Group #38743855 — subgroups

Subgroups — #38743855

Order 25216.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20263246	C₂	1	2	yes	maximal
2	#27160981	C₂	1	1	yes	maximal, characteristic
3	#27160841	C₂	1	2	yes	maximal, complemented
4	#27160834	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
7	C₂²×#13096802	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
9	C₂×#15691335	—	2	2	no	—
10	#20263246	—	2	4	no	—
11	#125×C₁₉₇	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂³×D₁₉₇	—	2	1	no	—
16	C₂×D₇₈₈	—	2	2	no	—
17	C₂×D₇₈₈	—	2	2	no	—
18	C₂×D₇₈₈	—	2	1	no	—
19	C₂×D₇₈₈	Q₈	1	2	yes	—
20	C₂²×Dic₁₉₇	—	2	1	no	—
21	C₂×#13096802	—	2	2	no	—
22	C₂×#13096802	—	2	2	no	—
23	C₂×#13096802	—	2	2	no	—
24	C₄⋊Dic₁₉₇	—	2	2	no	—
25	#15691335	—	4	4	no	—
26	C₂²×C₄×C₁₉₇	—	2	2	no	—
27	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
28	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₉₇	—	2	8	no	—
30	C₄⋊C₄×C₁₉₇	—	2	2	no	—
31	C₄⋊C₄×C₁₉₇	D₄	1	4	yes	—
32	C₄²×C₁₉₇	—	2	2	no	—
33	C₂²×D₁₉₇	—	2	2	no	—
34	C₂²×D₁₉₇	—	4	1	no	—
35	C₂²×D₁₉₇	—	4	1	no	—
36	C₂²×D₁₉₇	—	4	2	no	—
37	C₂²×D₁₉₇	—	4	2	no	—
38	D₇₈₈	—	2	4	no	—
39	D₇₈₈	—	4	2	no	—
40	C₂×Dic₁₉₇	—	4	1	no	—
41	C₂×Dic₁₉₇	—	4	2	no	—
42	#13096802	—	4	2	no	—
43	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₇	—	2	2	no	—
45	C₂×C₄×C₁₉₇	—	2	4	no	—
46	C₂×C₄×C₁₉₇	—	2	2	no	—
47	C₂×C₄×C₁₉₇	—	2	2	no	—
48	C₂×C₄×C₁₉₇	—	4	2	no	—
49	C₂×C₄×C₁₉₇	—	4	2	no	—
50	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	#41	1	2	yes	—
53	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
54	C₂²×C₁₉₇	#120	1	1	yes	characteristic
55	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₉₇	#124	1	1	yes	characteristic
57	C₂²×C₁₉₇	#135	1	2	yes	—
58	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₁₉₇	—	4	2	no	—
60	C₂×D₁₉₇	—	4	4	no	—
61	C₂×D₁₉₇	—	8	2	no	—
62	C₂×D₁₉₇	—	8	1	no	—
63	C₂×D₁₉₇	—	8	1	no	—
64	C₄×C₁₉₇	—	4	2	no	—
65	C₄×C₁₉₇	—	4	4	no	—
66	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₉₇	#124	1	2	yes	—
68	Dic₁₉₇	—	8	1	no	—
69	C₂×C₁₉₇	#393	1	1	yes	characteristic
70	C₂×C₁₉₇	#449	1	2	yes	—
71	C₂×C₁₉₇	#478	1	2	yes	—
72	C₂×C₁₉₇	#488	1	1	yes	characteristic
73	C₂×C₁₉₇	#490	1	1	yes	characteristic
74	D₁₉₇	—	8	2	no	—
75	C₁₉₇	#2025	1	1	yes	characteristic, complemented, Sylow
76	#2025	—	197	1	no	maximal, Sylow
77	C₂×#125	D₁₉₇	1	1	yes	characteristic, complemented
78	C₂×#102	—	197	2	no	—
79	#390	—	197	1	no	—
80	#340	—	197	2	no	—
81	#336	—	197	1	no	—
82	C₂²×D₄	—	197	1	no	—
83	C₂²×C₈	—	197	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	197	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
88	C₂×C₂²⋊C₄	—	394	2	no	—
89	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
90	#102	—	394	4	no	—
91	C₂⁴	—	394	1	no	—
92	C₂×D₄	—	394	2	no	—
93	C₂×D₄	—	394	1	no	—
94	C₂×D₄	—	394	2	no	—
95	C₂×D₄	—	197	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	394	1	no	—
98	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
100	C₂×C₈	—	394	2	no	—
101	C₂×C₈	—	394	2	no	—
102	C₂×C₈	—	394	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	394	2	no	—
106	C₄⋊C₄	C₂²⋊D₁₉₇	1	4	yes	—
107	C₂²⋊C₄	—	788	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	788	1	no	—
110	C₂³	—	788	1	no	—
111	C₂³	—	788	2	no	—
112	C₂³	—	788	2	no	—
113	C₂³	—	394	2	no	—
114	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
115	D₄	—	788	2	no	—
116	D₄	—	394	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	788	1	no	—
124	C₂×C₄	—	788	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
126	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
127	C₂×C₄	#15691359	1	2	yes	—
128	C₂×C₄	#15691359	1	1	yes	characteristic
129	C₈	—	788	2	no	—
130	C₂²	—	788	4	no	—
131	C₂²	—	788	2	no	—
132	C₂²	—	1576	1	no	—
133	C₂²	—	1576	1	no	—
134	C₂²	—	1576	2	no	—
135	C₂²	#20263295	1	1	yes	characteristic
136	C₂²	#20263302	1	1	yes	characteristic
137	C₂²	#20263315	1	1	yes	characteristic
138	C₂²	#20263346	1	2	yes	—
139	C₂²	#20263351	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	1576	1	no	—
143	C₄	#20263302	1	2	yes	—
144	C₄	#20263315	1	2	yes	—
145	C₂	—	1576	2	no	—
146	C₂	#27160990	1	1	yes	characteristic
147	C₂	#27161224	1	2	yes	—
148	C₂	#27161376	1	2	yes	—
149	C₂	#27161436	1	1	yes	characteristic
150	C₂	#27161445	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263246

C₂

yes

maximal

#27160981

C₂

yes

maximal, characteristic

#27160841

C₂

yes

maximal, complemented

#27160834

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₈₈

C₂²

yes

characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

characteristic

C₂×#15691335

—

#20263246

—

#125×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂³×D₁₉₇

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

C₂×D₇₈₈

Q₈

yes

—

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

#15691335

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

D₇₈₈

—

D₇₈₈

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

C₂²⋊Q₈

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

—

C₂²×C₁₉₇

AGL(1,ℤ/8)

yes

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

#124

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#393

yes

characteristic

C₂×C₁₉₇

#449

yes

—

C₂×C₁₉₇

#478

yes

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

D₁₉₇

—

C₁₉₇

#2025

yes

characteristic, complemented, Sylow

#2025

—

197

maximal, Sylow

C₂×#125

D₁₉₇

yes

characteristic, complemented

C₂×#102

—

197

—

#390

—

197

—

#340

—

197

—

#336

—

197

—

C₂²×D₄

—

197

—

C₂²×C₈

—

197

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₂²⋊C₄

—

394

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#102

—

394

—

C₂⁴

—

394

—

C₂×D₄

—

394

—

C₂×D₄

—

394

—

C₂×D₄

—

394

—

C₂×D₄

—

197

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

100

C₂×C₈

—

394

—

101

C₂×C₈

—

394

—

102

C₂×C₈

—

394

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

394

—

106

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

107

C₂²⋊C₄

—

788

—

108

C₄²

—

109

C₂³

—

788

—

110

C₂³

—

788

—

111

C₂³

—

788

—

112

C₂³

—

788

—

113

C₂³

—

394

—

114

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

115

D₄

—

788

—

116

D₄

—

394

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

788

—

124

C₂×C₄

—

788

—

125

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

126

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

127

C₂×C₄

#15691359

yes

—

128

C₂×C₄

#15691359

yes

characteristic

129

C₈

—

788

—

130

C₂²

—

788

—

131

C₂²

—

788

—

132

C₂²

—

1576

—

133

C₂²

—

1576

—

134

C₂²

—

1576

—

135

C₂²

#20263295

yes

characteristic

136

C₂²

#20263302

yes

characteristic

137

C₂²

#20263315

yes

characteristic

138

C₂²

#20263346

yes

—

139

C₂²

#20263351

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

1576

—

143

C₄

#20263302

yes

—

144

C₄

#20263315

yes

—

145

C₂

—

1576

—

146

C₂

#27160990

yes

characteristic

147

C₂

#27161224

yes

—

148

C₂

#27161376

yes

—

149

C₂

#27161436

yes

characteristic

150

C₂

#27161445

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#125×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂×D₄
prime radical	C₂²×D₇₈₈	C₂²
hypercenter	C₂×#125	D₁₉₇
hypocenter	C₁₉₇	#2025
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×#125×C₁₉₇

C₂

socle

C₂³×C₁₉₇

C₂×D₄

prime radical

C₂²×D₇₈₈

C₂²

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #27160841	C₂	semidirect
4. #27160834	C₂	semidirect
5. C₂×#125×C₁₉₇	C₂	semidirect
75. C₁₉₇	#2025	semidirect
77. C₂×#125	D₁₉₇	semidirect

H ≅ G/N

type

3. #27160841

C₂

semidirect

4. #27160834

C₂

semidirect

5. C₂×#125×C₁₉₇

C₂

semidirect

75. C₁₉₇

#2025

semidirect

77. C₂×#125

D₁₉₇

semidirect

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

subgroup	index
C₂×#20263246	2
#27160981	2
#27160841	2
#27160834	2
C₂×#125×C₁₉₇	2
#2025	197

subgroup

index

C₂×#125×C₁₉₇

197

Subgroups — #38743855

Conjugacy classes of subgroups

Complements

Maximal subgroups