Group #41013440 — subgroups

Subgroups — #41013440

Order 27008.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 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	2	yes	maximal, complemented
2	#27213110	C₂	1	2	yes	maximal
3	#27212911	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16809946	C₂²	1	4	yes	—
6	#20269913	—	2	4	no	—
7	C₂²⋊(C₂₁₁⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₁₁	—	2	4	no	—
10	C₂×M₁₆×C₂₁₁	C₄	1	4	yes	—
11	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	#16809946	—	2	4	no	—
14	#16809946	—	2	4	no	—
15	C₂×C₂₁₁⋊C₈	—	2	4	no	—
16	C₂⁴×C₂₁₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₁₁	—	2	2	no	—
18	C₂²×C₄×C₂₁₁	—	2	2	no	—
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₁₁	—	2	8	no	—
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
28	C₂₁₁⋊C₈	—	4	4	no	—
29	C₂³×C₂₁₁	—	2	2	no	—
30	C₂³×C₂₁₁	—	4	2	no	—
31	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	—	2	2	no	—
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	2	2	no	—
36	C₂×C₄×C₂₁₁	—	2	4	no	—
37	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	#41	1	2	yes	—
42	C₈×C₂₁₁	—	2	2	no	—
43	C₈×C₂₁₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₂₁₁	—	2	2	no	—
45	C₂²×C₂₁₁	—	2	2	no	—
46	C₂²×C₂₁₁	—	2	4	no	—
47	C₂²×C₂₁₁	—	4	4	no	—
48	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₂₁₁	—	2	2	no	—
51	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₂₁₁	#117	1	1	yes	characteristic
53	C₄×C₂₁₁	#121	1	2	yes	—
54	C₂×C₂₁₁	—	2	2	no	—
55	C₂×C₂₁₁	—	4	2	no	—
56	C₂×C₂₁₁	#380	1	1	yes	characteristic
57	C₂×C₂₁₁	#430	1	2	yes	—
58	C₂₁₁	#1811	1	1	yes	characteristic, complemented, Sylow
59	#1811	—	211	1	no	maximal, Sylow
60	C₂²×M₁₆	D₂₁₁	1	1	yes	characteristic
61	C₂×#108	—	211	2	no	—
62	#407	—	211	2	no	—
63	#341	—	211	2	no	—
64	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	—	211	4	no	—
67	C₂×M₁₆	Dic₂₁₁	1	4	yes	—
68	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
70	#108	—	422	4	no	—
71	C₂²⋊C₈	—	422	4	no	—
72	C₂⁴	Dic₄₂₂	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
76	C₂²×C₄	D₈₄₄	1	2	yes	—
77	C₂²×C₄	Dic₄₂₂	1	1	yes	characteristic
78	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	422	4	no	—
81	M₁₆	—	422	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	422	4	no	—
84	C₂×C₈	C₂×Dic₂₁₁	1	2	yes	—
85	C₂×C₈	C₂×Dic₂₁₁	1	2	yes	—
86	C₂×C₈	C₂²⋊D₂₁₁	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₂₁₁	1	2	yes	—
90	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
96	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
98	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
99	C₂×C₄	#16809962	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	844	4	no	—
102	C₈	C₂²⋊Dic₂₁₁	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
108	C₂²	#20269956	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
111	C₄	#20269936	1	1	yes	characteristic
112	C₄	#20269951	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#27213006	1	1	yes	characteristic
116	C₂	#27213195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269913

C₂

yes

maximal, complemented

#27213110

C₂

yes

maximal

#27212911

C₂

yes

maximal, complemented

C₂²×M₁₆×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

#20269913

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

—

C₂×M₁₆×C₂₁₁

C₄

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

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

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂₁₁⋊C₈

—

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

—

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

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

#430

yes

—

C₂₁₁

#1811

yes

characteristic, complemented, Sylow

#1811

—

211

maximal, Sylow

C₂²×M₁₆

D₂₁₁

yes

characteristic

C₂×#108

—

211

—

#407

—

211

—

#341

—

211

—

C₂³×C₄

C₂×D₂₁₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

211

—

C₂×M₁₆

Dic₂₁₁

yes

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#108

—

422

—

C₂²⋊C₈

—

422

—

C₂⁴

Dic₄₂₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

D₈₄₄

yes

—

C₂²×C₄

Dic₄₂₂

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

C₂×Dic₂₁₁

yes

—

C₂×C₈

C₂×Dic₂₁₁

yes

—

C₂×C₈

C₂²⋊D₂₁₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₂₁₁

yes

—

C₂³

C₂×Dic₄₂₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

—

C₂×C₄

C₂×D₈₄₄

yes

characteristic

C₂×C₄

#16809962

yes

—

100

C₈

—

101

C₈

—

844

—

102

C₈

C₂²⋊Dic₂₁₁

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₂₁₁

yes

characteristic

108

C₂²

#20269956

yes

—

109

C₄

—

110

C₄

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

111

C₄

#20269936

yes

characteristic

112

C₄

#20269951

yes

—

113

C₂

—

114

C₂

—

115

C₂

#27213006

yes

characteristic

116

C₂

#27213195

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂²×M₁₆×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₁₁	Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂²×M₁₆	D₂₁₁
hypocenter	C₂₁₁	#1811
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₈₄₄

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂²×M₁₆×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₁₁

Q₈

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂²×M₁₆

D₂₁₁

hypocenter

C₂₁₁

#1811

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#20269913	C₂	semidirect
3. #27212911	C₂	semidirect
58. C₂₁₁	#1811	semidirect

H ≅ G/N

type

1. C₂×#20269913

C₂

semidirect

3. #27212911

C₂

semidirect

58. C₂₁₁

#1811

semidirect

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

subgroup	index
C₂×#20269913	2
#27213110	2
#27212911	2
C₂²×M₁₆×C₂₁₁	2
#1811	211

subgroup

index

C₂²×M₁₆×C₂₁₁

211

Subgroups — #41013440

Conjugacy classes of subgroups

Complements

Maximal subgroups