Group #41013254 — subgroups

Subgroups — #41013254

Order 27008.

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

normal subgroups only

This group has 260 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	C₂×Dic₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#20269887	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#27212992	C₂	1	1	yes	maximal, characteristic
6	#27212901	C₂	1	1	yes	maximal, characteristic
7	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
12	Dic₂₁₁⋊C₈	C₄	1	4	yes	complemented
13	Dic₂₁₁⋊C₈	C₄	1	4	yes	complemented
14	#20269887	—	2	2	no	—
15	C₈×Dic₂₁₁	—	2	2	no	—
16	C₂₁₁⋊#95	C₄	1	2	yes	—
17	C₂₁₁⋊#95	C₄	1	2	yes	—
18	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂²×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
24	C₂×C₂₁₁⋊C₈	—	2	1	no	—
25	C₂×C₂₁₁⋊C₈	—	2	1	no	—
26	C₂×C₂₁₁⋊C₈	—	2	1	no	—
27	C₂×C₂₁₁⋊C₈	—	2	1	no	—
28	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
31	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
32	C₄×Dic₂₁₁	D₄	1	2	yes	—
33	C₄×Dic₂₁₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₂₁₁	—	2	1	no	—
37	C₂×C₈×C₂₁₁	—	2	1	no	—
38	C₂×C₈×C₂₁₁	—	2	1	no	—
39	C₂×C₈×C₂₁₁	—	2	1	no	—
40	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
42	C₄²×C₂₁₁	—	2	1	no	—
43	C₄²×C₂₁₁	—	2	1	no	—
44	C₂×Dic₂₁₁	—	2	2	no	—
45	C₂×Dic₂₁₁	—	2	2	no	—
46	C₂×Dic₂₁₁	—	2	2	no	—
47	C₂×Dic₂₁₁	—	2	2	no	—
48	C₂×Dic₂₁₁	C₄²	1	2	yes	—
49	C₂×Dic₂₁₁	C₄²	1	2	yes	—
50	C₂×Dic₂₁₁	C₄⋊C₄	1	4	yes	—
51	C₂₁₁⋊C₈	—	2	2	no	—
52	C₂₁₁⋊C₈	—	2	2	no	—
53	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	—	2	2	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	—	2	2	no	—
57	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
62	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
63	C₈×C₂₁₁	—	2	2	no	—
64	C₈×C₂₁₁	—	2	2	no	—
65	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₂₁₁	#117	1	1	yes	characteristic
68	C₂²×C₂₁₁	#131	1	2	yes	—
69	C₂²×C₂₁₁	#131	1	2	yes	—
70	C₄×C₂₁₁	—	4	2	no	—
71	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
72	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
73	C₄×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
75	Dic₂₁₁	—	2	2	no	—
76	Dic₂₁₁	—	2	2	no	—
77	Dic₂₁₁	—	2	4	no	—
78	C₂×C₂₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₂₁₁	#405	1	1	yes	characteristic
80	C₂×C₂₁₁	#405	1	1	yes	characteristic
81	C₂×C₂₁₁	#424	1	2	yes	—
82	C₂×C₂₁₁	#433	1	2	yes	—
83	C₂₁₁	#1768	1	1	yes	characteristic, complemented, Sylow
84	#1768	—	211	1	no	maximal, Sylow
85	C₂×C₄⋊C₈	—	211	1	no	—
86	C₂×C₄⋊C₈	—	211	1	no	—
87	C₂×#97	—	211	1	no	—
88	C₂×C₄×C₈	—	211	1	no	—
89	#376	—	211	1	no	—
90	#336	—	211	1	no	—
91	#336	D₂₁₁	1	1	yes	characteristic
92	C₂²×C₈	—	211	1	no	—
93	C₂²×C₈	—	211	1	no	—
94	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
96	C₂×C₄²	—	211	1	no	—
97	C₂×C₄²	—	211	1	no	—
98	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
99	C₄⋊C₈	—	211	4	no	—
100	C₄⋊C₈	—	211	4	no	—
101	#97	—	422	2	no	—
102	C₄×C₈	—	422	2	no	—
103	#95	—	211	2	no	—
104	#95	—	211	2	no	—
105	C₂²×C₄	—	211	1	no	—
106	C₂²×C₄	—	211	2	no	—
107	C₂²×C₄	—	211	1	no	—
108	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	422	1	no	—
115	C₂×C₈	—	422	1	no	—
116	C₂×C₈	—	422	1	no	—
117	C₂×C₈	—	422	1	no	—
118	C₂×C₈	—	211	2	no	—
119	C₂×C₈	—	211	2	no	—
120	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
121	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	211	2	no	—
125	C₄²	—	211	2	no	—
126	C₄²	—	211	2	no	—
127	C₄²	—	211	2	no	—
128	C₂³	C₂×C₄×D₂₁₁	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	—	422	2	no	—
133	C₂×C₄	—	422	2	no	—
134	C₂×C₄	—	422	2	no	—
135	C₂×C₄	—	422	2	no	—
136	C₂×C₄	—	211	2	no	—
137	C₂×C₄	—	211	2	no	—
138	C₂×C₄	—	211	4	no	—
139	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
141	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
143	C₂×C₄	#16809961	1	1	yes	characteristic
144	C₂×C₄	#16809961	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	422	2	no	—
148	C₈	—	422	2	no	—
149	C₂²	C₄²×D₂₁₁	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
151	C₂²	#20269933	1	1	yes	characteristic
152	C₂²	#20269983	1	2	yes	—
153	C₂²	#20269985	1	2	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	422	2	no	—
156	C₄	—	422	2	no	—
157	C₄	—	422	4	no	—
158	C₄	(C₂²⋊D₂₁₁)⋊C₄	1	1	yes	characteristic
159	C₄	(C₂²⋊D₂₁₁)⋊C₄	1	1	yes	characteristic
160	C₄	#20269942	1	1	yes	characteristic
161	C₄	#20269942	1	1	yes	characteristic
162	C₂	#27212997	1	1	yes	characteristic
163	C₂	#27213100	1	1	yes	characteristic
164	C₂	#27213103	1	1	yes	characteristic
165	C₂	#27213171	1	2	yes	—
166	C₂	#27213204	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₁₁⋊C₈

C₂

yes

maximal, characteristic

C₂×Dic₂₁₁⋊C₈

C₂

yes

maximal, characteristic

C₂×#20269887

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₁₁

C₂

yes

maximal, characteristic

#27212992

C₂

yes

maximal, characteristic

#27212901

C₂

yes

maximal, characteristic

#336×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

Dic₂₁₁⋊C₈

C₄

yes

complemented

Dic₂₁₁⋊C₈

C₄

yes

complemented

#20269887

—

C₈×Dic₂₁₁

—

C₂₁₁⋊#95

C₄

yes

—

C₂₁₁⋊#95

C₄

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

D₄

yes

—

C₄×Dic₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄²

yes

—

C₂×Dic₂₁₁

C₄²

yes

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄²

yes

characteristic

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₂²×C₂₁₁

#131

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×D₄

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

characteristic

C₄×C₂₁₁

C₄×Q₈

yes

characteristic

C₄×C₂₁₁

C₄×Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

C₄×C₄⋊C₄

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#433

yes

—

C₂₁₁

#1768

yes

characteristic, complemented, Sylow

#1768

—

211

maximal, Sylow

C₂×C₄⋊C₈

—

211

—

C₂×C₄⋊C₈

—

211

—

C₂×#97

—

211

—

C₂×C₄×C₈

—

211

—

#376

—

211

—

#336

—

211

—

#336

D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

211

—

100

C₄⋊C₈

—

211

—

101

#97

—

422

—

102

C₄×C₈

—

422

—

103

#95

—

211

—

104

#95

—

211

—

105

C₂²×C₄

—

211

—

106

C₂²×C₄

—

211

—

107

C₂²×C₄

—

211

—

108

C₂²×C₄

C₄×D₂₁₁

yes

—

109

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

422

—

115

C₂×C₈

—

422

—

116

C₂×C₈

—

422

—

117

C₂×C₈

—

422

—

118

C₂×C₈

—

211

—

119

C₂×C₈

—

211

—

120

C₂×C₈

C₄×D₂₁₁

yes

—

121

C₂×C₈

C₄×D₂₁₁

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

211

—

125

C₄²

—

211

—

126

C₄²

—

211

—

127

C₄²

—

211

—

128

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

422

—

133

C₂×C₄

—

422

—

134

C₂×C₄

—

422

—

135

C₂×C₄

—

422

—

136

C₂×C₄

—

211

—

137

C₂×C₄

—

211

—

138

C₂×C₄

—

211

—

139

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

141

C₂×C₄

D₄×D₂₁₁

yes

characteristic

142

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

143

C₂×C₄

#16809961

yes

characteristic

144

C₂×C₄

#16809961

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

422

—

148

C₈

—

422

—

149

C₂²

C₄²×D₂₁₁

yes

characteristic

150

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

151

C₂²

#20269933

yes

characteristic

152

C₂²

#20269983

yes

—

153

C₂²

#20269985

yes

—

154

C₄

—

155

C₄

—

422

—

156

C₄

—

422

—

157

C₄

—

422

—

158

C₄

(C₂²⋊D₂₁₁)⋊C₄

yes

characteristic

159

C₄

(C₂²⋊D₂₁₁)⋊C₄

yes

characteristic

160

C₄

#20269942

yes

characteristic

161

C₄

#20269942

yes

characteristic

162

C₂

#27212997

yes

characteristic

163

C₂

#27213100

yes

characteristic

164

C₂

#27213103

yes

characteristic

165

C₂

#27213171

yes

—

166

C₂

#27213204

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂×C₄²
Fitting	#336×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#336	D₂₁₁
hypocenter	C₂₁₁	#1768
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator

C₂²×C₂₁₁

C₂×C₄²

Fitting

#336×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂²×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
12. Dic₂₁₁⋊C₈	C₄	semidirect
13. Dic₂₁₁⋊C₈	C₄	semidirect
83. C₂₁₁	#1768	semidirect

H ≅ G/N

type

12. Dic₂₁₁⋊C₈

C₄

semidirect

13. Dic₂₁₁⋊C₈

C₄

semidirect

83. C₂₁₁

#1768

semidirect

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

subgroup	index
C₂×Dic₂₁₁⋊C₈	2
C₂×#20269887	2
C₂×C₈×Dic₂₁₁	2
#27212992	2
#27212901	2
#336×C₂₁₁	2
#1768	211

subgroup

index

C₂×Dic₂₁₁⋊C₈

C₂×C₈×Dic₂₁₁

#336×C₂₁₁

211

Subgroups — #41013254

Conjugacy classes of subgroups

Complements

Maximal subgroups