Group #41009735 — subgroups

Subgroups — #2115×C₂₁₁ #41009735

Order 27008.

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

normal subgroups only

This group has 220 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×M₃₂×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#349×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₂₁₁	C₄	1	1	yes	characteristic
5	#131×C₂₁₁	—	2	2	no	—
6	#131×C₂₁₁	—	2	2	no	—
7	C₂×M₁₆×C₂₁₁	C₄	1	1	yes	characteristic
8	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
9	C₂²×C₈×C₂₁₁	C₄	1	2	yes	—
10	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	M₃₂×C₂₁₁	—	2	2	no	—
12	M₃₂×C₂₁₁	C₂²	1	4	yes	—
13	C₂×C₁₆×C₂₁₁	—	2	2	no	—
14	#41×C₂₁₁	—	2	2	no	—
15	#41×C₂₁₁	—	2	2	no	—
16	C₂×Q₈×C₂₁₁	C₈	1	1	yes	characteristic
17	C₂×D₄×C₂₁₁	C₈	1	2	yes	—
18	C₂×D₄×C₂₁₁	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
21	M₁₆×C₂₁₁	—	2	2	no	—
22	M₁₆×C₂₁₁	—	2	4	no	—
23	C₂×C₈×C₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
28	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
29	C₂×C₈×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₁₆×C₂₁₁	—	2	4	no	—
31	C₂³×C₂₁₁	C₂×C₈	1	2	yes	—
32	C₂³×C₂₁₁	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₂₁₁	—	2	2	no	—
34	D₄×C₂₁₁	—	2	4	no	—
35	D₄×C₂₁₁	—	2	2	no	—
36	C₂×C₄×C₂₁₁	—	2	2	no	—
37	C₂×C₄×C₂₁₁	—	2	2	no	—
38	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₁₁	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	M₁₆	1	2	yes	—
42	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₂₁₁	—	2	2	no	—
44	C₈×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₈×C₂₁₁	C₂×D₄	1	2	yes	—
47	C₂²×C₂₁₁	—	2	2	no	—
48	C₂²×C₂₁₁	—	4	1	no	—
49	C₂²×C₂₁₁	—	4	2	no	—
50	C₂²×C₂₁₁	C₂²⋊C₈	1	2	yes	—
51	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₂₁₁	—	2	2	no	—
53	C₄×C₂₁₁	C₂²⋊C₈	1	2	yes	—
54	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₂₁₁	—	2	1	no	—
57	C₂×C₂₁₁	—	2	2	no	—
58	C₂×C₂₁₁	—	4	2	no	—
59	C₂×C₂₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	C₂₁₁	#2115	1	1	yes	characteristic, direct product factor, Sylow
61	#2115	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×#131	C₂×C₂₁₁	1	1	yes	characteristic
63	C₂×M₃₂	C₂×C₂₁₁	1	2	yes	complemented
64	#349	C₂×C₂₁₁	1	4	yes	complemented
65	C₂×#41	C₄×C₂₁₁	1	1	yes	characteristic
66	#131	—	2	2	no	—
67	#131	—	2	2	no	—
68	C₂×M₁₆	C₄×C₂₁₁	1	1	yes	characteristic
69	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
70	C₂²×C₈	C₄×C₂₁₁	1	2	yes	—
71	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
72	M₃₂	—	2	2	no	—
73	M₃₂	C₂²×C₂₁₁	1	4	yes	—
74	C₂×C₁₆	—	2	2	no	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	C₈×C₂₁₁	1	1	yes	characteristic
78	C₂×D₄	C₈×C₂₁₁	1	2	yes	—
79	C₂×D₄	C₈×C₂₁₁	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
81	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
87	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
88	C₂×C₈	D₄×C₂₁₁	1	2	yes	—
89	C₂×C₈	D₄×C₂₁₁	1	2	yes	—
90	C₂×C₈	C₂³×C₂₁₁	1	1	yes	characteristic
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×C₈×C₂₁₁	1	2	yes	—
93	C₂³	C₂×C₈×C₂₁₁	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	D₄	—	2	4	no	—
96	D₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
100	C₂×C₄	C₂×C₈×C₂₁₁	1	2	yes	—
101	C₂×C₄	C₂×C₈×C₂₁₁	1	1	yes	characteristic
102	C₂×C₄	M₁₆×C₂₁₁	1	2	yes	—
103	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
106	C₈	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
107	C₈	C₂×D₄×C₂₁₁	1	2	yes	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	C₂²⋊C₈×C₂₁₁	1	2	yes	—
112	C₂²	C₂²×C₈×C₂₁₁	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	C₂²⋊C₈×C₂₁₁	1	2	yes	—
115	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
116	C₄	C₂×M₁₆×C₂₁₁	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	4	2	no	—
120	C₂	C₂×C₂²⋊C₈×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#131×C₂₁₁

C₂

yes

maximal, characteristic

C₂×M₃₂×C₂₁₁

C₂

yes

maximal, complemented

#349×C₂₁₁

C₂

yes

maximal, complemented

C₂×#41×C₂₁₁

C₄

yes

characteristic

#131×C₂₁₁

—

#131×C₂₁₁

—

C₂×M₁₆×C₂₁₁

C₄

yes

characteristic

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₄

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

M₃₂×C₂₁₁

—

M₃₂×C₂₁₁

C₂²

yes

—

C₂×C₁₆×C₂₁₁

—

#41×C₂₁₁

—

#41×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₈

yes

characteristic

C₂×D₄×C₂₁₁

C₈

yes

—

C₂×D₄×C₂₁₁

C₈

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

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

D₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

C₂³

yes

characteristic

C₁₆×C₂₁₁

—

C₂³×C₂₁₁

C₂×C₈

yes

—

C₂³×C₂₁₁

C₂×C₈

yes

characteristic

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×C₈

yes

—

C₂×C₄×C₂₁₁

C₂×C₈

yes

characteristic

C₂×C₄×C₂₁₁

M₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₈×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₈×C₂₁₁

C₂×D₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂²⋊C₈

yes

—

C₂²×C₂₁₁

C₂²×C₈

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊C₈

yes

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

C₂×M₁₆

yes

characteristic

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂×C₂²⋊C₈

yes

characteristic

C₂₁₁

#2115

yes

characteristic, direct product factor, Sylow

#2115

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#131

C₂×C₂₁₁

yes

characteristic

C₂×M₃₂

C₂×C₂₁₁

yes

complemented

#349

C₂×C₂₁₁

yes

complemented

C₂×#41

C₄×C₂₁₁

yes

characteristic

#131

—

#131

—

C₂×M₁₆

C₄×C₂₁₁

yes

characteristic

C₂×M₁₆

C₂²×C₂₁₁

yes

—

C₂²×C₈

C₄×C₂₁₁

yes

—

C₂²×C₈

C₂²×C₂₁₁

yes

characteristic

M₃₂

—

M₃₂

C₂²×C₂₁₁

yes

—

C₂×C₁₆

—

#41

—

#41

—

C₂×Q₈

C₈×C₂₁₁

yes

characteristic

C₂×D₄

C₈×C₂₁₁

yes

—

C₂×D₄

C₈×C₂₁₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

characteristic

C₂×C₈

D₄×C₂₁₁

yes

—

C₂×C₈

D₄×C₂₁₁

yes

—

C₂×C₈

C₂³×C₂₁₁

yes

characteristic

C₁₆

—

C₂³

C₂×C₈×C₂₁₁

yes

—

C₂³

C₂×C₈×C₂₁₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

100

C₂×C₄

C₂×C₈×C₂₁₁

yes

—

101

C₂×C₄

C₂×C₈×C₂₁₁

yes

characteristic

102

C₂×C₄

M₁₆×C₂₁₁

yes

—

103

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

104

C₈

—

105

C₈

C₂²⋊C₄×C₂₁₁

yes

characteristic

106

C₈

C₂²⋊C₄×C₂₁₁

yes

characteristic

107

C₈

C₂×D₄×C₂₁₁

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊C₈×C₂₁₁

yes

—

112

C₂²

C₂²×C₈×C₂₁₁

yes

characteristic

113

C₄

—

114

C₄

C₂²⋊C₈×C₂₁₁

yes

—

115

C₄

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

116

C₄

C₂×M₁₆×C₂₁₁

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊C₈×C₂₁₁

yes

characteristic

	subgroup	G/H
center	C₈×C₂₁₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₂₁₁
Frattini, cosocle	C₂×C₈	C₂³×C₂₁₁
socle	C₂×C₂₁₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₂₁₁	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₁₁ ◁ C₂²×C₈×C₂₁₁ ◁ G

subgroup

G/H

center

C₈×C₂₁₁

C₂²⋊C₄

commutator

C₂²

C₂²×C₈×C₂₁₁

Frattini, cosocle

C₂×C₈

C₂³×C₂₁₁

socle

C₂×C₂₁₁

C₂×C₂²⋊C₈

prime radical

C₂×#41×C₂₁₁

C₄

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈×C₂₁₁ ◁ C₂²×C₈×C₂₁₁ ◁ G

N	H ≅ G/N	type
2. C₂×M₃₂×C₂₁₁	C₂	semidirect
3. #349×C₂₁₁	C₂	semidirect
60. C₂₁₁	#2115	direct (N × H)
61. #2115	C₂₁₁	direct (N × H)
63. C₂×M₃₂	C₂×C₂₁₁	semidirect
64. #349	C₂×C₂₁₁	semidirect

H ≅ G/N

type

2. C₂×M₃₂×C₂₁₁

C₂

semidirect

3. #349×C₂₁₁

C₂

semidirect

60. C₂₁₁

#2115

direct (N × H)

61. #2115

C₂₁₁

direct (N × H)

63. C₂×M₃₂

C₂×C₂₁₁

semidirect

64. #349

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×M₃₂×C₂₁₁	2
#349×C₂₁₁	2
#2115	211

subgroup

index

C₂×#131×C₂₁₁

C₂×M₃₂×C₂₁₁

#349×C₂₁₁

#2115

211

Subgroups — #2115×C₂₁₁ #41009735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2115×C211 #41009735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2115×C₂₁₁ #41009735