Group #44399951 — subgroups

Subgroups — #44399951

Order 29056.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	#28409510	C₂	1	1	yes	maximal, characteristic
2	#28409074	C₂	1	2	yes	maximal
3	#28408939	C₂	1	1	yes	maximal, characteristic
4	#28408793	C₂	1	2	yes	maximal
5	C₂×#101×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₂₂₇	—	2	2	no	—
7	Q₈×Dic₂₂₇	—	2	2	no	—
8	#20300247	—	2	2	no	—
9	#20300247	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
12	#20300222	—	2	2	no	—
13	#20300220	—	2	2	no	—
14	C₂₂₇⋊#95	—	2	2	no	—
15	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
17	#101×C₂₂₇	C₄	1	4	yes	complemented
18	C₂²×Dic₂₂₇	—	2	1	no	—
19	C₂²×Dic₂₂₇	—	2	1	no	—
20	C₄⋊Dic₂₂₇	—	2	2	no	—
21	C₄⋊Dic₂₂₇	—	2	2	no	—
22	C₄⋊Dic₂₂₇	—	2	2	no	—
23	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
24	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
25	C₂²⋊Dic₂₂₇	—	2	2	no	—
26	C₄×Dic₂₂₇	—	2	4	no	—
27	C₄×Dic₂₂₇	—	2	2	no	—
28	C₄×Dic₂₂₇	—	2	2	no	—
29	C₂×Q₈×C₂₂₇	—	2	2	no	—
30	C₂×Q₈×C₂₂₇	—	2	2	no	—
31	C₂×Q₈×C₂₂₇	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₂₂₇	Q₈	1	2	yes	—
33	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
34	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₂₇	—	2	2	no	—
36	M₁₆×C₂₂₇	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₂₂₇	—	2	2	no	—
38	C₂×Dic₂₂₇	—	2	2	no	—
39	C₂×Dic₂₂₇	—	2	2	no	—
40	C₂×Dic₂₂₇	—	2	2	no	—
41	C₂×Dic₂₂₇	—	4	1	no	—
42	C₂×Dic₂₂₇	—	4	1	no	—
43	C₂×Dic₂₂₇	—	4	2	no	—
44	C₂×Dic₂₂₇	—	4	1	no	—
45	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₂₇	—	2	4	no	—
47	Q₈×C₂₂₇	—	2	4	no	—
48	C₂×C₄×C₂₂₇	—	2	2	no	—
49	C₂×C₄×C₂₂₇	—	2	2	no	—
50	C₂×C₄×C₂₂₇	C₄⋊C₄	1	4	yes	—
51	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₂₂₇	#41	1	2	yes	—
55	C₈×C₂₂₇	—	2	4	no	—
56	C₂²×C₂₂₇	—	2	1	no	—
57	C₂²×C₂₂₇	—	2	1	no	—
58	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₂₇	#120	1	1	yes	characteristic
60	C₂²×C₂₂₇	#123	1	1	yes	characteristic
61	C₄×C₂₂₇	—	2	4	no	—
62	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
63	C₄×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
64	Dic₂₂₇	—	4	2	no	—
65	Dic₂₂₇	—	4	2	no	—
66	Dic₂₂₇	—	4	2	no	—
67	Dic₂₂₇	—	8	1	no	—
68	C₂×C₂₂₇	—	2	1	no	—
69	C₂×C₂₂₇	—	2	1	no	—
70	C₂×C₂₂₇	#385	1	1	yes	characteristic
71	C₂×C₂₂₇	#454	1	1	yes	characteristic
72	C₂×C₂₂₇	#456	1	1	yes	characteristic
73	C₂₂₇	#1857	1	1	yes	characteristic, complemented, Sylow
74	#1857	—	227	1	no	maximal, Sylow
75	#519	—	227	1	no	—
76	#428	—	227	2	no	—
77	C₂×#101	D₂₂₇	1	1	yes	characteristic
78	#391	—	227	1	no	—
79	#339	—	227	2	no	—
80	C₂²×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
82	C₄×Q₈	—	454	2	no	—
83	C₄×Q₈	—	454	2	no	—
84	#117	—	454	2	no	—
85	#117	—	227	2	no	—
86	C₂×C₄⋊C₄	—	227	1	no	—
87	C₂×C₄²	—	227	1	no	—
88	#107	—	454	2	no	—
89	#106	—	454	2	no	—
90	#101	Dic₂₂₇	1	4	yes	complemented
91	#95	—	454	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	C₂×Dic₂₂₇	1	2	yes	—
95	C₂×Q₈	Dic₄₅₄	1	2	yes	—
96	C₂²×C₄	—	454	1	no	—
97	C₂²×C₄	—	454	1	no	—
98	C₂²×C₄	D₉₀₈	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	C₂×Dic₂₂₇	1	4	yes	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	454	2	no	—
104	C₄⋊C₄	—	454	2	no	—
105	C₄⋊C₄	—	454	2	no	—
106	C₄⋊C₄	—	227	2	no	—
107	C₄⋊C₄	—	227	2	no	—
108	C₂²⋊C₄	—	454	2	no	—
109	C₄²	—	454	4	no	—
110	C₄²	—	454	2	no	—
111	C₄²	—	454	2	no	—
112	C₂³	C₂×D₉₀₈	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	908	2	no	—
118	C₂×C₄	—	908	1	no	—
119	C₂×C₄	—	908	1	no	—
120	C₂×C₄	—	908	1	no	—
121	C₂×C₄	—	454	2	no	—
122	C₂×C₄	—	454	2	no	—
123	C₂×C₄	—	454	2	no	—
124	C₂×C₄	C₄⋊Dic₂₂₇	1	4	yes	—
125	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
127	C₂×C₄	C₂×Dic₄₅₄	1	1	yes	characteristic
128	C₂×C₄	#16813883	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₄⋊Dic₂₂₇	1	1	yes	characteristic
133	C₂²	#20300258	1	1	yes	characteristic
134	C₂²	#20300272	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	1816	1	no	—
137	C₄	—	908	2	no	—
138	C₄	—	908	2	no	—
139	C₄	—	908	2	no	—
140	C₄	D₄×Dic₂₂₇	1	2	yes	—
141	C₄	#20300265	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#28408907	1	1	yes	characteristic
145	C₂	#28409207	1	1	yes	characteristic
146	C₂	#28409217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409510

C₂

yes

maximal, characteristic

#28409074

C₂

yes

maximal

#28408939

C₂

yes

maximal, characteristic

#28408793

C₂

yes

maximal

C₂×#101×C₂₂₇

C₂

yes

maximal, characteristic

Q₈×Dic₂₂₇

—

Q₈×Dic₂₂₇

—

#20300247

—

#20300247

C₂²

yes

—

C₂×C₄⋊Dic₂₂₇

C₂²

yes

characteristic

C₂×C₄×Dic₂₂₇

C₂²

yes

characteristic

#20300222

—

#20300220

—

C₂₂₇⋊#95

—

C₂²×Q₈×C₂₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₂₇

C₂²

yes

—

#101×C₂₂₇

C₄

yes

complemented

C₂²×Dic₂₂₇

—

C₂²×Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

D₄

yes

—

C₄⋊Dic₂₂₇

D₄

yes

—

C₂²⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂×Q₈×C₂₂₇

Q₈

yes

—

C₂²×C₄×C₂₂₇

D₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

M₁₆×C₂₂₇

C₂×C₄

yes

—

C₂×C₈×C₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

Q₈×C₂₂₇

—

Q₈×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

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₄⋊C₄

yes

C₂²×C₂₂₇

yes

C₂²×C₂₂₇

yes

C₄×C₂₂₇

—

C₄×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

C₂²⋊Q₈

yes

—

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂₂₇

yes

characteristic, complemented, Sylow

#1857

—

227

maximal, Sylow

#519

—

227

—

#428

—

227

—

C₂×#101

D₂₂₇

yes

characteristic

#391

—

227

—

#339

—

227

—

C₂²×Q₈

C₂×D₂₂₇

yes

characteristic

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₄×Q₈

—

454

—

C₄×Q₈

—

454

—

#117

—

454

—

#117

—

227

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄²

—

227

—

#107

—

454

—

#106

—

454

—

#101

Dic₂₂₇

yes

complemented

#95

—

454

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₂₇

yes

—

C₂×Q₈

Dic₄₅₄

yes

—

C₂²×C₄

—

454

—

C₂²×C₄

—

454

—

C₂²×C₄

D₉₀₈

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

100

M₁₆

—

101

M₁₆

C₂×Dic₂₂₇

yes

—

102

C₂×C₈

—

103

C₄⋊C₄

—

454

—

104

C₄⋊C₄

—

454

—

105

C₄⋊C₄

—

454

—

106

C₄⋊C₄

—

227

—

107

C₄⋊C₄

—

227

—

108

C₂²⋊C₄

—

454

—

109

C₄²

—

454

—

110

C₄²

—

454

—

111

C₄²

—

454

—

112

C₂³

C₂×D₉₀₈

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

908

—

118

C₂×C₄

—

908

—

119

C₂×C₄

—

908

—

120

C₂×C₄

—

908

—

121

C₂×C₄

—

454

—

122

C₂×C₄

—

454

—

123

C₂×C₄

—

454

—

124

C₂×C₄

C₄⋊Dic₂₂₇

yes

—

125

C₂×C₄

C₂²×Dic₂₂₇

yes

characteristic

126

C₂×C₄

D₄×D₂₂₇

yes

—

127

C₂×C₄

C₂×Dic₄₅₄

yes

characteristic

128

C₂×C₄

#16813883

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₄⋊Dic₂₂₇

yes

characteristic

133

C₂²

#20300258

yes

characteristic

134

C₂²

#20300272

yes

characteristic

135

C₄

—

136

C₄

—

1816

—

137

C₄

—

908

—

138

C₄

—

908

—

139

C₄

—

908

—

140

C₄

D₄×Dic₂₂₇

yes

—

141

C₄

#20300265

yes

—

142

C₂

—

143

C₂

—

144

C₂

#28408907

yes

characteristic

145

C₂

#28409207

yes

characteristic

146

C₂

#28409217

yes

characteristic

	subgroup	G/H
center	C₂²	#20300258
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	C₂×#101×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#120
prime radical	C₂³×C₂₂₇	C₂×D₄
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	C₂×#101	D₂₂₇
hypocenter	C₂₂₇	#1857
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#20300258

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

C₂×#101×C₂₂₇

C₂

socle

C₂²×C₂₂₇

#120

prime radical

C₂³×C₂₂₇

C₂×D₄

cosocle

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
17. #101×C₂₂₇	C₄	semidirect
73. C₂₂₇	#1857	semidirect
90. #101	Dic₂₂₇	semidirect

H ≅ G/N

type

17. #101×C₂₂₇

C₄

semidirect

73. C₂₂₇

#1857

semidirect

90. #101

Dic₂₂₇

semidirect

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

subgroup	index
#28409510	2
#28409074	2
#28408939	2
#28408793	2
C₂×#101×C₂₂₇	2
#1857	227

subgroup

index

C₂×#101×C₂₂₇

227

Subgroups — #44399951

Conjugacy classes of subgroups

Complements

Maximal subgroups