Group #27186015 — subgroups

Subgroups — #27186015

Order 12928.

190 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 48 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	#20266148	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692312	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108084	C₄	1	2	yes	—
5	C₂²×#11717581	—	2	1	no	—
6	#15692312	—	2	1	no	—
7	#15692312	—	2	1	no	—
8	#15692312	C₄	1	2	yes	—
9	C₂²⋊#11717581	—	2	1	no	—
10	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₀₁	—	2	1	no	—
12	C₂²⋊#11717579	C₄	1	1	yes	characteristic
13	#15692167	—	2	1	no	—
14	C₂²⋊C₈×C₁₀₁	C₄	1	1	yes	characteristic
15	#13108084	—	4	2	no	—
16	C₂×#11717581	—	2	2	no	—
17	C₂×#11717581	—	2	2	no	—
18	C₂×#11717581	—	4	2	no	—
19	C₂³×D₁₀₁	Q₈	1	1	yes	characteristic
20	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₁	—	4	2	no	—
22	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
26	C₂×#11717579	—	2	1	no	—
27	C₈×D₁₀₁	—	4	2	no	—
28	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₁	—	2	1	no	—
30	C₂³×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×D₁₀₁	—	2	1	no	—
32	C₂²×D₁₀₁	—	2	1	no	—
33	C₂²×D₁₀₁	—	2	2	no	—
34	C₂²×D₁₀₁	—	4	1	no	—
35	C₂²×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	—	4	1	no	—
38	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₁	—	2	2	no	—
41	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×D₁₀₁	—	2	2	no	—
44	C₄×D₁₀₁	—	2	2	no	—
45	C₄×D₁₀₁	—	4	1	no	—
46	C₄×D₁₀₁	—	4	1	no	—
47	#11717581	—	4	2	no	—
48	#11717581	—	4	2	no	—
49	C₈×C₁₀₁	—	4	1	no	—
50	#11717579	—	4	1	no	—
51	C₂²×C₁₀₁	—	2	1	no	—
52	C₂²×C₁₀₁	—	2	1	no	—
53	C₂²×C₁₀₁	#95	1	1	yes	characteristic
54	C₂²×C₁₀₁	#108	1	1	yes	characteristic
55	C₂²×C₁₀₁	#108	1	1	yes	characteristic
56	C₂×D₁₀₁	—	2	1	no	—
57	C₂×D₁₀₁	—	2	1	no	—
58	C₂×D₁₀₁	—	2	1	no	—
59	C₂×D₁₀₁	—	2	1	no	—
60	C₂×D₁₀₁	—	2	1	no	—
61	C₂×D₁₀₁	—	4	2	no	—
62	C₂×D₁₀₁	—	4	2	no	—
63	C₂×D₁₀₁	#108	1	2	yes	—
64	C₄×C₁₀₁	—	2	1	no	—
65	C₄×C₁₀₁	—	2	1	no	—
66	Dic₁₀₁	—	2	1	no	—
67	Dic₁₀₁	#99	1	2	yes	—
68	C₂×C₁₀₁	—	2	1	no	—
69	C₂×C₁₀₁	—	2	1	no	—
70	C₂×C₁₀₁	#341	1	1	yes	characteristic
71	C₂×C₁₀₁	#341	1	1	yes	characteristic
72	C₂×C₁₀₁	#342	1	1	yes	characteristic
73	D₁₀₁	—	2	2	no	—
74	D₁₀₁	—	4	1	no	—
75	#1298	—	101	1	no	maximal, Sylow
76	C₁₀₁	#1298	1	1	yes	characteristic, complemented, Sylow
77	#407	—	101	1	no	—
78	C₂×C₂²⋊C₈	—	101	1	no	—
79	C₂×C₂²⋊C₈	—	101	1	no	—
80	C₂³×C₄	—	101	1	no	—
81	C₂×M₁₆	—	101	2	no	—
82	C₂²×C₈	—	202	1	no	—
83	C₂²×C₈	—	202	1	no	—
84	C₂²⋊C₈	—	101	2	no	—
85	C₂²⋊C₈	—	101	1	no	—
86	C₂²⋊C₈	—	202	1	no	—
87	C₂²⋊C₈	—	202	1	no	—
88	C₂²⋊C₈	—	202	1	no	—
89	C₂²⋊C₈	—	202	1	no	—
90	C₂²⋊C₈	#87938	1	1	yes	characteristic
91	C₂⁴	—	101	1	no	—
92	C₂²×C₄	—	101	1	no	—
93	C₂²×C₄	—	101	1	no	—
94	C₂²×C₄	—	101	1	no	—
95	C₂²×C₄	—	101	1	no	—
96	C₂²×C₄	—	101	1	no	—
97	C₂²×C₄	—	404	2	no	—
98	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
99	M₁₆	—	404	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	404	2	no	—
102	C₂×C₈	—	404	2	no	—
103	C₂×C₈	—	202	2	no	—
104	C₂×C₈	—	202	2	no	—
105	C₂×C₈	—	202	1	no	—
106	C₂³	—	101	1	no	—
107	C₂³	—	101	1	no	—
108	C₂³	—	404	1	no	—
109	C₂³	—	202	2	no	—
110	C₂³	—	202	1	no	—
111	C₂³	—	202	1	no	—
112	C₂³	C₄⋊#87938	1	1	yes	characteristic
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	101	1	no	—
115	C₂×C₄	—	101	1	no	—
116	C₂×C₄	—	404	1	no	—
117	C₂×C₄	—	404	1	no	—
118	C₂×C₄	—	202	2	no	—
119	C₂×C₄	—	202	2	no	—
120	C₂×C₄	—	202	2	no	—
121	C₂×C₄	C₄×#87938	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	404	1	no	—
125	C₈	—	404	2	no	—
126	C₈	—	404	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	101	2	no	—
130	C₂²	—	404	2	no	—
131	C₂²	—	404	2	no	—
132	C₂²	—	202	1	no	—
133	C₂²	—	202	1	no	—
134	C₂²	—	202	1	no	—
135	C₂²	—	202	1	no	—
136	C₂²	—	202	1	no	—
137	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
138	C₂²	#15692325	1	1	yes	characteristic
139	C₂²	#15692326	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	101	2	no	—
143	C₄	—	202	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	—	404	1	no	—
147	C₂	—	202	2	no	—
148	C₂	#20266052	1	1	yes	characteristic
149	C₂	#20266053	1	1	yes	characteristic
150	C₂	#20266054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20266148

C₂

yes

maximal, characteristic

C₂×#15692312

C₂

yes

maximal, characteristic

C₂²⋊C₈×D₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108084

C₄

yes

—

—

—

—

C₄

yes

—

C₂²⋊#11717581

—

C₂²×C₄×D₁₀₁

C₂²

yes

characteristic

C₂×C₈×D₁₀₁

—

C₂²⋊#11717579

C₄

yes

characteristic

#15692167

—

C₂²⋊C₈×C₁₀₁

C₄

yes

—

—

—

—

C₂³×D₁₀₁

Q₈

yes

characteristic

C₂²×Dic₁₀₁

D₄

yes

characteristic

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

characteristic

C₂×#11717579

—

C₈×D₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₄⋊C₄

yes

characteristic

C₂²×D₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

#11717581

—

#11717581

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

#108

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#99

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

D₁₀₁

—

#1298

—

101

maximal, Sylow

C₁₀₁

#1298

yes

characteristic, complemented, Sylow

#407

—

101

—

C₂×C₂²⋊C₈

—

101

—

C₂×C₂²⋊C₈

—

101

—

C₂³×C₄

—

101

—

C₂×M₁₆

—

101

—

C₂²×C₈

—

202

—

C₂²×C₈

—

202

—

C₂²⋊C₈

—

101

—

C₂²⋊C₈

—

101

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

#87938

yes

characteristic

C₂⁴

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

404

—

C₂²×C₄

C₂×#87938

yes

characteristic

M₁₆

—

404

—

100

C₂×C₈

—

101

C₂×C₈

—

404

—

102

C₂×C₈

—

404

—

103

C₂×C₈

—

202

—

104

C₂×C₈

—

202

—

105

C₂×C₈

—

202

—

106

C₂³

—

101

—

107

C₂³

—

101

—

108

C₂³

—

404

—

109

C₂³

—

202

—

110

C₂³

—

202

—

111

C₂³

—

202

—

112

C₂³

C₄⋊#87938

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

101

—

115

C₂×C₄

—

101

—

116

C₂×C₄

—

404

—

117

C₂×C₄

—

404

—

118

C₂×C₄

—

202

—

119

C₂×C₄

—

202

—

120

C₂×C₄

—

202

—

121

C₂×C₄

C₄×#87938

yes

characteristic

122

C₂×C₄

C₂²⋊#87938

yes

characteristic

123

C₈

—

124

C₈

—

404

—

125

C₈

—

404

—

126

C₈

—

404

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

101

—

130

C₂²

—

404

—

131

C₂²

—

404

—

132

C₂²

—

202

—

133

C₂²

—

202

—

134

C₂²

—

202

—

135

C₂²

—

202

—

136

C₂²

—

202

—

137

C₂²

(C₂×#87938)⋊C₄

yes

characteristic

138

C₂²

#15692325

yes

characteristic

139

C₂²

#15692326

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

101

—

143

C₄

—

202

—

144

C₂

—

145

C₂

—

146

C₂

—

404

—

147

C₂

—

202

—

148

C₂

#20266052

yes

characteristic

149

C₂

#20266053

yes

characteristic

150

C₂

#20266054

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#87938)⋊C₄
commutator	C₂×C₄×C₁₀₁	C₄²
Frattini	C₂²×C₄	C₂×#87938
Fitting	C₂²⋊C₈×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#95
prime radical	C₂³×D₁₀₁	Q₈
cosocle	C₂²×C₄×D₁₀₁	C₂²
hypercenter	C₂²⋊C₈	#87938
hypocenter	C₁₀₁	#1298
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

(C₂×#87938)⋊C₄

commutator

C₂×C₄×C₁₀₁

C₄²

Frattini

C₂²×C₄

C₂×#87938

Fitting

C₂²⋊C₈×C₁₀₁

C₄

socle

C₂²×C₁₀₁

#95

prime radical

C₂³×D₁₀₁

Q₈

cosocle

C₂²×C₄×D₁₀₁

C₂²

C₂²⋊C₈

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
76. C₁₀₁	#1298	semidirect

H ≅ G/N

type

76. C₁₀₁

#1298

semidirect

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

subgroup	index
#20266148	2
C₂×#15692312	2
C₂²⋊C₈×D₁₀₁	2
#1298	101

subgroup

index

#20266148

C₂×#15692312

C₂²⋊C₈×D₁₀₁

#1298

101

Subgroups — #27186015

Conjugacy classes of subgroups

Complements

Maximal subgroups