Group #41009396 — subgroups

Subgroups — #1776×C₂₁₁ #41009396

Order 27008.

272 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 168 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×#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×C₂₁₁	C₂	1	2	yes	maximal
5	#340×C₂₁₁	C₂	1	2	yes	maximal
6	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₁₁	—	2	2	no	—
11	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	#107×C₂₁₁	C₄	1	4	yes	complemented
13	#106×C₂₁₁	C₄	1	4	yes	complemented
14	#97×C₂₁₁	C₄	1	4	yes	complemented
15	#95×C₂₁₁	—	2	2	no	—
16	C₂²×C₄×C₂₁₁	—	2	2	no	—
17	C₂²×C₄×C₂₁₁	—	2	2	no	—
18	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
19	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₂₁₁	—	2	1	no	—
21	C₂×C₈×C₂₁₁	—	2	1	no	—
22	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₄⋊C₄×C₂₁₁	—	2	2	no	—
27	C₄⋊C₄×C₂₁₁	—	2	2	no	—
28	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
29	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₄²×C₂₁₁	—	4	4	no	—
31	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
32	C₄²×C₂₁₁	Q₈	1	2	yes	—
33	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	2	2	no	—
36	C₂×C₄×C₂₁₁	—	2	4	no	—
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	—	4	2	no	—
39	C₂×C₄×C₂₁₁	—	4	2	no	—
40	C₂×C₄×C₂₁₁	—	4	2	no	—
41	C₂×C₄×C₂₁₁	—	4	2	no	—
42	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
48	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
49	C₈×C₂₁₁	—	2	2	no	—
50	C₈×C₂₁₁	C₄²	1	4	yes	complemented
51	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₂₁₁	#137	1	2	yes	—
56	C₄×C₂₁₁	—	2	4	no	—
57	C₄×C₂₁₁	—	4	4	no	—
58	C₄×C₂₁₁	—	4	4	no	—
59	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
60	C₄×C₂₁₁	#117	1	1	yes	characteristic
61	C₄×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
62	C₄×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₂₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₂₁₁	#428	1	2	yes	—
65	C₂×C₂₁₁	#440	1	2	yes	—
66	C₂×C₂₁₁	#441	1	1	yes	characteristic
67	C₂×C₂₁₁	#442	1	1	yes	characteristic
68	C₂₁₁	#1776	1	1	yes	characteristic, direct product factor, Sylow
69	#1776	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×#107	C₂×C₂₁₁	1	1	yes	characteristic
71	C₂×#106	C₂×C₂₁₁	1	1	yes	characteristic
72	C₂×#97	C₂×C₂₁₁	1	1	yes	characteristic
73	C₄×C₄⋊C₄	C₂×C₂₁₁	1	2	yes	—
74	#340	C₂×C₂₁₁	1	2	yes	—
75	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
79	C₂×C₄²	—	2	2	no	—
80	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
81	#107	C₄×C₂₁₁	1	4	yes	complemented
82	#106	C₄×C₂₁₁	1	4	yes	complemented
83	#97	C₄×C₂₁₁	1	4	yes	complemented
84	#95	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
88	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
92	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
93	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
94	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
98	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
99	C₄²	—	4	4	no	—
100	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
101	C₄²	Q₈×C₂₁₁	1	2	yes	—
102	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₄⋊C₄×C₂₁₁	1	4	yes	—
112	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
115	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
116	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
117	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	C₄²×C₂₁₁	1	4	yes	complemented
120	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
121	C₂²	C₄×D₄×C₂₁₁	1	1	yes	characteristic
122	C₂²	C₄×D₄×C₂₁₁	1	1	yes	characteristic
123	C₂²	AGL(1,ℤ/8)×C₂₁₁	1	2	yes	—
124	C₂²	#137×C₂₁₁	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	4	4	no	—
127	C₄	—	4	4	no	—
128	C₄	C₂×C₄²×C₂₁₁	1	1	yes	characteristic
129	C₄	#117×C₂₁₁	1	1	yes	characteristic
130	C₄	C₄×Q₈×C₂₁₁	1	1	yes	characteristic
131	C₄	C₄×Q₈×C₂₁₁	1	1	yes	characteristic
132	C₂	C₄×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
133	C₂	#428×C₂₁₁	1	2	yes	—
134	C₂	#440×C₂₁₁	1	2	yes	—
135	C₂	#441×C₂₁₁	1	1	yes	characteristic
136	C₂	#442×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#107×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#106×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#97×C₂₁₁

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₂₁₁

C₂

yes

maximal

#340×C₂₁₁

C₂

yes

maximal

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#107×C₂₁₁

C₄

yes

complemented

#106×C₂₁₁

C₄

yes

complemented

#97×C₂₁₁

C₄

yes

complemented

#95×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

Q₈

yes

—

C₂³×C₂₁₁

C₂×D₄

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

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₈×C₂₁₁

C₄²

yes

complemented

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×D₄

yes

characteristic

C₂²×C₂₁₁

C₄×D₄

yes

characteristic

C₂²×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄²

yes

characteristic

C₄×C₂₁₁

#117

yes

characteristic

C₄×C₂₁₁

C₄×Q₈

yes

characteristic

C₄×C₂₁₁

C₄×Q₈

yes

characteristic

C₂×C₂₁₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₁₁

#428

yes

—

C₂×C₂₁₁

#440

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#1776

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#107

C₂×C₂₁₁

yes

characteristic

C₂×#106

C₂×C₂₁₁

yes

characteristic

C₂×#97

C₂×C₂₁₁

yes

characteristic

C₄×C₄⋊C₄

C₂×C₂₁₁

yes

—

#340

C₂×C₂₁₁

yes

—

C₂²×C₈

C₂²×C₂₁₁

yes

characteristic

C₂²×C₈

C₂²×C₂₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

#107

C₄×C₂₁₁

yes

complemented

#106

C₄×C₂₁₁

yes

complemented

#97

C₄×C₂₁₁

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₁₁

yes

—

C₄⋊C₄

C₂×C₄×C₂₁₁

yes

—

C₄²

—

100

C₄²

C₂×C₄×C₂₁₁

yes

—

101

C₄²

Q₈×C₂₁₁

yes

—

102

C₂³

C₂×D₄×C₂₁₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₄⋊C₄×C₂₁₁

yes

—

112

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

113

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

114

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

115

C₂×C₄

C₂×Q₈×C₂₁₁

yes

characteristic

116

C₂×C₄

#41×C₂₁₁

yes

characteristic

117

C₂×C₄

#41×C₂₁₁

yes

characteristic

118

C₈

—

119

C₈

C₄²×C₂₁₁

yes

complemented

120

C₂²

C₂×C₄⋊C₄×C₂₁₁

yes

characteristic

121

C₂²

C₄×D₄×C₂₁₁

yes

characteristic

122

C₂²

C₄×D₄×C₂₁₁

yes

characteristic

123

C₂²

AGL(1,ℤ/8)×C₂₁₁

yes

—

124

C₂²

#137×C₂₁₁

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂×C₄²×C₂₁₁

yes

characteristic

129

C₄

#117×C₂₁₁

yes

characteristic

130

C₄

C₄×Q₈×C₂₁₁

yes

characteristic

131

C₄

C₄×Q₈×C₂₁₁

yes

characteristic

132

C₂

C₄×C₄⋊C₄×C₂₁₁

yes

characteristic

133

C₂

#428×C₂₁₁

yes

—

134

C₂

#440×C₂₁₁

yes

—

135

C₂

#441×C₂₁₁

yes

characteristic

136

C₂

#442×C₂₁₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₁₁

C₂×D₄

commutator

C₄

C₂×C₄²×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
12. #107×C₂₁₁	C₄	semidirect
13. #106×C₂₁₁	C₄	semidirect
14. #97×C₂₁₁	C₄	semidirect
50. C₈×C₂₁₁	C₄²	semidirect
68. C₂₁₁	#1776	direct (N × H)
69. #1776	C₂₁₁	direct (N × H)
81. #107	C₄×C₂₁₁	semidirect
82. #106	C₄×C₂₁₁	semidirect
83. #97	C₄×C₂₁₁	semidirect
119. C₈	C₄²×C₂₁₁	semidirect

H ≅ G/N

type

12. #107×C₂₁₁

C₄

semidirect

13. #106×C₂₁₁

C₄

semidirect

14. #97×C₂₁₁

C₄

semidirect

50. C₈×C₂₁₁

C₄²

semidirect

68. C₂₁₁

#1776

direct (N × H)

69. #1776

C₂₁₁

direct (N × H)

81. #107

C₄×C₂₁₁

semidirect

82. #106

C₄×C₂₁₁

semidirect

83. #97

C₄×C₂₁₁

semidirect

119. C₈

C₄²×C₂₁₁

semidirect

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

subgroup	index
C₂×#107×C₂₁₁	2
C₂×#106×C₂₁₁	2
C₂×#97×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
#340×C₂₁₁	2
#1776	211

subgroup

index

C₂×#107×C₂₁₁

C₂×#106×C₂₁₁

C₂×#97×C₂₁₁

C₄×C₄⋊C₄×C₂₁₁

#340×C₂₁₁

#1776

211

Subgroups — #1776×C₂₁₁ #41009396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1776×C211 #41009396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1776×C₂₁₁ #41009396