Group #44408601 — subgroups

Subgroups — #44408601

Order 29056.

456 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 181 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×Dic₉₀₈	C₂	1	1	yes	maximal, characteristic
2	#28409509	C₂	1	2	yes	maximal
3	#28409138	C₂	1	4	yes	maximal, complemented
4	#28409126	C₂	1	4	yes	maximal, complemented
5	#28409040	C₂	1	2	yes	maximal
6	#28408987	C₂	1	1	yes	maximal, characteristic
7	#427×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₄₅₄	C₂²	1	2	yes	—
9	C₂×Dic₉₀₈	C₄	1	8	yes	complemented
10	C₂×Dic₉₀₈	C₂²	1	2	yes	—
11	C₂×Dic₉₀₈	C₂²	1	2	yes	—
12	C₂×#16813868	C₂²	1	1	yes	characteristic
13	C₄×Dic₄₅₄	—	2	4	no	—
14	#20300253	C₂²	1	8	yes	—
15	#20300247	C₂²	1	1	yes	characteristic
16	#20300244	—	2	4	no	—
17	#20300211	C₂²	1	8	yes	—
18	#20300198	C₂²	1	2	yes	—
19	C₈×Dic₂₂₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
21	#117×C₂₂₇	C₂²	1	2	yes	—
22	#107×C₂₂₇	C₂²	1	2	yes	—
23	#106×C₂₂₇	C₂²	1	2	yes	—
24	C₂×Dic₄₅₄	—	2	2	no	—
25	C₂×Dic₄₅₄	—	2	4	no	—
26	C₂×Dic₄₅₄	C₂×C₄	1	8	yes	—
27	C₂×Dic₄₅₄	C₂³	1	4	yes	—
28	C₂²×Dic₂₂₇	—	2	2	no	—
29	Dic₉₀₈	C₂×C₄	1	8	yes	—
30	Dic₉₀₈	C₂×C₄	1	8	yes	—
31	#16813868	—	2	2	no	—
32	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
33	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
34	Dic₂₂₇⋊C₄	—	2	8	no	—
35	C₂²⋊Dic₂₂₇	D₄	1	2	yes	—
36	C₄×Dic₂₂₇	—	2	8	no	—
37	C₄×Dic₂₂₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₂₂₇	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₂₂₇	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
44	C₄²×C₂₂₇	—	2	2	no	—
45	Dic₄₅₄	—	2	8	no	—
46	Dic₄₅₄	—	2	4	no	—
47	Dic₄₅₄	C₂²×C₄	1	8	yes	—
48	C₂×Dic₂₂₇	—	2	8	no	—
49	C₂×Dic₂₂₇	—	2	4	no	—
50	C₂×Dic₂₂₇	C₂×D₄	1	4	yes	—
51	C₂₂₇⋊C₈	—	2	2	no	—
52	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₂₇	—	2	4	no	—
54	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₂₇	#41	1	2	yes	—
57	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
58	C₈×C₂₂₇	C₂²×C₄	1	2	yes	—
59	C₈×C₂₂₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₂₂₇	—	2	2	no	—
61	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
62	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₂₂₇	—	4	4	no	—
64	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
65	C₄×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
67	Dic₂₂₇	—	2	8	no	—
68	Dic₂₂₇	—	2	4	no	—
69	C₂×C₂₂₇	—	2	2	no	—
70	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₂₂₇	#578	1	2	yes	—
72	C₂₂₇	#2950	1	1	yes	characteristic, complemented, Sylow
73	#2950	—	227	1	no	maximal, Sylow
74	C₂²×Q₁₆	—	227	1	no	—
75	#519	—	227	2	no	—
76	#441	—	227	4	no	—
77	C₄×Q₁₆	—	227	4	no	—
78	#427	D₂₂₇	1	1	yes	characteristic
79	#419	—	227	2	no	—
80	#405	—	227	1	no	—
81	C₂²×Q₈	—	227	2	no	—
82	C₂×Q₁₆	—	227	8	no	—
83	C₂×Q₁₆	—	227	2	no	—
84	C₂×Q₁₆	—	227	2	no	—
85	C₂×M₁₆	—	227	1	no	—
86	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
87	C₄×Q₈	—	454	4	no	—
88	C₄×Q₈	—	227	8	no	—
89	#117	—	454	4	no	—
90	#117	—	227	1	no	—
91	#117	C₂×D₂₂₇	1	2	yes	—
92	#107	C₂×D₂₂₇	1	2	yes	—
93	#106	C₂×D₂₂₇	1	2	yes	—
94	#103	—	227	8	no	—
95	#97	—	227	2	no	—
96	C₄×C₈	—	227	2	no	—
97	C₂×Q₈	—	454	2	no	—
98	C₂×Q₈	—	454	4	no	—
99	C₂×Q₈	—	227	8	no	—
100	C₂×Q₈	—	227	4	no	—
101	C₂²×C₄	—	454	2	no	—
102	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
103	Q₁₆	—	227	8	no	—
104	Q₁₆	—	227	8	no	—
105	M₁₆	—	454	2	no	—
106	C₂×C₈	—	227	2	no	—
107	C₂×C₈	C₄×D₂₂₇	1	4	yes	—
108	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
110	C₄⋊C₄	—	454	8	no	—
111	C₄⋊C₄	—	227	2	no	—
112	C₄⋊C₄	C₂²×D₂₂₇	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	227	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	454	8	no	—
117	C₄²	—	227	2	no	—
118	C₂³	D₄×D₂₂₇	1	1	yes	characteristic
119	Q₈	—	454	4	no	—
120	Q₈	—	454	8	no	—
121	Q₈	—	227	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	454	4	no	—
124	C₂×C₄	—	454	8	no	—
125	C₂×C₄	—	227	4	no	—
126	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
127	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
128	C₂×C₄	#16813885	1	2	yes	—
129	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
130	C₈	—	454	2	no	—
131	C₈	C₂×C₄×D₂₂₇	1	2	yes	—
132	C₈	C₂×C₄×D₂₂₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#20300250	1	2	yes	—
135	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	454	8	no	—
138	C₄	—	454	4	no	—
139	C₄	#20300250	1	2	yes	—
140	C₄	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
141	C₄	C₂×#16813885	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#20300250	1	1	yes	characteristic
144	C₂	#28409846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₉₀₈

C₂

yes

maximal, characteristic

#28409509

C₂

yes

maximal

#28409138

C₂

yes

maximal, complemented

#28409126

C₂

yes

maximal, complemented

#28409040

C₂

yes

maximal

#28408987

C₂

yes

maximal, characteristic

#427×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₄₅₄

C₂²

yes

—

C₂×Dic₉₀₈

C₄

yes

complemented

C₂×Dic₉₀₈

C₂²

yes

—

C₂×Dic₉₀₈

C₂²

yes

—

C₂×#16813868

C₂²

yes

characteristic

C₄×Dic₄₅₄

—

#20300253

C₂²

yes

—

#20300247

C₂²

yes

characteristic

#20300244

—

#20300211

C₂²

yes

—

#20300198

C₂²

yes

—

C₈×Dic₂₂₇

C₂²

yes

—

C₂²×C₈×C₂₂₇

C₂²

yes

characteristic

#117×C₂₂₇

C₂²

yes

—

#107×C₂₂₇

C₂²

yes

—

#106×C₂₂₇

C₂²

yes

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

C₂×C₄

yes

—

C₂×Dic₄₅₄

C₂³

yes

—

C₂²×Dic₂₂₇

—

Dic₉₀₈

C₂×C₄

yes

—

Dic₉₀₈

C₂×C₄

yes

—

#16813868

—

C₂×C₂₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₂₂₇

D₄

yes

—

Dic₂₂₇⋊C₄

—

C₂²⋊Dic₂₂₇

D₄

yes

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₂×C₈×C₂₂₇

C₂×C₄

yes

—

C₂×C₈×C₂₂₇

C₂³

yes

characteristic

C₂×C₈×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

C₂³

yes

—

C₂²⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

C₂²×C₄

yes

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

C₂×D₄

yes

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

—

C₂×C₄×C₂₂₇

C₂⁴

yes

characteristic

C₈×C₂₂₇

C₂²×C₄

yes

—

C₈×C₂₂₇

C₂²×C₄

yes

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₄×D₄

yes

—

C₂²×C₂₂₇

C₂²×D₄

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

C₂³×C₄

yes

characteristic

C₄×C₂₂₇

C₂×#41

yes

characteristic

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

C₂×C₄×D₄

yes

characteristic

C₂×C₂₂₇

#578

yes

—

C₂₂₇

#2950

yes

characteristic, complemented, Sylow

#2950

—

227

maximal, Sylow

C₂²×Q₁₆

—

227

—

#519

—

227

—

#441

—

227

—

C₄×Q₁₆

—

227

—

#427

D₂₂₇

yes

characteristic

#419

—

227

—

#405

—

227

—

C₂²×Q₈

—

227

—

C₂×Q₁₆

—

227

—

C₂×Q₁₆

—

227

—

C₂×Q₁₆

—

227

—

C₂×M₁₆

—

227

—

C₂²×C₈

C₂×D₂₂₇

yes

characteristic

C₄×Q₈

—

454

—

C₄×Q₈

—

227

—

#117

—

454

—

#117

—

227

—

#117

C₂×D₂₂₇

yes

—

#107

C₂×D₂₂₇

yes

—

#106

C₂×D₂₂₇

yes

—

#103

—

227

—

#97

—

227

—

C₄×C₈

—

227

—

C₂×Q₈

—

454

—

C₂×Q₈

—

454

—

C₂×Q₈

—

227

—

100

C₂×Q₈

—

227

—

101

C₂²×C₄

—

454

—

102

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

103

Q₁₆

—

227

—

104

Q₁₆

—

227

—

105

M₁₆

—

454

—

106

C₂×C₈

—

227

—

107

C₂×C₈

C₄×D₂₂₇

yes

—

108

C₂×C₈

C₂²×D₂₂₇

yes

characteristic

109

C₂×C₈

C₂²×D₂₂₇

yes

characteristic

110

C₄⋊C₄

—

454

—

111

C₄⋊C₄

—

227

—

112

C₄⋊C₄

C₂²×D₂₂₇

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

227

—

115

C₄²

—

116

C₄²

—

454

—

117

C₄²

—

227

—

118

C₂³

D₄×D₂₂₇

yes

characteristic

119

Q₈

—

454

—

120

Q₈

—

454

—

121

Q₈

—

227

—

122

C₂×C₄

—

123

C₂×C₄

—

454

—

124

C₂×C₄

—

454

—

125

C₂×C₄

—

227

—

126

C₂×C₄

C₂×C₄×D₂₂₇

yes

—

127

C₂×C₄

D₄×D₂₂₇

yes

characteristic

128

C₂×C₄

#16813885

yes

—

129

C₂×C₄

C₂³×D₂₂₇

yes

characteristic

130

C₈

—

454

—

131

C₈

C₂×C₄×D₂₂₇

yes

—

132

C₈

C₂×C₄×D₂₂₇

yes

—

133

C₂²

—

134

C₂²

#20300250

yes

—

135

C₂²

C₂×D₄×D₂₂₇

yes

characteristic

136

C₄

—

137

C₄

—

454

—

138

C₄

—

454

—

139

C₄

#20300250

yes

—

140

C₄

C₂²×C₄×D₂₂₇

yes

characteristic

141

C₄

C₂×#16813885

yes

characteristic

142

C₂

—

143

C₂

C₂×#20300250

yes

characteristic

144

C₂

#28409846

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₂₇
commutator	C₄×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	#427×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂²×D₄
prime radical	C₂³×C₂₂₇	C₂×D₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#427	D₂₂₇
hypocenter	C₂₂₇	#2950
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₂₂₇

commutator

C₄×C₂₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₂₇

Fitting

#427×C₂₂₇

C₂

socle

C₂²×C₂₂₇

C₂²×D₄

prime radical

C₂³×C₂₂₇

C₂×D₄

cosocle

C₂×C₄×C₂₂₇

C₂⁴

D₂₂₇

C₂₂₇

G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. #28409138	C₂	semidirect
4. #28409126	C₂	semidirect
9. C₂×Dic₉₀₈	C₄	semidirect
72. C₂₂₇	#2950	semidirect

H ≅ G/N

type

3. #28409138

C₂

semidirect

4. #28409126

C₂

semidirect

9. C₂×Dic₉₀₈

C₄

semidirect

72. C₂₂₇

#2950

semidirect

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

subgroup	index
C₂²×Dic₉₀₈	2
#28409509	2
#28409138	2
#28409126	2
#28409040	2
#28408987	2
#427×C₂₂₇	2
#2950	227

subgroup

index

C₂²×Dic₉₀₈

#427×C₂₂₇

227

Subgroups — #44408601

Conjugacy classes of subgroups

Complements

Maximal subgroups