Group #27171982 — subgroups

Subgroups — #27171982

Order 12928.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20265201	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692174	C₂	1	2	yes	maximal, complemented
3	C₂×#15692172	C₂	1	2	yes	maximal
4	C₄×#13108058	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₂×#13108058	C₂²	1	2	yes	—
8	C₄⋊#11717579	—	2	2	no	—
9	#15692174	—	2	4	no	—
10	#15692172	—	2	4	no	—
11	#15692166	—	2	1	no	—
12	C₄×#11717579	—	2	1	no	—
13	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	#124×C₁₀₁	—	2	1	no	—
16	#124×C₁₀₁	—	2	1	no	—
17	#124×C₁₀₁	—	2	2	no	—
18	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	4	yes	—
19	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
20	#13108058	—	2	2	no	—
21	#13108058	—	2	2	no	—
22	#13108058	—	2	2	no	—
23	#13108058	D₄	1	4	yes	—
24	C₂×#11717579	—	2	2	no	—
25	C₂×#11717579	—	2	2	no	—
26	C₂⁴×C₁₀₁	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₁₀₁	—	2	1	no	—
28	C₂×Q₈×C₁₀₁	—	2	1	no	—
29	C₂×Q₈×C₁₀₁	—	2	2	no	—
30	C₂×Q₈×C₁₀₁	—	2	2	no	—
31	C₂×D₄×C₁₀₁	—	2	1	no	—
32	C₂×D₄×C₁₀₁	—	2	1	no	—
33	C₂×D₄×C₁₀₁	—	2	2	no	—
34	C₂×D₄×C₁₀₁	—	2	2	no	—
35	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
38	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
39	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
40	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
41	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
42	C₄²×C₁₀₁	D₄	1	2	yes	—
43	C₂³×C₁₀₁	—	2	2	no	—
44	C₂³×C₁₀₁	—	2	4	no	—
45	C₂³×C₁₀₁	—	4	4	no	—
46	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₀₁	—	4	2	no	—
48	Q₈×C₁₀₁	—	4	1	no	—
49	Q₈×C₁₀₁	—	4	1	no	—
50	D₄×C₁₀₁	—	4	2	no	—
51	D₄×C₁₀₁	—	4	1	no	—
52	D₄×C₁₀₁	—	4	1	no	—
53	C₂×C₄×C₁₀₁	—	2	2	no	—
54	C₂×C₄×C₁₀₁	—	2	4	no	—
55	C₂×C₄×C₁₀₁	—	4	2	no	—
56	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₁	#41	1	2	yes	—
62	#11717579	—	2	4	no	—
63	#11717579	—	4	2	no	—
64	C₂²×C₁₀₁	—	2	2	no	—
65	C₂²×C₁₀₁	—	4	2	no	—
66	C₂²×C₁₀₁	—	4	4	no	—
67	C₂²×C₁₀₁	—	8	4	no	—
68	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
70	C₄×C₁₀₁	—	2	2	no	—
71	C₄×C₁₀₁	—	2	2	no	—
72	C₄×C₁₀₁	—	4	2	no	—
73	C₄×C₁₀₁	#121	1	2	yes	—
74	C₄×C₁₀₁	#124	1	1	yes	characteristic
75	C₄×C₁₀₁	#127	1	1	yes	characteristic
76	C₂×C₁₀₁	—	2	2	no	—
77	C₂×C₁₀₁	—	8	2	no	—
78	C₂×C₁₀₁	#390	1	1	yes	characteristic
79	C₂×C₁₀₁	#413	1	2	yes	—
80	#1967	—	101	1	no	maximal, Sylow
81	C₁₀₁	#1967	1	1	yes	characteristic, complemented, Sylow
82	C₂×#124	D₁₀₁	1	1	yes	characteristic
83	#423	—	101	1	no	—
84	C₂×#101	—	101	2	no	—
85	C₂×#100	—	101	2	no	—
86	C₄×M₁₆	—	101	1	no	—
87	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
88	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
89	C₂×M₁₆	—	101	2	no	—
90	C₂×M₁₆	—	101	2	no	—
91	#124	—	2	1	no	—
92	#124	—	2	1	no	—
93	#124	—	2	2	no	—
94	C₂×C₂²⋊C₄	Dic₁₀₁	1	4	yes	—
95	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
96	C₄⋊C₈	—	202	2	no	—
97	#101	—	202	4	no	—
98	#100	—	202	4	no	—
99	#97	—	202	1	no	—
100	C₄×C₈	—	202	1	no	—
101	C₂⁴	C₂×Dic₁₀₁	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
113	M₁₆	—	101	4	no	—
114	M₁₆	—	202	2	no	—
115	M₁₆	—	202	2	no	—
116	M₁₆	—	202	2	no	—
117	C₂×C₈	—	202	2	no	—
118	C₂×C₈	—	202	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
123	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
128	Q₈	—	4	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
138	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
139	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
142	C₂×C₄	#13108073	1	2	yes	—
143	C₈	—	404	2	no	—
144	C₈	—	202	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
150	C₂²	D₄×Dic₁₀₁	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#15692226	1	2	yes	—
155	C₄	#15692242	1	1	yes	characteristic
156	C₄	#15692251	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	8	2	no	—
159	C₂	#20265080	1	1	yes	characteristic
160	C₂	#20265161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265201

C₂

yes

maximal, characteristic, complemented

C₂×#15692174

C₂

yes

maximal, complemented

C₂×#15692172

C₂

yes

maximal

C₄×#13108058

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂×#13108058

C₂²

yes

—

—

—

—

—

—

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

#124×C₁₀₁

—

#124×C₁₀₁

—

#124×C₁₀₁

—

C₂×C₂²⋊C₄×C₁₀₁

C₄

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

—

—

D₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂⁴×C₁₀₁

C₂×C₄

yes

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

#413

yes

—

#1967

—

101

maximal, Sylow

C₁₀₁

#1967

yes

characteristic, complemented, Sylow

C₂×#124

D₁₀₁

yes

characteristic

#423

—

101

—

C₂×#101

—

101

—

C₂×#100

—

101

—

C₄×M₁₆

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

Dic₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

#101

—

202

—

#100

—

202

—

#97

—

202

—

100

C₄×C₈

—

202

—

101

C₂⁴

C₂×Dic₁₀₁

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×Dic₁₀₁

yes

—

111

C₂²×C₄

C₂×Dic₁₀₁

yes

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

113

M₁₆

—

101

—

114

M₁₆

—

202

—

115

M₁₆

—

202

—

116

M₁₆

—

202

—

117

C₂×C₈

—

202

—

118

C₂×C₈

—

202

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₁₀₁

yes

characteristic

122

C₄²

C₂²⋊D₁₀₁

yes

characteristic

123

C₄²

C₂²⋊D₁₀₁

yes

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²×Dic₁₀₁

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

138

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

139

C₂×C₄

D₄×D₁₀₁

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

142

C₂×C₄

#13108073

yes

—

143

C₈

—

404

—

144

C₈

—

202

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

150

C₂²

D₄×Dic₁₀₁

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#15692226

yes

—

155

C₄

#15692242

yes

characteristic

156

C₄

#15692251

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

#20265080

yes

characteristic

160

C₂

#20265161

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	#1967
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₁

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#124×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #20265201	C₂	semidirect
2. C₂×#15692174	C₂	semidirect
4. C₄×#13108058	C₂	semidirect
81. C₁₀₁	#1967	semidirect

H ≅ G/N

type

1. #20265201

C₂

semidirect

2. C₂×#15692174

C₂

semidirect

4. C₄×#13108058

C₂

semidirect

81. C₁₀₁

#1967

semidirect

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

subgroup	index
#20265201	2
C₂×#15692174	2
C₂×#15692172	2
C₄×#13108058	2
C₂×#124×C₁₀₁	2
#1967	101

subgroup

index

C₂×#124×C₁₀₁

101

Subgroups — #27171982

Conjugacy classes of subgroups

Complements

Maximal subgroups