Group #38742507 — subgroups

Subgroups — #38742507

Order 25216.

296 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20263291	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263244	C₂	1	2	yes	maximal
3	M₁₆×Dic₁₉₇	C₂	1	2	yes	maximal
4	#27160848	C₂	1	1	yes	maximal, characteristic
5	#343×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	C₂×#15691342	C₂²	1	2	yes	—
8	#20263291	—	2	2	no	—
9	#20263291	—	2	2	no	—
10	C₂×#15691337	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
13	#20263244	C₄	1	8	yes	complemented
14	#20263236	—	2	2	no	—
15	C₈×Dic₁₉₇	—	2	2	no	—
16	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₉₄	—	2	2	no	—
19	C₂×Dic₃₉₄	—	2	1	no	—
20	C₂×Dic₃₉₄	—	2	1	no	—
21	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	—
22	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
25	#15691342	—	2	2	no	—
26	#15691342	C₂×C₄	1	4	yes	—
27	C₂×#13096802	—	2	2	no	—
28	#15691337	—	2	4	no	—
29	C₄×Dic₁₉₇	—	2	2	no	—
30	C₄×Dic₁₉₇	—	2	2	no	—
31	C₄×Dic₁₉₇	D₄	1	2	yes	—
32	C₄×Dic₁₉₇	D₄	1	2	yes	—
33	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₉₇	—	2	2	no	—
36	M₁₆×C₁₉₇	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₁₉₇	—	2	2	no	—
38	C₄⋊C₄×C₁₉₇	—	2	2	no	—
39	Dic₃₉₄	—	2	4	no	—
40	Dic₃₉₄	—	2	4	no	—
41	C₂×Dic₁₉₇	—	2	2	no	—
42	C₂×Dic₁₉₇	—	2	2	no	—
43	C₂×Dic₁₉₇	—	2	2	no	—
44	C₂×Dic₁₉₇	—	4	1	no	—
45	C₂×Dic₁₉₇	C₄²	1	4	yes	—
46	C₂×Dic₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Dic₁₉₇	C₂²⋊C₄	1	2	yes	—
48	C₂×Dic₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
49	#13096802	—	2	4	no	—
50	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	—	2	2	no	—
52	C₂×C₄×C₁₉₇	—	4	2	no	—
53	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
57	C₈×C₁₉₇	—	2	4	no	—
58	C₂²×C₁₉₇	—	2	1	no	—
59	C₂²×C₁₉₇	—	2	1	no	—
60	C₂²×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₉₇	#117	1	1	yes	characteristic
63	C₄×C₁₉₇	—	4	2	no	—
64	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
66	Dic₁₉₇	—	2	4	no	—
67	Dic₁₉₇	—	2	1	no	—
68	Dic₁₉₇	—	2	2	no	—
69	Dic₁₉₇	#101	1	2	yes	complemented
70	C₂×C₁₉₇	—	2	1	no	—
71	C₂×C₁₉₇	—	2	1	no	—
72	C₂×C₁₉₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₁₉₇	C₂×#101	1	1	yes	characteristic
74	C₂×C₁₉₇	#413	1	1	yes	characteristic
75	C₁₉₇	C₄×#101	1	1	yes	characteristic, complemented, Sylow
76	C₄×#101	—	197	1	no	maximal, Sylow
77	C₂×C₄×Q₈	—	197	1	no	—
78	C₂×#101	—	197	2	no	—
79	C₄×M₁₆	—	197	2	no	—
80	#343	—	197	1	no	—
81	#343	D₁₉₇	1	1	yes	characteristic
82	C₂²×Q₈	—	197	1	no	—
83	C₂×M₁₆	—	197	2	no	—
84	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
85	C₄×Q₈	—	394	2	no	—
86	C₄×Q₈	—	394	2	no	—
87	C₂×C₄⋊C₄	—	197	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
89	C₂×C₄²	—	197	2	no	—
90	C₂×C₄²	—	197	1	no	—
91	#101	—	197	8	no	—
92	#97	—	394	2	no	—
93	C₄×C₈	—	394	2	no	—
94	C₂×Q₈	—	394	2	no	—
95	C₂×Q₈	—	394	1	no	—
96	C₂×Q₈	—	394	1	no	—
97	C₂×Q₈	—	197	4	no	—
98	C₂²×C₄	—	197	1	no	—
99	C₂²×C₄	—	197	1	no	—
100	C₂²×C₄	—	197	2	no	—
101	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	394	2	no	—
105	M₁₆	—	197	4	no	—
106	M₁₆	C₄×D₁₉₇	1	4	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	394	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	394	4	no	—
111	C₄²	—	394	2	no	—
112	C₄²	—	394	2	no	—
113	C₄²	—	197	2	no	—
114	C₄²	—	197	2	no	—
115	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
116	Q₈	—	394	4	no	—
117	Q₈	—	394	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	788	1	no	—
121	C₂×C₄	—	394	2	no	—
122	C₂×C₄	—	394	2	no	—
123	C₂×C₄	—	394	2	no	—
124	C₂×C₄	—	197	4	no	—
125	C₂×C₄	—	197	1	no	—
126	C₂×C₄	—	197	2	no	—
127	C₂×C₄	—	197	1	no	—
128	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
129	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
130	C₂×C₄	#15691357	1	1	yes	characteristic
131	C₂×C₄	#15691359	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	—	394	4	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₄²×D₁₉₇	1	1	yes	characteristic
137	C₂²	C₂²⋊C₄×D₁₉₇	1	1	yes	characteristic
138	C₂²	#20263284	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	394	4	no	—
141	C₄	—	394	1	no	—
142	C₄	—	394	2	no	—
143	C₄	—	197	2	no	—
144	C₄	#20263286	1	2	yes	—
145	C₄	#20263288	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#27160926	1	1	yes	characteristic
149	C₂	#101×D₁₉₇	1	1	yes	characteristic
150	C₂	#27161060	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263291

C₂

yes

maximal, characteristic

C₂×#20263244

C₂

yes

maximal

M₁₆×Dic₁₉₇

C₂

yes

maximal

#27160848

C₂

yes

maximal, characteristic

#343×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₄

C₂²

yes

characteristic

C₂×#15691342

C₂²

yes

—

#20263291

—

#20263291

—

C₂×#15691337

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263244

C₄

yes

complemented

#20263236

—

C₈×Dic₁₉₇

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

C₂×C₄

yes

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

#15691342

—

#15691342

C₂×C₄

yes

—

C₂×#13096802

—

#15691337

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

D₄

yes

—

C₄×Dic₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

C₂×C₄

yes

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

Dic₃₉₄

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄²

yes

—

C₂×Dic₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₉₇

C₂²⋊C₄

yes

—

C₂×Dic₁₉₇

C₂²⋊C₄

yes

characteristic

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

yes

C₂×C₄×C₁₉₇

yes

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄²

yes

characteristic

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

C₄×D₄

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

#101

yes

complemented

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₄×C₂²⋊C₄

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

C₄×#101

—

197

maximal, Sylow

C₂×C₄×Q₈

—

197

—

C₂×#101

—

197

—

C₄×M₁₆

—

197

—

#343

—

197

—

#343

D₁₉₇

yes

characteristic

C₂²×Q₈

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₄×Q₈

—

394

—

C₄×Q₈

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

characteristic

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

#101

—

197

—

#97

—

394

—

C₄×C₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

100

C₂²×C₄

—

197

—

101

C₂²×C₄

C₄×D₁₉₇

yes

—

102

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

103

M₁₆

—

104

M₁₆

—

394

—

105

M₁₆

—

197

—

106

M₁₆

C₄×D₁₉₇

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

394

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

394

—

111

C₄²

—

394

—

112

C₄²

—

394

—

113

C₄²

—

197

—

114

C₄²

—

197

—

115

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

116

Q₈

—

394

—

117

Q₈

—

394

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

788

—

121

C₂×C₄

—

394

—

122

C₂×C₄

—

394

—

123

C₂×C₄

—

394

—

124

C₂×C₄

—

197

—

125

C₂×C₄

—

197

—

126

C₂×C₄

—

197

—

127

C₂×C₄

—

197

—

128

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

129

C₂×C₄

D₄×D₁₉₇

yes

—

130

C₂×C₄

#15691357

yes

characteristic

131

C₂×C₄

#15691359

yes

characteristic

132

C₈

—

133

C₈

—

394

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₄²×D₁₉₇

yes

characteristic

137

C₂²

C₂²⋊C₄×D₁₉₇

yes

characteristic

138

C₂²

#20263284

yes

characteristic

139

C₄

—

140

C₄

—

394

—

141

C₄

—

394

—

142

C₄

—

394

—

143

C₄

—

197

—

144

C₄

#20263286

yes

—

145

C₄

#20263288

yes

—

146

C₂

—

147

C₂

—

148

C₂

#27160926

yes

characteristic

149

C₂

#101×D₁₉₇

yes

characteristic

150

C₂

#27161060

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₇
commutator	C₂²×C₁₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#343×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#343	D₁₉₇
hypocenter	C₁₉₇	C₄×#101
LCS	G ▷ C₂²×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₇

commutator

C₂²×C₁₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#343×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂²×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
13. #20263244	C₄	semidirect
69. Dic₁₉₇	#101	semidirect
75. C₁₉₇	C₄×#101	semidirect

H ≅ G/N

type

13. #20263244

C₄

semidirect

69. Dic₁₉₇

#101

semidirect

75. C₁₉₇

C₄×#101

semidirect

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

subgroup	index
C₂×#20263291	2
C₂×#20263244	2
M₁₆×Dic₁₉₇	2
#27160848	2
#343×C₁₉₇	2
C₄×#101	197

subgroup

index

M₁₆×Dic₁₉₇

#343×C₁₉₇

197

Subgroups — #38742507

Conjugacy classes of subgroups

Complements

Maximal subgroups