Group #41014666 — subgroups

Subgroups — #41014666

Order 27008.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic
2	#27213154	C₂	1	2	yes	maximal
3	C₂×#20269895	C₂	1	2	yes	maximal
4	#27212911	C₂	1	1	yes	maximal, characteristic
5	#424×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₂	C₂²	1	2	yes	—
7	C₂×Dic₈₄₄	—	2	4	no	—
8	C₂×Dic₈₄₄	—	2	2	no	—
9	C₂×#16809946	C₂²	1	2	yes	—
10	#20269900	—	2	4	no	—
11	#20269895	—	2	4	no	—
12	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	#117×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₂₁₁	—	2	2	no	—
16	C₂×Dic₄₂₂	—	2	2	no	—
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	—	2	2	no	—
19	C₂×Dic₄₂₂	—	2	2	no	—
20	C₂×Dic₄₂₂	—	2	2	no	—
21	C₂×Dic₄₂₂	D₄	1	4	yes	—
22	C₂²×Dic₂₁₁	—	2	2	no	—
23	Dic₈₄₄	—	2	8	no	—
24	#16809946	—	2	2	no	—
25	#16809946	—	2	2	no	—
26	C₂×C₂₁₁⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₁₁	—	2	2	no	—
29	C₂×C₈×C₂₁₁	—	2	2	no	—
30	C₂×C₈×C₂₁₁	—	2	1	no	—
31	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
32	C₄⋊C₄×C₂₁₁	D₄	1	2	yes	—
33	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
34	C₄²×C₂₁₁	—	2	1	no	—
35	Dic₄₂₂	—	2	4	no	—
36	Dic₄₂₂	—	2	4	no	—
37	Dic₄₂₂	—	4	2	no	—
38	Dic₄₂₂	—	4	2	no	—
39	C₂×Dic₂₁₁	—	2	4	no	—
40	C₂×Dic₂₁₁	—	4	2	no	—
41	C₂×Dic₂₁₁	—	4	2	no	—
42	C₂₁₁⋊C₈	—	4	2	no	—
43	C₂³×C₂₁₁	#41	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₁₁	#41	1	2	yes	—
49	C₈×C₂₁₁	—	2	2	no	—
50	C₈×C₂₁₁	—	4	1	no	—
51	C₂²×C₂₁₁	—	2	2	no	—
52	C₂²×C₂₁₁	#121	1	1	yes	characteristic
53	C₂²×C₂₁₁	#123	1	2	yes	—
54	C₄×C₂₁₁	—	4	2	no	—
55	C₄×C₂₁₁	#120	1	1	yes	characteristic
56	C₄×C₂₁₁	#121	1	2	yes	—
57	C₄×C₂₁₁	#124	1	1	yes	characteristic
58	Dic₂₁₁	—	4	4	no	—
59	C₂×C₂₁₁	—	2	2	no	—
60	C₂×C₂₁₁	#394	1	1	yes	characteristic
61	C₂×C₂₁₁	#473	1	2	yes	—
62	C₂₁₁	#2047	1	1	yes	characteristic, complemented, Sylow
63	#2047	—	211	1	no	maximal, Sylow
64	C₂²×Q₁₆	—	211	1	no	—
65	#424	D₂₁₁	1	1	yes	characteristic
66	#419	—	211	2	no	—
67	C₂×#101	—	211	2	no	—
68	#341	—	211	1	no	—
69	C₂²×Q₈	—	211	2	no	—
70	C₂×Q₁₆	—	422	4	no	—
71	C₂×Q₁₆	—	422	2	no	—
72	C₂×M₁₆	—	211	2	no	—
73	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
75	#117	C₂×D₂₁₁	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	422	4	no	—
78	#101	—	422	4	no	—
79	C₂×Q₈	—	422	2	no	—
80	C₂×Q₈	—	422	2	no	—
81	C₂×Q₈	—	422	2	no	—
82	C₂×Q₈	—	422	2	no	—
83	C₂×Q₈	—	422	2	no	—
84	C₂×Q₈	—	211	4	no	—
85	C₂²×C₄	—	422	2	no	—
86	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
87	Q₁₆	—	422	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	422	2	no	—
90	M₁₆	—	422	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	422	2	no	—
94	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
95	C₄⋊C₄	D₈₄₄	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#16809962	1	1	yes	characteristic
99	Q₈	—	844	2	no	—
100	Q₈	—	844	2	no	—
101	Q₈	—	422	4	no	—
102	Q₈	—	422	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	844	2	no	—
105	C₂×C₄	—	844	2	no	—
106	C₂×C₄	—	422	4	no	—
107	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
110	C₂×C₄	#16809963	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	844	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#20269951	1	1	yes	characteristic
116	C₂²	#20269962	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	844	4	no	—
119	C₄	#20269945	1	1	yes	characteristic
120	C₄	#20269952	1	2	yes	—
121	C₄	#20269966	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#27213063	1	1	yes	characteristic
124	C₂	#27213421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₈₄₄

C₂

yes

maximal, characteristic

#27213154

C₂

yes

maximal

C₂×#20269895

C₂

yes

maximal

#27212911

C₂

yes

maximal, characteristic

#424×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

—

C₂×Dic₈₄₄

—

C₂×Dic₈₄₄

—

C₂×#16809946

C₂²

yes

—

#20269900

—

#20269895

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

D₄

yes

—

C₂²×Dic₂₁₁

—

Dic₈₄₄

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

D₄

yes

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₂²×C₂₁₁

#123

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#120

yes

characteristic

C₄×C₂₁₁

#121

yes

—

C₄×C₂₁₁

#124

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#394

yes

characteristic

C₂×C₂₁₁

#473

yes

—

C₂₁₁

#2047

yes

characteristic, complemented, Sylow

#2047

—

211

maximal, Sylow

C₂²×Q₁₆

—

211

—

#424

D₂₁₁

yes

characteristic

#419

—

211

—

C₂×#101

—

211

—

#341

—

211

—

C₂²×Q₈

—

211

—

C₂×Q₁₆

—

422

—

C₂×Q₁₆

—

422

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#117

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

#103

—

422

—

#101

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

Q₁₆

—

422

—

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

C₂²⋊D₂₁₁

yes

—

C₄⋊C₄

D₈₄₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#16809962

yes

characteristic

Q₈

—

844

—

100

Q₈

—

844

—

101

Q₈

—

422

—

102

Q₈

—

422

—

103

C₂×C₄

—

104

C₂×C₄

—

844

—

105

C₂×C₄

—

844

—

106

C₂×C₄

—

422

—

107

C₂×C₄

D₄×D₂₁₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

109

C₂×C₄

C₂×D₈₄₄

yes

characteristic

110

C₂×C₄

#16809963

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

844

—

114

C₂²

—

115

C₂²

#20269951

yes

characteristic

116

C₂²

#20269962

yes

—

117

C₄

—

118

C₄

—

844

—

119

C₄

#20269945

yes

characteristic

120

C₄

#20269952

yes

—

121

C₄

#20269966

yes

characteristic

122

C₂

—

123

C₂

#27213063

yes

characteristic

124

C₂

#27213421

yes

—

	subgroup	G/H
center	C₂²	#20269951
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#424×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#121
prime radical	C₂³×C₂₁₁	#41
hypercenter	#424	D₂₁₁
hypocenter	C₂₁₁	#2047
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#20269951

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#424×C₂₁₁

C₂

C₂²×C₂₁₁

C₂³×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
62. C₂₁₁	#2047	semidirect

H ≅ G/N

type

62. C₂₁₁

#2047

semidirect

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

subgroup	index
C₂²×Dic₈₄₄	2
#27213154	2
C₂×#20269895	2
#27212911	2
#424×C₂₁₁	2
#2047	211

subgroup

index

C₂²×Dic₈₄₄

#424×C₂₁₁

211

Subgroups — #41014666

Conjugacy classes of subgroups

Complements

Maximal subgroups