Group #41013434 — subgroups

Subgroups — #41013434

Order 27008.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269912	C₂	1	2	yes	maximal, complemented
3	#27213110	C₂	1	1	yes	maximal, characteristic
4	#27212910	C₂	1	2	yes	maximal, complemented
5	#407×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	—	2	4	no	—
8	C₂²×C₂₁₁⋊C₈	C₄	1	4	yes	—
9	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	#20269912	—	2	4	no	—
12	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₂₁₁	—	2	2	no	—
16	#16809946	—	2	2	no	—
17	#16809946	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	8	no	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
24	C₂⁴×C₂₁₁	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₂₁₁	—	2	2	no	—
26	C₂²×C₄×C₂₁₁	—	2	2	no	—
27	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
29	C₂²×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	M₁₆×C₂₁₁	—	2	2	no	—
32	M₁₆×C₂₁₁	—	2	2	no	—
33	C₂×C₈×C₂₁₁	—	2	2	no	—
34	C₂₁₁⋊C₈	—	2	2	no	—
35	C₂₁₁⋊C₈	—	4	2	no	—
36	C₂₁₁⋊C₈	C₂²⋊C₄	1	4	yes	—
37	C₂³×C₂₁₁	—	2	4	no	—
38	C₂³×C₂₁₁	—	2	2	no	—
39	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	—	2	2	no	—
42	C₂×C₄×C₂₁₁	—	2	2	no	—
43	C₂×C₄×C₂₁₁	—	2	2	no	—
44	C₂×C₄×C₂₁₁	—	2	4	no	—
45	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₁₁	#41	1	2	yes	—
50	C₈×C₂₁₁	—	4	2	no	—
51	C₂²×C₂₁₁	—	2	8	no	—
52	C₂²×C₂₁₁	—	2	2	no	—
53	C₂²×C₂₁₁	—	2	2	no	—
54	C₂²×C₂₁₁	—	2	2	no	—
55	C₂²×C₂₁₁	#108	1	4	yes	complemented
56	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₁₁	—	2	2	no	—
59	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₁₁	#117	1	1	yes	characteristic
61	C₄×C₂₁₁	#121	1	2	yes	—
62	C₂×C₂₁₁	—	2	4	no	—
63	C₂×C₂₁₁	—	2	2	no	—
64	C₂×C₂₁₁	#380	1	1	yes	characteristic
65	C₂×C₂₁₁	C₂×#108	1	2	yes	—
66	C₂₁₁	#1810	1	1	yes	characteristic, complemented, Sylow
67	#1810	—	211	1	no	maximal, Sylow
68	C₂³×C₈	—	211	1	no	—
69	C₂×#108	—	211	2	no	—
70	#407	—	211	1	no	—
71	#407	D₂₁₁	1	1	yes	characteristic
72	#341	—	211	2	no	—
73	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
74	C₂×M₁₆	—	211	2	no	—
75	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
76	C₂²×C₈	—	422	4	no	—
77	C₂²×C₈	—	211	4	no	—
78	C₂²×C₈	—	211	1	no	—
79	C₂²×C₈	—	211	1	no	—
80	#108	—	422	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	422	2	no	—
83	C₂⁴	Dic₄₂₂	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
88	C₂²×C₄	Dic₄₂₂	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	422	2	no	—
93	M₁₆	—	422	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	422	2	no	—
96	C₂×C₈	—	422	2	no	—
97	C₂×C₈	—	422	8	no	—
98	C₂×C₈	—	211	2	no	—
99	C₂×C₈	—	211	2	no	—
100	C₂×C₈	—	211	4	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	Dic₂₁₁⋊C₄	1	2	yes	—
104	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
113	C₂×C₄	#16809961	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	844	2	no	—
116	C₈	—	422	2	no	—
117	C₈	—	211	4	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#20269912	1	4	yes	complemented
123	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
124	C₂²	#20269954	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₂₁₁	1	1	yes	characteristic
127	C₄	#20269933	1	1	yes	characteristic
128	C₄	#20269949	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#27213004	1	1	yes	characteristic
132	C₂	C₂×#20269912	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

C₂×#20269912

C₂

yes

maximal, complemented

#27213110

C₂

yes

maximal, characteristic

#27212910

C₂

yes

maximal, complemented

#407×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

—

C₂²×C₂₁₁⋊C₈

C₄

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

#20269912

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²⋊C₈×C₂₁₁

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂⁴×C₂₁₁

Q₈

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

Q₈

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×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

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#108

yes

complemented

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

#380

yes

characteristic

C₂×C₂₁₁

C₂×#108

yes

—

C₂₁₁

#1810

yes

characteristic, complemented, Sylow

#1810

—

211

maximal, Sylow

C₂³×C₈

—

211

—

C₂×#108

—

211

—

#407

—

211

—

#407

D₂₁₁

yes

characteristic

#341

—

211

—

C₂³×C₄

C₂×D₂₁₁

yes

characteristic

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

—

422

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

#108

—

422

—

C₂²⋊C₈

—

C₂²⋊C₈

—

422

—

C₂⁴

Dic₄₂₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₁₁

yes

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

Dic₄₂₂

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

100

C₂×C₈

—

211

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₂₁₁⋊C₄

yes

—

104

C₂³

C₂×Dic₄₂₂

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

111

C₂×C₄

D₄×D₂₁₁

yes

—

112

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

113

C₂×C₄

#16809961

yes

—

114

C₈

—

115

C₈

—

844

—

116

C₈

—

422

—

117

C₈

—

211

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#20269912

yes

complemented

123

C₂²

C₂×Dic₂₁₁⋊C₄

yes

characteristic

124

C₂²

#20269954

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₂₁₁

yes

characteristic

127

C₄

#20269933

yes

characteristic

128

C₄

#20269949

yes

—

129

C₂

—

130

C₂

—

131

C₂

#27213004

yes

characteristic

132

C₂

C₂×#20269912

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#407×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₁₁	Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#407	D₂₁₁
hypocenter	C₂₁₁	#1810
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#407×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₁₁

Q₈

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#20269912	C₂	semidirect
4. #27212910	C₂	semidirect
55. C₂²×C₂₁₁	#108	semidirect
66. C₂₁₁	#1810	semidirect
122. C₂²	#20269912	semidirect

H ≅ G/N

type

2. C₂×#20269912

C₂

semidirect

4. #27212910

C₂

semidirect

55. C₂²×C₂₁₁

#108

semidirect

66. C₂₁₁

#1810

semidirect

122. C₂²

#20269912

semidirect

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

subgroup	index
C₂³×C₂₁₁⋊C₈	2
C₂×#20269912	2
#27213110	2
#27212910	2
#407×C₂₁₁	2
#1810	211

subgroup

index

C₂³×C₂₁₁⋊C₈

#407×C₂₁₁

211

Subgroups — #41013434

Conjugacy classes of subgroups

Complements

Maximal subgroups