Group #38743561 — subgroups

Subgroups — #38743561

Order 25216.

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

normal subgroups only

This group has 312 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	C₂×#20263314	C₂	1	1	yes	maximal, characteristic
2	#27161103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20263244	C₂	1	2	yes	maximal, complemented
4	C₂×#20263242	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
8	C₂×#15691342	C₂²	1	2	yes	—
9	#20263314	—	2	1	no	—
10	#20263314	—	2	1	no	—
11	#20263314	—	2	2	no	—
12	C₂×#15691335	C₄	1	4	yes	—
13	C₄⋊#13096802	—	2	2	no	—
14	#20263244	—	2	4	no	—
15	#20263242	—	2	4	no	—
16	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
18	#97×C₁₉₇	—	2	1	no	—
19	C₄×C₈×C₁₉₇	—	2	1	no	—
20	C₂³×D₁₉₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₉₄	—	2	2	no	—
22	C₂×Dic₃₉₄	—	2	1	no	—
23	C₂×Dic₃₉₄	—	2	1	no	—
24	C₂×Dic₃₉₄	—	2	2	no	—
25	C₂×D₇₈₈	—	2	2	no	—
26	C₂×D₇₈₈	—	2	1	no	—
27	C₂×D₇₈₈	—	2	1	no	—
28	C₂×D₇₈₈	—	2	2	no	—
29	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
30	#15691342	—	2	2	no	—
31	#15691342	—	2	2	no	—
32	C₂×#13096802	—	2	2	no	—
33	#15691335	—	2	4	no	—
34	#15691335	—	2	4	no	—
35	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₉₇	—	2	2	no	—
38	M₁₆×C₁₉₇	D₄	1	4	yes	—
39	C₂×C₈×C₁₉₇	—	2	2	no	—
40	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
41	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
42	C₄²×C₁₉₇	D₄	1	2	yes	—
43	C₂²×D₁₉₇	—	2	2	no	—
44	C₂²×D₁₉₇	—	2	4	no	—
45	C₂²×D₁₉₇	—	4	4	no	—
46	Dic₃₉₄	—	4	2	no	—
47	Dic₃₉₄	—	4	1	no	—
48	Dic₃₉₄	—	4	1	no	—
49	D₇₈₈	—	4	2	no	—
50	D₇₈₈	—	4	1	no	—
51	D₇₈₈	—	4	1	no	—
52	C₂×Dic₁₉₇	—	2	2	no	—
53	C₂×Dic₁₉₇	—	2	4	no	—
54	#13096802	—	4	2	no	—
55	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₇	—	4	2	no	—
57	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	#41	1	2	yes	—
63	C₈×C₁₉₇	—	2	4	no	—
64	C₂²×C₁₉₇	—	2	2	no	—
65	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
67	C₂×D₁₉₇	—	4	2	no	—
68	C₂×D₁₉₇	—	4	4	no	—
69	C₂×D₁₉₇	—	8	4	no	—
70	C₄×C₁₉₇	—	2	2	no	—
71	C₄×C₁₉₇	—	2	2	no	—
72	C₄×C₁₉₇	#121	1	2	yes	—
73	C₄×C₁₉₇	#124	1	1	yes	characteristic
74	C₄×C₁₉₇	#127	1	1	yes	characteristic
75	Dic₁₉₇	—	4	2	no	—
76	C₂×C₁₉₇	—	2	2	no	—
77	C₂×C₁₉₇	#390	1	1	yes	characteristic
78	C₂×C₁₉₇	#413	1	2	yes	—
79	D₁₉₇	—	8	2	no	—
80	C₁₉₇	#1967	1	1	yes	characteristic, complemented, Sylow
81	#1967	—	197	1	no	maximal, Sylow
82	C₂×#124	—	197	1	no	—
83	#423	—	197	1	no	—
84	C₂×#101	—	197	2	no	—
85	C₂×#100	—	197	2	no	—
86	C₄×M₁₆	D₁₉₇	1	1	yes	characteristic, complemented
87	C₂²×Q₈	—	197	1	no	—
88	C₂²×D₄	—	197	1	no	—
89	C₂×M₁₆	—	197	2	no	—
90	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
91	#124	—	394	1	no	—
92	#124	—	394	1	no	—
93	#124	—	394	2	no	—
94	C₂×C₂²⋊C₄	—	197	4	no	—
95	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
96	C₄⋊C₈	—	394	2	no	—
97	#101	—	394	4	no	—
98	#100	—	394	4	no	—
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₂⁴	—	197	2	no	—
102	C₂×Q₈	—	394	1	no	—
103	C₂×Q₈	—	394	2	no	—
104	C₂×Q₈	—	394	1	no	—
105	C₂×Q₈	—	394	2	no	—
106	C₂×D₄	—	394	1	no	—
107	C₂×D₄	—	394	2	no	—
108	C₂×D₄	—	394	1	no	—
109	C₂×D₄	—	394	2	no	—
110	C₂²×C₄	—	197	2	no	—
111	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	394	2	no	—
115	M₁₆	—	394	2	no	—
116	M₁₆	D₇₈₈	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	394	2	no	—
119	C₂²⋊C₄	—	394	4	no	—
120	C₂²⋊C₄	—	394	4	no	—
121	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
123	C₄²	D₇₈₈	1	2	yes	—
124	C₂³	—	788	4	no	—
125	C₂³	—	394	2	no	—
126	C₂³	—	394	4	no	—
127	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
128	Q₈	—	788	2	no	—
129	Q₈	—	788	1	no	—
130	Q₈	—	788	1	no	—
131	D₄	—	788	2	no	—
132	D₄	—	788	1	no	—
133	D₄	—	788	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	394	2	no	—
136	C₂×C₄	—	394	4	no	—
137	C₂×C₄	#15691335	1	2	yes	—
138	C₂×C₄	#15691335	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
140	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
141	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
142	C₂×C₄	#15691358	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	788	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	788	2	no	—
147	C₂²	—	788	4	no	—
148	C₂²	—	1576	4	no	—
149	C₂²	C₂×#15691335	1	1	yes	characteristic
150	C₂²	C₄×D₇₈₈	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	788	2	no	—
154	C₄	#20263300	1	2	yes	—
155	C₄	#20263314	1	1	yes	characteristic
156	C₄	#20263323	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	1576	2	no	—
159	C₂	#27160981	1	1	yes	characteristic
160	C₂	#27161061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263314

C₂

yes

maximal, characteristic

#27161103

C₂

yes

maximal, characteristic, complemented

C₂×#20263244

C₂

yes

maximal, complemented

C₂×#20263242

C₂

yes

maximal

C₄×M₁₆×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₃₉₄

C₂²

yes

characteristic

C₂²×D₇₈₈

C₂²

yes

characteristic

C₂×#15691342

C₂²

yes

—

—

—

—

C₄

yes

—

C₄⋊#13096802

—

#20263244

—

#20263242

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

#97×C₁₉₇

—

C₄×C₈×C₁₉₇

—

C₂³×D₁₉₇

C₂×C₄

yes

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

D₄

yes

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

Dic₃₉₄

—

Dic₃₉₄

—

Dic₃₉₄

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄×D₄

yes

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#121

yes

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#390

yes

characteristic

C₂×C₁₉₇

#413

yes

—

D₁₉₇

—

C₁₉₇

#1967

yes

characteristic, complemented, Sylow

#1967

—

197

maximal, Sylow

C₂×#124

—

197

—

#423

—

197

—

C₂×#101

—

197

—

C₂×#100

—

197

—

C₄×M₁₆

D₁₉₇

yes

characteristic, complemented

C₂²×Q₈

—

197

—

C₂²×D₄

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

#124

—

394

—

#124

—

394

—

#124

—

394

—

C₂×C₂²⋊C₄

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

#101

—

394

—

#100

—

394

—

#97

—

100

C₄×C₈

—

101

C₂⁴

—

197

—

102

C₂×Q₈

—

394

—

103

C₂×Q₈

—

394

—

104

C₂×Q₈

—

394

—

105

C₂×Q₈

—

394

—

106

C₂×D₄

—

394

—

107

C₂×D₄

—

394

—

108

C₂×D₄

—

394

—

109

C₂×D₄

—

394

—

110

C₂²×C₄

—

197

—

111

C₂²×C₄

C₄×D₁₉₇

yes

—

112

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

394

—

115

M₁₆

—

394

—

116

M₁₆

D₇₈₈

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

394

—

119

C₂²⋊C₄

—

394

—

120

C₂²⋊C₄

—

394

—

121

C₄²

C₂²⋊D₁₉₇

yes

characteristic

122

C₄²

C₂²⋊D₁₉₇

yes

characteristic

123

C₄²

D₇₈₈

yes

—

124

C₂³

—

788

—

125

C₂³

—

394

—

126

C₂³

—

394

—

127

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

128

Q₈

—

788

—

129

Q₈

—

788

—

130

Q₈

—

788

—

131

D₄

—

788

—

132

D₄

—

788

—

133

D₄

—

788

—

134

C₂×C₄

—

135

C₂×C₄

—

394

—

136

C₂×C₄

—

394

—

137

C₂×C₄

#15691335

yes

—

138

C₂×C₄

#15691335

yes

—

139

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

140

C₂×C₄

C₂×D₇₈₈

yes

—

141

C₂×C₄

C₂×D₇₈₈

yes

characteristic

142

C₂×C₄

#15691358

yes

—

143

C₈

—

144

C₈

—

788

—

145

C₂²

—

146

C₂²

—

788

—

147

C₂²

—

788

—

148

C₂²

—

1576

—

149

C₂²

C₂×#15691335

yes

characteristic

150

C₂²

C₄×D₇₈₈

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

788

—

154

C₄

#20263300

yes

—

155

C₄

#20263314

yes

characteristic

156

C₄

#20263323

yes

characteristic

157

C₂

—

158

C₂

—

1576

—

159

C₂

#27160981

yes

characteristic

160

C₂

#27161061

yes

—

	subgroup	G/H
center	C₂²	C₂×#15691335
commutator	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₄×M₁₆×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₇₈₈	C₂²
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₄×M₁₆	D₁₉₇
hypocenter	C₁₉₇	#1967
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#15691335

commutator

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₄×M₁₆×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₇₈₈

C₂²

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₄×M₁₆

D₁₉₇

hypocenter

C₁₉₇

#1967

LCS

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #27161103	C₂	semidirect
3. C₂×#20263244	C₂	semidirect
5. C₄×M₁₆×C₁₉₇	C₂	semidirect
80. C₁₉₇	#1967	semidirect
86. C₄×M₁₆	D₁₉₇	semidirect

H ≅ G/N

type

2. #27161103

C₂

semidirect

3. C₂×#20263244

C₂

semidirect

5. C₄×M₁₆×C₁₉₇

C₂

semidirect

80. C₁₉₇

#1967

semidirect

86. C₄×M₁₆

D₁₉₇

semidirect

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

subgroup	index
C₂×#20263314	2
#27161103	2
C₂×#20263244	2
C₂×#20263242	2
C₄×M₁₆×C₁₉₇	2
#1967	197

subgroup

index

C₄×M₁₆×C₁₉₇

197

Subgroups — #38743561

Conjugacy classes of subgroups

Complements

Maximal subgroups