Group #44398853 — subgroups

Subgroups — #44398853

Order 29056.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#28409768	C₂	1	1	yes	maximal, characteristic
2	#28409324	C₂	1	2	yes	maximal, complemented
3	#28408776	C₂	1	1	yes	maximal, characteristic
4	#28408764	C₂	1	2	yes	maximal, complemented
5	#492×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	#20300275	—	2	2	no	—
7	#20300269	—	2	2	no	—
8	C₄×Dic₄₅₄	C₂²	1	2	yes	—
9	D₄×Dic₂₂₇	—	2	2	no	—
10	#20300221	—	2	2	no	—
11	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
12	#20300211	—	2	2	no	—
13	C₂×SD₁₆×C₂₂₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
15	#127×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₅₄	D₄	1	2	yes	—
18	C₂²×Dic₂₂₇	—	2	2	no	—
19	C₂×C₂₂₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
21	Dic₂₂₇⋊C₄	—	2	2	no	—
22	C₂²⋊Dic₂₂₇	—	2	4	no	—
23	C₂²⋊Dic₂₂₇	—	2	2	no	—
24	C₄×Dic₂₂₇	—	2	2	no	—
25	C₂×Q₈×C₂₂₇	—	2	1	no	—
26	C₂×D₄×C₂₂₇	—	2	2	no	—
27	C₂×D₄×C₂₂₇	D₄	1	2	yes	—
28	SD₁₆×C₂₂₇	—	4	4	no	—
29	C₂×C₈×C₂₂₇	—	2	1	no	—
30	C₄⋊C₄×C₂₂₇	—	2	2	no	—
31	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
32	Dic₄₅₄	—	2	2	no	—
33	Dic₄₅₄	—	2	2	no	—
34	C₂×Dic₂₂₇	—	2	2	no	—
35	C₂×Dic₂₂₇	—	2	2	no	—
36	C₂×Dic₂₂₇	—	8	2	no	—
37	C₂₂₇⋊C₈	—	4	2	no	—
38	C₂³×C₂₂₇	—	2	2	no	—
39	Q₈×C₂₂₇	—	4	2	no	—
40	D₄×C₂₂₇	—	4	2	no	—
41	D₄×C₂₂₇	—	4	2	no	—
42	D₄×C₂₂₇	—	4	2	no	—
43	C₂×C₄×C₂₂₇	—	4	1	no	—
44	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
46	C₈×C₂₂₇	—	2	2	no	—
47	C₂²×C₂₂₇	—	4	4	no	—
48	C₂²×C₂₂₇	—	4	2	no	—
49	C₂²×C₂₂₇	#120	1	1	yes	characteristic
50	C₄×C₂₂₇	—	2	1	no	—
51	C₄×C₂₂₇	—	8	1	no	—
52	C₄×C₂₂₇	#135	1	2	yes	—
53	C₄×C₂₂₇	#137	1	2	yes	—
54	Dic₂₂₇	—	4	2	no	—
55	Dic₂₂₇	—	4	2	no	—
56	C₂×C₂₂₇	—	8	2	no	—
57	C₂×C₂₂₇	#452	1	2	yes	—
58	C₂×C₂₂₇	#453	1	1	yes	characteristic
59	C₂₂₇	#1630	1	1	yes	characteristic, complemented, Sylow
60	#1630	—	227	1	no	maximal, Sylow
61	#559	—	227	1	no	—
62	#492	D₂₂₇	1	1	yes	characteristic
63	#477	—	227	2	no	—
64	#331	—	227	1	no	—
65	Q₈⋊C₈	—	227	2	no	—
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
68	#127	C₂×D₂₂₇	1	1	yes	characteristic
69	#124	—	454	2	no	—
70	#123	—	454	2	no	—
71	C₄×Q₈	—	227	2	no	—
72	C₄×D₄	—	454	2	no	—
73	#107	—	454	2	no	—
74	C₄⋊C₈	—	227	2	no	—
75	#103	—	454	2	no	—
76	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	227	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₂₂₇	1	2	yes	—
81	C₂²×C₄	—	454	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	454	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	454	2	no	—
87	C₄⋊C₄	—	227	2	no	—
88	C₂²⋊C₄	—	454	4	no	—
89	C₂²⋊C₄	—	454	2	no	—
90	C₄²	—	454	2	no	—
91	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	454	2	no	—
95	Q₈	—	454	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	1816	2	no	—
101	C₂×C₄	—	454	2	no	—
102	C₂×C₄	—	454	2	no	—
103	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	908	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#20300256	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	908	2	no	—
113	C₄	—	908	2	no	—
114	C₄	#20300309	1	2	yes	—
115	C₄	#20300317	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#28409191	1	2	yes	—
118	C₂	#28409199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409768

C₂

yes

maximal, characteristic

#28409324

C₂

yes

maximal, complemented

#28408776

C₂

yes

maximal, characteristic

#28408764

C₂

yes

maximal, complemented

#492×C₂₂₇

C₂

yes

maximal, characteristic

#20300275

—

#20300269

—

C₄×Dic₄₅₄

C₂²

yes

—

D₄×Dic₂₂₇

—

#20300221

—

C₄⋊(C₂₂₇⋊C₈)

C₂²

yes

—

#20300211

—

C₂×SD₁₆×C₂₂₇

—

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

#127×C₂₂₇

C₂²

yes

characteristic

C₄×C₈×C₂₂₇

C₂²

yes

characteristic

C₂×Dic₄₅₄

D₄

yes

—

C₂²×Dic₂₂₇

—

C₂×C₂₂₇⋊C₈

—

C₄⋊Dic₂₂₇

D₄

yes

—

Dic₂₂₇⋊C₄

—

C₂²⋊Dic₂₂₇

—

C₂²⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×D₄×C₂₂₇

—

C₂×D₄×C₂₂₇

D₄

yes

—

SD₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

—

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

#120

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#135

yes

—

C₄×C₂₂₇

#137

yes

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#452

yes

—

C₂×C₂₂₇

#453

yes

characteristic

C₂₂₇

#1630

yes

characteristic, complemented, Sylow

#1630

—

227

maximal, Sylow

#559

—

227

—

#492

D₂₂₇

yes

characteristic

#477

—

227

—

#331

—

227

—

Q₈⋊C₈

—

227

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

#127

C₂×D₂₂₇

yes

characteristic

#124

—

454

—

#123

—

454

—

C₄×Q₈

—

227

—

C₄×D₄

—

454

—

#107

—

454

—

C₄⋊C₈

—

227

—

#103

—

454

—

C₄×C₈

C₂×D₂₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

227

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₂₇

yes

—

C₂²×C₄

—

454

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

454

—

C₄⋊C₄

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

227

—

C₂²⋊C₄

—

454

—

C₂²⋊C₄

—

454

—

C₄²

—

454

—

C₄²

C₂²×D₂₂₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

454

—

Q₈

—

454

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

1816

—

101

C₂×C₄

—

454

—

102

C₂×C₄

—

454

—

103

C₂×C₄

D₄×D₂₂₇

yes

—

104

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

105

C₈

—

106

C₈

—

908

—

107

C₂²

—

108

C₂²

—

109

C₂²

#20300256

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

908

—

113

C₄

—

908

—

114

C₄

#20300309

yes

—

115

C₄

#20300317

yes

—

116

C₂

—

117

C₂

#28409191

yes

—

118

C₂

#28409199

yes

characteristic

	subgroup	G/H
center	C₂²	#20300256
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#492×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#120
prime radical	#127×C₂₂₇	C₂²
hypercenter	#492	D₂₂₇
hypocenter	C₂₂₇	#1630
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20300256

commutator, cosocle

C₄²×C₂₂₇

C₂³

Frattini

C₄²

C₂²×D₂₂₇

Fitting

#492×C₂₂₇

C₂

socle

C₂²×C₂₂₇

#120

prime radical

#127×C₂₂₇

C₂²

D₂₂₇

C₂₂₇

G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #28409324	C₂	semidirect
4. #28408764	C₂	semidirect
59. C₂₂₇	#1630	semidirect

H ≅ G/N

type

2. #28409324

C₂

semidirect

4. #28408764

C₂

semidirect

59. C₂₂₇

#1630

semidirect

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

subgroup	index
#28409768	2
#28409324	2
#28408776	2
#28408764	2
#492×C₂₂₇	2
#1630	227

subgroup

index

#492×C₂₂₇

227

Subgroups — #44398853

Conjugacy classes of subgroups

Complements

Maximal subgroups