Group #27173717 — subgroups

Subgroups — #27173717

Order 12928.

784 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 229 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	C₂³×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692211	C₂	1	2	yes	maximal
3	#20265081	C₂	1	8	yes	maximal, complemented
4	#20265040	C₂	1	1	yes	maximal, characteristic
5	#20265038	C₂	1	2	yes	maximal
6	#381×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₀₂	—	2	8	no	—
8	C₂²×Dic₂₀₂	C₄	1	8	yes	complemented
9	C₂²×Dic₂₀₂	C₂²	1	4	yes	—
10	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	#15692211	—	2	8	no	—
13	C₂×#13108053	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
16	#15692162	C₂²	1	8	yes	—
17	#15692162	C₂²	1	4	yes	—
18	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
20	#95×C₁₀₁	C₂²	1	4	yes	—
21	C₂×Dic₂₀₂	—	2	32	no	—
22	C₂×Dic₂₀₂	—	2	8	no	—
23	C₂×Dic₂₀₂	—	2	8	no	—
24	C₂×Dic₂₀₂	C₂×C₄	1	16	yes	—
25	C₂²×Dic₁₀₁	—	2	4	no	—
26	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
27	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
28	C₂²×Dic₁₀₁	D₄	1	8	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
30	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
31	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
32	#13108053	—	2	8	no	—
33	C₂²⋊Dic₁₀₁	—	2	4	no	—
34	C₄×Dic₁₀₁	—	2	8	no	—
35	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₁	—	2	4	no	—
37	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
40	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
41	C₂³×C₁₀₁	—	2	4	no	—
42	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
43	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
44	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	—	2	8	no	—
46	C₂×C₄×C₁₀₁	—	2	4	no	—
47	C₂×C₄×C₁₀₁	—	2	4	no	—
48	C₂×C₄×C₁₀₁	—	2	8	no	—
49	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
50	C₂×Dic₁₀₁	—	2	8	no	—
51	C₂×Dic₁₀₁	—	2	4	no	—
52	C₂×Dic₁₀₁	—	2	4	no	—
53	C₂×Dic₁₀₁	—	2	8	no	—
54	C₂×Dic₁₀₁	C₂²⋊C₄	1	16	yes	—
55	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
56	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
57	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
58	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
59	Dic₂₀₂	—	2	32	no	—
60	C₂²×C₁₀₁	—	2	4	no	—
61	C₂²×C₁₀₁	—	2	8	no	—
62	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
63	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
64	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
65	C₂²×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
66	C₄×C₁₀₁	—	2	4	no	—
67	C₄×C₁₀₁	—	4	4	no	—
68	Dic₁₀₁	—	2	4	no	—
69	Dic₁₀₁	—	4	4	no	—
70	Dic₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
71	C₂×C₁₀₁	—	2	4	no	—
72	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₁₀₁	#519	1	2	yes	—
74	C₂×C₁₀₁	#536	1	4	yes	—
75	#2310	—	101	1	no	maximal, Sylow
76	C₁₀₁	#2310	1	1	yes	characteristic, complemented, Sylow
77	C₂³×Q₈	—	101	1	no	—
78	C₂×#117	—	101	2	no	—
79	#391	—	101	8	no	—
80	#381	—	101	1	no	—
81	#381	—	101	2	no	—
82	#381	D₁₀₁	1	1	yes	characteristic
83	C₂²×Q₈	—	101	8	no	—
84	C₂²×Q₈	—	101	4	no	—
85	C₂²×Q₈	—	202	8	no	—
86	C₂³×C₄	—	101	1	no	—
87	C₂³×C₄	—	101	1	no	—
88	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
89	#117	—	202	8	no	—
90	C₂×C₄⋊C₄	—	101	4	no	—
91	C₂×C₂²⋊C₄	—	101	2	no	—
92	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
93	C₂×C₄²	—	101	4	no	—
94	#95	—	101	8	no	—
95	#95	—	101	4	no	—
96	#95	C₂×D₁₀₁	1	4	yes	—
97	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₂×Q₈	—	101	16	no	—
99	C₂×Q₈	—	202	8	no	—
100	C₂×Q₈	—	202	8	no	—
101	C₂×Q₈	—	202	32	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	101	4	no	—
104	C₂²×C₄	—	101	8	no	—
105	C₂²×C₄	—	101	2	no	—
106	C₂²×C₄	—	101	4	no	—
107	C₂²×C₄	—	101	2	no	—
108	C₂²×C₄	—	101	4	no	—
109	C₂²×C₄	—	202	4	no	—
110	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
111	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
112	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
113	C₄⋊C₄	—	202	8	no	—
114	C₂²⋊C₄	—	2	4	no	—
115	C₂²⋊C₄	—	202	4	no	—
116	C₄²	—	202	8	no	—
117	C₂³	—	2	4	no	—
118	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
119	C₂³	D₄×D₁₀₁	1	4	yes	—
120	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
121	Q₈	—	202	32	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	8	no	—
126	C₂×C₄	—	101	4	no	—
127	C₂×C₄	—	101	4	no	—
128	C₂×C₄	—	101	4	no	—
129	C₂×C₄	—	101	16	no	—
130	C₂×C₄	—	101	4	no	—
131	C₂×C₄	—	202	4	no	—
132	C₂×C₄	—	202	8	no	—
133	C₂×C₄	—	202	4	no	—
134	C₂×C₄	—	202	8	no	—
135	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	4	no	—
138	C₂²	C₂²⋊C₄×D₁₀₁	1	4	yes	—
139	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
140	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
141	C₂²	#15692302	1	4	yes	—
142	C₄	—	2	4	no	—
143	C₄	—	4	4	no	—
144	C₄	—	101	8	no	—
145	C₄	—	404	4	no	—
146	C₄	—	202	4	no	—
147	C₂	—	2	4	no	—
148	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
149	C₂	#20265654	1	2	yes	—
150	C₂	#20265749	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₂₀₂

C₂

yes

maximal, characteristic

C₂×#15692211

C₂

yes

maximal

#20265081

C₂

yes

maximal, complemented

#20265040

C₂

yes

maximal, characteristic

#20265038

C₂

yes

maximal

#381×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

—

C₂²×Dic₂₀₂

C₄

yes

complemented

C₂²×Dic₂₀₂

C₂²

yes

—

C₂³×Dic₁₀₁

C₂²

yes

characteristic

C₂³×Dic₁₀₁

C₂²

yes

characteristic

#15692211

—

C₂×#13108053

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

#15692162

C₂²

yes

—

#15692162

C₂²

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₁

C₂²

yes

—

#95×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

D₄

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

—

Dic₂₀₂

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₂²×C₁₀₁

C₂²×D₄

yes

—

C₂²×C₁₀₁

2₋¹⁺⁴

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₁

#519

yes

—

C₂×C₁₀₁

#536

yes

—

#2310

—

101

maximal, Sylow

C₁₀₁

#2310

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

101

—

C₂×#117

—

101

—

#391

—

101

—

#381

—

101

—

#381

—

101

—

#381

D₁₀₁

yes

characteristic

C₂²×Q₈

—

101

—

C₂²×Q₈

—

101

—

C₂²×Q₈

—

202

—

C₂³×C₄

—

101

—

C₂³×C₄

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

—

101

—

#95

—

101

—

#95

—

101

—

#95

C₂×D₁₀₁

yes

—

C₂⁴

C₂²×D₁₀₁

yes

characteristic

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

100

C₂×Q₈

—

202

—

101

C₂×Q₈

—

202

—

102

C₂²×C₄

—

103

C₂²×C₄

—

101

—

104

C₂²×C₄

—

101

—

105

C₂²×C₄

—

101

—

106

C₂²×C₄

—

101

—

107

C₂²×C₄

—

101

—

108

C₂²×C₄

—

101

—

109

C₂²×C₄

—

202

—

110

C₂²×C₄

C₄×D₁₀₁

yes

—

111

C₂²×C₄

C₂²×D₁₀₁

yes

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

—

113

C₄⋊C₄

—

202

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

202

—

116

C₄²

—

202

—

117

C₂³

—

118

C₂³

C₂×C₄×D₁₀₁

yes

—

119

C₂³

D₄×D₁₀₁

yes

—

120

C₂³

C₂³×D₁₀₁

yes

characteristic

121

Q₈

—

202

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

101

—

127

C₂×C₄

—

101

—

128

C₂×C₄

—

101

—

129

C₂×C₄

—

101

—

130

C₂×C₄

—

101

—

131

C₂×C₄

—

202

—

132

C₂×C₄

—

202

—

133

C₂×C₄

—

202

—

134

C₂×C₄

—

202

—

135

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²⋊C₄×D₁₀₁

yes

—

139

C₂²

C₂²×C₄×D₁₀₁

yes

characteristic

140

C₂²

C₂×D₄×D₁₀₁

yes

—

141

C₂²

#15692302

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

101

—

145

C₄

—

404

—

146

C₄

—

202

—

147

C₂

—

148

C₂

C₂×C₂²⋊C₄×D₁₀₁

yes

characteristic

149

C₂

#20265654

yes

—

150

C₂

#20265749

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#381×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	#381	D₁₀₁
hypocenter	C₁₀₁	#2310
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #381

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator

C₂²×C₁₀₁

C₂³×C₄

Fitting

#381×C₁₀₁

C₂

socle, cosocle

C₂³×C₁₀₁

C₂⁴

prime radical

C₂⁴×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ #381

N	H ≅ G/N	type
3. #20265081	C₂	semidirect
8. C₂²×Dic₂₀₂	C₄	semidirect
76. C₁₀₁	#2310	semidirect

H ≅ G/N

type

3. #20265081

C₂

semidirect

8. C₂²×Dic₂₀₂

C₄

semidirect

76. C₁₀₁

#2310

semidirect

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

subgroup	index
C₂³×Dic₂₀₂	2
C₂×#15692211	2
#20265081	2
#20265040	2
#20265038	2
#381×C₁₀₁	2
#2310	101

subgroup

index

C₂³×Dic₂₀₂

#381×C₁₀₁

101

Subgroups — #27173717

Conjugacy classes of subgroups

Complements

Maximal subgroups