Group #41011248 — subgroups

Subgroups — #41011248

Order 27008.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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	#27213199	C₂	1	1	yes	maximal, characteristic
2	C₄×#16809946	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16809946	C₄	1	2	yes	—
5	C₄⋊(C₂₁₁⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₂₁₁⋊C₈)	C₄	1	2	yes	—
7	#20269888	—	2	1	no	—
8	C₄×C₂₁₁⋊C₈	—	2	1	no	—
9	C₄×C₂₁₁⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₂₁₁	C₄	1	2	yes	—
11	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₁₁	—	2	1	no	—
13	C₄⋊C₈×C₂₁₁	—	2	1	no	—
14	#16809946	—	2	2	no	—
15	#16809946	C₈	1	4	yes	complemented
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₁₁	—	2	2	no	—
22	C₂×C₈×C₂₁₁	—	4	2	no	—
23	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
24	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
25	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
26	C₄²×C₂₁₁	Q₈	1	1	yes	characteristic
27	C₂₁₁⋊C₈	—	2	4	no	—
28	C₂₁₁⋊C₈	—	4	2	no	—
29	C₂³×C₂₁₁	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	—	2	1	no	—
31	C₂×C₄×C₂₁₁	—	2	1	no	—
32	C₂×C₄×C₂₁₁	—	4	1	no	—
33	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₂₁₁	M₁₆	1	2	yes	—
41	C₈×C₂₁₁	—	4	2	no	—
42	C₂²×C₂₁₁	—	2	1	no	—
43	C₂²×C₂₁₁	—	2	1	no	—
44	C₂²×C₂₁₁	#95	1	1	yes	characteristic
45	C₂²×C₂₁₁	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₂₁₁	#97	1	1	yes	characteristic
47	C₄×C₂₁₁	—	2	1	no	—
48	C₄×C₂₁₁	—	2	1	no	—
49	C₄×C₂₁₁	—	2	1	no	—
50	C₄×C₂₁₁	C₂²⋊C₈	1	2	yes	—
51	C₄×C₂₁₁	#100	1	1	yes	characteristic
52	C₄×C₂₁₁	#101	1	1	yes	characteristic
53	C₄×C₂₁₁	C₄⋊C₈	1	2	yes	—
54	C₂×C₂₁₁	—	2	1	no	—
55	C₂×C₂₁₁	—	2	1	no	—
56	C₂×C₂₁₁	#336	1	1	yes	characteristic
57	C₂×C₂₁₁	#343	1	1	yes	characteristic
58	C₂×C₂₁₁	#344	1	1	yes	characteristic
59	C₂₁₁	#1283	1	1	yes	characteristic, complemented, Sylow
60	#1283	—	211	1	no	maximal, Sylow
61	#431	—	211	1	no	—
62	C₂×C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
63	C₄×M₁₆	—	211	1	no	—
64	C₂×M₁₆	—	211	2	no	—
65	C₂²×C₈	Dic₂₁₁	1	2	yes	—
66	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	422	1	no	—
70	C₂²⋊C₈	—	211	2	no	—
71	#97	—	422	1	no	—
72	C₄×C₈	—	422	1	no	—
73	C₄×C₈	—	422	1	no	—
74	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
77	M₁₆	—	422	2	no	—
78	M₁₆	—	211	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	422	2	no	—
82	C₂×C₈	—	422	2	no	—
83	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
84	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
85	C₄²	D₈₄₄	1	1	yes	characteristic
86	C₄²	Dic₄₂₂	1	1	yes	characteristic
87	C₂³	C₄×Dic₂₁₁	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
94	C₂×C₄	Dic₂₁₁⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₂₁₁⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₂₁₁	1	1	yes	characteristic
97	C₂×C₄	C₈×D₂₁₁	1	2	yes	—
98	C₂×C₄	#16809945	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	844	2	no	—
101	C₈	—	422	4	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₂₁₁⋊#95	1	1	yes	characteristic
105	C₂²	C₈×Dic₂₁₁	1	1	yes	characteristic
106	C₂²	#20269887	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	(C₂×D₂₁₁)⋊C₈	1	2	yes	—
111	C₄	#20269894	1	1	yes	characteristic
112	C₄	#20269896	1	1	yes	characteristic
113	C₄	Dic₂₁₁⋊C₈	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#27212900	1	1	yes	characteristic
117	C₂	#27212915	1	1	yes	characteristic
118	C₂	#27212917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213199

C₂

yes

maximal, characteristic

C₄×#16809946

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₄

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

C₂²⋊(C₂₁₁⋊C₈)

C₄

yes

—

#20269888

—

C₄×C₂₁₁⋊C₈

—

C₄×C₂₁₁⋊C₈

—

C₂²×C₈×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₄⋊C₈×C₂₁₁

—

#16809946

—

#16809946

C₈

yes

complemented

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

Q₈

yes

characteristic

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₄²

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×C₈

yes

—

C₂×C₄×C₂₁₁

M₁₆

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#95

yes

characteristic

C₂²×C₂₁₁

C₄×C₈

yes

characteristic

C₂²×C₂₁₁

#97

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊C₈

yes

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

C₄⋊C₈

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1283

—

211

maximal, Sylow

#431

—

211

—

C₂×C₄⋊C₈

D₂₁₁

yes

characteristic

C₄×M₁₆

—

211

—

C₂×M₁₆

—

211

—

C₂²×C₈

Dic₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

422

—

C₂²⋊C₈

—

211

—

#97

—

422

—

C₄×C₈

—

422

—

C₄×C₈

—

422

—

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

M₁₆

—

422

—

M₁₆

—

211

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₄²

C₂²⋊D₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

characteristic

C₄²

D₈₄₄

yes

characteristic

C₄²

Dic₄₂₂

yes

characteristic

C₂³

C₄×Dic₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₁₁

yes

characteristic

C₂×C₄

Dic₂₁₁⋊C₄

yes

characteristic

C₂×C₄

Dic₂₁₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₁₁

yes

characteristic

C₂×C₄

C₈×D₂₁₁

yes

—

C₂×C₄

#16809945

yes

—

C₈

—

100

C₈

—

844

—

101

C₈

—

422

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂₁₁⋊#95

yes

characteristic

105

C₂²

C₈×Dic₂₁₁

yes

characteristic

106

C₂²

#20269887

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

(C₂×D₂₁₁)⋊C₈

yes

—

111

C₄

#20269894

yes

characteristic

112

C₄

#20269896

yes

characteristic

113

C₄

Dic₂₁₁⋊C₈

yes

—

114

C₂

—

115

C₂

—

116

C₂

#27212900

yes

characteristic

117

C₂

#27212915

yes

characteristic

118

C₂

#27212917

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₁₁
commutator	C₂²×C₂₁₁	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₁₁
Fitting	C₂×C₄⋊C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#95
prime radical	C₂³×C₂₁₁	C₄²
cosocle	C₂×C₄²×C₂₁₁	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#1283
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₂₁₁

commutator

C₂²×C₂₁₁

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₁₁

Fitting

C₂×C₄⋊C₈×C₂₁₁

C₂

socle

C₂²×C₂₁₁

#95

prime radical

C₂³×C₂₁₁

C₄²

cosocle

C₂×C₄²×C₂₁₁

C₂²

hypercenter

C₂×C₄⋊C₈

D₂₁₁

hypocenter

C₂₁₁

#1283

LCS

G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
15. #16809946	C₈	semidirect
59. C₂₁₁	#1283	semidirect

H ≅ G/N

type

15. #16809946

C₈

semidirect

59. C₂₁₁

#1283

semidirect

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

subgroup	index
#27213199	2
C₄×#16809946	2
C₂×C₄⋊C₈×C₂₁₁	2
#1283	211

subgroup

index

#27213199

C₄×#16809946

C₂×C₄⋊C₈×C₂₁₁

#1283

211

Subgroups — #41011248

Conjugacy classes of subgroups

Complements

Maximal subgroups