Group #41014868 — subgroups

Subgroups — #41014868

Order 27008.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27213194	C₂	1	1	yes	maximal, characteristic
2	#27213156	C₂	1	1	yes	maximal, characteristic
3	#27213154	C₂	1	1	yes	maximal, characteristic
4	C₂×#20269895	C₂	1	1	yes	maximal, characteristic
5	#27212918	C₂	1	1	yes	maximal, characteristic
6	#27212902	C₂	1	1	yes	maximal, characteristic
7	#344×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₄₂₂	C₂²	1	1	yes	characteristic
9	C₂×#16809946	C₂²	1	1	yes	characteristic
10	C₂×#16809946	C₂²	1	1	yes	characteristic
11	#20269936	C₂²	1	1	yes	characteristic
12	#20269912	—	2	2	no	—
13	#20269902	—	2	2	no	—
14	#20269900	—	2	2	no	—
15	#20269895	—	2	2	no	—
16	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
18	#117×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₂₂	—	2	1	no	—
20	C₂×Dic₄₂₂	—	2	1	no	—
21	C₂×Dic₄₂₂	—	2	1	no	—
22	C₂×Dic₄₂₂	—	2	1	no	—
23	C₂×Dic₄₂₂	—	2	1	no	—
24	C₂×Dic₄₂₂	D₄	1	2	yes	—
25	C₂²×Dic₂₁₁	—	2	1	no	—
26	#16809946	—	2	1	no	—
27	#16809946	—	2	1	no	—
28	#16809946	—	2	1	no	—
29	#16809946	—	2	1	no	—
30	C₂×C₂₁₁⋊C₈	—	2	1	no	—
31	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
32	C₄⋊Dic₂₁₁	—	2	1	no	—
33	C₂²⋊Dic₂₁₁	—	2	1	no	—
34	C₄×Dic₂₁₁	—	2	1	no	—
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₁₁	—	2	1	no	—
37	M₁₆×C₂₁₁	—	2	1	no	—
38	M₁₆×C₂₁₁	—	2	1	no	—
39	M₁₆×C₂₁₁	—	2	1	no	—
40	C₂×C₈×C₂₁₁	—	2	1	no	—
41	C₂×C₈×C₂₁₁	—	2	1	no	—
42	C₄⋊C₄×C₂₁₁	—	2	1	no	—
43	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
44	C₄²×C₂₁₁	—	2	1	no	—
45	Dic₄₂₂	—	2	2	no	—
46	Dic₄₂₂	—	2	2	no	—
47	Dic₄₂₂	—	4	1	no	—
48	Dic₄₂₂	—	4	1	no	—
49	C₂×Dic₂₁₁	—	2	2	no	—
50	C₂×Dic₂₁₁	—	4	1	no	—
51	C₂×Dic₂₁₁	—	4	1	no	—
52	C₂×Dic₂₁₁	—	4	1	no	—
53	C₂₁₁⋊C₈	—	2	2	no	—
54	C₂₁₁⋊C₈	—	4	1	no	—
55	C₂³×C₂₁₁	#41	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	—	4	1	no	—
57	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
60	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
61	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
62	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
63	C₈×C₂₁₁	—	4	1	no	—
64	C₈×C₂₁₁	—	4	1	no	—
65	C₂²×C₂₁₁	—	4	1	no	—
66	C₂²×C₂₁₁	#123	1	1	yes	characteristic
67	C₂²×C₂₁₁	#124	1	1	yes	characteristic
68	C₂²×C₂₁₁	#126	1	1	yes	characteristic
69	C₄×C₂₁₁	—	8	1	no	—
70	C₄×C₂₁₁	#121	1	1	yes	characteristic
71	C₄×C₂₁₁	#123	1	1	yes	characteristic
72	C₄×C₂₁₁	#123	1	1	yes	characteristic
73	C₄×C₂₁₁	#126	1	1	yes	characteristic
74	Dic₂₁₁	—	4	2	no	—
75	Dic₂₁₁	—	8	1	no	—
76	C₂×C₂₁₁	—	2	1	no	—
77	C₂×C₂₁₁	—	2	1	no	—
78	C₂×C₂₁₁	—	2	1	no	—
79	C₂×C₂₁₁	#397	1	1	yes	characteristic
80	C₂₁₁	#2082	1	1	yes	characteristic, complemented, Sylow
81	#2082	—	211	1	no	maximal, Sylow
82	#430	—	211	1	no	—
83	#419	—	211	1	no	—
84	#419	—	211	1	no	—
85	C₂×#101	—	211	1	no	—
86	#344	—	211	1	no	—
87	#344	D₂₁₁	1	1	yes	characteristic
88	H(3,ℤ/4)	—	211	1	no	—
89	C₂²×Q₈	—	211	1	no	—
90	C₂×M₁₆	—	211	1	no	—
91	C₂×M₁₆	—	211	1	no	—
92	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
93	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
94	#117	—	211	1	no	—
95	#117	C₂×D₂₁₁	1	1	yes	characteristic
96	#108	—	422	2	no	—
97	#103	—	422	2	no	—
98	#103	—	422	2	no	—
99	#101	—	422	2	no	—
100	C₂×Q₈	—	422	1	no	—
101	C₂×Q₈	—	422	1	no	—
102	C₂×Q₈	—	422	1	no	—
103	C₂×Q₈	—	422	1	no	—
104	C₂×Q₈	—	422	1	no	—
105	C₂×Q₈	—	211	2	no	—
106	C₂²×C₄	—	422	1	no	—
107	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	422	1	no	—
113	M₁₆	—	422	1	no	—
114	M₁₆	—	422	1	no	—
115	M₁₆	—	422	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	422	1	no	—
119	C₂×C₈	—	211	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	422	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	422	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	422	1	no	—
126	C₂³	#16809962	1	1	yes	characteristic
127	Q₈	—	844	1	no	—
128	Q₈	—	844	1	no	—
129	Q₈	—	422	2	no	—
130	Q₈	—	422	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	844	1	no	—
133	C₂×C₄	—	844	1	no	—
134	C₂×C₄	—	844	1	no	—
135	C₂×C₄	—	422	2	no	—
136	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
137	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
138	C₂×C₄	#16809961	1	1	yes	characteristic
139	C₂×C₄	#16809962	1	1	yes	characteristic
140	C₂×C₄	#16809962	1	1	yes	characteristic
141	C₂×C₄	#16809963	1	1	yes	characteristic
142	C₈	—	4	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	844	1	no	—
145	C₈	—	422	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#20269962	1	1	yes	characteristic
148	C₂²	#20269963	1	1	yes	characteristic
149	C₂²	#20269971	1	1	yes	characteristic
150	C₄	—	8	1	no	—
151	C₄	—	844	2	no	—
152	C₄	—	1688	1	no	—
153	C₄	#20269950	1	1	yes	characteristic
154	C₄	#20269959	1	1	yes	characteristic
155	C₄	#20269962	1	1	yes	characteristic
156	C₄	#20269972	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#27213075	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213194

C₂

yes

maximal, characteristic

#27213156

C₂

yes

maximal, characteristic

#27213154

C₂

yes

maximal, characteristic

C₂×#20269895

C₂

yes

maximal, characteristic

#27212918

C₂

yes

maximal, characteristic

#27212902

C₂

yes

maximal, characteristic

#344×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

characteristic

C₂×#16809946

C₂²

yes

characteristic

C₂×#16809946

C₂²

yes

characteristic

#20269936

C₂²

yes

—

—

—

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

D₄

yes

—

C₂²×Dic₂₁₁

—

—

—

—

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₁₁

—

C₂²⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#397

yes

characteristic

C₂₁₁

#2082

yes

characteristic, complemented, Sylow

#2082

—

211

maximal, Sylow

#430

—

211

—

#419

—

211

—

#419

—

211

—

C₂×#101

—

211

—

#344

—

211

—

#344

D₂₁₁

yes

characteristic

H(3,ℤ/4)

—

211

—

C₂²×Q₈

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

#117

—

211

—

#117

C₂×D₂₁₁

yes

characteristic

#108

—

422

—

#103

—

422

—

#103

—

422

—

#101

—

422

—

100

C₂×Q₈

—

422

—

101

C₂×Q₈

—

422

—

102

C₂×Q₈

—

422

—

103

C₂×Q₈

—

422

—

104

C₂×Q₈

—

422

—

105

C₂×Q₈

—

211

—

106

C₂²×C₄

—

422

—

107

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

422

—

113

M₁₆

—

422

—

114

M₁₆

—

422

—

115

M₁₆

—

422

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

422

—

119

C₂×C₈

—

211

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

422

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

422

—

124

C₄²

—

125

C₄²

—

422

—

126

C₂³

#16809962

yes

characteristic

127

Q₈

—

844

—

128

Q₈

—

844

—

129

Q₈

—

422

—

130

Q₈

—

422

—

131

C₂×C₄

—

132

C₂×C₄

—

844

—

133

C₂×C₄

—

844

—

134

C₂×C₄

—

844

—

135

C₂×C₄

—

422

—

136

C₂×C₄

D₄×D₂₁₁

yes

characteristic

137

C₂×C₄

D₄×D₂₁₁

yes

characteristic

138

C₂×C₄

#16809961

yes

characteristic

139

C₂×C₄

#16809962

yes

characteristic

140

C₂×C₄

#16809962

yes

characteristic

141

C₂×C₄

#16809963

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

844

—

145

C₈

—

422

—

146

C₂²

—

147

C₂²

#20269962

yes

characteristic

148

C₂²

#20269963

yes

characteristic

149

C₂²

#20269971

yes

characteristic

150

C₄

—

151

C₄

—

844

—

152

C₄

—

1688

—

153

C₄

#20269950

yes

characteristic

154

C₄

#20269959

yes

characteristic

155

C₄

#20269962

yes

characteristic

156

C₄

#20269972

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#27213075

yes

characteristic

	subgroup	G/H
center	C₂	#27213075
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#344×C₂₁₁	C₂
socle	C₂×C₂₁₁	#397
prime radical	C₂³×C₂₁₁	#41
hypercenter	#344	D₂₁₁
hypocenter	C₂₁₁	#2082
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#27213075

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#344×C₂₁₁

C₂

C₂×C₂₁₁

C₂³×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
80. C₂₁₁	#2082	semidirect

H ≅ G/N

type

80. C₂₁₁

#2082

semidirect

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

subgroup	index
#27213194	2
#27213156	2
#27213154	2
C₂×#20269895	2
#27212918	2
#27212902	2
#344×C₂₁₁	2
#2082	211

subgroup

index

#344×C₂₁₁

211

Subgroups — #41014868

Conjugacy classes of subgroups

Complements

Maximal subgroups