Group #44399778 — subgroups

Subgroups — #44399778

Order 29056.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20300295	C₂	1	1	yes	maximal, characteristic
2	#28409068	C₂	1	2	yes	maximal, complemented
3	#28409006	C₂	1	1	yes	maximal, characteristic
4	#28408788	C₂	1	2	yes	maximal, complemented
5	#410×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	#20300295	—	2	2	no	—
7	#20300295	—	2	2	no	—
8	C₂×#16813868	C₄	1	2	yes	complemented
9	C₂×#16813868	C₂²	1	1	yes	characteristic
10	C₂²×C₂₂₇⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	#20300247	C₂²	1	2	yes	—
13	#20300221	—	2	2	no	—
14	#20300219	—	2	2	no	—
15	#20300203	—	2	2	no	—
16	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
17	#117×C₂₂₇	C₂²	1	2	yes	—
18	#99×C₂₂₇	—	2	2	no	—
19	#16813868	—	2	4	no	—
20	#16813868	—	2	2	no	—
21	C₂×C₂₂₇⋊C₈	—	2	2	no	—
22	C₂×C₂₂₇⋊C₈	—	2	2	no	—
23	C₂×C₂₂₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
27	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₂₂₇	—	2	2	no	—
29	C₂²⋊Dic₂₂₇	—	2	2	no	—
30	C₄×Dic₂₂₇	—	2	2	no	—
31	#41×C₂₂₇	—	2	2	no	—
32	#41×C₂₂₇	—	2	2	no	—
33	C₂×Q₈×C₂₂₇	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₂₂₇	D₄	1	2	yes	—
35	C₂×D₄×C₂₂₇	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₂₇	—	2	2	no	—
39	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
40	C₄²×C₂₂₇	—	2	2	no	—
41	C₂×Dic₂₂₇	—	4	2	no	—
42	C₂₂₇⋊C₈	—	2	2	no	—
43	C₂₂₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂₂₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₂₂₇	C₄⋊C₄	1	2	yes	—
47	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₂₂₇	—	2	2	no	—
49	D₄×C₂₂₇	—	2	4	no	—
50	D₄×C₂₂₇	—	2	2	no	—
51	C₂×C₄×C₂₂₇	—	2	2	no	—
52	C₂×C₄×C₂₂₇	—	2	2	no	—
53	C₂×C₄×C₂₂₇	—	4	2	no	—
54	C₂×C₄×C₂₂₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₂₇	#41	1	2	yes	—
59	C₂²×C₂₂₇	—	2	2	no	—
60	C₂²×C₂₂₇	—	4	1	no	—
61	C₂²×C₂₂₇	—	4	2	no	—
62	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₂₂₇	#121	1	2	yes	—
64	C₄×C₂₂₇	—	2	2	no	—
65	C₄×C₂₂₇	—	8	2	no	—
66	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₂₂₇	#117	1	1	yes	characteristic
68	C₄×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
69	Dic₂₂₇	—	8	2	no	—
70	C₂×C₂₂₇	—	2	2	no	—
71	C₂×C₂₂₇	—	2	1	no	—
72	C₂×C₂₂₇	—	4	2	no	—
73	C₂×C₂₂₇	#380	1	1	yes	characteristic
74	C₂₂₇	#1812	1	1	yes	characteristic, complemented, Sylow
75	#1812	—	227	1	no	maximal, Sylow
76	C₂×#131	—	227	1	no	—
77	#427	—	227	2	no	—
78	#410	—	227	1	no	—
79	#410	D₂₂₇	1	1	yes	characteristic
80	H(3,ℤ/4)	—	227	2	no	—
81	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
82	#131	—	454	2	no	—
83	#131	—	454	2	no	—
84	C₂×M₁₆	—	227	2	no	—
85	C₂×M₁₆	—	227	1	no	—
86	C₂²×C₈	—	227	2	no	—
87	C₂²×C₈	—	227	1	no	—
88	#117	—	227	2	no	—
89	#117	C₂×D₂₂₇	1	2	yes	—
90	#107	—	454	2	no	—
91	#106	—	454	2	no	—
92	#99	—	2	2	no	—
93	#99	—	454	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₄₅₄	1	1	yes	characteristic
97	C₂×D₄	C₂²⋊D₂₂₇	1	2	yes	—
98	C₂×D₄	Dic₄₅₄	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₂₂₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
101	M₁₆	—	454	2	no	—
102	M₁₆	—	454	4	no	—
103	C₂×C₈	—	454	2	no	—
104	C₂×C₈	—	454	2	no	—
105	C₂×C₈	—	227	2	no	—
106	C₂×C₈	—	227	1	no	—
107	C₂×C₈	—	227	1	no	—
108	C₂×C₈	—	227	2	no	—
109	C₂×C₈	—	227	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	454	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	454	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	454	2	no	—
116	C₂³	Dic₂₂₇⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	908	2	no	—
125	C₂×C₄	Dic₂₂₇⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
128	C₂×C₄	C₂×Dic₄₅₄	1	1	yes	characteristic
129	C₂×C₄	#16813883	1	2	yes	—
130	C₈	—	454	2	no	—
131	C₈	—	227	1	no	—
132	C₈	—	227	1	no	—
133	C₈	—	227	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₂₂₇⋊C₄	1	1	yes	characteristic
138	C₂²	#20300260	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	1816	2	no	—
142	C₄	C₂²⋊C₄×D₂₂₇	1	1	yes	characteristic
143	C₄	#20300244	1	1	yes	characteristic
144	C₄	#20300265	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#28408890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300295

C₂

yes

maximal, characteristic

#28409068

C₂

yes

maximal, complemented

#28409006

C₂

yes

maximal, characteristic

#28408788

C₂

yes

maximal, complemented

#410×C₂₂₇

C₂

yes

maximal, characteristic

#20300295

—

#20300295

—

C₂×#16813868

C₄

yes

complemented

C₂×#16813868

C₂²

yes

characteristic

C₂²×C₂₂₇⋊C₈

C₄

yes

complemented

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

#20300247

C₂²

yes

—

#20300221

—

#20300219

—

#20300203

—

C₂×#41×C₂₂₇

C₂²

yes

characteristic

#117×C₂₂₇

C₂²

yes

—

#99×C₂₂₇

—

#16813868

—

#16813868

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₂₂₇

—

C₂²⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

#41×C₂₂₇

—

#41×C₂₂₇

—

C₂×Q₈×C₂₂₇

Q₈

yes

characteristic

C₂×D₄×C₂₂₇

D₄

yes

—

C₂×D₄×C₂₂₇

Q₈

yes

characteristic

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

—

C₂²⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₂₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₂₇

C₄⋊C₄

yes

—

C₂³×C₂₂₇

C₂×D₄

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

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

#121

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₂₇

#117

yes

characteristic

C₄×C₂₂₇

C₂²⋊Q₈

yes

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#380

yes

characteristic

C₂₂₇

#1812

yes

characteristic, complemented, Sylow

#1812

—

227

maximal, Sylow

C₂×#131

—

227

—

#427

—

227

—

#410

—

227

—

#410

D₂₂₇

yes

characteristic

H(3,ℤ/4)

—

227

—

C₂×#41

C₂×D₂₂₇

yes

characteristic

#131

—

454

—

#131

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

—

227

—

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

#117

—

227

—

#117

C₂×D₂₂₇

yes

—

#107

—

454

—

#106

—

454

—

#99

—

#99

—

454

—

#41

—

#41

—

C₂×Q₈

Dic₄₅₄

yes

characteristic

C₂×D₄

C₂²⋊D₂₂₇

yes

—

C₂×D₄

Dic₄₅₄

yes

characteristic

C₂²×C₄

C₄×D₂₂₇

yes

—

100

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

101

M₁₆

—

454

—

102

M₁₆

—

454

—

103

C₂×C₈

—

454

—

104

C₂×C₈

—

454

—

105

C₂×C₈

—

227

—

106

C₂×C₈

—

227

—

107

C₂×C₈

—

227

—

108

C₂×C₈

—

227

—

109

C₂×C₈

—

227

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

454

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

454

—

114

C₄²

—

115

C₄²

—

454

—

116

C₂³

Dic₂₂₇⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₂₂₇

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

908

—

125

C₂×C₄

Dic₂₂₇⋊C₄

yes

—

126

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

127

C₂×C₄

D₄×D₂₂₇

yes

—

128

C₂×C₄

C₂×Dic₄₅₄

yes

characteristic

129

C₂×C₄

#16813883

yes

—

130

C₈

—

454

—

131

C₈

—

227

—

132

C₈

—

227

—

133

C₈

—

227

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₂₂₇⋊C₄

yes

characteristic

138

C₂²

#20300260

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

1816

—

142

C₄

C₂²⋊C₄×D₂₂₇

yes

characteristic

143

C₄

#20300244

yes

characteristic

144

C₄

#20300265

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#28408890

yes

characteristic

	subgroup	G/H
center	C₄	#20300244
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#410×C₂₂₇	C₂
socle	C₂×C₂₂₇	#380
prime radical	C₂×#41×C₂₂₇	C₂²
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#410	D₂₂₇
hypocenter	C₂₂₇	#1812
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#20300244

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#410×C₂₂₇

C₂

socle

C₂×C₂₂₇

#380

prime radical

C₂×#41×C₂₂₇

C₂²

cosocle

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #28409068	C₂	semidirect
4. #28408788	C₂	semidirect
8. C₂×#16813868	C₄	semidirect
10. C₂²×C₂₂₇⋊C₈	C₄	semidirect
74. C₂₂₇	#1812	semidirect

H ≅ G/N

type

2. #28409068

C₂

semidirect

4. #28408788

C₂

semidirect

8. C₂×#16813868

C₄

semidirect

10. C₂²×C₂₂₇⋊C₈

C₄

semidirect

74. C₂₂₇

#1812

semidirect

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

subgroup	index
C₂×#20300295	2
#28409068	2
#28409006	2
#28408788	2
#410×C₂₂₇	2
#1812	227

subgroup

index

#410×C₂₂₇

227

Subgroups — #44399778

Conjugacy classes of subgroups

Complements

Maximal subgroups