Group #41009488 — subgroups

Subgroups — #1868×C₂₁₁ #41009488

Order 27008.

260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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	#568×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#424×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#410×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#408×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#344×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
7	#131×C₂₁₁	—	2	4	no	—
8	C₂×M₁₆×C₂₁₁	—	2	2	no	—
9	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	#117×C₂₁₁	C₂²	1	2	yes	—
13	C₄⋊C₈×C₂₁₁	—	2	4	no	—
14	#99×C₂₁₁	C₄	1	4	yes	—
15	C₂²⋊C₈×C₂₁₁	—	2	2	no	—
16	#41×C₂₁₁	—	2	2	no	—
17	#41×C₂₁₁	—	2	2	no	—
18	C₂×Q₈×C₂₁₁	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₂₁₁	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₂₁₁	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
22	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₁₁	—	2	4	no	—
24	M₁₆×C₂₁₁	—	2	2	no	—
25	M₁₆×C₂₁₁	—	2	2	no	—
26	M₁₆×C₂₁₁	—	2	2	no	—
27	C₂×C₈×C₂₁₁	—	2	4	no	—
28	C₂×C₈×C₂₁₁	—	2	1	no	—
29	C₂×C₈×C₂₁₁	—	2	1	no	—
30	C₂×C₈×C₂₁₁	—	2	2	no	—
31	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
32	C₄⋊C₄×C₂₁₁	—	2	2	no	—
33	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
34	C₄²×C₂₁₁	—	2	2	no	—
35	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
36	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₂₁₁	—	2	2	no	—
38	D₄×C₂₁₁	—	2	4	no	—
39	D₄×C₂₁₁	—	2	2	no	—
40	C₂×C₄×C₂₁₁	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	2	2	no	—
42	C₂×C₄×C₂₁₁	—	2	4	no	—
43	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	#41	1	2	yes	—
48	C₈×C₂₁₁	—	2	4	no	—
49	C₈×C₂₁₁	—	4	1	no	—
50	C₈×C₂₁₁	—	4	1	no	—
51	C₂²×C₂₁₁	—	2	2	no	—
52	C₂²×C₂₁₁	—	4	1	no	—
53	C₂²×C₂₁₁	—	4	2	no	—
54	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₁₁	C₄×D₄	1	2	yes	—
56	C₄×C₂₁₁	—	2	2	no	—
57	C₄×C₂₁₁	—	4	4	no	—
58	C₄×C₂₁₁	#120	1	1	yes	characteristic
59	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₂₁₁	#123	1	1	yes	characteristic
61	C₂×C₂₁₁	—	2	2	no	—
62	C₂×C₂₁₁	—	2	1	no	—
63	C₂×C₂₁₁	—	4	2	no	—
64	C₂×C₂₁₁	#385	1	1	yes	characteristic
65	C₂₁₁	#1868	1	1	yes	characteristic, direct product factor, Sylow
66	#1868	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	#568	C₂×C₂₁₁	1	1	yes	characteristic
68	#424	C₂×C₂₁₁	1	2	yes	complemented
69	#410	C₂×C₂₁₁	1	1	yes	characteristic
70	#408	C₂×C₂₁₁	1	1	yes	characteristic
71	#344	C₂×C₂₁₁	1	2	yes	complemented
72	C₂×#41	C₂²×C₂₁₁	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
76	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
77	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
78	#117	C₂²×C₂₁₁	1	2	yes	—
79	C₄⋊C₈	—	2	4	no	—
80	#99	C₄×C₂₁₁	1	4	yes	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Q₈×C₂₁₁	1	1	yes	characteristic
85	C₂×D₄	C₂×C₄×C₂₁₁	1	2	yes	—
86	C₂×D₄	Q₈×C₂₁₁	1	1	yes	characteristic
87	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
88	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	D₄×C₂₁₁	1	4	yes	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₂³	C₄⋊C₄×C₂₁₁	1	2	yes	—
102	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
110	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
111	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
112	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
113	C₂×C₄	#41×C₂₁₁	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	4	1	no	—
116	C₈	—	4	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
121	C₂²	C₄×D₄×C₂₁₁	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	#120×C₂₁₁	1	1	yes	characteristic
125	C₄	C₂²⋊Q₈×C₂₁₁	1	2	yes	—
126	C₄	#123×C₂₁₁	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#385×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#568×C₂₁₁

C₂

yes

maximal, characteristic

#424×C₂₁₁

C₂

yes

maximal, complemented

#410×C₂₁₁

C₂

yes

maximal, characteristic

#408×C₂₁₁

C₂

yes

maximal, characteristic

#344×C₂₁₁

C₂

yes

maximal, complemented

C₂×#41×C₂₁₁

C₂²

yes

characteristic

#131×C₂₁₁

—

C₂×M₁₆×C₂₁₁

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

—

C₄⋊C₈×C₂₁₁

—

#99×C₂₁₁

C₄

yes

—

C₂²⋊C₈×C₂₁₁

—

#41×C₂₁₁

—

#41×C₂₁₁

—

C₂×Q₈×C₂₁₁

Q₈

yes

characteristic

C₂×D₄×C₂₁₁

C₂×C₄

yes

—

C₂×D₄×C₂₁₁

Q₈

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#120

yes

characteristic

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#123

yes

characteristic

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#385

yes

characteristic

C₂₁₁

#1868

yes

characteristic, direct product factor, Sylow

#1868

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

#568

C₂×C₂₁₁

yes

characteristic

#424

C₂×C₂₁₁

yes

complemented

#410

C₂×C₂₁₁

yes

characteristic

#408

C₂×C₂₁₁

yes

characteristic

#344

C₂×C₂₁₁

yes

complemented

C₂×#41

C₂²×C₂₁₁

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×C₂₁₁

yes

characteristic

C₂×M₁₆

C₂²×C₂₁₁

yes

—

C₂²×C₈

C₂²×C₂₁₁

yes

characteristic

#117

C₂²×C₂₁₁

yes

—

C₄⋊C₈

—

#99

C₄×C₂₁₁

yes

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Q₈×C₂₁₁

yes

characteristic

C₂×D₄

C₂×C₄×C₂₁₁

yes

—

C₂×D₄

Q₈×C₂₁₁

yes

characteristic

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₁₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

C₂×C₄×C₂₁₁

yes

—

100

C₄²

—

101

C₂³

C₄⋊C₄×C₂₁₁

yes

—

102

C₂³

C₂²×C₄×C₂₁₁

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊C₄×C₂₁₁

yes

—

110

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

111

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

112

C₂×C₄

C₂×Q₈×C₂₁₁

yes

characteristic

113

C₂×C₄

#41×C₂₁₁

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄⋊C₄×C₂₁₁

yes

characteristic

121

C₂²

C₄×D₄×C₂₁₁

yes

—

122

C₄

—

123

C₄

—

124

C₄

#120×C₂₁₁

yes

characteristic

125

C₄

C₂²⋊Q₈×C₂₁₁

yes

—

126

C₄

#123×C₂₁₁

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#385×C₂₁₁

yes

characteristic

	subgroup	G/H
center	C₄×C₂₁₁	#120
commutator	C₂³	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂×C₂₁₁	#385
prime radical	C₂×#41×C₂₁₁	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₁₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

subgroup

G/H

center

C₄×C₂₁₁

#120

commutator

C₂³

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

socle

C₂×C₂₁₁

#385

prime radical

C₂×#41×C₂₁₁

C₂²

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₂₁₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

N	H ≅ G/N	type
2. #424×C₂₁₁	C₂	semidirect
5. #344×C₂₁₁	C₂	semidirect
65. C₂₁₁	#1868	direct (N × H)
66. #1868	C₂₁₁	direct (N × H)
68. #424	C₂×C₂₁₁	semidirect
71. #344	C₂×C₂₁₁	semidirect

H ≅ G/N

type

2. #424×C₂₁₁

C₂

semidirect

5. #344×C₂₁₁

C₂

semidirect

65. C₂₁₁

#1868

direct (N × H)

66. #1868

C₂₁₁

direct (N × H)

68. #424

C₂×C₂₁₁

semidirect

71. #344

C₂×C₂₁₁

semidirect

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

subgroup	index
#568×C₂₁₁	2
#424×C₂₁₁	2
#410×C₂₁₁	2
#408×C₂₁₁	2
#344×C₂₁₁	2
#1868	211

subgroup

index

#568×C₂₁₁

#424×C₂₁₁

#410×C₂₁₁

#408×C₂₁₁

#344×C₂₁₁

#1868

211

Subgroups — #1868×C₂₁₁ #41009488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1868×C211 #41009488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1868×C₂₁₁ #41009488