Group #41014721 — subgroups

Subgroups — #41014721

Order 27008.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269902	C₂	1	2	yes	maximal
3	#27213052	C₂	1	2	yes	maximal
4	#27212909	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₂	C₂²	1	2	yes	—
7	C₂×Dic₈₄₄	—	2	4	no	—
8	C₂×Dic₈₄₄	—	2	2	no	—
9	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	#20269902	—	2	4	no	—
11	C₂₁₁⋊#95	—	2	2	no	—
12	C₂₁₁⋊#95	—	2	2	no	—
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₁₁	—	2	2	no	—
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	—	2	2	no	—
19	C₂×Dic₄₂₂	—	2	2	no	—
20	C₂×Dic₄₂₂	—	2	2	no	—
21	C₂×Dic₄₂₂	—	2	2	no	—
22	C₂×Dic₄₂₂	D₄	1	4	yes	—
23	C₂²×Dic₂₁₁	—	2	2	no	—
24	C₂²×Dic₂₁₁	—	2	2	no	—
25	Dic₈₄₄	—	2	8	no	—
26	C₄⋊Dic₂₁₁	—	2	2	no	—
27	C₄⋊Dic₂₁₁	—	2	2	no	—
28	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₁₁	—	2	2	no	—
32	C₂×C₈×C₂₁₁	—	2	1	no	—
33	C₂×C₈×C₂₁₁	—	2	2	no	—
34	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
35	C₄²×C₂₁₁	—	2	1	no	—
36	C₄²×C₂₁₁	—	2	1	no	—
37	Dic₄₂₂	—	2	4	no	—
38	Dic₄₂₂	—	2	4	no	—
39	Dic₄₂₂	—	4	2	no	—
40	Dic₄₂₂	—	4	2	no	—
41	C₂×Dic₂₁₁	—	2	4	no	—
42	C₂×Dic₂₁₁	—	4	2	no	—
43	C₂×Dic₂₁₁	—	4	2	no	—
44	C₂×Dic₂₁₁	—	4	2	no	—
45	C₂×Dic₂₁₁	—	4	2	no	—
46	C₂×Dic₂₁₁	—	4	2	no	—
47	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₁₁	—	2	1	no	—
49	C₂×C₄×C₂₁₁	—	2	1	no	—
50	C₂×C₄×C₂₁₁	—	2	1	no	—
51	C₂×C₄×C₂₁₁	—	2	2	no	—
52	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	complemented
53	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	#41	1	2	yes	—
57	C₈×C₂₁₁	—	2	2	no	—
58	C₈×C₂₁₁	—	4	1	no	—
59	C₂²×C₂₁₁	#121	1	1	yes	characteristic
60	C₂²×C₂₁₁	#121	1	2	yes	—
61	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₂₁₁	#135	1	1	yes	characteristic
63	C₂²×C₂₁₁	#137	1	2	yes	—
64	C₄×C₂₁₁	—	2	2	no	—
65	C₄×C₂₁₁	—	4	1	no	—
66	C₄×C₂₁₁	#120	1	1	yes	characteristic
67	C₄×C₂₁₁	#123	1	2	yes	—
68	C₄×C₂₁₁	#124	1	1	yes	characteristic
69	Dic₂₁₁	—	4	4	no	—
70	Dic₂₁₁	—	8	2	no	—
71	C₂×C₂₁₁	#394	1	1	yes	characteristic
72	C₂×C₂₁₁	#462	1	2	yes	—
73	C₂×C₂₁₁	#464	1	2	yes	—
74	C₂×C₂₁₁	#467	1	1	yes	characteristic
75	C₂×C₂₁₁	#470	1	1	yes	characteristic
76	C₂₁₁	#2056	1	1	yes	characteristic, complemented, Sylow
77	#2056	—	211	1	no	maximal, Sylow
78	C₂²×Q₁₆	—	211	1	no	—
79	C₂×C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
80	C₂×#103	—	211	2	no	—
81	#391	—	211	2	no	—
82	#340	—	211	1	no	—
83	C₂²×Q₈	—	211	2	no	—
84	C₂×Q₁₆	—	422	4	no	—
85	C₂×Q₁₆	—	422	2	no	—
86	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	211	2	no	—
89	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	422	4	no	—
92	#95	—	422	2	no	—
93	#95	—	422	2	no	—
94	C₂×Q₈	—	422	2	no	—
95	C₂×Q₈	—	422	2	no	—
96	C₂×Q₈	—	422	2	no	—
97	C₂×Q₈	—	422	2	no	—
98	C₂×Q₈	—	422	2	no	—
99	C₂×Q₈	—	211	4	no	—
100	C₂²×C₄	—	422	2	no	—
101	C₂²×C₄	—	422	2	no	—
102	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
103	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
105	Q₁₆	—	422	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
110	C₄⋊C₄	—	422	2	no	—
111	C₄⋊C₄	—	422	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
115	Q₈	—	844	2	no	—
116	Q₈	—	844	2	no	—
117	Q₈	—	422	4	no	—
118	Q₈	—	422	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	844	2	no	—
124	C₂×C₄	—	844	2	no	—
125	C₂×C₄	—	844	2	no	—
126	C₂×C₄	—	844	2	no	—
127	C₂×C₄	—	844	2	no	—
128	C₂×C₄	—	422	4	no	—
129	C₂×C₄	Dic₈₄₄	1	2	yes	complemented
130	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
132	C₂×C₄	#16809962	1	1	yes	characteristic
133	C₂×C₄	#16809963	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#20269951	1	1	yes	characteristic
137	C₂²	#20269952	1	2	yes	—
138	C₂²	C₂×Dic₈₄₄	1	1	yes	characteristic
139	C₂²	#20269999	1	1	yes	characteristic
140	C₂²	#20270010	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	844	4	no	—
144	C₄	—	1688	2	no	—
145	C₄	#20269945	1	1	yes	characteristic
146	C₄	#20269962	1	2	yes	—
147	C₄	#20269966	1	1	yes	characteristic
148	C₂	#27213063	1	1	yes	characteristic
149	C₂	#27213357	1	2	yes	—
150	C₂	#27213369	1	2	yes	—
151	C₂	#27213387	1	1	yes	characteristic
152	C₂	#27213404	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₈₄₄

C₂

yes

maximal, characteristic

C₂×#20269902

C₂

yes

maximal

#27213052

C₂

yes

maximal

#27212909

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

—

C₂×Dic₈₄₄

—

C₂×Dic₈₄₄

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269902

—

C₂₁₁⋊#95

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

D₄

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

Dic₈₄₄

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

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

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

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

Q₁₆

yes

complemented

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₂²×C₂₁₁

#121

yes

—

C₂²×C₂₁₁

C₂×Q₁₆

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#120

yes

characteristic

C₄×C₂₁₁

#123

yes

—

C₄×C₂₁₁

#124

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#394

yes

characteristic

C₂×C₂₁₁

#462

yes

—

C₂×C₂₁₁

#464

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#2056

—

211

maximal, Sylow

C₂²×Q₁₆

—

211

—

C₂×C₄⋊C₈

D₂₁₁

yes

characteristic

C₂×#103

—

211

—

#391

—

211

—

#340

—

211

—

C₂²×Q₈

—

211

—

C₂×Q₁₆

—

422

—

C₂×Q₁₆

—

422

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

#103

—

422

—

#95

—

422

—

#95

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

100

C₂²×C₄

—

422

—

101

C₂²×C₄

—

422

—

102

C₂²×C₄

D₈₄₄

yes

characteristic

103

C₂²×C₄

D₈₄₄

yes

characteristic

104

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

105

Q₁₆

—

422

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₂₁₁

yes

—

110

C₄⋊C₄

—

422

—

111

C₄⋊C₄

—

422

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₈₄₄

yes

characteristic

115

Q₈

—

844

—

116

Q₈

—

844

—

117

Q₈

—

422

—

118

Q₈

—

422

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

844

—

124

C₂×C₄

—

844

—

125

C₂×C₄

—

844

—

126

C₂×C₄

—

844

—

127

C₂×C₄

—

844

—

128

C₂×C₄

—

422

—

129

C₂×C₄

Dic₈₄₄

yes

complemented

130

C₂×C₄

D₄×D₂₁₁

yes

—

131

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

132

C₂×C₄

#16809962

yes

characteristic

133

C₂×C₄

#16809963

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#20269951

yes

characteristic

137

C₂²

#20269952

yes

—

138

C₂²

C₂×Dic₈₄₄

yes

characteristic

139

C₂²

#20269999

yes

characteristic

140

C₂²

#20270010

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

844

—

144

C₄

—

1688

—

145

C₄

#20269945

yes

characteristic

146

C₄

#20269962

yes

—

147

C₄

#20269966

yes

characteristic

148

C₂

#27213063

yes

characteristic

149

C₂

#27213357

yes

—

150

C₂

#27213369

yes

—

151

C₂

#27213387

yes

characteristic

152

C₂

#27213404

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄⋊C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#2056
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄⋊C₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₂₁₁

hypocenter

C₂₁₁

#2056

LCS

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
52. C₂×C₄×C₂₁₁	Q₁₆	semidirect
76. C₂₁₁	#2056	semidirect
129. C₂×C₄	Dic₈₄₄	semidirect

H ≅ G/N

type

52. C₂×C₄×C₂₁₁

Q₁₆

semidirect

76. C₂₁₁

#2056

semidirect

129. C₂×C₄

Dic₈₄₄

semidirect

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

subgroup	index
C₂²×Dic₈₄₄	2
C₂×#20269902	2
#27213052	2
#27212909	2
C₂×C₄⋊C₈×C₂₁₁	2
#2056	211

subgroup

index

C₂²×Dic₈₄₄

C₂×C₄⋊C₈×C₂₁₁

211

Subgroups — #41014721

Conjugacy classes of subgroups

Complements

Maximal subgroups