Group #41013443 — subgroups

Subgroups — #41013443

Order 27008.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20269985	C₂	1	1	yes	maximal, characteristic
2	#27213182	C₂	1	2	yes	maximal, complemented
3	#27213119	C₂	1	2	yes	maximal
4	#27212903	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16809962	C₂²	1	1	yes	characteristic
7	#20269985	—	2	2	no	—
8	#20269985	—	2	2	no	—
9	C₂×#16809945	C₄	1	2	yes	complemented
10	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₁₁	C₄	1	2	yes	complemented
12	#20269936	C₂²	1	2	yes	—
13	#20269910	—	2	2	no	—
14	#20269908	—	2	2	no	—
15	#20269891	—	2	4	no	—
16	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
17	#117×C₂₁₁	C₂²	1	2	yes	—
18	#16809962	—	2	2	no	—
19	#16809962	—	2	2	no	—
20	C₂×Dic₄₂₂	Q₈	1	1	yes	characteristic
21	C₂×D₈₄₄	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₂₁₁	D₄	1	2	yes	—
23	C₂×C₄×D₂₁₁	C₂×C₄	1	2	yes	—
24	#16809945	—	2	4	no	—
25	C₂×C₂₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₂₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
28	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
29	C₈×D₂₁₁	—	2	2	no	—
30	C₈×D₂₁₁	—	2	2	no	—
31	C₄⋊Dic₂₁₁	—	2	2	no	—
32	C₂²⋊Dic₂₁₁	—	2	2	no	—
33	C₄×Dic₂₁₁	—	2	2	no	—
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₁₁	—	2	2	no	—
36	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₂₁₁	—	2	2	no	—
38	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
39	C₄²×C₂₁₁	—	2	2	no	—
40	C₂²×D₂₁₁	C₄⋊C₄	1	2	yes	—
41	Dic₄₂₂	—	2	2	no	—
42	D₈₄₄	—	2	2	no	—
43	C₂²⋊D₂₁₁	—	2	4	no	—
44	C₂×Dic₂₁₁	—	4	2	no	—
45	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
46	C₄×D₂₁₁	—	2	2	no	—
47	C₄×D₂₁₁	—	2	2	no	—
48	C₂₁₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂₁₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂₁₁⋊C₈	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	—	4	2	no	—
53	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	#41	1	2	yes	—
57	C₈×C₂₁₁	—	2	2	no	—
58	C₂²×C₂₁₁	—	4	1	no	—
59	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₁₁	#121	1	2	yes	—
61	C₂×D₂₁₁	—	2	2	no	—
62	C₂×D₂₁₁	—	4	2	no	—
63	C₄×C₂₁₁	—	8	2	no	—
64	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₁₁	#117	1	1	yes	characteristic
66	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
67	Dic₂₁₁	—	2	2	no	—
68	Dic₂₁₁	—	8	2	no	—
69	C₂×C₂₁₁	—	2	2	no	—
70	C₂×C₂₁₁	—	2	1	no	—
71	C₂×C₂₁₁	#380	1	1	yes	characteristic
72	D₂₁₁	—	4	2	no	—
73	C₂₁₁	#1812	1	1	yes	characteristic, complemented, Sylow
74	#1812	—	211	1	no	maximal, Sylow
75	C₂×#131	—	211	1	no	—
76	#427	—	211	2	no	—
77	#410	—	211	2	no	—
78	H(3,ℤ/4)	—	211	1	no	—
79	H(3,ℤ/4)	D₂₁₁	1	1	yes	characteristic, complemented
80	C₂×#41	—	211	1	no	—
81	#131	—	422	2	no	—
82	#131	—	422	2	no	—
83	C₂×M₁₆	—	211	2	no	—
84	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
85	C₂²×C₈	—	211	2	no	—
86	C₂²×C₈	—	211	1	no	—
87	#117	—	211	2	no	—
88	#117	C₂×D₂₁₁	1	2	yes	—
89	#107	—	422	2	no	—
90	#106	—	422	2	no	—
91	#99	—	422	4	no	—
92	#41	—	422	2	no	—
93	#41	—	422	2	no	—
94	C₂×Q₈	—	211	1	no	—
95	C₂×D₄	—	211	2	no	—
96	C₂×D₄	—	211	1	no	—
97	C₂²×C₄	—	211	2	no	—
98	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	422	4	no	—
101	C₂×C₈	—	422	2	no	—
102	C₂×C₈	—	422	2	no	—
103	C₂×C₈	—	211	1	no	—
104	C₂×C₈	—	211	1	no	—
105	C₂×C₈	—	211	2	no	—
106	C₂×C₈	—	211	2	no	—
107	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	422	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	422	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	422	2	no	—
114	C₂³	—	211	2	no	—
115	C₂³	D₄×D₂₁₁	1	1	yes	characteristic
116	Q₈	—	422	2	no	—
117	D₄	—	422	4	no	—
118	D₄	—	422	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	844	2	no	—
121	C₂×C₄	—	422	2	no	—
122	C₂×C₄	—	422	2	no	—
123	C₂×C₄	—	211	2	no	—
124	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
125	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
126	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
127	C₂×C₄	#16809962	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	211	1	no	—
130	C₈	—	211	1	no	—
131	C₈	—	211	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	844	2	no	—
134	C₂²	—	422	2	no	—
135	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
136	C₂²	#20269950	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	1688	2	no	—
139	C₄	—	422	2	no	—
140	C₄	C₂²⋊C₄×D₂₁₁	1	1	yes	characteristic
141	C₄	#20269934	1	1	yes	characteristic
142	C₄	#20269955	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	844	2	no	—
146	C₂	#27213005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269985

C₂

yes

maximal, characteristic

#27213182

C₂

yes

maximal, complemented

#27213119

C₂

yes

maximal

#27212903

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×#16809962

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₁₁

C₄

yes

complemented

#20269936

C₂²

yes

—

#20269910

—

#20269908

—

#20269891

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

—

#16809962

—

#16809962

—

C₂×Dic₄₂₂

Q₈

yes

characteristic

C₂×D₈₄₄

Q₈

yes

characteristic

C₂×C₂²⋊D₂₁₁

D₄

yes

—

C₂×C₄×D₂₁₁

C₂×C₄

yes

—

#16809945

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₈×D₂₁₁

—

C₈×D₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂²×D₂₁₁

C₄⋊C₄

yes

—

Dic₄₂₂

—

D₈₄₄

—

C₂²⋊D₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₂₁₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₁₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₁₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

#121

yes

—

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

#117

yes

characteristic

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#380

yes

characteristic

D₂₁₁

—

C₂₁₁

#1812

yes

characteristic, complemented, Sylow

#1812

—

211

maximal, Sylow

C₂×#131

—

211

—

#427

—

211

—

#410

—

211

—

H(3,ℤ/4)

—

211

—

H(3,ℤ/4)

D₂₁₁

yes

characteristic, complemented

C₂×#41

—

211

—

#131

—

422

—

#131

—

422

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

#117

—

211

—

#117

C₂×D₂₁₁

yes

—

#107

—

422

—

#106

—

422

—

#99

—

422

—

#41

—

422

—

#41

—

422

—

C₂×Q₈

—

211

—

C₂×D₄

—

211

—

C₂×D₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

100

M₁₆

—

422

—

101

C₂×C₈

—

422

—

102

C₂×C₈

—

422

—

103

C₂×C₈

—

211

—

104

C₂×C₈

—

211

—

105

C₂×C₈

—

211

—

106

C₂×C₈

—

211

—

107

C₂×C₈

C₄×D₂₁₁

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

422

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

422

—

112

C₄²

—

113

C₄²

—

422

—

114

C₂³

—

211

—

115

C₂³

D₄×D₂₁₁

yes

characteristic

116

Q₈

—

422

—

117

D₄

—

422

—

118

D₄

—

422

—

119

C₂×C₄

—

120

C₂×C₄

—

844

—

121

C₂×C₄

—

422

—

122

C₂×C₄

—

422

—

123

C₂×C₄

—

211

—

124

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

125

C₂×C₄

D₄×D₂₁₁

yes

—

126

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

127

C₂×C₄

#16809962

yes

—

128

C₈

—

129

C₈

—

211

—

130

C₈

—

211

—

131

C₈

—

211

—

132

C₂²

—

133

C₂²

—

844

—

134

C₂²

—

422

—

135

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

136

C₂²

#20269950

yes

—

137

C₄

—

138

C₄

—

1688

—

139

C₄

—

422

—

140

C₄

C₂²⋊C₄×D₂₁₁

yes

characteristic

141

C₄

#20269934

yes

characteristic

142

C₄

#20269955

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

844

—

146

C₂

#27213005

yes

characteristic

	subgroup	G/H
center	C₄	#20269934
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	H(3,ℤ/4)×C₂₁₁	C₂
socle	C₂×C₂₁₁	#380
prime radical	C₂×#16809962	C₂²
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	H(3,ℤ/4)	D₂₁₁
hypocenter	C₂₁₁	#1812
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20269934

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

H(3,ℤ/4)×C₂₁₁

C₂

C₂×C₂₁₁

C₂²

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #27213182	C₂	semidirect
4. #27212903	C₂	semidirect
5. H(3,ℤ/4)×C₂₁₁	C₂	semidirect
9. C₂×#16809945	C₄	semidirect
11. C₂×C₈×D₂₁₁	C₄	semidirect
73. C₂₁₁	#1812	semidirect
79. H(3,ℤ/4)	D₂₁₁	semidirect

H ≅ G/N

type

2. #27213182

C₂

semidirect

4. #27212903

C₂

semidirect

5. H(3,ℤ/4)×C₂₁₁

C₂

semidirect

9. C₂×#16809945

C₄

semidirect

11. C₂×C₈×D₂₁₁

C₄

semidirect

73. C₂₁₁

#1812

semidirect

79. H(3,ℤ/4)

D₂₁₁

semidirect

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

subgroup	index
C₂×#20269985	2
#27213182	2
#27213119	2
#27212903	2
H(3,ℤ/4)×C₂₁₁	2
#1812	211

subgroup

index

H(3,ℤ/4)×C₂₁₁

211

Subgroups — #41013443

Conjugacy classes of subgroups

Complements

Maximal subgroups