Group #41013435 — subgroups

Subgroups — #41013435

Order 27008.

292 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 89 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×C₈×D₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269912	C₂	1	2	yes	maximal, complemented
3	#27213109	C₂	1	2	yes	maximal
4	#27212911	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×#16809946	C₂²	1	2	yes	—
8	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₂₁₁	—	2	4	no	—
10	C₂×C₈×D₂₁₁	C₄	1	4	yes	—
11	#20269912	—	2	4	no	—
12	(C₂×D₂₁₁)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂³×D₂₁₁	Q₈	1	1	yes	characteristic
16	C₂²×Dic₂₁₁	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₂₁₁	—	2	2	no	—
18	C₂×C₄×D₂₁₁	—	2	2	no	—
19	C₂×C₄×D₂₁₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₂₁₁	D₄	1	2	yes	—
21	#16809946	—	2	2	no	—
22	#16809946	—	2	2	no	—
23	C₂×C₂₁₁⋊C₈	—	2	2	no	—
24	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
26	C₈×D₂₁₁	—	2	8	no	—
27	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₁₁	—	2	2	no	—
29	M₁₆×C₂₁₁	—	2	2	no	—
30	C₂×C₈×C₂₁₁	—	2	2	no	—
31	C₂×C₈×C₂₁₁	—	2	2	no	—
32	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
33	C₂²×D₂₁₁	—	2	2	no	—
34	C₂²×D₂₁₁	—	2	4	no	—
35	C₂²×D₂₁₁	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₁₁	—	2	2	no	—
37	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
38	C₄×D₂₁₁	—	2	2	no	—
39	C₄×D₂₁₁	—	2	2	no	—
40	C₄×D₂₁₁	—	2	4	no	—
41	C₂₁₁⋊C₈	—	4	2	no	—
42	C₂₁₁⋊C₈	C₂²⋊C₄	1	4	yes	—
43	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	#41	1	2	yes	—
48	C₈×C₂₁₁	—	2	2	no	—
49	C₈×C₂₁₁	—	4	2	no	—
50	C₂²×C₂₁₁	—	2	2	no	—
51	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₂₁₁	—	2	8	no	—
54	C₂×D₂₁₁	—	2	2	no	—
55	C₂×D₂₁₁	—	2	2	no	—
56	C₂×D₂₁₁	#108	1	4	yes	complemented
57	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₁₁	#117	1	1	yes	characteristic
59	C₄×C₂₁₁	#121	1	2	yes	—
60	Dic₂₁₁	—	2	2	no	—
61	C₂×C₂₁₁	—	2	2	no	—
62	C₂×C₂₁₁	#380	1	1	yes	characteristic
63	C₂×C₂₁₁	C₂×#108	1	2	yes	—
64	D₂₁₁	—	2	4	no	—
65	C₂₁₁	#1810	1	1	yes	characteristic, complemented, Sylow
66	#1810	—	211	1	no	maximal, Sylow
67	C₂³×C₈	—	211	1	no	—
68	C₂×#108	—	211	2	no	—
69	#407	—	211	2	no	—
70	#341	—	211	1	no	—
71	#341	D₂₁₁	1	1	yes	characteristic, complemented
72	C₂³×C₄	—	211	1	no	—
73	C₂×M₁₆	—	211	2	no	—
74	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
75	C₂²×C₈	—	422	4	no	—
76	C₂²×C₈	—	211	4	no	—
77	C₂²×C₈	—	211	1	no	—
78	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
79	#108	—	422	4	no	—
80	C₂²⋊C₈	—	422	4	no	—
81	C₂⁴	—	211	1	no	—
82	C₂²×C₄	—	422	2	no	—
83	C₂²×C₄	—	422	2	no	—
84	C₂²×C₄	—	211	1	no	—
85	C₂²×C₄	—	211	2	no	—
86	C₂²×C₄	—	211	2	no	—
87	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	422	2	no	—
91	M₁₆	—	422	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	422	8	no	—
95	C₂×C₈	—	422	2	no	—
96	C₂×C₈	—	211	2	no	—
97	C₂×C₈	—	211	4	no	—
98	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
99	C₂³	—	422	4	no	—
100	C₂³	—	422	2	no	—
101	C₂³	—	211	2	no	—
102	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
103	C₂×C₄	—	422	2	no	—
104	C₂×C₄	—	422	2	no	—
105	C₂×C₄	—	422	2	no	—
106	C₂×C₄	—	422	4	no	—
107	C₂×C₄	—	211	2	no	—
108	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
109	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
110	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
111	C₂×C₄	#16809962	1	2	yes	—
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	844	2	no	—
115	C₈	—	211	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	422	8	no	—
118	C₂²	—	422	2	no	—
119	C₂²	—	422	2	no	—
120	C₂²	—	211	4	no	—
121	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
122	C₂²	#20269955	1	2	yes	—
123	C₄	—	422	2	no	—
124	C₄	C₂²⋊C₄×D₂₁₁	1	1	yes	characteristic
125	C₄	#20269934	1	1	yes	characteristic
126	C₄	#20269950	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	422	4	no	—
129	C₂	#27213005	1	1	yes	characteristic
130	C₂	#108×D₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₂₁₁

C₂

yes

maximal, characteristic

C₂×#20269912

C₂

yes

maximal, complemented

#27213109

C₂

yes

maximal

#27212911

C₂

yes

maximal, characteristic, complemented

#341×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₁₁

C₂²

yes

characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₁₁

—

C₂×C₈×D₂₁₁

C₄

yes

—

#20269912

—

(C₂×D₂₁₁)⋊C₈

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂³×D₂₁₁

Q₈

yes

characteristic

C₂²×Dic₂₁₁

Q₈

yes

characteristic

C₂×C₄×D₂₁₁

—

C₂×C₄×D₂₁₁

—

C₂×C₄×D₂₁₁

C₂×C₄

yes

—

C₂×C₄×D₂₁₁

D₄

yes

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₈×D₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂²×D₂₁₁

—

C₂²×D₂₁₁

—

C₂²×D₂₁₁

C₄⋊C₄

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₄×D₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

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

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₂×D₂₁₁

#108

yes

complemented

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#380

yes

characteristic

C₂×C₂₁₁

C₂×#108

yes

—

D₂₁₁

—

C₂₁₁

#1810

yes

characteristic, complemented, Sylow

#1810

—

211

maximal, Sylow

C₂³×C₈

—

211

—

C₂×#108

—

211

—

#407

—

211

—

#341

—

211

—

#341

D₂₁₁

yes

characteristic, complemented

C₂³×C₄

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

—

422

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#108

—

422

—

C₂²⋊C₈

—

422

—

C₂⁴

—

211

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₂³

—

422

—

100

C₂³

—

422

—

101

C₂³

—

211

—

102

C₂³

Q₈×D₂₁₁

yes

characteristic

103

C₂×C₄

—

422

—

104

C₂×C₄

—

422

—

105

C₂×C₄

—

422

—

106

C₂×C₄

—

422

—

107

C₂×C₄

—

211

—

108

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

109

C₂×C₄

D₄×D₂₁₁

yes

—

110

C₂×C₄

D₄×D₂₁₁

yes

characteristic

111

C₂×C₄

#16809962

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

844

—

115

C₈

—

211

—

116

C₂²

—

117

C₂²

—

422

—

118

C₂²

—

422

—

119

C₂²

—

422

—

120

C₂²

—

211

—

121

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

122

C₂²

#20269955

yes

—

123

C₄

—

422

—

124

C₄

C₂²⋊C₄×D₂₁₁

yes

characteristic

125

C₄

#20269934

yes

characteristic

126

C₄

#20269950

yes

—

127

C₂

—

128

C₂

—

422

—

129

C₂

#27213005

yes

characteristic

130

C₂

#108×D₂₁₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#341×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×D₂₁₁	Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#341	D₂₁₁
hypocenter	C₂₁₁	#1810
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#341×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄⋊C₄

prime radical

C₂³×D₂₁₁

Q₈

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#20269912	C₂	semidirect
4. #27212911	C₂	semidirect
5. #341×C₂₁₁	C₂	semidirect
56. C₂×D₂₁₁	#108	semidirect
65. C₂₁₁	#1810	semidirect
71. #341	D₂₁₁	semidirect

H ≅ G/N

type

2. C₂×#20269912

C₂

semidirect

4. #27212911

C₂

semidirect

5. #341×C₂₁₁

C₂

semidirect

56. C₂×D₂₁₁

#108

semidirect

65. C₂₁₁

#1810

semidirect

71. #341

D₂₁₁

semidirect

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

subgroup	index
C₂²×C₈×D₂₁₁	2
C₂×#20269912	2
#27213109	2
#27212911	2
#341×C₂₁₁	2
#1810	211

subgroup

index

C₂²×C₈×D₂₁₁

#341×C₂₁₁

211

Subgroups — #41013435

Conjugacy classes of subgroups

Complements

Maximal subgroups