Group #44401978 — subgroups

Subgroups — #44401978

Order 29056.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#28409433	C₂	1	1	yes	maximal, characteristic
2	#28409410	C₂	1	1	yes	maximal, characteristic
3	#28408837	C₂	1	1	yes	maximal, characteristic
4	#28408836	C₂	1	1	yes	maximal, characteristic
5	#28408835	C₂	1	1	yes	maximal, characteristic
6	#28408807	C₂	1	1	yes	maximal, characteristic
7	#358×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₉₀₈	C₂²	1	1	yes	characteristic
9	C₂×#16813874	—	2	1	no	—
10	#20300286	—	2	1	no	—
11	#20300279	—	2	1	no	—
12	C₂×C₂₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#20300222	C₂²	1	1	yes	characteristic
14	#20300219	—	2	1	no	—
15	C₈×Dic₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
18	#107×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₅₄	—	2	1	no	—
20	Dic₉₀₈	—	2	1	no	—
21	Dic₉₀₈	—	2	1	no	—
22	#16813874	—	4	2	no	—
23	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₂₂₇	—	2	1	no	—
25	C₄⋊Dic₂₂₇	—	4	1	no	—
26	Dic₂₂₇⋊C₄	—	4	1	no	—
27	Dic₂₂₇⋊C₄	—	4	1	no	—
28	Dic₂₂₇⋊C₄	—	4	1	no	—
29	C₄×Dic₂₂₇	D₄	1	1	yes	characteristic
30	C₂₂₇⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₂₂₇	—	2	1	no	—
32	Q₁₆×C₂₂₇	—	2	1	no	—
33	Q₁₆×C₂₂₇	—	2	1	no	—
34	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₂₇	—	2	1	no	—
36	C₁₆×C₂₂₇	—	2	1	no	—
37	Dic₄₅₄	—	4	1	no	—
38	Dic₄₅₄	—	4	1	no	—
39	C₂×Dic₂₂₇	—	4	1	no	—
40	C₂×Dic₂₂₇	—	4	1	no	—
41	C₂×Dic₂₂₇	D₈	1	1	yes	characteristic
42	C₂×Dic₂₂₇	D₈	1	1	yes	characteristic
43	C₂₂₇⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₂₂₇	—	4	1	no	—
45	Q₈×C₂₂₇	—	4	1	no	—
46	C₂×C₄×C₂₂₇	—	4	1	no	—
47	C₂×C₄×C₂₂₇	—	4	1	no	—
48	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₂₇	#41	1	1	yes	characteristic
50	C₈×C₂₂₇	#41	1	1	yes	characteristic
51	C₂²×C₂₂₇	C₂×D₈	1	1	yes	characteristic
52	C₄×C₂₂₇	—	8	1	no	—
53	C₄×C₂₂₇	—	8	1	no	—
54	C₄×C₂₂₇	#124	1	1	yes	characteristic
55	C₄×C₂₂₇	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₂₂₇	—	2	1	no	—
57	Dic₂₂₇	—	2	1	no	—
58	Dic₂₂₇	—	8	1	no	—
59	Dic₂₂₇	—	8	1	no	—
60	C₂×C₂₂₇	#486	1	1	yes	characteristic
61	C₂×C₂₂₇	#510	1	1	yes	characteristic
62	C₂×C₂₂₇	#510	1	1	yes	characteristic
63	C₂₂₇	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	227	1	no	maximal, Sylow
65	#499	—	227	1	no	—
66	#494	—	227	1	no	—
67	#358	—	227	1	no	—
68	#358	—	227	1	no	—
69	#358	—	227	1	no	—
70	#358	D₂₂₇	1	1	yes	characteristic
71	#346	—	227	1	no	—
72	C₂×Q₁₆	—	454	1	no	—
73	C₂×Q₁₆	—	227	1	no	—
74	C₂×Q₁₆	C₂×D₂₂₇	1	1	yes	characteristic
75	C₄⋊Q₈	—	454	1	no	—
76	#125	—	454	1	no	—
77	C₂×C₁₆	—	227	1	no	—
78	C₂×C₁₆	C₂×D₂₂₇	1	1	yes	characteristic
79	#107	—	227	1	no	—
80	#107	C₂×D₂₂₇	1	1	yes	characteristic
81	#106	—	454	1	no	—
82	C₄×C₈	—	227	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	454	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	908	2	no	—
88	Q₁₆	—	454	1	no	—
89	Q₁₆	—	454	1	no	—
90	C₂×C₈	—	227	1	no	—
91	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	908	1	no	—
94	C₄⋊C₄	—	908	1	no	—
95	C₄⋊C₄	—	908	1	no	—
96	C₄⋊C₄	—	908	1	no	—
97	C₄⋊C₄	—	454	1	no	—
98	C₄²	—	227	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	454	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	908	1	no	—
104	Q₈	—	908	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	908	1	no	—
108	C₂×C₄	—	908	1	no	—
109	C₂×C₄	—	227	1	no	—
110	C₂×C₄	—	227	1	no	—
111	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
112	C₈	—	227	2	no	—
113	C₈	#16813883	1	1	yes	characteristic
114	C₈	#16813884	1	1	yes	characteristic
115	C₂²	D₈×D₂₂₇	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	1816	1	no	—
119	C₄	—	1816	1	no	—
120	C₄	—	454	1	no	—
121	C₄	—	454	1	no	—
122	C₄	#20300274	1	1	yes	characteristic
123	C₄	SD₁₆×D₂₂₇	1	1	yes	characteristic
124	C₂	#28409377	1	1	yes	characteristic
125	C₂	#28409471	1	1	yes	characteristic
126	C₂	#28409474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409433

C₂

yes

maximal, characteristic

#28409410

C₂

yes

maximal, characteristic

#28408837

C₂

yes

maximal, characteristic

#28408836

C₂

yes

maximal, characteristic

#28408835

C₂

yes

maximal, characteristic

#28408807

C₂

yes

maximal, characteristic

#358×C₂₂₇

C₂

yes

maximal, characteristic

C₂×Dic₉₀₈

C₂²

yes

—

—

—

C₂×C₂₂₇⋊C₁₆

C₂²

yes

characteristic

#20300222

C₂²

yes

characteristic

#20300219

—

C₈×Dic₂₂₇

C₂²

yes

characteristic

C₂×Q₁₆×C₂₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₂₂₇

C₂²

yes

characteristic

#107×C₂₂₇

C₂²

yes

characteristic

C₂×Dic₄₅₄

—

Dic₉₀₈

—

Dic₉₀₈

—

#16813874

—

C₂×C₂₂₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

Dic₂₂₇⋊C₄

—

Dic₂₂₇⋊C₄

—

Dic₂₂₇⋊C₄

—

C₄×Dic₂₂₇

D₄

yes

characteristic

C₂₂₇⋊C₁₆

—

C₂×Q₈×C₂₂₇

—

Q₁₆×C₂₂₇

—

Q₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

—

C₁₆×C₂₂₇

—

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

D₈

yes

characteristic

C₂×Dic₂₂₇

D₈

yes

characteristic

C₂₂₇⋊C₈

SD₁₆

yes

—

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

C₈×C₂₂₇

yes

C₈×C₂₂₇

yes

C₂²×C₂₂₇

C₂×D₈

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#124

yes

characteristic

C₄×C₂₂₇

C₂×SD₁₆

yes

characteristic

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂₂₇

yes

characteristic, complemented, Sylow

#2245

—

227

maximal, Sylow

#499

—

227

—

#494

—

227

—

#358

—

227

—

#358

—

227

—

#358

—

227

—

#358

D₂₂₇

yes

characteristic

#346

—

227

—

C₂×Q₁₆

—

454

—

C₂×Q₁₆

—

227

—

C₂×Q₁₆

C₂×D₂₂₇

yes

characteristic

C₄⋊Q₈

—

454

—

#125

—

454

—

C₂×C₁₆

—

227

—

C₂×C₁₆

C₂×D₂₂₇

yes

characteristic

#107

—

227

—

#107

C₂×D₂₂₇

yes

characteristic

#106

—

454

—

C₄×C₈

—

227

—

C₂×Q₈

—

C₂×Q₈

—

454

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

908

—

Q₁₆

—

454

—

Q₁₆

—

454

—

C₂×C₈

—

227

—

C₂×C₈

C₂²×D₂₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

908

—

C₄⋊C₄

—

908

—

C₄⋊C₄

—

908

—

C₄⋊C₄

—

908

—

C₄⋊C₄

—

454

—

C₄²

—

227

—

C₁₆

—

100

C₁₆

—

454

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

908

—

104

Q₈

—

908

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

908

—

108

C₂×C₄

—

908

—

109

C₂×C₄

—

227

—

110

C₂×C₄

—

227

—

111

C₂×C₄

D₄×D₂₂₇

yes

characteristic

112

C₈

—

227

—

113

C₈

#16813883

yes

characteristic

114

C₈

#16813884

yes

characteristic

115

C₂²

D₈×D₂₂₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

1816

—

119

C₄

—

1816

—

120

C₄

—

454

—

121

C₄

—

454

—

122

C₄

#20300274

yes

characteristic

123

C₄

SD₁₆×D₂₂₇

yes

characteristic

124

C₂

#28409377

yes

characteristic

125

C₂

#28409471

yes

characteristic

126

C₂

#28409474

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₂₇
commutator, cosocle	C₂×C₈×C₂₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₂₂₇
Fitting	#358×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	C₂×D₈
hypercenter	#358	D₂₂₇
hypocenter	C₂₂₇	#2245
LCS	G ▷ C₂×C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₂₂₇

commutator, cosocle

C₂×C₈×C₂₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₂₂₇

Fitting

#358×C₂₂₇

C₂

socle, prime radical

C₂²×C₂₂₇

C₂×D₈

D₂₂₇

C₂₂₇

G ▷ C₂×C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₂₂₇	#2245	semidirect

H ≅ G/N

type

63. C₂₂₇

#2245

semidirect

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

subgroup	index
#28409433	2
#28409410	2
#28408837	2
#28408836	2
#28408835	2
#28408807	2
#358×C₂₂₇	2
#2245	227

subgroup

index

#358×C₂₂₇

227

Subgroups — #44401978

Conjugacy classes of subgroups

Complements

Maximal subgroups