Group #41014845 — subgroups

Subgroups — #41014845

Order 27008.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#20269909	C₂	1	1	yes	maximal, characteristic, complemented
2	#27213148	C₂	1	2	yes	maximal
3	#27213112	C₂	1	1	yes	maximal, characteristic
4	#27212914	C₂	1	2	yes	maximal, complemented
5	#341×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16809962	C₂²	1	1	yes	characteristic
7	C₂×#16809946	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	#20269909	—	2	2	no	—
10	#20269902	—	2	2	no	—
11	#20269899	—	2	2	no	—
12	(C₂×D₂₁₁)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	#16809962	—	2	2	no	—
16	#16809962	—	2	2	no	—
17	C₂×Dic₄₂₂	D₄	1	1	yes	characteristic
18	C₂×D₈₄₄	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₁₁	—	2	1	no	—
20	C₂²×Dic₂₁₁	—	2	2	no	—
21	C₂×C₄×D₂₁₁	—	2	1	no	—
22	#16809946	—	2	2	no	—
23	C₂×C₂₁₁⋊C₈	—	2	1	no	—
24	C₄⋊Dic₂₁₁	—	2	2	no	—
25	C₄⋊Dic₂₁₁	—	2	2	no	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₁₁	—	2	2	no	—
28	M₁₆×C₂₁₁	—	2	2	no	—
29	C₂×C₈×C₂₁₁	—	2	2	no	—
30	C₂×C₈×C₂₁₁	—	2	2	no	—
31	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
32	C₂²×D₂₁₁	—	2	1	no	—
33	Dic₄₂₂	—	2	2	no	—
34	D₈₄₄	—	2	2	no	—
35	C₂²⋊D₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	2	1	no	—
37	C₂×Dic₂₁₁	—	4	2	no	—
38	C₂×Dic₂₁₁	—	8	2	no	—
39	C₄×D₂₁₁	—	4	1	no	—
40	C₄×D₂₁₁	—	4	1	no	—
41	C₂₁₁⋊C₈	—	4	1	no	—
42	C₂³×C₂₁₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	#41	1	2	yes	—
46	C₂×C₄×C₂₁₁	#41	1	2	yes	—
47	C₈×C₂₁₁	—	2	2	no	—
48	C₈×C₂₁₁	—	4	2	no	—
49	C₂²×C₂₁₁	—	2	2	no	—
50	C₂²×C₂₁₁	#121	1	1	yes	characteristic
51	C₂²×C₂₁₁	#126	1	2	yes	—
52	C₂×D₂₁₁	—	4	1	no	—
53	C₂×D₂₁₁	—	4	2	no	—
54	C₄×C₂₁₁	#123	1	2	yes	—
55	C₄×C₂₁₁	#123	1	1	yes	characteristic
56	C₄×C₂₁₁	#124	1	1	yes	characteristic
57	Dic₂₁₁	—	4	1	no	—
58	Dic₂₁₁	—	8	2	no	—
59	C₂×C₂₁₁	—	2	2	no	—
60	C₂×C₂₁₁	#397	1	1	yes	characteristic
61	C₂×C₂₁₁	#471	1	2	yes	—
62	D₂₁₁	—	8	1	no	—
63	C₂₁₁	#2079	1	1	yes	characteristic, complemented, Sylow
64	#2079	—	211	1	no	maximal, Sylow
65	C₂×#106	—	211	1	no	—
66	#417	—	211	2	no	—
67	#408	—	211	1	no	—
68	#343	—	211	2	no	—
69	#341	D₂₁₁	1	1	yes	characteristic, complemented
70	C₂×#41	—	211	1	no	—
71	C₂×M₁₆	—	211	1	no	—
72	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
73	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	211	2	no	—
75	#106	—	422	2	no	—
76	#103	—	422	2	no	—
77	#102	—	422	2	no	—
78	C₂²⋊C₈	—	422	2	no	—
79	#41	—	422	2	no	—
80	#41	—	422	2	no	—
81	C₂×Q₈	—	211	1	no	—
82	C₂×D₄	—	422	1	no	—
83	C₂×D₄	—	211	1	no	—
84	C₂²×C₄	—	422	1	no	—
85	C₂²×C₄	—	422	2	no	—
86	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	422	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	422	1	no	—
93	C₂×C₈	D₈₄₄	1	2	yes	—
94	C₄⋊C₄	—	422	2	no	—
95	C₄⋊C₄	—	422	2	no	—
96	C₂³	—	422	1	no	—
97	C₂³	#16809963	1	1	yes	characteristic
98	Q₈	—	422	2	no	—
99	D₄	—	844	2	no	—
100	D₄	—	422	2	no	—
101	C₂×C₄	—	844	1	no	—
102	C₂×C₄	—	844	1	no	—
103	C₂×C₄	—	844	2	no	—
104	C₂×C₄	—	1688	2	no	—
105	C₂×C₄	—	422	1	no	—
106	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
107	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
108	C₂×C₄	#16809961	1	2	yes	—
109	C₂×C₄	#16809962	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	844	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	844	1	no	—
115	C₂²	—	844	2	no	—
116	C₂²	#20269952	1	1	yes	characteristic
117	C₂²	#20269972	1	2	yes	—
118	C₄	—	844	1	no	—
119	C₄	—	1688	2	no	—
120	C₄	#20269959	1	2	yes	—
121	C₄	#20269961	1	1	yes	characteristic
122	C₄	#20269965	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	1688	1	no	—
125	C₂	#27213073	1	1	yes	characteristic
126	C₂	#27213410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269909

C₂

yes

maximal, characteristic, complemented

#27213148

C₂

yes

maximal

#27213112

C₂

yes

maximal, characteristic

#27212914

C₂

yes

maximal, complemented

#341×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×#16809962

C₂²

yes

characteristic

C₂×#16809946

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269909

—

#20269902

—

#20269899

—

(C₂×D₂₁₁)⋊C₈

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#16809962

—

#16809962

—

C₂×Dic₄₂₂

D₄

yes

characteristic

C₂×D₈₄₄

D₄

yes

characteristic

C₂×C₂²⋊D₂₁₁

—

C₂²×Dic₂₁₁

—

C₂×C₄×D₂₁₁

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂²×D₂₁₁

—

Dic₄₂₂

—

D₈₄₄

—

C₂²⋊D₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₂²×C₂₁₁

#126

yes

—

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₄×C₂₁₁

#123

yes

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#397

yes

characteristic

C₂×C₂₁₁

#471

yes

—

D₂₁₁

—

C₂₁₁

#2079

yes

characteristic, complemented, Sylow

#2079

—

211

maximal, Sylow

C₂×#106

—

211

—

#417

—

211

—

#408

—

211

—

#343

—

211

—

#341

D₂₁₁

yes

characteristic, complemented

C₂×#41

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

#106

—

422

—

#103

—

422

—

#102

—

422

—

C₂²⋊C₈

—

422

—

#41

—

422

—

#41

—

422

—

C₂×Q₈

—

211

—

C₂×D₄

—

422

—

C₂×D₄

—

211

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

D₈₄₄

yes

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₂³

—

422

—

C₂³

#16809963

yes

characteristic

Q₈

—

422

—

D₄

—

844

—

100

D₄

—

422

—

101

C₂×C₄

—

844

—

102

C₂×C₄

—

844

—

103

C₂×C₄

—

844

—

104

C₂×C₄

—

1688

—

105

C₂×C₄

—

422

—

106

C₂×C₄

D₄×D₂₁₁

yes

characteristic

107

C₂×C₄

C₂×D₈₄₄

yes

characteristic

108

C₂×C₄

#16809961

yes

—

109

C₂×C₄

#16809962

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

844

—

113

C₂²

—

114

C₂²

—

844

—

115

C₂²

—

844

—

116

C₂²

#20269952

yes

characteristic

117

C₂²

#20269972

yes

—

118

C₄

—

844

—

119

C₄

—

1688

—

120

C₄

#20269959

yes

—

121

C₄

#20269961

yes

characteristic

122

C₄

#20269965

yes

characteristic

123

C₂

—

124

C₂

—

1688

—

125

C₂

#27213073

yes

characteristic

126

C₂

#27213410

yes

—

	subgroup	G/H
center	C₂²	#20269952
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#341×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#121
prime radical	C₂×#16809962	C₂²
hypercenter	#341	D₂₁₁
hypocenter	C₂₁₁	#2079
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#20269952

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#341×C₂₁₁

C₂

C₂²×C₂₁₁

C₂²

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#20269909	C₂	semidirect
4. #27212914	C₂	semidirect
5. #341×C₂₁₁	C₂	semidirect
63. C₂₁₁	#2079	semidirect
69. #341	D₂₁₁	semidirect

H ≅ G/N

type

1. C₂×#20269909

C₂

semidirect

4. #27212914

C₂

semidirect

5. #341×C₂₁₁

C₂

semidirect

63. C₂₁₁

#2079

semidirect

69. #341

D₂₁₁

semidirect

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

subgroup	index
C₂×#20269909	2
#27213148	2
#27213112	2
#27212914	2
#341×C₂₁₁	2
#2079	211

subgroup

index

#341×C₂₁₁

211

Subgroups — #41014845

Conjugacy classes of subgroups

Complements

Maximal subgroups