Group #41014147 — subgroups

Subgroups — #41014147

Order 27008.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#20269913	C₂	1	1	yes	maximal, characteristic
2	#27213173	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#27212905	C₂	1	2	yes	maximal
5	#427×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	#20269913	—	2	2	no	—
10	C₄⋊(C₂₁₁⋊C₈)	—	2	4	no	—
11	C₈×Dic₂₁₁	—	2	2	no	—
12	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	#117×C₂₁₁	C₂²	1	2	yes	—
14	#107×C₂₁₁	C₄	1	2	yes	complemented
15	#106×C₂₁₁	C₄	1	2	yes	complemented
16	C₂²×Dic₂₁₁	—	2	1	no	—
17	#16809946	—	2	2	no	—
18	#16809946	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	—	2	1	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	—	2	1	no	—
22	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₂₁₁	—	2	1	no	—
24	C₄×Dic₂₁₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
29	C₂×C₈×C₂₁₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
32	C₄²×C₂₁₁	—	2	2	no	—
33	C₂×Dic₂₁₁	—	2	4	no	—
34	C₂×Dic₂₁₁	—	2	2	no	—
35	C₂₁₁⋊C₈	—	2	2	no	—
36	C₂₁₁⋊C₈	—	4	2	no	—
37	C₂³×C₂₁₁	#41	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	—	2	4	no	—
39	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
45	C₈×C₂₁₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₁₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₁₁	—	2	2	no	—
48	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₁₁	#125	1	1	yes	characteristic
51	C₄×C₂₁₁	—	4	4	no	—
52	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₁₁	#121	1	1	yes	characteristic
55	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₂₁₁	—	2	4	no	—
57	C₂×C₂₁₁	—	2	1	no	—
58	C₂×C₂₁₁	—	2	1	no	—
59	C₂×C₂₁₁	#389	1	1	yes	characteristic
60	C₂×C₂₁₁	#443	1	2	yes	—
61	C₂₁₁	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	211	1	no	maximal, Sylow
63	C₂×#108	—	211	1	no	—
64	#427	D₂₁₁	1	1	yes	characteristic
65	#424	—	211	2	no	—
66	C₂×C₄×C₈	—	211	1	no	—
67	#339	—	211	2	no	—
68	C₂×M₁₆	—	211	2	no	—
69	C₂²×C₈	—	211	1	no	—
70	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
71	#117	C₂×D₂₁₁	1	2	yes	—
72	C₂×C₄²	—	211	1	no	—
73	#108	—	422	2	no	—
74	#107	Dic₂₁₁	1	2	yes	complemented
75	#106	Dic₂₁₁	1	2	yes	complemented
76	C₄⋊C₈	—	422	4	no	—
77	C₄×C₈	—	422	2	no	—
78	C₂²×C₄	—	422	1	no	—
79	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
80	M₁₆	—	422	2	no	—
81	M₁₆	—	422	2	no	—
82	C₂×C₈	—	422	2	no	—
83	C₂×C₈	—	422	1	no	—
84	C₂×C₈	—	422	1	no	—
85	C₂×C₈	—	211	2	no	—
86	C₂×C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
88	C₂×C₈	D₈₄₄	1	2	yes	—
89	C₂×C₈	Dic₄₂₂	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	422	1	no	—
94	C₄²	—	211	2	no	—
95	C₂³	#16809961	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	422	4	no	—
98	C₂×C₄	—	422	2	no	—
99	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
104	C₂×C₄	#16809963	1	1	yes	characteristic
105	C₈	—	844	2	no	—
106	C₈	—	422	2	no	—
107	C₈	C₄⋊Dic₂₁₁	1	2	yes	—
108	C₈	C₄⋊Dic₂₁₁	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₂₁₁	1	1	yes	characteristic
111	C₂²	#20269957	1	1	yes	characteristic
112	C₂²	#20269968	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	422	4	no	—
115	C₄	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
116	C₄	Q₈×Dic₂₁₁	1	1	yes	characteristic
117	C₄	#20269952	1	1	yes	characteristic
118	C₄	#20269976	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#27213045	1	1	yes	characteristic
122	C₂	#27213263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269913

C₂

yes

maximal, characteristic

#27213173

C₂

yes

maximal

C₂×C₈×Dic₂₁₁

C₂

yes

maximal, characteristic

#27212905

C₂

yes

maximal

#427×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

#20269913

—

C₄⋊(C₂₁₁⋊C₈)

—

C₈×Dic₂₁₁

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

—

#107×C₂₁₁

C₄

yes

complemented

#106×C₂₁₁

C₄

yes

complemented

C₂²×Dic₂₁₁

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

Q₈

yes

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

C₄⋊C₄

yes

—

C₈×C₂₁₁

C₄⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₄×D₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₁₁

#125

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₁₁

C₄×Q₈

yes

characteristic

C₄×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#389

yes

characteristic

C₂×C₂₁₁

#443

yes

—

C₂₁₁

#1945

yes

characteristic, complemented, Sylow

#1945

—

211

maximal, Sylow

C₂×#108

—

211

—

#427

D₂₁₁

yes

characteristic

#424

—

211

—

C₂×C₄×C₈

—

211

—

#339

—

211

—

C₂×M₁₆

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#117

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

#108

—

422

—

#107

Dic₂₁₁

yes

complemented

#106

Dic₂₁₁

yes

complemented

C₄⋊C₈

—

422

—

C₄×C₈

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

C₂×Dic₂₁₁

yes

characteristic

C₂×C₈

C₂×Dic₂₁₁

yes

characteristic

C₂×C₈

D₈₄₄

yes

—

C₂×C₈

Dic₄₂₂

yes

—

C₄⋊C₄

C₂×Dic₂₁₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

422

—

C₄²

—

211

—

C₂³

#16809961

yes

characteristic

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

100

C₂×C₄

D₄×D₂₁₁

yes

characteristic

101

C₂×C₄

C₂×D₈₄₄

yes

characteristic

102

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

103

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

104

C₂×C₄

#16809963

yes

characteristic

105

C₈

—

844

—

106

C₈

—

422

—

107

C₈

C₄⋊Dic₂₁₁

yes

—

108

C₈

C₄⋊Dic₂₁₁

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₂₁₁

yes

characteristic

111

C₂²

#20269957

yes

characteristic

112

C₂²

#20269968

yes

characteristic

113

C₄

—

114

C₄

—

422

—

115

C₄

C₂×C₄⋊Dic₂₁₁

yes

characteristic

116

C₄

Q₈×Dic₂₁₁

yes

characteristic

117

C₄

#20269952

yes

characteristic

118

C₄

#20269976

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#27213045

yes

characteristic

122

C₂

#27213263

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#427×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₄×D₄
prime radical	C₂³×C₂₁₁	#41
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#427	D₂₁₁
hypocenter	C₂₁₁	#1945
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#427×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₄×D₄

prime radical

C₂³×C₂₁₁

#41

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₂₁₁	C₄	semidirect
15. #106×C₂₁₁	C₄	semidirect
61. C₂₁₁	#1945	semidirect
74. #107	Dic₂₁₁	semidirect
75. #106	Dic₂₁₁	semidirect

H ≅ G/N

type

14. #107×C₂₁₁

C₄

semidirect

15. #106×C₂₁₁

C₄

semidirect

61. C₂₁₁

#1945

semidirect

74. #107

Dic₂₁₁

semidirect

75. #106

Dic₂₁₁

semidirect

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

subgroup	index
C₂×#20269913	2
#27213173	2
C₂×C₈×Dic₂₁₁	2
#27212905	2
#427×C₂₁₁	2
#1945	211

subgroup

index

C₂×C₈×Dic₂₁₁

#427×C₂₁₁

211

Subgroups — #41014147

Conjugacy classes of subgroups

Complements

Maximal subgroups