Group #44400856 — subgroups

Subgroups — #44400856

Order 29056.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂×#20300321	C₂	1	2	yes	maximal
2	C₂²×Dic₉₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20300206	C₂	1	2	yes	maximal, complemented
4	#28408998	C₂	1	1	yes	maximal, characteristic
5	#341×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16813884	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₅₄	C₂²	1	2	yes	—
8	#20300321	—	2	4	no	—
9	#20300321	—	2	4	no	—
10	C₂×Dic₉₀₈	—	2	4	no	—
11	C₂×Dic₉₀₈	—	2	2	no	—
12	C₂×Dic₉₀₈	—	2	2	no	—
13	C₂×#16813873	—	2	2	no	—
14	C₂×#16813868	C₂²	1	1	yes	characteristic
15	#20300206	—	2	4	no	—
16	(C₂×D₂₂₇)⋊C₈	—	2	2	no	—
17	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
19	#16813884	—	2	2	no	—
20	#16813884	—	2	2	no	—
21	C₂×Dic₄₅₄	—	2	2	no	—
22	C₂×Dic₄₅₄	—	2	2	no	—
23	C₂×Dic₄₅₄	—	2	2	no	—
24	C₂×Dic₄₅₄	—	2	2	no	—
25	C₂×Dic₄₅₄	—	2	2	no	—
26	C₂×Dic₄₅₄	D₄	1	1	yes	characteristic
27	C₂×Dic₄₅₄	D₄	1	4	yes	—
28	C₂×D₉₀₈	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊D₂₂₇	—	2	1	no	—
30	C₂²×Dic₂₂₇	—	2	2	no	—
31	C₂×C₄×D₂₂₇	—	2	1	no	—
32	Dic₉₀₈	—	2	8	no	—
33	Dic₉₀₈	—	2	8	no	—
34	#16813873	—	2	8	no	—
35	#16813868	—	2	2	no	—
36	C₂×C₂₂₇⋊C₈	—	2	1	no	—
37	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
38	M₁₆×C₂₂₇	—	2	2	no	—
39	M₁₆×C₂₂₇	D₄	1	4	yes	—
40	C₂×C₈×C₂₂₇	—	2	2	no	—
41	C₂×C₈×C₂₂₇	—	2	2	no	—
42	C₂×C₈×C₂₂₇	D₄	1	2	yes	—
43	C₂²×D₂₂₇	—	2	1	no	—
44	Dic₄₅₄	—	2	4	no	—
45	Dic₄₅₄	—	2	4	no	—
46	Dic₄₅₄	—	2	2	no	—
47	Dic₄₅₄	—	4	2	no	—
48	Dic₄₅₄	—	4	2	no	—
49	D₉₀₈	—	2	2	no	—
50	C₂²⋊D₂₂₇	—	4	2	no	—
51	C₂×Dic₂₂₇	—	2	1	no	—
52	C₂×Dic₂₂₇	—	2	4	no	—
53	C₂×Dic₂₂₇	—	4	2	no	—
54	C₂×Dic₂₂₇	—	4	2	no	—
55	C₄×D₂₂₇	—	4	1	no	—
56	C₄×D₂₂₇	—	4	1	no	—
57	C₂₂₇⋊C₈	—	4	1	no	—
58	C₂³×C₂₂₇	#41	1	1	yes	characteristic
59	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
63	C₈×C₂₂₇	—	2	2	no	—
64	C₈×C₂₂₇	—	2	4	no	—
65	C₂²×C₂₂₇	—	2	2	no	—
66	C₂²×C₂₂₇	#121	1	2	yes	—
67	C₂²×C₂₂₇	#121	1	1	yes	characteristic
68	C₂×D₂₂₇	—	4	1	no	—
69	C₂×D₂₂₇	—	4	2	no	—
70	C₄×C₂₂₇	#120	1	1	yes	characteristic
71	C₄×C₂₂₇	#120	1	2	yes	—
72	C₄×C₂₂₇	#127	1	1	yes	characteristic
73	Dic₂₂₇	—	4	1	no	—
74	Dic₂₂₇	—	4	4	no	—
75	C₂×C₂₂₇	—	2	2	no	—
76	C₂×C₂₂₇	#392	1	1	yes	characteristic
77	C₂×C₂₂₇	#473	1	2	yes	—
78	D₂₂₇	—	8	1	no	—
79	C₂₂₇	#2020	1	1	yes	characteristic, complemented, Sylow
80	#2020	—	227	1	no	maximal, Sylow
81	C₂×#137	—	227	2	no	—
82	C₂²×Q₁₆	—	227	1	no	—
83	C₂×#101	—	227	2	no	—
84	#408	—	227	1	no	—
85	#341	D₂₂₇	1	1	yes	characteristic, complemented
86	C₂×#41	—	227	1	no	—
87	C₂²×Q₈	—	227	2	no	—
88	#137	—	454	4	no	—
89	#137	—	454	4	no	—
90	C₂×Q₁₆	—	454	4	no	—
91	C₂×Q₁₆	—	454	2	no	—
92	C₂×Q₁₆	—	454	2	no	—
93	C₂×SD₁₆	—	454	2	no	—
94	C₂×M₁₆	—	227	1	no	—
95	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
96	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
97	#101	—	454	4	no	—
98	C₂²⋊C₈	—	454	2	no	—
99	#41	—	454	2	no	—
100	#41	—	454	2	no	—
101	C₂×Q₈	—	454	2	no	—
102	C₂×Q₈	—	454	2	no	—
103	C₂×Q₈	—	454	2	no	—
104	C₂×Q₈	—	454	2	no	—
105	C₂×Q₈	—	454	2	no	—
106	C₂×Q₈	—	227	1	no	—
107	C₂×Q₈	—	227	4	no	—
108	C₂×D₄	—	454	1	no	—
109	C₂×D₄	—	227	1	no	—
110	C₂²×C₄	—	454	2	no	—
111	C₂²×C₄	—	454	1	no	—
112	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
113	Q₁₆	—	454	8	no	—
114	Q₁₆	—	454	8	no	—
115	SD₁₆	—	454	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	454	2	no	—
118	M₁₆	D₉₀₈	1	4	yes	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	454	1	no	—
122	C₂×C₈	D₉₀₈	1	2	yes	—
123	C₂³	—	454	1	no	—
124	C₂³	#16813885	1	1	yes	characteristic
125	Q₈	—	908	2	no	—
126	Q₈	—	908	2	no	—
127	Q₈	—	454	4	no	—
128	Q₈	—	454	4	no	—
129	Q₈	—	454	2	no	—
130	D₄	—	908	2	no	—
131	D₄	—	454	2	no	—
132	C₂×C₄	—	908	2	no	—
133	C₂×C₄	—	908	2	no	—
134	C₂×C₄	—	908	1	no	—
135	C₂×C₄	—	908	1	no	—
136	C₂×C₄	—	454	4	no	—
137	C₂×C₄	—	454	1	no	—
138	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
139	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
140	C₂×C₄	C₂×D₉₀₈	1	2	yes	—
141	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	—	908	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	908	1	no	—
147	C₂²	—	908	2	no	—
148	C₂²	#20300263	1	2	yes	—
149	C₂²	#20300263	1	1	yes	characteristic
150	C₄	—	908	1	no	—
151	C₄	—	908	4	no	—
152	C₄	#20300258	1	1	yes	characteristic
153	C₄	#20300258	1	2	yes	—
154	C₄	#20300285	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	1816	1	no	—
157	C₂	#28408942	1	1	yes	characteristic
158	C₂	#28409308	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300321

C₂

yes

maximal

C₂²×Dic₉₀₈

C₂

yes

maximal, characteristic, complemented

C₂×#20300206

C₂

yes

maximal, complemented

#28408998

C₂

yes

maximal, characteristic

#341×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#16813884

C₂²

yes

characteristic

C₂²×Dic₄₅₄

C₂²

yes

—

#20300321

—

#20300321

—

C₂×Dic₉₀₈

—

C₂×Dic₉₀₈

—

C₂×Dic₉₀₈

—

C₂×#16813873

—

C₂×#16813868

C₂²

yes

characteristic

#20300206

—

(C₂×D₂₂₇)⋊C₈

—

C₂×M₁₆×C₂₂₇

C₂²

yes

—

C₂²×C₈×C₂₂₇

C₂²

yes

characteristic

#16813884

—

#16813884

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

D₄

yes

characteristic

C₂×Dic₄₅₄

D₄

yes

—

C₂×D₉₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₂₂₇

—

C₂²×Dic₂₂₇

—

C₂×C₄×D₂₂₇

—

Dic₉₀₈

—

Dic₉₀₈

—

#16813873

—

#16813868

—

C₂×C₂₂₇⋊C₈

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

M₁₆×C₂₂₇

D₄

yes

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

D₄

yes

—

C₂²×D₂₂₇

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

D₉₀₈

—

C₂²⋊D₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

#41

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₈×C₂₂₇

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

#121

yes

—

C₂²×C₂₂₇

#121

yes

characteristic

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

#120

yes

characteristic

C₄×C₂₂₇

#120

yes

—

C₄×C₂₂₇

#127

yes

characteristic

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#392

yes

characteristic

C₂×C₂₂₇

#473

yes

—

D₂₂₇

—

C₂₂₇

#2020

yes

characteristic, complemented, Sylow

#2020

—

227

maximal, Sylow

C₂×#137

—

227

—

C₂²×Q₁₆

—

227

—

C₂×#101

—

227

—

#408

—

227

—

#341

D₂₂₇

yes

characteristic, complemented

C₂×#41

—

227

—

C₂²×Q₈

—

227

—

#137

—

454

—

#137

—

454

—

C₂×Q₁₆

—

454

—

C₂×Q₁₆

—

454

—

C₂×Q₁₆

—

454

—

C₂×SD₁₆

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₂²×C₈

C₂×D₂₂₇

yes

characteristic

#101

—

454

—

C₂²⋊C₈

—

454

—

#41

—

454

—

100

#41

—

454

—

101

C₂×Q₈

—

454

—

102

C₂×Q₈

—

454

—

103

C₂×Q₈

—

454

—

104

C₂×Q₈

—

454

—

105

C₂×Q₈

—

454

—

106

C₂×Q₈

—

227

—

107

C₂×Q₈

—

227

—

108

C₂×D₄

—

454

—

109

C₂×D₄

—

227

—

110

C₂²×C₄

—

454

—

111

C₂²×C₄

—

454

—

112

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

113

Q₁₆

—

454

—

114

Q₁₆

—

454

—

115

SD₁₆

—

454

—

116

M₁₆

—

117

M₁₆

—

454

—

118

M₁₆

D₉₀₈

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

454

—

122

C₂×C₈

D₉₀₈

yes

—

123

C₂³

—

454

—

124

C₂³

#16813885

yes

characteristic

125

Q₈

—

908

—

126

Q₈

—

908

—

127

Q₈

—

454

—

128

Q₈

—

454

—

129

Q₈

—

454

—

130

D₄

—

908

—

131

D₄

—

454

—

132

C₂×C₄

—

908

—

133

C₂×C₄

—

908

—

134

C₂×C₄

—

908

—

135

C₂×C₄

—

908

—

136

C₂×C₄

—

454

—

137

C₂×C₄

—

454

—

138

C₂×C₄

D₄×D₂₂₇

yes

—

139

C₂×C₄

D₄×D₂₂₇

yes

characteristic

140

C₂×C₄

C₂×D₉₀₈

yes

—

141

C₂×C₄

C₂×D₉₀₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

908

—

145

C₂²

—

146

C₂²

—

908

—

147

C₂²

—

908

—

148

C₂²

#20300263

yes

—

149

C₂²

#20300263

yes

characteristic

150

C₄

—

908

—

151

C₄

—

908

—

152

C₄

#20300258

yes

characteristic

153

C₄

#20300258

yes

—

154

C₄

#20300285

yes

characteristic

155

C₂

—

156

C₂

—

1816

—

157

C₂

#28408942

yes

characteristic

158

C₂

#28409308

yes

—

	subgroup	G/H
center	C₂²	#20300263
commutator, cosocle	C₂²×C₄×C₂₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#341×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#121
prime radical	C₂×#16813884	C₂²
hypercenter	#341	D₂₂₇
hypocenter	C₂₂₇	#2020
LCS	G ▷ C₂²×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#20300263

commutator, cosocle

C₂²×C₄×C₂₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#341×C₂₂₇

C₂

C₂²×C₂₂₇

C₂²

D₂₂₇

C₂₂₇

G ▷ C₂²×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂²×Dic₉₀₈	C₂	semidirect
3. C₂×#20300206	C₂	semidirect
5. #341×C₂₂₇	C₂	semidirect
79. C₂₂₇	#2020	semidirect
85. #341	D₂₂₇	semidirect

H ≅ G/N

type

2. C₂²×Dic₉₀₈

C₂

semidirect

3. C₂×#20300206

C₂

semidirect

5. #341×C₂₂₇

C₂

semidirect

79. C₂₂₇

#2020

semidirect

85. #341

D₂₂₇

semidirect

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

subgroup	index
C₂×#20300321	2
C₂²×Dic₉₀₈	2
C₂×#20300206	2
#28408998	2
#341×C₂₂₇	2
#2020	227

subgroup

index

C₂²×Dic₉₀₈

#341×C₂₂₇

227

Subgroups — #44400856

Conjugacy classes of subgroups

Complements

Maximal subgroups