Group #44400565 — subgroups

Subgroups — #44400565

Order 29056.

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

normal subgroups only

This group has 320 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	#28409582	C₂	1	1	yes	maximal, characteristic
2	C₂×#20300214	C₂	1	4	yes	maximal, complemented
3	#28408878	C₂	1	1	yes	maximal, characteristic, complemented
4	#423×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16813884	C₂²	1	2	yes	—
6	#20300288	C₄	1	1	yes	characteristic, complemented
7	#20300285	C₄	1	1	yes	characteristic, complemented
8	#20300276	C₄	1	2	yes	complemented
9	#20300262	—	2	2	no	—
10	C₄×D₉₀₈	—	2	2	no	—
11	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
12	#20300214	—	2	8	no	—
13	C₂₂₇⋊#95	—	2	2	no	—
14	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₂₇	—	2	2	no	—
17	#16813884	—	2	4	no	—
18	#16813884	—	2	4	no	—
19	C₂×Dic₄₅₄	C₂×C₄	1	2	yes	—
20	C₂×D₉₀₈	—	2	2	no	—
21	C₂×D₉₀₈	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₂₂₇	—	2	2	no	—
23	C₂²×Dic₂₂₇	—	2	2	no	—
24	C₂×C₄×D₂₂₇	—	2	2	no	—
25	C₄⋊Dic₂₂₇	—	2	2	no	—
26	(C₂×D₂₂₇)⋊C₄	—	2	4	no	—
27	C₄×Dic₂₂₇	—	2	2	no	—
28	C₄×Dic₂₂₇	D₄	1	4	yes	complemented
29	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
30	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₂₂₇	—	2	4	no	—
32	C₂×C₈×C₂₂₇	—	2	2	no	—
33	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₂₂₇	D₄	1	2	yes	—
36	C₂²×D₂₂₇	—	2	2	no	—
37	Dic₄₅₄	—	2	4	no	—
38	D₉₀₈	—	2	4	no	—
39	D₉₀₈	—	4	4	no	—
40	C₂²⋊D₂₂₇	—	4	4	no	—
41	C₂×Dic₂₂₇	—	2	2	no	—
42	C₂×Dic₂₂₇	—	2	4	no	—
43	C₂×Dic₂₂₇	—	4	4	no	—
44	C₄×D₂₂₇	—	4	2	no	—
45	C₄×D₂₂₇	—	4	2	no	—
46	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₂₇	—	4	2	no	—
48	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₂₇	#41	1	2	yes	—
54	C₈×C₂₂₇	—	4	2	no	—
55	C₂²×C₂₂₇	—	2	2	no	—
56	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₂₇	#121	1	2	yes	—
58	C₂×D₂₂₇	—	4	2	no	—
59	C₂×D₂₂₇	—	4	4	no	—
60	C₄×C₂₂₇	—	2	2	no	—
61	C₄×C₂₂₇	—	2	2	no	—
62	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
63	C₄×C₂₂₇	#124	1	1	yes	characteristic
64	C₄×C₂₂₇	#127	1	1	yes	characteristic
65	Dic₂₂₇	—	4	4	no	—
66	Dic₂₂₇	—	4	2	no	—
67	C₂×C₂₂₇	—	2	2	no	—
68	C₂×C₂₂₇	#390	1	1	yes	characteristic
69	C₂×C₂₂₇	#421	1	2	yes	—
70	D₂₂₇	—	8	2	no	—
71	C₂₂₇	#1963	1	1	yes	characteristic, complemented, Sylow
72	#1963	—	227	1	no	maximal, Sylow
73	#532	—	227	1	no	—
74	#423	D₂₂₇	1	1	yes	characteristic, complemented
75	C₂×#104	—	227	4	no	—
76	#376	—	227	1	no	—
77	C₂×#41	—	227	2	no	—
78	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
79	C₄⋊Q₈	—	227	1	no	—
80	#127	—	227	1	no	—
81	#124	—	227	2	no	—
82	#121	—	454	2	no	—
83	C₄×D₄	—	454	2	no	—
84	C₂×C₄²	—	227	2	no	—
85	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#104	—	454	8	no	—
88	#95	—	454	2	no	—
89	#41	—	454	4	no	—
90	#41	—	454	4	no	—
91	C₂×Q₈	—	227	2	no	—
92	C₂×D₄	—	454	2	no	—
93	C₂×D₄	—	454	2	no	—
94	C₂×D₄	—	227	2	no	—
95	C₂²×C₄	—	454	2	no	—
96	C₂²×C₄	—	454	2	no	—
97	C₂²×C₄	D₉₀₈	1	2	yes	—
98	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	454	2	no	—
102	C₂²⋊C₄	—	454	4	no	—
103	C₄²	—	454	2	no	—
104	C₄²	—	227	4	no	—
105	C₄²	C₄×D₂₂₇	1	1	yes	characteristic
106	C₄²	C₄×D₂₂₇	1	1	yes	characteristic
107	C₄²	C₂²⋊D₂₂₇	1	2	yes	—
108	C₂³	—	454	2	no	—
109	C₂³	C₂×D₉₀₈	1	1	yes	characteristic
110	Q₈	—	454	4	no	—
111	D₄	—	908	4	no	—
112	D₄	—	908	4	no	—
113	D₄	—	454	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	908	4	no	—
116	C₂×C₄	—	908	2	no	—
117	C₂×C₄	—	908	2	no	—
118	C₂×C₄	—	454	2	no	—
119	C₂×C₄	—	454	4	no	—
120	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
123	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
125	C₂×C₄	#16813885	1	2	yes	—
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	908	2	no	—
129	C₂²	—	908	4	no	—
130	C₂²	C₂×(C₂×D₂₂₇)⋊C₄	1	1	yes	characteristic
131	C₂²	#20300263	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	908	4	no	—
135	C₄	—	908	2	no	—
136	C₄	#20300250	1	2	yes	—
137	C₄	#20300277	1	1	yes	characteristic
138	C₄	#20300284	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	1816	2	no	—
141	C₂	#28408933	1	1	yes	characteristic
142	C₂	#28409048	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#28409582

C₂

yes

maximal, characteristic

C₂×#20300214

C₂

yes

maximal, complemented

#28408878

C₂

yes

maximal, characteristic, complemented

#423×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#16813884

C₂²

yes

—

#20300288

C₄

yes

characteristic, complemented

#20300285

C₄

yes

characteristic, complemented

#20300276

C₄

yes

complemented

#20300262

—

C₄×D₉₀₈

—

C₂×C₄×Dic₂₂₇

C₂²

yes

—

#20300214

—

C₂₂₇⋊#95

—

C₂×M₁₆×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

—

#16813884

—

#16813884

—

C₂×Dic₄₅₄

C₂×C₄

yes

—

C₂×D₉₀₈

—

C₂×D₉₀₈

C₂×C₄

yes

—

C₂×C₂²⋊D₂₂₇

—

C₂²×Dic₂₂₇

—

C₂×C₄×D₂₂₇

—

C₄⋊Dic₂₂₇

—

(C₂×D₂₂₇)⋊C₄

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

D₄

yes

complemented

C₂²×C₄×C₂₂₇

D₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

D₄

yes

—

C₂²×D₂₂₇

—

Dic₄₅₄

—

D₉₀₈

—

D₉₀₈

—

C₂²⋊D₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₂₇

#121

yes

—

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#390

yes

characteristic

C₂×C₂₂₇

#421

yes

—

D₂₂₇

—

C₂₂₇

#1963

yes

characteristic, complemented, Sylow

#1963

—

227

maximal, Sylow

#532

—

227

—

#423

D₂₂₇

yes

characteristic, complemented

C₂×#104

—

227

—

#376

—

227

—

C₂×#41

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₄⋊Q₈

—

227

—

#127

—

227

—

#124

—

227

—

#121

—

454

—

C₄×D₄

—

454

—

C₂×C₄²

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

—

#104

—

454

—

#95

—

454

—

#41

—

454

—

#41

—

454

—

C₂×Q₈

—

227

—

C₂×D₄

—

454

—

C₂×D₄

—

454

—

C₂×D₄

—

227

—

C₂²×C₄

—

454

—

C₂²×C₄

—

454

—

C₂²×C₄

D₉₀₈

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

454

—

102

C₂²⋊C₄

—

454

—

103

C₄²

—

454

—

104

C₄²

—

227

—

105

C₄²

C₄×D₂₂₇

yes

characteristic

106

C₄²

C₄×D₂₂₇

yes

characteristic

107

C₄²

C₂²⋊D₂₂₇

yes

—

108

C₂³

—

454

—

109

C₂³

C₂×D₉₀₈

yes

characteristic

110

Q₈

—

454

—

111

D₄

—

908

—

112

D₄

—

908

—

113

D₄

—

454

—

114

C₂×C₄

—

115

C₂×C₄

—

908

—

116

C₂×C₄

—

908

—

117

C₂×C₄

—

908

—

118

C₂×C₄

—

454

—

119

C₂×C₄

—

454

—

120

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

123

C₂×C₄

D₄×D₂₂₇

yes

—

124

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

125

C₂×C₄

#16813885

yes

—

126

C₈

—

127

C₂²

—

128

C₂²

—

908

—

129

C₂²

—

908

—

130

C₂²

C₂×(C₂×D₂₂₇)⋊C₄

yes

characteristic

131

C₂²

#20300263

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

908

—

135

C₄

—

908

—

136

C₄

#20300250

yes

—

137

C₄

#20300277

yes

characteristic

138

C₄

#20300284

yes

characteristic

139

C₂

—

140

C₂

—

1816

—

141

C₂

#28408933

yes

characteristic

142

C₂

#28409048

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₂₇
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#423×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×C₂²⋊C₄
prime radical	#28409582	C₂
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#423	D₂₂₇
hypocenter	C₂₂₇	#1963
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₂₇

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#423×C₂₂₇

C₂

socle

C₂²×C₂₂₇

C₂×C₂²⋊C₄

prime radical

#28409582

C₂

cosocle

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#20300214	C₂	semidirect
3. #28408878	C₂	semidirect
4. #423×C₂₂₇	C₂	semidirect
6. #20300288	C₄	semidirect
7. #20300285	C₄	semidirect
8. #20300276	C₄	semidirect
28. C₄×Dic₂₂₇	D₄	semidirect
71. C₂₂₇	#1963	semidirect
74. #423	D₂₂₇	semidirect

H ≅ G/N

type

2. C₂×#20300214

C₂

semidirect

3. #28408878

C₂

semidirect

4. #423×C₂₂₇

C₂

semidirect

6. #20300288

C₄

semidirect

7. #20300285

C₄

semidirect

8. #20300276

C₄

semidirect

28. C₄×Dic₂₂₇

D₄

semidirect

71. C₂₂₇

#1963

semidirect

74. #423

D₂₂₇

semidirect

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

subgroup	index
#28409582	2
C₂×#20300214	2
#28408878	2
#423×C₂₂₇	2
#1963	227

subgroup

index

#423×C₂₂₇

227

Subgroups — #44400565

Conjugacy classes of subgroups

Complements

Maximal subgroups