Group #41009437 — subgroups

Subgroups — #1817×C₂₁₁ #41009437

Order 27008.

380 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 160 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×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	2	yes	maximal
3	#336×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄×C₂₁₁	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₁₁	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₁₁	C₂²	1	4	yes	—
8	C₂×C₄²×C₂₁₁	—	2	4	no	—
9	C₂×C₄²×C₂₁₁	C₄	1	2	yes	—
10	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
11	C₂²⋊C₈×C₂₁₁	—	2	4	no	—
12	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₁₁	—	2	2	no	—
14	C₂²×C₄×C₂₁₁	—	2	2	no	—
15	C₂²×C₄×C₂₁₁	—	2	2	no	—
16	C₂²×C₄×C₂₁₁	—	2	4	no	—
17	C₂²×C₄×C₂₁₁	—	2	4	no	—
18	C₂²×C₄×C₂₁₁	C₈	1	4	yes	complemented
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
23	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₁₁	—	2	4	no	—
25	C₂×C₈×C₂₁₁	—	2	8	no	—
26	C₄²×C₂₁₁	—	2	4	no	—
27	C₄²×C₂₁₁	—	2	4	no	—
28	C₂³×C₂₁₁	—	2	2	no	—
29	C₂³×C₂₁₁	—	2	2	no	—
30	C₂³×C₂₁₁	C₂²⋊C₄	1	2	yes	—
31	C₂³×C₂₁₁	C₂×C₈	1	2	yes	—
32	C₂³×C₂₁₁	M₁₆	1	2	yes	—
33	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	—	2	8	no	—
35	C₂×C₄×C₂₁₁	—	2	8	no	—
36	C₂×C₄×C₂₁₁	—	2	2	no	—
37	C₂×C₄×C₂₁₁	—	2	2	no	—
38	C₂×C₄×C₂₁₁	—	2	2	no	—
39	C₂×C₄×C₂₁₁	—	2	2	no	—
40	C₂×C₄×C₂₁₁	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	2	8	no	—
42	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₁₁	C₂×C₈	1	4	yes	—
44	C₂×C₄×C₂₁₁	M₁₆	1	4	yes	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	#41	1	4	yes	—
47	C₈×C₂₁₁	—	4	4	no	—
48	C₂²×C₂₁₁	—	2	2	no	—
49	C₂²×C₂₁₁	—	2	2	no	—
50	C₂²×C₂₁₁	—	2	4	no	—
51	C₂²×C₂₁₁	—	2	4	no	—
52	C₂²×C₂₁₁	C₂²⋊C₈	1	4	yes	—
53	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₁₁	#117	1	2	yes	—
55	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₂×M₁₆	1	2	yes	—
58	C₄×C₂₁₁	—	2	2	no	—
59	C₄×C₂₁₁	—	2	4	no	—
60	C₄×C₂₁₁	—	2	4	no	—
61	C₄×C₂₁₁	#123	1	4	yes	—
62	C₂×C₂₁₁	—	2	4	no	—
63	C₂×C₂₁₁	#381	1	1	yes	characteristic
64	C₂×C₂₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₂₁₁	#407	1	1	yes	characteristic
66	C₂×C₂₁₁	#431	1	2	yes	—
67	C₂×C₂₁₁	#432	1	2	yes	—
68	C₂₁₁	#1817	1	1	yes	characteristic, direct product factor, Sylow
69	#1817	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic
71	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	2	yes	—
72	#336	C₂×C₂₁₁	1	4	yes	complemented
73	C₂³×C₄	C₄×C₂₁₁	1	1	yes	characteristic
74	C₂³×C₄	C₄×C₂₁₁	1	1	yes	characteristic
75	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₂₁₁	1	4	yes	—
77	C₂×C₄²	—	2	4	no	—
78	C₂×C₄²	C₄×C₂₁₁	1	2	yes	—
79	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
80	C₂²⋊C₈	—	2	4	no	—
81	C₂⁴	C₂×C₄×C₂₁₁	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	4	no	—
87	C₂²×C₄	C₈×C₂₁₁	1	4	yes	complemented
88	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
89	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
90	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
91	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
92	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	8	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	C₂²⋊C₄×C₂₁₁	1	2	yes	—
100	C₂³	C₂×C₈×C₂₁₁	1	2	yes	—
101	C₂³	M₁₆×C₂₁₁	1	2	yes	—
102	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	8	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
112	C₂×C₄	C₂×C₈×C₂₁₁	1	4	yes	—
113	C₂×C₄	M₁₆×C₂₁₁	1	4	yes	—
114	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
115	C₂×C₄	#41×C₂₁₁	1	4	yes	—
116	C₈	—	4	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	4	no	—
121	C₂²	C₂²⋊C₈×C₂₁₁	1	4	yes	—
122	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
123	C₂²	#117×C₂₁₁	1	2	yes	—
124	C₂²	C₂²×C₈×C₂₁₁	1	1	yes	characteristic
125	C₂²	C₂×M₁₆×C₂₁₁	1	1	yes	characteristic
126	C₂²	C₂×M₁₆×C₂₁₁	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	2	4	no	—
129	C₄	—	2	4	no	—
130	C₄	#123×C₂₁₁	1	4	yes	—
131	C₂	—	2	4	no	—
132	C₂	#381×C₂₁₁	1	1	yes	characteristic
133	C₂	C₂×C₂²⋊C₈×C₂₁₁	1	1	yes	characteristic
134	C₂	#407×C₂₁₁	1	1	yes	characteristic
135	C₂	#431×C₂₁₁	1	2	yes	—
136	C₂	#432×C₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄²×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₂₁₁

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, complemented

C₂³×C₄×C₂₁₁

C₄

yes

characteristic

C₂³×C₄×C₂₁₁

C₄

yes

characteristic

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

—

C₂²⋊C₈×C₂₁₁

—

C₂⁴×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₈

yes

complemented

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₂²⋊C₄

yes

—

C₂³×C₂₁₁

C₂×C₈

yes

—

C₂³×C₂₁₁

M₁₆

yes

—

C₂³×C₂₁₁

C₂²×C₄

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

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×C₈

yes

—

C₂×C₄×C₂₁₁

M₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂²⋊C₈

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

#117

yes

—

C₂²×C₂₁₁

C₂²×C₈

yes

characteristic

C₂²×C₂₁₁

C₂×M₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×M₁₆

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#123

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#381

yes

characteristic

C₂×C₂₁₁

C₂×C₂²⋊C₈

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#432

yes

—

C₂₁₁

#1817

yes

characteristic, direct product factor, Sylow

#1817

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄²

C₂×C₂₁₁

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₂₁₁

yes

—

#336

C₂×C₂₁₁

yes

complemented

C₂³×C₄

C₄×C₂₁₁

yes

characteristic

C₂³×C₄

C₄×C₂₁₁

yes

characteristic

C₂³×C₄

C₂²×C₂₁₁

yes

characteristic

C₂²×C₈

C₂²×C₂₁₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₄×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

—

C₂²⋊C₈

—

C₂⁴

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₈×C₂₁₁

yes

complemented

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₂₁₁

yes

—

100

C₂³

C₂×C₈×C₂₁₁

yes

—

101

C₂³

M₁₆×C₂₁₁

yes

—

102

C₂³

C₂²×C₄×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

C₂²⋊C₄×C₂₁₁

yes

—

112

C₂×C₄

C₂×C₈×C₂₁₁

yes

—

113

C₂×C₄

M₁₆×C₂₁₁

yes

—

114

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

115

C₂×C₄

#41×C₂₁₁

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²⋊C₈×C₂₁₁

yes

—

122

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

123

C₂²

#117×C₂₁₁

yes

—

124

C₂²

C₂²×C₈×C₂₁₁

yes

characteristic

125

C₂²

C₂×M₁₆×C₂₁₁

yes

characteristic

126

C₂²

C₂×M₁₆×C₂₁₁

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#123×C₂₁₁

yes

—

131

C₂

—

132

C₂

#381×C₂₁₁

yes

characteristic

133

C₂

C₂×C₂²⋊C₈×C₂₁₁

yes

characteristic

134

C₂

#407×C₂₁₁

yes

characteristic

135

C₂

#431×C₂₁₁

yes

—

136

C₂

#432×C₂₁₁

yes

—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂²×C₈×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂³×C₂₁₁	C₂²×C₄
prime radical	C₂⁴×C₂₁₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₁₁

C₂³

commutator

C₂²

C₂²×C₈×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

socle

C₂³×C₂₁₁

C₂²×C₄

prime radical

C₂⁴×C₂₁₁

C₂×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

N	H ≅ G/N	type
3. #336×C₂₁₁	C₂	semidirect
18. C₂²×C₄×C₂₁₁	C₈	semidirect
68. C₂₁₁	#1817	direct (N × H)
69. #1817	C₂₁₁	direct (N × H)
72. #336	C₂×C₂₁₁	semidirect
87. C₂²×C₄	C₈×C₂₁₁	semidirect

H ≅ G/N

type

3. #336×C₂₁₁

C₂

semidirect

18. C₂²×C₄×C₂₁₁

C₈

semidirect

68. C₂₁₁

#1817

direct (N × H)

69. #1817

C₂₁₁

direct (N × H)

72. #336

C₂×C₂₁₁

semidirect

87. C₂²×C₄

C₈×C₂₁₁

semidirect

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

subgroup	index
C₂²×C₄²×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
#336×C₂₁₁	2
#1817	211

subgroup

index

C₂²×C₄²×C₂₁₁

C₂×C₂²⋊C₈×C₂₁₁

#336×C₂₁₁

#1817

211

Subgroups — #1817×C₂₁₁ #41009437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1817×C211 #41009437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1817×C₂₁₁ #41009437