Group #44399779 — subgroups

Subgroups — #44399779

Order 29056.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20300296	C₂	1	1	yes	maximal, characteristic
2	#28409068	C₂	1	2	yes	maximal, complemented
3	#28409005	C₂	1	2	yes	maximal
4	#28408789	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16813884	C₂²	1	1	yes	characteristic
7	#20300296	—	2	2	no	—
8	#20300296	—	2	2	no	—
9	C₂×#16813867	C₄	1	2	yes	complemented
10	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₂₇	C₄	1	2	yes	complemented
12	#20300247	C₂²	1	2	yes	—
13	#20300221	—	2	2	no	—
14	#20300219	—	2	2	no	—
15	#20300202	—	2	4	no	—
16	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
17	#117×C₂₂₇	C₂²	1	2	yes	—
18	#16813884	—	2	2	no	—
19	#16813884	—	2	2	no	—
20	C₂×Dic₄₅₄	Q₈	1	1	yes	characteristic
21	C₂×D₉₀₈	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₂₂₇	D₄	1	2	yes	—
23	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
24	#16813867	—	2	4	no	—
25	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
28	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
29	C₈×D₂₂₇	—	2	2	no	—
30	C₈×D₂₂₇	—	2	2	no	—
31	C₄⋊Dic₂₂₇	—	2	2	no	—
32	C₂²⋊Dic₂₂₇	—	2	2	no	—
33	C₄×Dic₂₂₇	—	2	2	no	—
34	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₂₇	—	2	2	no	—
36	C₂×C₈×C₂₂₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₂₂₇	—	2	2	no	—
38	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
39	C₄²×C₂₂₇	—	2	2	no	—
40	C₂²×D₂₂₇	C₄⋊C₄	1	2	yes	—
41	Dic₄₅₄	—	2	2	no	—
42	D₉₀₈	—	2	2	no	—
43	C₂²⋊D₂₂₇	—	2	4	no	—
44	C₂×Dic₂₂₇	—	4	2	no	—
45	C₂×Dic₂₂₇	C₄⋊C₄	1	2	yes	—
46	C₄×D₂₂₇	—	2	2	no	—
47	C₄×D₂₂₇	—	2	2	no	—
48	C₂₂₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂₂₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₂₇	—	4	2	no	—
53	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₂₇	#41	1	2	yes	—
57	C₈×C₂₂₇	—	2	2	no	—
58	C₂²×C₂₂₇	—	4	1	no	—
59	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₂₇	#121	1	2	yes	—
61	C₂×D₂₂₇	—	2	2	no	—
62	C₂×D₂₂₇	—	4	2	no	—
63	C₄×C₂₂₇	—	8	2	no	—
64	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₂₇	#117	1	1	yes	characteristic
66	C₄×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₂₂₇	—	2	2	no	—
68	Dic₂₂₇	—	8	2	no	—
69	C₂×C₂₂₇	—	2	2	no	—
70	C₂×C₂₂₇	—	2	1	no	—
71	C₂×C₂₂₇	#380	1	1	yes	characteristic
72	D₂₂₇	—	4	2	no	—
73	C₂₂₇	#1812	1	1	yes	characteristic, complemented, Sylow
74	#1812	—	227	1	no	maximal, Sylow
75	C₂×#131	—	227	1	no	—
76	#427	—	227	2	no	—
77	#410	—	227	2	no	—
78	H(3,ℤ/4)	—	227	1	no	—
79	H(3,ℤ/4)	D₂₂₇	1	1	yes	characteristic, complemented
80	C₂×#41	—	227	1	no	—
81	#131	—	454	2	no	—
82	#131	—	454	2	no	—
83	C₂×M₁₆	—	227	2	no	—
84	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
85	C₂²×C₈	—	227	2	no	—
86	C₂²×C₈	—	227	1	no	—
87	#117	—	227	2	no	—
88	#117	C₂×D₂₂₇	1	2	yes	—
89	#107	—	454	2	no	—
90	#106	—	454	2	no	—
91	#99	—	454	4	no	—
92	#41	—	454	2	no	—
93	#41	—	454	2	no	—
94	C₂×Q₈	—	227	1	no	—
95	C₂×D₄	—	227	2	no	—
96	C₂×D₄	—	227	1	no	—
97	C₂²×C₄	—	227	2	no	—
98	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	454	4	no	—
101	C₂×C₈	—	454	2	no	—
102	C₂×C₈	—	454	2	no	—
103	C₂×C₈	—	227	1	no	—
104	C₂×C₈	—	227	1	no	—
105	C₂×C₈	—	227	2	no	—
106	C₂×C₈	—	227	2	no	—
107	C₂×C₈	C₄×D₂₂₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	454	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	454	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	454	2	no	—
114	C₂³	—	227	2	no	—
115	C₂³	D₄×D₂₂₇	1	1	yes	characteristic
116	Q₈	—	454	2	no	—
117	D₄	—	454	4	no	—
118	D₄	—	454	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	908	2	no	—
121	C₂×C₄	—	454	2	no	—
122	C₂×C₄	—	454	2	no	—
123	C₂×C₄	—	227	2	no	—
124	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
126	C₂×C₄	Q₈×D₂₂₇	1	1	yes	characteristic
127	C₂×C₄	#16813884	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	227	1	no	—
130	C₈	—	227	1	no	—
131	C₈	—	227	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	908	2	no	—
134	C₂²	—	454	2	no	—
135	C₂²	C₄⋊C₄×D₂₂₇	1	1	yes	characteristic
136	C₂²	#20300261	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	1816	2	no	—
139	C₄	—	454	2	no	—
140	C₄	C₂²⋊C₄×D₂₂₇	1	1	yes	characteristic
141	C₄	#20300245	1	1	yes	characteristic
142	C₄	#20300266	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	908	2	no	—
146	C₂	#28408891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300296

C₂

yes

maximal, characteristic

#28409068

C₂

yes

maximal, complemented

#28409005

C₂

yes

maximal

#28408789

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#16813884

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₂₇

C₄

yes

complemented

#20300247

C₂²

yes

—

#20300221

—

#20300219

—

#20300202

—

C₂×M₁₆×C₂₂₇

C₂²

yes

characteristic

#117×C₂₂₇

C₂²

yes

—

#16813884

—

#16813884

—

C₂×Dic₄₅₄

Q₈

yes

characteristic

C₂×D₉₀₈

Q₈

yes

characteristic

C₂×C₂²⋊D₂₂₇

D₄

yes

—

C₂×C₄×D₂₂₇

C₂×C₄

yes

—

#16813867

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₈×D₂₂₇

—

C₈×D₂₂₇

—

C₄⋊Dic₂₂₇

—

C₂²⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₂₂₇

—

C₂²⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

—

C₂²×D₂₂₇

C₄⋊C₄

yes

—

Dic₄₅₄

—

D₉₀₈

—

C₂²⋊D₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

C₄⋊C₄

yes

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₂₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₂₇

#121

yes

—

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₂₇

#117

yes

characteristic

C₄×C₂₂₇

C₂²⋊Q₈

yes

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#380

yes

characteristic

D₂₂₇

—

C₂₂₇

#1812

yes

characteristic, complemented, Sylow

#1812

—

227

maximal, Sylow

C₂×#131

—

227

—

#427

—

227

—

#410

—

227

—

H(3,ℤ/4)

—

227

—

H(3,ℤ/4)

D₂₂₇

yes

characteristic, complemented

C₂×#41

—

227

—

#131

—

454

—

#131

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

characteristic

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

#117

—

227

—

#117

C₂×D₂₂₇

yes

—

#107

—

454

—

#106

—

454

—

#99

—

454

—

#41

—

454

—

#41

—

454

—

C₂×Q₈

—

227

—

C₂×D₄

—

227

—

C₂×D₄

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

M₁₆

—

100

M₁₆

—

454

—

101

C₂×C₈

—

454

—

102

C₂×C₈

—

454

—

103

C₂×C₈

—

227

—

104

C₂×C₈

—

227

—

105

C₂×C₈

—

227

—

106

C₂×C₈

—

227

—

107

C₂×C₈

C₄×D₂₂₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

454

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

454

—

112

C₄²

—

113

C₄²

—

454

—

114

C₂³

—

227

—

115

C₂³

D₄×D₂₂₇

yes

characteristic

116

Q₈

—

454

—

117

D₄

—

454

—

118

D₄

—

454

—

119

C₂×C₄

—

120

C₂×C₄

—

908

—

121

C₂×C₄

—

454

—

122

C₂×C₄

—

454

—

123

C₂×C₄

—

227

—

124

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

125

C₂×C₄

D₄×D₂₂₇

yes

—

126

C₂×C₄

Q₈×D₂₂₇

yes

characteristic

127

C₂×C₄

#16813884

yes

—

128

C₈

—

129

C₈

—

227

—

130

C₈

—

227

—

131

C₈

—

227

—

132

C₂²

—

133

C₂²

—

908

—

134

C₂²

—

454

—

135

C₂²

C₄⋊C₄×D₂₂₇

yes

characteristic

136

C₂²

#20300261

yes

—

137

C₄

—

138

C₄

—

1816

—

139

C₄

—

454

—

140

C₄

C₂²⋊C₄×D₂₂₇

yes

characteristic

141

C₄

#20300245

yes

characteristic

142

C₄

#20300266

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

908

—

146

C₂

#28408891

yes

characteristic

	subgroup	G/H
center	C₄	#20300245
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	H(3,ℤ/4)×C₂₂₇	C₂
socle	C₂×C₂₂₇	#380
prime radical	C₂×#16813884	C₂²
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	H(3,ℤ/4)	D₂₂₇
hypocenter	C₂₂₇	#1812
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20300245

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

H(3,ℤ/4)×C₂₂₇

C₂

C₂×C₂₂₇

C₂²

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #28409068	C₂	semidirect
4. #28408789	C₂	semidirect
5. H(3,ℤ/4)×C₂₂₇	C₂	semidirect
9. C₂×#16813867	C₄	semidirect
11. C₂×C₈×D₂₂₇	C₄	semidirect
73. C₂₂₇	#1812	semidirect
79. H(3,ℤ/4)	D₂₂₇	semidirect

H ≅ G/N

type

2. #28409068

C₂

semidirect

4. #28408789

C₂

semidirect

5. H(3,ℤ/4)×C₂₂₇

C₂

semidirect

9. C₂×#16813867

C₄

semidirect

11. C₂×C₈×D₂₂₇

C₄

semidirect

73. C₂₂₇

#1812

semidirect

79. H(3,ℤ/4)

D₂₂₇

semidirect

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

subgroup	index
C₂×#20300296	2
#28409068	2
#28409005	2
#28408789	2
H(3,ℤ/4)×C₂₂₇	2
#1812	227

subgroup

index

H(3,ℤ/4)×C₂₂₇

227

Subgroups — #44399779

Conjugacy classes of subgroups

Complements

Maximal subgroups