Group #41013583 — subgroups

Subgroups — #41013583

Order 27008.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂²×#16809946	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₁₁⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#27212901	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	—	2	4	no	—
7	C₂×#16809946	—	2	2	no	—
8	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₂₁₁⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₂₁₁	C₄	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
16	#95×C₂₁₁	C₄	1	1	yes	characteristic
17	#95×C₂₁₁	C₄	1	1	yes	characteristic
18	#16809946	—	2	8	no	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	—	2	1	no	—
22	C₂×C₂₁₁⋊C₈	—	2	1	no	—
23	C₂×C₂₁₁⋊C₈	—	2	2	no	—
24	C₂×C₂₁₁⋊C₈	—	2	2	no	—
25	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
26	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₁₁	—	2	2	no	—
28	C₂²×C₄×C₂₁₁	—	2	2	no	—
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
36	C₄²×C₂₁₁	—	2	2	no	—
37	C₄²×C₂₁₁	—	2	2	no	—
38	C₂₁₁⋊C₈	—	2	4	no	—
39	C₂₁₁⋊C₈	—	4	1	no	—
40	C₂₁₁⋊C₈	—	4	1	no	—
41	C₂³×C₂₁₁	—	2	2	no	—
42	C₂³×C₂₁₁	—	2	2	no	—
43	C₂³×C₂₁₁	—	2	1	no	—
44	C₂³×C₂₁₁	—	2	1	no	—
45	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
46	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	—	2	2	no	—
48	C₂×C₄×C₂₁₁	—	2	2	no	—
49	C₂×C₄×C₂₁₁	—	2	2	no	—
50	C₂×C₄×C₂₁₁	—	2	2	no	—
51	C₂×C₄×C₂₁₁	—	2	4	no	—
52	C₂×C₄×C₂₁₁	—	2	2	no	—
53	C₂×C₄×C₂₁₁	—	2	2	no	—
54	C₂×C₄×C₂₁₁	—	2	2	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₁₁	M₁₆	1	4	yes	complemented
58	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₂₁₁	#41	1	2	yes	—
62	C₂²×C₂₁₁	—	2	2	no	—
63	C₂²×C₂₁₁	—	2	2	no	—
64	C₂²×C₂₁₁	—	2	2	no	—
65	C₂²×C₂₁₁	—	4	2	no	—
66	C₂²×C₂₁₁	—	4	2	no	—
67	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₁₁	C₄×D₄	1	2	yes	—
69	C₂²×C₂₁₁	C₂×M₁₆	1	2	yes	—
70	C₂²×C₂₁₁	#131	1	1	yes	characteristic
71	C₂²×C₂₁₁	#131	1	1	yes	characteristic
72	C₄×C₂₁₁	—	2	2	no	—
73	C₄×C₂₁₁	—	2	4	no	—
74	C₄×C₂₁₁	—	4	2	no	—
75	C₄×C₂₁₁	#120	1	1	yes	characteristic
76	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
77	C₄×C₂₁₁	#123	1	1	yes	characteristic
78	C₂×C₂₁₁	—	4	2	no	—
79	C₂×C₂₁₁	#385	1	1	yes	characteristic
80	C₂×C₂₁₁	#423	1	1	yes	characteristic
81	C₂×C₂₁₁	#424	1	1	yes	characteristic
82	C₂×C₂₁₁	#435	1	2	yes	—
83	C₂×C₂₁₁	#436	1	2	yes	—
84	C₂₁₁	#1851	1	1	yes	characteristic, complemented, Sylow
85	#1851	—	211	1	no	maximal, Sylow
86	C₂²×M₁₆	—	211	1	no	—
87	C₂×C₄⋊C₈	—	211	2	no	—
88	C₂×C₂²⋊C₈	—	211	1	no	—
89	C₄×C₂²⋊C₄	D₂₁₁	1	1	yes	characteristic
90	#336	—	211	2	no	—
91	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
92	C₂×M₁₆	—	422	2	no	—
93	C₂×M₁₆	—	422	4	no	—
94	C₂²×C₈	—	211	1	no	—
95	C₂²×C₈	—	211	1	no	—
96	C₂²×C₈	—	211	2	no	—
97	C₂×C₂²⋊C₄	Dic₂₁₁	1	2	yes	—
98	C₂×C₄²	C₂×D₂₁₁	1	2	yes	—
99	C₄⋊C₈	—	422	4	no	—
100	C₂²⋊C₈	—	422	2	no	—
101	#95	Dic₂₁₁	1	1	yes	characteristic
102	#95	Dic₂₁₁	1	1	yes	characteristic
103	C₂⁴	C₂×Dic₂₁₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
107	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
108	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
109	C₂²×C₄	D₈₄₄	1	2	yes	—
110	C₂²×C₄	Dic₄₂₂	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
112	M₁₆	—	422	8	no	—
113	C₂×C₈	—	422	2	no	—
114	C₂×C₈	—	422	1	no	—
115	C₂×C₈	—	422	1	no	—
116	C₂×C₈	—	422	2	no	—
117	C₂×C₈	—	422	2	no	—
118	C₂×C₈	—	422	2	no	—
119	C₂×C₈	—	211	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	C₄⋊Dic₂₁₁	1	2	yes	—
128	C₂³	C₂²×Dic₂₁₁	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
139	C₂×C₄	#16809946	1	4	yes	complemented
140	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
141	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
143	C₂×C₄	#16809961	1	2	yes	—
144	C₈	—	844	1	no	—
145	C₈	—	844	1	no	—
146	C₈	—	422	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
153	C₂²	D₄×Dic₂₁₁	1	2	yes	—
154	C₂²	C₂×#16809946	1	2	yes	—
155	C₂²	#20269984	1	1	yes	characteristic
156	C₂²	#20269984	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	#20269947	1	1	yes	characteristic
161	C₄	#20269954	1	2	yes	—
162	C₄	#20269961	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#27213021	1	1	yes	characteristic
165	C₂	#27213170	1	1	yes	characteristic
166	C₂	#27213173	1	1	yes	characteristic
167	C₂	#27213218	1	2	yes	—
168	C₂	#27213224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16809946

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, characteristic

#27212901

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

—

C₂×#16809946

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

—

#95×C₂₁₁

C₄

yes

characteristic

#95×C₂₁₁

C₄

yes

characteristic

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂⁴×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

M₁₆

yes

complemented

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×D₄

yes

—

C₂²×C₂₁₁

C₂×M₁₆

yes

—

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#120

yes

characteristic

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#123

yes

characteristic

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#436

yes

—

C₂₁₁

#1851

yes

characteristic, complemented, Sylow

#1851

—

211

maximal, Sylow

C₂²×M₁₆

—

211

—

C₂×C₄⋊C₈

—

211

—

C₂×C₂²⋊C₈

—

211

—

C₄×C₂²⋊C₄

D₂₁₁

yes

characteristic

#336

—

211

—

C₂³×C₄

C₂×D₂₁₁

yes

characteristic

C₂×M₁₆

—

422

—

C₂×M₁₆

—

422

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂×C₂²⋊C₄

Dic₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

—

C₄⋊C₈

—

422

—

100

C₂²⋊C₈

—

422

—

101

#95

Dic₂₁₁

yes

characteristic

102

#95

Dic₂₁₁

yes

characteristic

103

C₂⁴

C₂×Dic₂₁₁

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×Dic₂₁₁

yes

—

107

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

108

C₂²×C₄

C₂×Dic₂₁₁

yes

—

109

C₂²×C₄

D₈₄₄

yes

—

110

C₂²×C₄

Dic₄₂₂

yes

—

111

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

112

M₁₆

—

422

—

113

C₂×C₈

—

422

—

114

C₂×C₈

—

422

—

115

C₂×C₈

—

422

—

116

C₂×C₈

—

422

—

117

C₂×C₈

—

422

—

118

C₂×C₈

—

422

—

119

C₂×C₈

—

211

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₂₁₁

yes

—

128

C₂³

C₂²×Dic₂₁₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₄⋊Dic₂₁₁

yes

—

139

C₂×C₄

#16809946

yes

complemented

140

C₂×C₄

D₄×D₂₁₁

yes

—

141

C₂×C₄

C₂×D₈₄₄

yes

characteristic

142

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

143

C₂×C₄

#16809961

yes

—

144

C₈

—

844

—

145

C₈

—

844

—

146

C₈

—

422

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₄⋊Dic₂₁₁

yes

characteristic

153

C₂²

D₄×Dic₂₁₁

yes

—

154

C₂²

C₂×#16809946

yes

—

155

C₂²

#20269984

yes

characteristic

156

C₂²

#20269984

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#20269947

yes

characteristic

161

C₄

#20269954

yes

—

162

C₄

#20269961

yes

characteristic

163

C₂

—

164

C₂

#27213021

yes

characteristic

165

C₂

#27213170

yes

characteristic

166

C₂

#27213173

yes

characteristic

167

C₂

#27213218

yes

—

168

C₂

#27213224

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator, socle	C₂³×C₂₁₁	C₂²×C₄
Fitting	C₄×C₂²⋊C₄×C₂₁₁	C₂
prime radical	C₂⁴×C₂₁₁	C₂×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₄×C₂²⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#1851
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator, socle

C₂³×C₂₁₁

C₂²×C₄

Fitting

C₄×C₂²⋊C₄×C₂₁₁

C₂

prime radical

C₂⁴×C₂₁₁

C₂×C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₄×C₂²⋊C₄

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₂₁₁⋊C₈)	C₂	semidirect
4. #27212901	C₂	semidirect
57. C₂×C₄×C₂₁₁	M₁₆	semidirect
84. C₂₁₁	#1851	semidirect
139. C₂×C₄	#16809946	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

semidirect

4. #27212901

C₂

semidirect

57. C₂×C₄×C₂₁₁

M₁₆

semidirect

84. C₂₁₁

#1851

semidirect

139. C₂×C₄

#16809946

semidirect

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

subgroup	index
C₂²×#16809946	2
C₂×C₄⋊(C₂₁₁⋊C₈)	2
C₂×C₂²⋊(C₂₁₁⋊C₈)	2
#27212901	2
C₄×C₂²⋊C₄×C₂₁₁	2
#1851	211

subgroup

index

C₂²×#16809946

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂×C₂²⋊(C₂₁₁⋊C₈)

#27212901

C₄×C₂²⋊C₄×C₂₁₁

#1851

211

Subgroups — #41013583

Conjugacy classes of subgroups

Complements

Maximal subgroups