Group #41009433 — subgroups

Subgroups — #1813×C₂₁₁ #41009433

Order 27008.

252 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 116 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×#131×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#413×C₂₁₁	C₂	1	2	yes	maximal
4	#338×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
6	#131×C₂₁₁	—	2	4	no	—
7	C₂×M₁₆×C₂₁₁	C₄	1	2	yes	—
8	C₂×M₁₆×C₂₁₁	C₂²	1	4	yes	—
9	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₁₁	C₄	1	2	yes	—
11	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	#108×C₂₁₁	—	2	4	no	—
13	#101×C₂₁₁	—	2	2	no	—
14	#100×C₂₁₁	—	2	2	no	—
15	#41×C₂₁₁	—	2	2	no	—
16	#41×C₂₁₁	—	2	2	no	—
17	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
18	C₂×D₄×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂×D₄×C₂₁₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₂₁₁	—	2	2	no	—
23	M₁₆×C₂₁₁	—	2	4	no	—
24	M₁₆×C₂₁₁	—	2	4	no	—
25	M₁₆×C₂₁₁	—	2	4	no	—
26	C₂×C₈×C₂₁₁	—	2	4	no	—
27	C₂×C₈×C₂₁₁	—	2	4	no	—
28	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
32	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
33	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
34	Q₈×C₂₁₁	—	2	2	no	—
35	D₄×C₂₁₁	—	2	4	no	—
36	D₄×C₂₁₁	—	2	2	no	—
37	C₂×C₄×C₂₁₁	—	2	2	no	—
38	C₂×C₄×C₂₁₁	—	2	2	no	—
39	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	#41	1	2	yes	—
44	C₈×C₂₁₁	—	2	2	no	—
45	C₈×C₂₁₁	—	4	4	no	—
46	C₈×C₂₁₁	C₂²⋊C₄	1	2	yes	—
47	C₈×C₂₁₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₂₁₁	—	2	2	no	—
49	C₂²×C₂₁₁	—	4	1	no	—
50	C₂²×C₂₁₁	—	4	2	no	—
51	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₂₁₁	—	2	2	no	—
54	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₂₁₁	#117	1	1	yes	characteristic
56	C₄×C₂₁₁	#121	1	2	yes	—
57	C₂×C₂₁₁	—	2	1	no	—
58	C₂×C₂₁₁	—	2	2	no	—
59	C₂×C₂₁₁	—	4	2	no	—
60	C₂×C₂₁₁	#380	1	1	yes	characteristic
61	C₂₁₁	#1813	1	1	yes	characteristic, direct product factor, Sylow
62	#1813	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×#131	C₂×C₂₁₁	1	1	yes	characteristic
64	C₂×#108	C₂×C₂₁₁	1	2	yes	complemented
65	#413	C₂×C₂₁₁	1	2	yes	—
66	#338	C₂×C₂₁₁	1	2	yes	complemented
67	C₂×#41	C₂²×C₂₁₁	1	1	yes	characteristic
68	#131	—	2	4	no	—
69	C₂×M₁₆	C₄×C₂₁₁	1	2	yes	—
70	C₂×M₁₆	C₂²×C₂₁₁	1	4	yes	—
71	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
72	C₂²×C₈	C₄×C₂₁₁	1	2	yes	—
73	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
74	#108	—	2	4	no	—
75	#101	—	2	2	no	—
76	#100	—	2	2	no	—
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₄×C₂₁₁	1	1	yes	characteristic
80	C₂×D₄	D₄×C₂₁₁	1	1	yes	characteristic
81	C₂×D₄	Q₈×C₂₁₁	1	2	yes	—
82	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
83	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	2	4	no	—
87	M₁₆	—	2	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
91	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
92	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
93	C₂×C₈	D₄×C₂₁₁	1	4	yes	—
94	C₂³	C₄⋊C₄×C₂₁₁	1	2	yes	—
95	C₂³	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
102	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
103	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
104	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
105	C₂×C₄	#41×C₂₁₁	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	4	4	no	—
108	C₈	C₂²⋊C₄×C₂₁₁	1	2	yes	—
109	C₈	C₂²⋊C₄×C₂₁₁	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
114	C₂²	C₂²⋊Q₈×C₂₁₁	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
117	C₄	#117×C₂₁₁	1	1	yes	characteristic
118	C₄	#121×C₂₁₁	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#380×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#131×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#108×C₂₁₁

C₂

yes

maximal, complemented

#413×C₂₁₁

C₂

yes

maximal

#338×C₂₁₁

C₂

yes

maximal, complemented

C₂×#41×C₂₁₁

C₂²

yes

characteristic

#131×C₂₁₁

—

C₂×M₁₆×C₂₁₁

C₄

yes

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₄

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#108×C₂₁₁

—

#101×C₂₁₁

—

#100×C₂₁₁

—

#41×C₂₁₁

—

#41×C₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

characteristic

C₂×D₄×C₂₁₁

D₄

yes

characteristic

C₂×D₄×C₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

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×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#380

yes

characteristic

C₂₁₁

#1813

yes

characteristic, direct product factor, Sylow

#1813

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#131

C₂×C₂₁₁

yes

characteristic

C₂×#108

C₂×C₂₁₁

yes

complemented

#413

C₂×C₂₁₁

yes

—

#338

C₂×C₂₁₁

yes

complemented

C₂×#41

C₂²×C₂₁₁

yes

characteristic

#131

—

C₂×M₁₆

C₄×C₂₁₁

yes

—

C₂×M₁₆

C₂²×C₂₁₁

yes

—

C₂×M₁₆

C₂²×C₂₁₁

yes

characteristic

C₂²×C₈

C₄×C₂₁₁

yes

—

C₂²×C₈

C₂²×C₂₁₁

yes

—

—

—

—

—

C₂×Q₈

D₄×C₂₁₁

yes

characteristic

C₂×D₄

D₄×C₂₁₁

yes

characteristic

C₂×D₄

Q₈×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

characteristic

C₂×C₈

C₂×C₄×C₂₁₁

yes

characteristic

C₂×C₈

D₄×C₂₁₁

yes

—

C₂³

C₄⋊C₄×C₂₁₁

yes

—

C₂³

C₂×Q₈×C₂₁₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊C₄×C₂₁₁

yes

—

102

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

103

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

104

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

105

C₂×C₄

#41×C₂₁₁

yes

—

106

C₈

—

107

C₈

—

108

C₈

C₂²⋊C₄×C₂₁₁

yes

—

109

C₈

C₂²⋊C₄×C₂₁₁

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄⋊C₄×C₂₁₁

yes

characteristic

114

C₂²

C₂²⋊Q₈×C₂₁₁

yes

—

115

C₄

—

116

C₄

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

117

C₄

#117×C₂₁₁

yes

characteristic

118

C₄

#121×C₂₁₁

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#380×C₂₁₁

yes

characteristic

	subgroup	G/H
center	C₄×C₂₁₁	#117
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂×C₂₁₁	#380
prime radical	C₂×#41×C₂₁₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₁₁ ◁ C₂×M₁₆×C₂₁₁ ◁ G

subgroup

G/H

center

C₄×C₂₁₁

#117

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

socle

C₂×C₂₁₁

#380

prime radical

C₂×#41×C₂₁₁

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₂₁₁ ◁ C₂×M₁₆×C₂₁₁ ◁ G

N	H ≅ G/N	type
2. C₂×#108×C₂₁₁	C₂	semidirect
4. #338×C₂₁₁	C₂	semidirect
61. C₂₁₁	#1813	direct (N × H)
62. #1813	C₂₁₁	direct (N × H)
64. C₂×#108	C₂×C₂₁₁	semidirect
66. #338	C₂×C₂₁₁	semidirect

H ≅ G/N

type

2. C₂×#108×C₂₁₁

C₂

semidirect

4. #338×C₂₁₁

C₂

semidirect

61. C₂₁₁

#1813

direct (N × H)

62. #1813

C₂₁₁

direct (N × H)

64. C₂×#108

C₂×C₂₁₁

semidirect

66. #338

C₂×C₂₁₁

semidirect

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

subgroup	index
C₂×#131×C₂₁₁	2
C₂×#108×C₂₁₁	2
#413×C₂₁₁	2
#338×C₂₁₁	2
#1813	211

subgroup

index

C₂×#131×C₂₁₁

C₂×#108×C₂₁₁

#413×C₂₁₁

#338×C₂₁₁

#1813

211

Subgroups — #1813×C₂₁₁ #41009433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1813×C211 #41009433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1813×C₂₁₁ #41009433