Group #44400855 — subgroups

Subgroups — #44400855

Order 29056.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20300318	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	#28408797	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₅₄	C₂²	1	2	yes	—
7	#20300318	—	2	4	no	—
8	#20300318	—	2	4	no	—
9	C₂×Dic₉₀₈	—	2	4	no	—
10	C₂×Dic₉₀₈	—	2	2	no	—
11	C₂×#16813874	—	2	2	no	—
12	C₂×#16813872	—	2	2	no	—
13	C₂×#16813868	C₂²	1	2	yes	—
14	#20300206	—	2	4	no	—
15	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₂₂₇	—	2	2	no	—
19	C₂×Dic₄₅₄	—	2	2	no	—
20	C₂×Dic₄₅₄	—	2	2	no	—
21	C₂×Dic₄₅₄	—	2	2	no	—
22	C₂×Dic₄₅₄	—	2	2	no	—
23	C₂×Dic₄₅₄	—	2	2	no	—
24	C₂×Dic₄₅₄	D₄	1	4	yes	—
25	C₂²×Dic₂₂₇	—	2	2	no	—
26	Dic₉₀₈	—	2	8	no	—
27	#16813874	—	2	8	no	—
28	#16813872	—	2	8	no	—
29	#16813868	—	2	2	no	—
30	#16813868	D₄	1	4	yes	—
31	C₂×C₂₂₇⋊C₈	—	2	2	no	—
32	#41×C₂₂₇	—	2	2	no	—
33	#41×C₂₂₇	—	2	2	no	—
34	C₂×Q₈×C₂₂₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₂₂₇	—	2	1	no	—
36	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₂₇	—	2	1	no	—
38	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
39	M₁₆×C₂₂₇	—	2	2	no	—
40	C₂×C₈×C₂₂₇	—	2	2	no	—
41	C₂×C₈×C₂₂₇	—	2	1	no	—
42	C₂×C₈×C₂₂₇	D₄	1	2	yes	—
43	Dic₄₅₄	—	2	4	no	—
44	Dic₄₅₄	—	2	4	no	—
45	Dic₄₅₄	—	4	2	no	—
46	Dic₄₅₄	—	4	2	no	—
47	C₂×Dic₂₂₇	—	2	4	no	—
48	C₂×Dic₂₂₇	—	4	2	no	—
49	C₂×Dic₂₂₇	—	4	2	no	—
50	C₂₂₇⋊C₈	—	2	4	no	—
51	C₂³×C₂₂₇	—	2	1	no	—
52	C₂³×C₂₂₇	#41	1	1	yes	characteristic
53	Q₈×C₂₂₇	—	2	2	no	—
54	D₄×C₂₂₇	—	2	2	no	—
55	D₄×C₂₂₇	—	4	2	no	—
56	C₂×C₄×C₂₂₇	—	2	1	no	—
57	C₂×C₄×C₂₂₇	—	4	1	no	—
58	C₂×C₄×C₂₂₇	—	4	1	no	—
59	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
63	C₈×C₂₂₇	—	2	2	no	—
64	C₈×C₂₂₇	—	4	1	no	—
65	C₂²×C₂₂₇	—	2	2	no	—
66	C₂²×C₂₂₇	—	4	1	no	—
67	C₂²×C₂₂₇	—	4	2	no	—
68	C₂²×C₂₂₇	#121	1	2	yes	—
69	C₂²×C₂₂₇	#121	1	1	yes	characteristic
70	C₄×C₂₂₇	—	4	1	no	—
71	C₄×C₂₂₇	#120	1	1	yes	characteristic
72	C₄×C₂₂₇	#120	1	2	yes	—
73	C₄×C₂₂₇	#127	1	1	yes	characteristic
74	Dic₂₂₇	—	4	4	no	—
75	C₂×C₂₂₇	—	2	2	no	—
76	C₂×C₂₂₇	—	8	1	no	—
77	C₂×C₂₂₇	#392	1	1	yes	characteristic
78	C₂×C₂₂₇	#473	1	2	yes	—
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	D₂₂₇	1	1	yes	characteristic
85	#341	—	227	1	no	—
86	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
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	2	no	—
95	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
97	#101	—	454	4	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	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	4	no	—
107	C₂×Q₈	D₉₀₈	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	D₉₀₈	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	454	2	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	—	227	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	454	2	no	—
122	C₂×C₈	C₂²⋊D₂₂₇	1	2	yes	—
123	C₂³	—	2	1	no	—
124	C₂³	#16813884	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	908	2	no	—
127	Q₈	—	908	2	no	—
128	Q₈	—	454	4	no	—
129	Q₈	—	454	4	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	908	2	no	—
136	C₂×C₄	—	908	2	no	—
137	C₂×C₄	—	454	4	no	—
138	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
139	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	454	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	#20300261	1	2	yes	—
149	C₂²	#20300262	1	1	yes	characteristic
150	C₄	—	4	1	no	—
151	C₄	—	908	4	no	—
152	C₄	#20300256	1	1	yes	characteristic
153	C₄	#20300258	1	2	yes	—
154	C₄	#20300284	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	8	1	no	—
157	C₂	#28408941	1	1	yes	characteristic
158	C₂	#28409307	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300318

C₂

yes

maximal

C₂²×Dic₉₀₈

C₂

yes

maximal, characteristic, complemented

C₂×#20300206

C₂

yes

maximal, complemented

#28408797

C₂

yes

maximal, characteristic, complemented

#408×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₄₅₄

C₂²

yes

—

#20300318

—

#20300318

—

C₂×Dic₉₀₈

—

C₂×Dic₉₀₈

—

C₂×#16813874

—

C₂×#16813872

—

C₂×#16813868

C₂²

yes

—

#20300206

—

C₂×#41×C₂₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₂₇

C₂²

yes

characteristic

C₂²×C₈×C₂₂₇

C₂²

yes

characteristic

C₂²⋊C₈×C₂₂₇

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

D₄

yes

—

C₂²×Dic₂₂₇

—

Dic₉₀₈

—

—

—

—

D₄

yes

—

C₂×C₂₂₇⋊C₈

—

#41×C₂₂₇

—

#41×C₂₂₇

—

C₂×Q₈×C₂₂₇

D₄

yes

characteristic

C₂×D₄×C₂₂₇

—

C₂×D₄×C₂₂₇

D₄

yes

characteristic

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

D₄

yes

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

—

C₂³×C₂₂₇

#41

yes

characteristic

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₈×C₂₂₇

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

#121

yes

—

C₂²×C₂₂₇

#121

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

#120

yes

characteristic

C₄×C₂₂₇

#120

yes

—

C₄×C₂₂₇

#127

yes

characteristic

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#392

yes

characteristic

C₂×C₂₂₇

#473

yes

—

C₂₂₇

#2020

yes

characteristic, complemented, Sylow

#2020

—

227

maximal, Sylow

C₂×#137

—

227

—

C₂²×Q₁₆

—

227

—

C₂×#101

—

227

—

#408

D₂₂₇

yes

characteristic

#341

—

227

—

C₂×#41

C₂×D₂₂₇

yes

characteristic

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

characteristic

C₂²×C₈

C₂×D₂₂₇

yes

characteristic

#101

—

454

—

C₂²⋊C₈

—

#41

—

100

#41

—

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

D₉₀₈

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

D₉₀₈

yes

characteristic

110

C₂²×C₄

—

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

—

227

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

454

—

122

C₂×C₈

C₂²⋊D₂₂₇

yes

—

123

C₂³

—

124

C₂³

#16813884

yes

characteristic

125

Q₈

—

126

Q₈

—

908

—

127

Q₈

—

908

—

128

Q₈

—

454

—

129

Q₈

—

454

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

908

—

136

C₂×C₄

—

908

—

137

C₂×C₄

—

454

—

138

C₂×C₄

D₄×D₂₂₇

yes

—

139

C₂×C₄

D₄×D₂₂₇

yes

—

140

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

141

C₂×C₄

C₂×D₉₀₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

454

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#20300261

yes

—

149

C₂²

#20300262

yes

characteristic

150

C₄

—

151

C₄

—

908

—

152

C₄

#20300256

yes

characteristic

153

C₄

#20300258

yes

—

154

C₄

#20300284

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#28408941

yes

characteristic

158

C₂

#28409307

yes

—

	subgroup	G/H
center	C₂²	#20300262
commutator, cosocle	C₂²×C₄×C₂₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#408×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#121
prime radical	C₂×#41×C₂₂₇	C₂²
hypercenter	#408	D₂₂₇
hypocenter	C₂₂₇	#2020
LCS	G ▷ C₂²×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#20300262

commutator, cosocle

C₂²×C₄×C₂₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#408×C₂₂₇

C₂

socle

C₂²×C₂₂₇

#121

prime radical

C₂×#41×C₂₂₇

C₂²

D₂₂₇

C₂₂₇

G ▷ C₂²×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. C₂²×Dic₉₀₈	C₂	semidirect
3. C₂×#20300206	C₂	semidirect
4. #28408797	C₂	semidirect
79. C₂₂₇	#2020	semidirect

H ≅ G/N

type

2. C₂²×Dic₉₀₈

C₂

semidirect

3. C₂×#20300206

C₂

semidirect

4. #28408797

C₂

semidirect

79. C₂₂₇

#2020

semidirect

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

subgroup	index
C₂×#20300318	2
C₂²×Dic₉₀₈	2
C₂×#20300206	2
#28408797	2
#408×C₂₂₇	2
#2020	227

subgroup

index

C₂²×Dic₉₀₈

#408×C₂₂₇

227

Subgroups — #44400855

Conjugacy classes of subgroups

Complements

Maximal subgroups