Group #27186837 — subgroups

Subgroups — #27186837

Order 12928.

388 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 130 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×D₁₀₁⋊M₁₆	C₂	1	2	yes	maximal
2	#20266057	C₂	1	4	yes	maximal, complemented
3	C₂×#15692212	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108084	—	2	2	no	—
5	D₁₀₁⋊M₁₆	—	2	4	no	—
6	D₁₀₁⋊M₁₆	C₄	1	8	yes	complemented
7	C₂×#13108083	C₂²	1	4	yes	—
8	C₂×D₁₀₁⋊C₈	—	2	2	no	—
9	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
10	#15692212	—	2	2	no	—
11	#15692212	—	2	2	no	—
12	C₂×#13108053	C₂²	1	2	yes	—
13	C₂×#13108051	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₀₁	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₁	C₄	1	1	yes	characteristic
16	#13108084	—	2	8	no	—
17	#13108083	—	2	4	no	—
18	#13108083	C₂×C₄	1	8	yes	—
19	D₁₀₁⋊C₈	—	2	8	no	—
20	C₂×#11717581	—	2	4	no	—
21	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
24	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
25	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
26	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
27	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
28	C₂×C₄×D₁₀₁	Q₈	1	2	yes	—
29	#13108053	—	2	4	no	—
30	#13108051	—	2	4	no	—
31	C₄×Dic₁₀₁	—	2	1	no	—
32	C₄×Dic₁₀₁	—	2	1	no	—
33	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₀₁	—	2	1	no	—
36	C₄²×C₁₀₁	—	2	1	no	—
37	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×D₁₀₁	—	4	2	no	—
39	C₂²×D₁₀₁	C₄²	1	2	yes	—
40	C₂²×D₁₀₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×D₁₀₁	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	—	4	2	no	—
44	C₂×C₄×C₁₀₁	C₄²	1	2	yes	—
45	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
48	C₂×Dic₁₀₁	—	2	2	no	—
49	C₂×Dic₁₀₁	—	4	2	no	—
50	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
51	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
53	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
54	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
55	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
56	C₄×D₁₀₁	C₄⋊C₄	1	4	yes	—
57	C₄×D₁₀₁	C₄⋊C₄	1	4	yes	—
58	#11717581	—	2	8	no	—
59	C₂²×C₁₀₁	—	2	2	no	—
60	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₁₀₁	—	2	4	no	—
64	C₂×D₁₀₁	—	4	2	no	—
65	C₂×D₁₀₁	—	4	2	no	—
66	C₂×D₁₀₁	#95	1	4	yes	—
67	C₄×C₁₀₁	—	4	2	no	—
68	C₄×C₁₀₁	#95	1	2	yes	—
69	C₄×C₁₀₁	#95	1	2	yes	—
70	Dic₁₀₁	—	4	2	no	—
71	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
72	Dic₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₁₀₁	—	2	1	no	—
74	C₂×C₁₀₁	—	2	1	no	—
75	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
76	C₂×C₁₀₁	#413	1	2	yes	—
77	D₁₀₁	—	2	4	no	—
78	#1742	—	101	1	no	maximal, Sylow
79	C₁₀₁	#1742	1	1	yes	characteristic, complemented, Sylow
80	C₂²×M₁₆	—	101	2	no	—
81	C₂×#117	—	101	1	no	—
82	#343	—	101	4	no	—
83	C₂³×C₄	—	101	1	no	—
84	C₂×M₁₆	—	101	4	no	—
85	C₂×M₁₆	—	101	8	no	—
86	C₂×M₁₆	—	202	4	no	—
87	C₂×M₁₆	—	202	2	no	—
88	C₂²×C₈	—	202	2	no	—
89	#117	—	202	2	no	—
90	#117	—	202	2	no	—
91	C₂×C₄⋊C₄	—	101	2	no	—
92	C₂×C₂²⋊C₄	—	101	2	no	—
93	C₂×C₄²	—	101	1	no	—
94	C₂×C₄²	#87938	1	1	yes	characteristic
95	C₂⁴	—	101	1	no	—
96	C₂²×C₄	—	101	1	no	—
97	C₂²×C₄	—	101	4	no	—
98	C₂²×C₄	—	101	2	no	—
99	C₂²×C₄	—	101	2	no	—
100	C₂²×C₄	—	101	2	no	—
101	C₂²×C₄	—	101	2	no	—
102	C₂²×C₄	—	101	2	no	—
103	C₂²×C₄	C₂×#87938	1	2	yes	—
104	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
105	M₁₆	—	101	8	no	—
106	M₁₆	—	202	4	no	—
107	M₁₆	—	202	8	no	—
108	C₂×C₈	—	202	4	no	—
109	C₂×C₈	—	202	8	no	—
110	C₄⋊C₄	—	202	4	no	—
111	C₂²⋊C₄	—	202	4	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	202	1	no	—
115	C₄²	—	202	1	no	—
116	C₂³	—	101	2	no	—
117	C₂³	—	101	2	no	—
118	C₂³	—	101	2	no	—
119	C₂³	—	404	2	no	—
120	C₂³	C₂²×#87938	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	101	2	no	—
124	C₂×C₄	—	101	2	no	—
125	C₂×C₄	—	101	1	no	—
126	C₂×C₄	—	101	1	no	—
127	C₂×C₄	—	101	4	no	—
128	C₂×C₄	—	101	4	no	—
129	C₂×C₄	—	101	4	no	—
130	C₂×C₄	—	101	4	no	—
131	C₂×C₄	—	404	2	no	—
132	C₂×C₄	—	202	2	no	—
133	C₂×C₄	C₄×#87938	1	2	yes	—
134	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
136	C₂×C₄	C₄⋊#87938	1	2	yes	—
137	C₈	—	202	8	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	101	4	no	—
140	C₂²	—	404	2	no	—
141	C₂²	—	404	2	no	—
142	C₂²	—	202	4	no	—
143	C₂²	C₂×C₄×#87938	1	1	yes	characteristic
144	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
145	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
146	C₄	—	4	2	no	—
147	C₄	—	101	2	no	—
148	C₄	—	101	2	no	—
149	C₄	—	404	2	no	—
150	C₄	(C₂×#87938)⋊C₄	1	2	yes	—
151	C₄	(C₂×#87938)⋊C₄	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	—	202	4	no	—
155	C₂	C₂×(C₂×#87938)⋊C₄	1	1	yes	characteristic
156	C₂	#20266163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₁⋊M₁₆

C₂

yes

maximal

#20266057

C₂

yes

maximal, complemented

C₂×#15692212

C₂

yes

maximal, characteristic

C₂×#13108084

—

D₁₀₁⋊M₁₆

—

D₁₀₁⋊M₁₆

C₄

yes

complemented

C₂×#13108083

C₂²

yes

—

C₂×D₁₀₁⋊C₈

—

C₂²×C₄×D₁₀₁

C₂²

yes

—

—

C₂²

yes

—

C₂×#13108051

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₄

yes

characteristic

C₂×C₄²×C₁₀₁

C₄

yes

—

—

C₂×C₄

yes

—

D₁₀₁⋊C₈

—

C₂×#11717581

—

C₂³×D₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

Q₈

yes

—

#13108053

—

#13108051

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₄²

yes

—

C₂²×D₁₀₁

C₂²⋊C₄

yes

—

C₂²×D₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

C₂×Q₈

yes

characteristic

C₄×D₁₀₁

C₂²⋊C₄

yes

—

C₄×D₁₀₁

C₂²⋊C₄

yes

—

C₄×D₁₀₁

C₄⋊C₄

yes

—

C₄×D₁₀₁

C₄⋊C₄

yes

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄²

yes

characteristic

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

#95

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#95

yes

—

C₄×C₁₀₁

#95

yes

—

Dic₁₀₁

—

Dic₁₀₁

C₂×C₂²⋊C₄

yes

—

Dic₁₀₁

C₂×C₄⋊C₄

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#95

yes

characteristic

C₂×C₁₀₁

#413

yes

—

D₁₀₁

—

#1742

—

101

maximal, Sylow

C₁₀₁

#1742

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

101

—

C₂×#117

—

101

—

#343

—

101

—

C₂³×C₄

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

202

—

C₂×M₁₆

—

202

—

C₂²×C₈

—

202

—

#117

—

202

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

#87938

yes

characteristic

C₂⁴

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

100

C₂²×C₄

—

101

—

101

C₂²×C₄

—

101

—

102

C₂²×C₄

—

101

—

103

C₂²×C₄

C₂×#87938

yes

—

104

C₂²×C₄

C₂×#87938

yes

characteristic

105

M₁₆

—

101

—

106

M₁₆

—

202

—

107

M₁₆

—

202

—

108

C₂×C₈

—

202

—

109

C₂×C₈

—

202

—

110

C₄⋊C₄

—

202

—

111

C₂²⋊C₄

—

202

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

202

—

115

C₄²

—

202

—

116

C₂³

—

101

—

117

C₂³

—

101

—

118

C₂³

—

101

—

119

C₂³

—

404

—

120

C₂³

C₂²×#87938

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

101

—

124

C₂×C₄

—

101

—

125

C₂×C₄

—

101

—

126

C₂×C₄

—

101

—

127

C₂×C₄

—

101

—

128

C₂×C₄

—

101

—

129

C₂×C₄

—

101

—

130

C₂×C₄

—

101

—

131

C₂×C₄

—

404

—

132

C₂×C₄

—

202

—

133

C₂×C₄

C₄×#87938

yes

—

134

C₂×C₄

C₂²⋊#87938

yes

characteristic

135

C₂×C₄

C₂²⋊#87938

yes

characteristic

136

C₂×C₄

C₄⋊#87938

yes

—

137

C₈

—

202

—

138

C₂²

—

139

C₂²

—

101

—

140

C₂²

—

404

—

141

C₂²

—

404

—

142

C₂²

—

202

—

143

C₂²

C₂×C₄×#87938

yes

characteristic

144

C₂²

C₂×C₂²⋊#87938

yes

characteristic

145

C₂²

C₂×C₄⋊#87938

yes

characteristic

146

C₄

—

147

C₄

—

101

—

148

C₄

—

101

—

149

C₄

—

404

—

150

C₄

(C₂×#87938)⋊C₄

yes

—

151

C₄

(C₂×#87938)⋊C₄

yes

—

152

C₂

—

153

C₂

—

154

C₂

—

202

—

155

C₂

C₂×(C₂×#87938)⋊C₄

yes

characteristic

156

C₂

#20266163

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87938
commutator	C₂²×C₁₀₁	C₂×C₄²
Frattini	C₂³	C₂²×#87938
Fitting	C₂×C₄²×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂³×D₁₀₁	C₂×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂×C₄²	#87938
hypocenter	C₁₀₁	#1742
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₂²⋊#87938

commutator

C₂²×C₁₀₁

C₂×C₄²

Frattini

C₂³

C₂²×#87938

Fitting

C₂×C₄²×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂³×D₁₀₁

C₂×C₄

cosocle

C₂²×Dic₁₀₁

C₂³

C₂×C₄²

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
2. #20266057	C₂	semidirect
6. D₁₀₁⋊M₁₆	C₄	semidirect
79. C₁₀₁	#1742	semidirect

H ≅ G/N

type

2. #20266057

C₂

semidirect

6. D₁₀₁⋊M₁₆

C₄

semidirect

79. C₁₀₁

#1742

semidirect

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

subgroup	index
C₂×D₁₀₁⋊M₁₆	2
#20266057	2
C₂×#15692212	2
#1742	101

subgroup

index

C₂×D₁₀₁⋊M₁₆

#20266057

C₂×#15692212

#1742

101

Subgroups — #27186837

Conjugacy classes of subgroups

Complements

Maximal subgroups