Group #44400022 — subgroups

Subgroups — #44400022

Order 29056.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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	#28409799	C₂	1	1	yes	maximal, characteristic
2	#28409059	C₂	1	2	yes	maximal, complemented
3	#28408999	C₂	1	1	yes	maximal, characteristic
4	#28408802	C₂	1	2	yes	maximal, complemented
5	#410×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	#20300295	—	2	4	no	—
7	C₂×#16813868	—	2	2	no	—
8	C₂×#16813868	C₂²	1	1	yes	characteristic
9	C₂×#16813868	C₂²	1	2	yes	—
10	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₂₂₇⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₂₂₇⋊C₈)	—	2	2	no	—
13	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
14	#117×C₂₂₇	C₂²	1	2	yes	—
15	#99×C₂₂₇	C₄	1	4	yes	—
16	#16813868	—	2	4	no	—
17	#16813868	—	2	2	no	—
18	#16813868	—	2	2	no	—
19	#16813868	—	2	2	no	—
20	C₂×C₂₂₇⋊C₈	—	2	4	no	—
21	C₂×C₂₂₇⋊C₈	—	2	1	no	—
22	C₂×C₂₂₇⋊C₈	—	2	2	no	—
23	C₂×C₂₂₇⋊C₈	—	2	1	no	—
24	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	—
25	#41×C₂₂₇	—	2	2	no	—
26	#41×C₂₂₇	—	2	2	no	—
27	C₂×Q₈×C₂₂₇	Q₈	1	1	yes	characteristic
28	C₂×D₄×C₂₂₇	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₂₂₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
31	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₂₇	—	2	2	no	—
33	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
34	C₄²×C₂₂₇	—	2	2	no	—
35	C₂₂₇⋊C₈	—	2	4	no	—
36	C₂₂₇⋊C₈	—	4	1	no	—
37	C₂₂₇⋊C₈	—	4	1	no	—
38	C₂³×C₂₂₇	C₄⋊C₄	1	2	yes	—
39	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₂₂₇	—	2	2	no	—
41	D₄×C₂₂₇	—	2	4	no	—
42	D₄×C₂₂₇	—	2	2	no	—
43	C₂×C₄×C₂₂₇	—	2	2	no	—
44	C₂×C₄×C₂₂₇	—	2	2	no	—
45	C₂×C₄×C₂₂₇	—	2	4	no	—
46	C₂×C₄×C₂₂₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₂₇	#41	1	2	yes	—
51	C₂²×C₂₂₇	—	2	2	no	—
52	C₂²×C₂₂₇	—	4	1	no	—
53	C₂²×C₂₂₇	—	4	2	no	—
54	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
56	C₄×C₂₂₇	—	2	2	no	—
57	C₄×C₂₂₇	—	4	4	no	—
58	C₄×C₂₂₇	#120	1	1	yes	characteristic
59	C₄×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₂₂₇	#123	1	1	yes	characteristic
61	C₂×C₂₂₇	—	2	2	no	—
62	C₂×C₂₂₇	—	2	1	no	—
63	C₂×C₂₂₇	—	4	2	no	—
64	C₂×C₂₂₇	#385	1	1	yes	characteristic
65	C₂₂₇	#1868	1	1	yes	characteristic, complemented, Sylow
66	#1868	—	227	1	no	maximal, Sylow
67	#568	—	227	1	no	—
68	#424	—	227	2	no	—
69	#410	D₂₂₇	1	1	yes	characteristic
70	#408	—	227	1	no	—
71	#344	—	227	2	no	—
72	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
73	#131	—	454	4	no	—
74	C₂×M₁₆	—	454	2	no	—
75	C₂×M₁₆	—	227	1	no	—
76	C₂×M₁₆	—	227	2	no	—
77	C₂²×C₈	—	227	1	no	—
78	#117	C₂×D₂₂₇	1	2	yes	—
79	C₄⋊C₈	—	454	4	no	—
80	#99	Dic₂₂₇	1	4	yes	—
81	C₂²⋊C₈	—	454	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₄₅₄	1	1	yes	characteristic
85	C₂×D₄	C₂×Dic₂₂₇	1	2	yes	—
86	C₂×D₄	Dic₄₅₄	1	1	yes	characteristic
87	C₂²×C₄	D₉₀₈	1	2	yes	—
88	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
89	M₁₆	—	454	2	no	—
90	M₁₆	—	454	4	no	—
91	M₁₆	—	454	2	no	—
92	M₁₆	—	454	2	no	—
93	C₂×C₈	—	454	2	no	—
94	C₂×C₈	—	454	1	no	—
95	C₂×C₈	—	454	1	no	—
96	C₂×C₈	—	454	4	no	—
97	C₂×C₈	—	227	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₂³	C₄⋊Dic₂₂₇	1	2	yes	—
102	C₂³	C₂²×Dic₂₂₇	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄⋊Dic₂₂₇	1	2	yes	—
110	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
111	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₄₅₄	1	1	yes	characteristic
113	C₂×C₄	#16813883	1	2	yes	—
114	C₈	—	908	1	no	—
115	C₈	—	908	1	no	—
116	C₈	—	454	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×C₄⋊Dic₂₂₇	1	1	yes	characteristic
121	C₂²	D₄×Dic₂₂₇	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	#20300258	1	1	yes	characteristic
125	C₄	#20300265	1	2	yes	—
126	C₄	#20300272	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#28408907	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409799

C₂

yes

maximal, characteristic

#28409059

C₂

yes

maximal, complemented

#28408999

C₂

yes

maximal, characteristic

#28408802

C₂

yes

maximal, complemented

#410×C₂₂₇

C₂

yes

maximal, characteristic

#20300295

—

C₂×#16813868

—

C₂×#16813868

C₂²

yes

characteristic

C₂×#16813868

C₂²

yes

—

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₂₂₇⋊C₈)

—

C₂²⋊(C₂₂₇⋊C₈)

—

C₂×#41×C₂₂₇

C₂²

yes

characteristic

#117×C₂₂₇

C₂²

yes

—

#99×C₂₂₇

C₄

yes

—

—

—

—

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

#41×C₂₂₇

—

#41×C₂₂₇

—

C₂×Q₈×C₂₂₇

Q₈

yes

characteristic

C₂×D₄×C₂₂₇

C₂×C₄

yes

—

C₂×D₄×C₂₂₇

Q₈

yes

characteristic

C₂²×C₄×C₂₂₇

D₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

—

C₂²⋊C₄×C₂₂₇

C₂×C₄

yes

—

C₄²×C₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₄⋊C₄

yes

—

C₂³×C₂₂₇

C₂²×C₄

yes

characteristic

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#120

yes

characteristic

C₄×C₂₂₇

C₂²⋊Q₈

yes

—

C₄×C₂₂₇

#123

yes

characteristic

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#385

yes

characteristic

C₂₂₇

#1868

yes

characteristic, complemented, Sylow

#1868

—

227

maximal, Sylow

#568

—

227

—

#424

—

227

—

#410

D₂₂₇

yes

characteristic

#408

—

227

—

#344

—

227

—

C₂×#41

C₂×D₂₂₇

yes

characteristic

#131

—

454

—

C₂×M₁₆

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

—

227

—

C₂²×C₈

—

227

—

#117

C₂×D₂₂₇

yes

—

C₄⋊C₈

—

454

—

#99

Dic₂₂₇

yes

—

C₂²⋊C₈

—

454

—

#41

—

#41

—

C₂×Q₈

Dic₄₅₄

yes

characteristic

C₂×D₄

C₂×Dic₂₂₇

yes

—

C₂×D₄

Dic₄₅₄

yes

characteristic

C₂²×C₄

D₉₀₈

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

M₁₆

—

454

—

M₁₆

—

454

—

M₁₆

—

454

—

M₁₆

—

454

—

C₂×C₈

—

454

—

C₂×C₈

—

454

—

C₂×C₈

—

454

—

C₂×C₈

—

454

—

C₂×C₈

—

227

—

C₄⋊C₄

—

C₂²⋊C₄

C₂×Dic₂₂₇

yes

—

100

C₄²

—

101

C₂³

C₄⋊Dic₂₂₇

yes

—

102

C₂³

C₂²×Dic₂₂₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊Dic₂₂₇

yes

—

110

C₂×C₄

D₄×D₂₂₇

yes

—

111

C₂×C₄

C₂×D₉₀₈

yes

characteristic

112

C₂×C₄

C₂×Dic₄₅₄

yes

characteristic

113

C₂×C₄

#16813883

yes

—

114

C₈

—

908

—

115

C₈

—

908

—

116

C₈

—

454

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄⋊Dic₂₂₇

yes

characteristic

121

C₂²

D₄×Dic₂₂₇

yes

—

122

C₄

—

123

C₄

—

124

C₄

#20300258

yes

characteristic

125

C₄

#20300265

yes

—

126

C₄

#20300272

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#28408907

yes

characteristic

	subgroup	G/H
center	C₄	#20300258
commutator	C₂³×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#410×C₂₂₇	C₂
socle	C₂×C₂₂₇	#385
prime radical	C₂×#41×C₂₂₇	C₂²
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#410	D₂₂₇
hypocenter	C₂₂₇	#1868
LCS	G ▷ C₂³×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#20300258

commutator

C₂³×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#410×C₂₂₇

C₂

socle

C₂×C₂₂₇

#385

prime radical

C₂×#41×C₂₂₇

C₂²

cosocle

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂³×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #28409059	C₂	semidirect
4. #28408802	C₂	semidirect
65. C₂₂₇	#1868	semidirect

H ≅ G/N

type

2. #28409059

C₂

semidirect

4. #28408802

C₂

semidirect

65. C₂₂₇

#1868

semidirect

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

subgroup	index
#28409799	2
#28409059	2
#28408999	2
#28408802	2
#410×C₂₂₇	2
#1868	227

subgroup

index

#410×C₂₂₇

227

Subgroups — #44400022

Conjugacy classes of subgroups

Complements

Maximal subgroups