Group #41013510 — subgroups

Subgroups — #41013510

Order 27008.

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

normal subgroups only

This group has 236 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	#27213624	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269896	C₂	1	1	yes	maximal, characteristic
3	#27212918	C₂	1	2	yes	maximal
4	#27212917	C₂	1	2	yes	maximal
5	C₂×#101×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	Q₈×Dic₂₁₁	—	2	2	no	—
8	Q₈×Dic₂₁₁	—	2	2	no	—
9	#20269936	C₄	1	4	yes	—
10	#20269936	C₂²	1	1	yes	characteristic
11	#20269936	C₂²	1	1	yes	characteristic
12	#20269896	—	2	2	no	—
13	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
15	#101×C₂₁₁	—	2	2	no	—
16	#16809946	—	2	2	no	—
17	#16809946	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₄⋊Dic₂₁₁	—	2	4	no	—
20	C₄⋊Dic₂₁₁	—	2	1	no	—
21	C₄⋊Dic₂₁₁	—	2	1	no	—
22	C₂²⋊Dic₂₁₁	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₂₁₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₁₁	—	2	4	no	—
25	C₄×Dic₂₁₁	—	2	1	no	—
26	C₄×Dic₂₁₁	—	2	1	no	—
27	C₂×Q₈×C₂₁₁	—	2	2	no	—
28	C₂×Q₈×C₂₁₁	—	2	2	no	—
29	C₂×Q₈×C₂₁₁	D₄	1	2	yes	—
30	C₂×Q₈×C₂₁₁	D₄	1	2	yes	—
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₂₁₁	—	2	2	no	—
34	M₁₆×C₂₁₁	—	2	2	no	—
35	C₂×C₈×C₂₁₁	—	2	2	no	—
36	C₂×Dic₂₁₁	—	2	2	no	—
37	C₂×Dic₂₁₁	—	2	2	no	—
38	C₂₁₁⋊C₈	—	4	2	no	—
39	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₂₁₁	—	2	4	no	—
41	Q₈×C₂₁₁	—	2	4	no	—
42	C₂×C₄×C₂₁₁	—	2	2	no	—
43	C₂×C₄×C₂₁₁	—	2	2	no	—
44	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	#41	1	2	yes	—
48	C₂×C₄×C₂₁₁	#41	1	2	yes	—
49	C₈×C₂₁₁	—	4	2	no	—
50	C₂²×C₂₁₁	—	4	1	no	—
51	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₁₁	#117	1	1	yes	characteristic
53	C₂²×C₂₁₁	#117	1	1	yes	characteristic
54	C₄×C₂₁₁	—	2	4	no	—
55	C₄×C₂₁₁	#123	1	2	yes	—
56	C₄×C₂₁₁	#123	1	2	yes	—
57	Dic₂₁₁	—	4	2	no	—
58	Dic₂₁₁	—	4	2	no	—
59	C₂×C₂₁₁	—	2	1	no	—
60	C₂×C₂₁₁	—	2	1	no	—
61	C₂×C₂₁₁	—	2	1	no	—
62	C₂×C₂₁₁	#381	1	1	yes	characteristic
63	C₂₁₁	#1830	1	1	yes	characteristic, complemented, Sylow
64	#1830	—	211	1	no	maximal, Sylow
65	#519	—	211	1	no	—
66	C₂×#101	—	211	1	no	—
67	C₂×#101	D₂₁₁	1	1	yes	characteristic
68	#344	—	211	2	no	—
69	#344	—	211	2	no	—
70	C₂²×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
71	C₂×M₁₆	—	211	2	no	—
72	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
73	C₄×Q₈	—	422	2	no	—
74	C₄×Q₈	—	422	2	no	—
75	#117	—	211	4	no	—
76	#117	—	211	1	no	—
77	#117	—	211	1	no	—
78	#101	—	2	2	no	—
79	#101	—	422	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	C₂²⋊D₂₁₁	1	2	yes	—
83	C₂×Q₈	D₈₄₄	1	2	yes	—
84	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	422	2	no	—
89	M₁₆	—	422	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	422	2	no	—
92	C₄⋊C₄	—	422	4	no	—
93	C₄⋊C₄	—	422	1	no	—
94	C₄⋊C₄	—	422	1	no	—
95	C₂²⋊C₄	—	211	2	no	—
96	C₂²⋊C₄	—	211	2	no	—
97	C₄²	—	422	4	no	—
98	C₄²	—	422	1	no	—
99	C₄²	—	422	1	no	—
100	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	422	2	no	—
106	C₂×C₄	—	422	2	no	—
107	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
110	C₂×C₄	#16809961	1	2	yes	—
111	C₂×C₄	#16809961	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	844	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
116	C₂²	#20269933	1	1	yes	characteristic
117	C₂²	#20269933	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	844	2	no	—
120	C₄	—	844	2	no	—
121	C₄	#20269958	1	2	yes	—
122	C₄	#20269961	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#27213007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213624

C₂

yes

maximal, characteristic

C₂×#20269896

C₂

yes

maximal, characteristic

#27212918

C₂

yes

maximal

#27212917

C₂

yes

maximal

C₂×#101×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

Q₈×Dic₂₁₁

—

Q₈×Dic₂₁₁

—

#20269936

C₄

yes

—

#20269936

C₂²

yes

characteristic

#20269936

C₂²

yes

characteristic

#20269896

—

C₂²×Q₈×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

C₂²

yes

—

#101×C₂₁₁

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²⋊Dic₂₁₁

C₂×C₄

yes

—

C₂²⋊Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

—

C₂×Q₈×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

#123

yes

—

C₄×C₂₁₁

#123

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#381

yes

characteristic

C₂₁₁

#1830

yes

characteristic, complemented, Sylow

#1830

—

211

maximal, Sylow

#519

—

211

—

C₂×#101

—

211

—

C₂×#101

D₂₁₁

yes

characteristic

#344

—

211

—

#344

—

211

—

C₂²×Q₈

C₂×D₂₁₁

yes

characteristic

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₄×Q₈

—

422

—

C₄×Q₈

—

422

—

#117

—

211

—

#117

—

211

—

#117

—

211

—

#101

—

#101

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₁₁

yes

—

C₂×Q₈

D₈₄₄

yes

—

C₂²×C₄

C₄×D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₂²⋊C₄

—

211

—

C₂²⋊C₄

—

211

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

—

422

—

100

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

422

—

106

C₂×C₄

—

422

—

107

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

108

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

109

C₂×C₄

C₂×D₈₄₄

yes

characteristic

110

C₂×C₄

#16809961

yes

—

111

C₂×C₄

#16809961

yes

—

112

C₈

—

113

C₈

—

844

—

114

C₂²

—

115

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

116

C₂²

#20269933

yes

characteristic

117

C₂²

#20269933

yes

characteristic

118

C₄

—

119

C₄

—

844

—

120

C₄

—

844

—

121

C₄

#20269958

yes

—

122

C₄

#20269961

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#27213007

yes

characteristic

	subgroup	G/H
center	C₂	#27213007
commutator, prime radical	C₂³×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#101×C₂₁₁	C₂
socle	C₂×C₂₁₁	#381
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#101	D₂₁₁
hypocenter	C₂₁₁	#1830
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#27213007

commutator, prime radical

C₂³×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×#101×C₂₁₁

C₂

socle

C₂×C₂₁₁

#381

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
63. C₂₁₁	#1830	semidirect

H ≅ G/N

type

63. C₂₁₁

#1830

semidirect

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

subgroup	index
#27213624	2
C₂×#20269896	2
#27212918	2
#27212917	2
C₂×#101×C₂₁₁	2
#1830	211

subgroup

index

C₂×#101×C₂₁₁

211

Subgroups — #41013510

Conjugacy classes of subgroups

Complements

Maximal subgroups