Group #41014540 — subgroups

Subgroups — #41014540

Order 27008.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#20269977	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269902	C₂	1	2	yes	maximal
3	#27213052	C₂	1	1	yes	maximal, characteristic
4	#27212909	C₂	1	2	yes	maximal
5	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₂	—	2	1	no	—
7	C₂²×Dic₄₂₂	C₂²	1	1	yes	characteristic
8	#20269977	—	2	4	no	—
9	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
12	#20269902	—	2	4	no	—
13	C₂₁₁⋊#95	—	2	1	no	—
14	C₂₁₁⋊#95	—	2	1	no	—
15	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₂₂	—	2	4	no	—
18	C₂×Dic₄₂₂	—	2	2	no	—
19	C₂×Dic₄₂₂	—	2	1	no	—
20	C₂×Dic₄₂₂	—	2	2	no	—
21	C₂×Dic₄₂₂	—	4	2	no	—
22	C₂×Dic₄₂₂	—	4	2	no	—
23	C₂×Dic₄₂₂	Q₈	1	2	yes	complemented
24	C₂²×Dic₂₁₁	—	2	1	no	—
25	C₂²×Dic₂₁₁	—	2	2	no	—
26	C₂²×Dic₂₁₁	—	2	1	no	—
27	C₄⋊Dic₂₁₁	—	2	4	no	—
28	C₄⋊Dic₂₁₁	—	2	2	no	—
29	C₄⋊Dic₂₁₁	—	2	2	no	—
30	C₄⋊Dic₂₁₁	D₄	1	4	yes	—
31	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₂₁₁	—	2	2	no	—
35	C₂×C₈×C₂₁₁	—	2	2	no	—
36	C₂×C₈×C₂₁₁	—	2	2	no	—
37	C₄²×C₂₁₁	—	2	2	no	—
38	Dic₄₂₂	—	2	4	no	—
39	Dic₄₂₂	—	4	8	no	—
40	Dic₄₂₂	—	4	2	no	—
41	C₂×Dic₂₁₁	—	2	4	no	—
42	C₂×Dic₂₁₁	—	2	2	no	—
43	C₂×Dic₂₁₁	—	4	1	no	—
44	C₂×Dic₂₁₁	—	4	1	no	—
45	C₂×Dic₂₁₁	—	4	2	no	—
46	C₂×Dic₂₁₁	—	4	2	no	—
47	C₂×Dic₂₁₁	—	4	1	no	—
48	C₂×Dic₂₁₁	—	4	1	no	—
49	C₂×Dic₂₁₁	—	4	1	no	—
50	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	—	2	1	no	—
52	C₂×C₄×C₂₁₁	—	2	1	no	—
53	C₂×C₄×C₂₁₁	—	2	1	no	—
54	C₂×C₄×C₂₁₁	—	2	1	no	—
55	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
56	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
57	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
60	C₂×C₄×C₂₁₁	#41	1	2	yes	—
61	C₈×C₂₁₁	—	4	2	no	—
62	C₂²×C₂₁₁	#120	1	1	yes	characteristic
63	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₂₁₁	#124	1	1	yes	characteristic
65	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₂₁₁	#137	1	1	yes	characteristic
69	C₄×C₂₁₁	—	2	2	no	—
70	C₄×C₂₁₁	—	2	2	no	—
71	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₂₁₁	#124	1	2	yes	—
73	Dic₂₁₁	—	4	2	no	—
74	Dic₂₁₁	—	4	4	no	—
75	Dic₂₁₁	—	8	1	no	—
76	C₂×C₂₁₁	#393	1	1	yes	characteristic
77	C₂×C₂₁₁	#450	1	1	yes	characteristic
78	C₂×C₂₁₁	#451	1	1	yes	characteristic
79	C₂×C₂₁₁	#475	1	1	yes	characteristic
80	C₂×C₂₁₁	#477	1	1	yes	characteristic
81	C₂×C₂₁₁	#487	1	1	yes	characteristic
82	C₂×C₂₁₁	#489	1	1	yes	characteristic
83	C₂₁₁	#2024	1	1	yes	characteristic, complemented, Sylow
84	#2024	—	211	1	no	maximal, Sylow
85	C₂×C₄⋊Q₈	—	211	1	no	—
86	C₂×#103	—	211	2	no	—
87	#391	—	211	1	no	—
88	#340	—	211	2	no	—
89	#336	D₂₁₁	1	1	yes	characteristic
90	C₂²×Q₈	—	422	1	no	—
91	C₂²×Q₈	—	211	1	no	—
92	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
93	C₄⋊Q₈	—	422	4	no	—
94	C₂×C₄⋊C₄	—	422	1	no	—
95	C₂×C₄⋊C₄	—	211	1	no	—
96	C₂×C₄⋊C₄	—	211	2	no	—
97	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
98	#103	—	422	4	no	—
99	#95	—	422	1	no	—
100	#95	—	422	1	no	—
101	C₂×Q₈	—	844	2	no	—
102	C₂×Q₈	—	844	2	no	—
103	C₂×Q₈	—	422	4	no	—
104	C₂×Q₈	—	422	2	no	—
105	C₂×Q₈	—	422	2	no	—
106	C₂×Q₈	—	422	1	no	—
107	C₂×Q₈	—	211	2	no	—
108	C₂²×C₄	—	422	1	no	—
109	C₂²×C₄	—	422	1	no	—
110	C₂²×C₄	—	422	2	no	—
111	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
112	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	422	4	no	—
118	C₄⋊C₄	—	422	2	no	—
119	C₄⋊C₄	—	422	2	no	—
120	C₄⋊C₄	—	211	4	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
123	Q₈	—	844	8	no	—
124	Q₈	—	844	2	no	—
125	Q₈	—	422	4	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	844	1	no	—
131	C₂×C₄	—	844	1	no	—
132	C₂×C₄	—	844	2	no	—
133	C₂×C₄	—	844	2	no	—
134	C₂×C₄	—	844	1	no	—
135	C₂×C₄	—	844	1	no	—
136	C₂×C₄	—	844	1	no	—
137	C₂×C₄	—	422	4	no	—
138	C₂×C₄	—	422	2	no	—
139	C₂×C₄	#16809951	1	2	yes	—
140	C₂×C₄	Dic₈₄₄	1	2	yes	—
141	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
142	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
143	C₂×C₄	#16809961	1	1	yes	characteristic
144	C₂×C₄	#16809962	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	#20269947	1	1	yes	characteristic
147	C₂²	#20269957	1	1	yes	characteristic
148	C₂²	#20269965	1	1	yes	characteristic
149	C₂²	C₂×#16809951	1	1	yes	characteristic
150	C₂²	C₂×Dic₈₄₄	1	1	yes	characteristic
151	C₂²	#20270005	1	1	yes	characteristic
152	C₂²	#20270010	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	844	2	no	—
156	C₄	—	844	4	no	—
157	C₄	—	1688	1	no	—
158	C₄	#20269955	1	2	yes	—
159	C₄	#20269963	1	2	yes	—
160	C₂	#27213059	1	1	yes	characteristic
161	C₂	#27213299	1	1	yes	characteristic
162	C₂	#27213304	1	1	yes	characteristic
163	C₂	#27213432	1	1	yes	characteristic
164	C₂	#27213442	1	1	yes	characteristic
165	C₂	#27213497	1	1	yes	characteristic
166	C₂	#27213506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269977

C₂

yes

maximal, characteristic

C₂×#20269902

C₂

yes

maximal

#27213052

C₂

yes

maximal, characteristic

#27212909

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

—

C₂²×Dic₄₂₂

C₂²

yes

characteristic

#20269977

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269902

—

C₂₁₁⋊#95

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

Q₈

yes

complemented

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

#120

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₁₁

#124

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#124

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#2024

—

211

maximal, Sylow

C₂×C₄⋊Q₈

—

211

—

C₂×#103

—

211

—

#391

—

211

—

#340

—

211

—

#336

D₂₁₁

yes

characteristic

C₂²×Q₈

—

422

—

C₂²×Q₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₄⋊Q₈

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

#95

—

422

—

100

#95

—

422

—

101

C₂×Q₈

—

844

—

102

C₂×Q₈

—

844

—

103

C₂×Q₈

—

422

—

104

C₂×Q₈

—

422

—

105

C₂×Q₈

—

422

—

106

C₂×Q₈

—

422

—

107

C₂×Q₈

—

211

—

108

C₂²×C₄

—

422

—

109

C₂²×C₄

—

422

—

110

C₂²×C₄

—

422

—

111

C₂²×C₄

D₈₄₄

yes

characteristic

112

C₂²×C₄

D₈₄₄

yes

characteristic

113

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

422

—

118

C₄⋊C₄

—

422

—

119

C₄⋊C₄

—

422

—

120

C₄⋊C₄

—

211

—

121

C₄²

—

122

C₂³

C₂×D₈₄₄

yes

characteristic

123

Q₈

—

844

—

124

Q₈

—

844

—

125

Q₈

—

422

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

844

—

131

C₂×C₄

—

844

—

132

C₂×C₄

—

844

—

133

C₂×C₄

—

844

—

134

C₂×C₄

—

844

—

135

C₂×C₄

—

844

—

136

C₂×C₄

—

844

—

137

C₂×C₄

—

422

—

138

C₂×C₄

—

422

—

139

C₂×C₄

#16809951

yes

—

140

C₂×C₄

Dic₈₄₄

yes

—

141

C₂×C₄

D₄×D₂₁₁

yes

—

142

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

143

C₂×C₄

#16809961

yes

characteristic

144

C₂×C₄

#16809962

yes

—

145

C₈

—

146

C₂²

#20269947

yes

characteristic

147

C₂²

#20269957

yes

characteristic

148

C₂²

#20269965

yes

characteristic

149

C₂²

C₂×#16809951

yes

characteristic

150

C₂²

C₂×Dic₈₄₄

yes

characteristic

151

C₂²

#20270005

yes

characteristic

152

C₂²

#20270010

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

844

—

156

C₄

—

844

—

157

C₄

—

1688

—

158

C₄

#20269955

yes

—

159

C₄

#20269963

yes

—

160

C₂

#27213059

yes

characteristic

161

C₂

#27213299

yes

characteristic

162

C₂

#27213304

yes

characteristic

163

C₂

#27213432

yes

characteristic

164

C₂

#27213442

yes

characteristic

165

C₂

#27213497

yes

characteristic

166

C₂

#27213506

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#336×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
hypercenter	#336	D₂₁₁
hypocenter	C₂₁₁	#2024
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#336×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
23. C₂×Dic₄₂₂	Q₈	semidirect
83. C₂₁₁	#2024	semidirect

H ≅ G/N

type

23. C₂×Dic₄₂₂

Q₈

semidirect

83. C₂₁₁

#2024

semidirect

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

subgroup	index
C₂×#20269977	2
C₂×#20269902	2
#27213052	2
#27212909	2
#336×C₂₁₁	2
#2024	211

subgroup

index

#336×C₂₁₁

211

Subgroups — #41014540

Conjugacy classes of subgroups

Complements

Maximal subgroups