
55	C₂×C₄×C₁₉₇	—	8	2	no	—
56	C₂×C₄×C₁₉₇	—	8	2	no	—
57	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉₇	#41	1	2	yes	—
61	C₂²×C₁₉₇	—	2	1	no	—
62	C₂²×C₁₉₇	—	2	1	no	—
63	C₂²×C₁₉₇	—	2	4	no	—
64	C₂²×C₁₉₇	—	4	4	no	—
65	C₂²×C₁₉₇	—	4	2	no	—
66	C₂²×C₁₉₇	—	4	2	no	—
67	C₂²×C₁₉₇	—	8	2	no	—
68	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₉₇	#120	1	1	yes	characteristic
70	C₂²×C₁₉₇	#123	1	1	yes	characteristic
71	C₄×C₁₉₇	—	4	2	no	—
72	C₄×C₁₉₇	—	4	2	no	—
73	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
74	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
75	Dic₁₉₇	—	4	2	no	—
76	Dic₁₉₇	—	8	1	no	—
77	C₂×C₁₉₇	—	2	1	no	—
78	C₂×C₁₉₇	—	2	1	no	—
79	C₂×C₁₉₇	—	4	4	no	—
80	C₂×C₁₉₇	#385	1	1	yes	characteristic
81	C₂×C₁₉₇	#453	1	1	yes	characteristic
82	C₂×C₁₉₇	#455	1	1	yes	characteristic
83	C₁₉₇	#1856	1	1	yes	characteristic, complemented, Sylow
84	#1856	—	197	1	no	maximal, Sylow
85	#518	D₁₉₇	1	1	yes	characteristic
86	#428	—	197	2	no	—
87	C₂×#100	—	197	1	no	—
88	#390	—	197	1	no	—
89	#339	—	197	2	no	—
90	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
91	C₂×M₁₆	—	197	2	no	—
92	C₄×D₄	—	2	2	no	—
93	C₄×D₄	—	2	2	no	—
94	#117	C₂×D₁₉₇	1	2	yes	—
95	C₂×C₄⋊C₄	—	197	1	no	—
96	C₂×C₂²⋊C₄	—	2	2	no	—
97	C₂×C₂²⋊C₄	—	394	2	no	—
98	C₂×C₄²	—	197	1	no	—
99	#107	—	394	2	no	—
100	#106	—	394	2	no	—
101	#100	—	197	4	no	—
102	C₂⁴	C₂²⋊D₁₉₇	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	C₄×D₁₉₇	1	2	yes	—
106	C₂×D₄	Dic₃₉₄	1	2	yes	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	394	1	no	—
110	C₂²×C₄	—	394	1	no	—
111	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
112	M₁₆	—	394	2	no	—
113	M₁₆	—	197	4	no	—
114	C₂×C₈	—	394	2	no	—
115	C₄⋊C₄	—	394	2	no	—
116	C₄⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
117	C₄⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
118	C₂²⋊C₄	—	2	4	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	788	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	394	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	#15691337	1	4	yes	—
128	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	788	1	no	—
136	C₂×C₄	—	788	1	no	—
137	C₂×C₄	—	788	2	no	—
138	C₂×C₄	—	788	1	no	—
139	C₂×C₄	—	394	2	no	—
140	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
142	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
143	C₂×C₄	#15691358	1	2	yes	—
144	C₈	—	394	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	2	no	—
152	C₂²	C₂×#15691337	1	1	yes	characteristic
153	C₂²	#20263295	1	1	yes	characteristic
154	C₂²	#20263309	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	4	2	no	—
157	C₄	—	788	2	no	—
158	C₄	—	1576	1	no	—
159	C₄	C₄×C₂²⋊D₁₉₇	1	2	yes	—
160	C₄	#20263305	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	4	no	—
164	C₂	#27160952	1	1	yes	characteristic
165	C₂	#27161247	1	1	yes	characteristic
166	C₂	#27161257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161122

C₂

yes

maximal, complemented

C₂×#20263243

C₂

yes

maximal, characteristic

#27160982

C₂

yes

maximal, characteristic

#27160838

C₂

yes

maximal, complemented

#518×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₉₇

—

C₂×C₄×Dic₁₉₇

C₂²

yes

—

—

C₄

yes

complemented

C₂²×D₄×C₁₉₇

C₂²

yes

characteristic

C₄×D₄×C₁₉₇

—

C₄×D₄×C₁₉₇

—

#117×C₁₉₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₉₇

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

#15691342

—

#15691342

C₂×C₄

yes

complemented

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂⁴×C₁₉₇

D₄

yes

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

C₂×C₄

yes

—

C₂×D₄×C₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₂²⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₄⋊C₄

yes

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1856

—

197

maximal, Sylow

#518

D₁₉₇

yes

characteristic

#428

—

197

—

C₂×#100

—

197

—

#390

—

197

—

#339

—

197

—

C₂²×D₄

C₂×D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₄×D₄

—

C₄×D₄

—

#117

C₂×D₁₉₇

yes

—

C₂×C₄⋊C₄

—

197

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

394

—

C₂×C₄²

—

197

—

#107

—

394

—

100

#106

—

394

—

101

#100

—

197

—

102

C₂⁴

C₂²⋊D₁₉₇

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₄×D₁₉₇

yes

—

106

C₂×D₄

Dic₃₉₄

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

394

—

110

C₂²×C₄

—

394

—

111

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

112

M₁₆

—

394

—

113

M₁₆

—

197

—

114

C₂×C₈

—

394

—

115

C₄⋊C₄

—

394

—

116

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

117

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

788

—

121

C₄²

—

122

C₄²

—

394

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

#15691337

yes

—

128

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

788

—

136

C₂×C₄

—

788

—

137

C₂×C₄

—

788

—

138

C₂×C₄

—

788

—

139

C₂×C₄

—

394

—

140

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

142

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

143

C₂×C₄

#15691358

yes

—

144

C₈

—

394

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×#15691337

yes

characteristic

153

C₂²

#20263295

yes

characteristic

154

C₂²

#20263309

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

788

—

158

C₄

—

1576

—

159

C₄

C₄×C₂²⋊D₁₉₇

yes

—

160

C₄

#20263305

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#27160952

yes

characteristic

165

C₂

#27161247

yes

characteristic

166

C₂

#27161257

yes

characteristic

	subgroup	G/H
center	C₂²	#20263295
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#518×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
prime radical	C₂²×D₄×C₁₉₇	C₂²
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#518	D₁₉₇
hypocenter	C₁₉₇	#1856
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #518

subgroup

G/H

center

C₂²

#20263295

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#518×C₁₉₇

C₂

socle

C₂²×C₁₉₇

#120

prime radical

C₂²×D₄×C₁₉₇

C₂²

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #518

N	H ≅ G/N	type
1. #27161122	C₂	semidirect
4. #27160838	C₂	semidirect
12. #20263243	C₄	semidirect
21. #15691342	C₂×C₄	semidirect
83. C₁₉₇	#1856	semidirect

H ≅ G/N

type

1. #27161122

C₂

semidirect

4. #27160838

C₂

semidirect

12. #20263243

C₄

semidirect

21. #15691342

C₂×C₄

semidirect

83. C₁₉₇

#1856

semidirect

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

subgroup	index
#27161122	2
C₂×#20263243	2
#27160982	2
#27160838	2
#518×C₁₉₇	2
#1856	197

subgroup

index

#518×C₁₉₇

197

Subgroups — #38742922

Conjugacy classes of subgroups

Complements

Maximal subgroups