Group #44551196 — subgroups

Subgroups — #44551196

Order 32128.

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	#30632237	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548641	C₂	1	4	yes	maximal, complemented
3	#30631533	C₂	1	1	yes	maximal, characteristic, complemented
4	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16860810	C₂²	1	2	yes	—
6	#22548715	C₄	1	1	yes	characteristic, complemented
7	#22548712	C₄	1	1	yes	characteristic, complemented
8	#22548703	C₄	1	2	yes	complemented
9	#22548689	—	2	2	no	—
10	C₄×D₁₀₀₄	—	2	2	no	—
11	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
12	#22548641	—	2	8	no	—
13	C₂₅₁⋊#95	—	2	2	no	—
14	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₅₁	—	2	2	no	—
17	#16860810	—	2	4	no	—
18	#16860810	—	2	4	no	—
19	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
20	C₂×D₁₀₀₄	—	2	2	no	—
21	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
23	C₂²×Dic₂₅₁	—	2	2	no	—
24	C₂×C₄×D₂₅₁	—	2	2	no	—
25	C₄⋊Dic₂₅₁	—	2	2	no	—
26	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
27	C₄×Dic₂₅₁	—	2	2	no	—
28	C₄×Dic₂₅₁	D₄	1	4	yes	complemented
29	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
30	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
31	M₁₆×C₂₅₁	—	2	4	no	—
32	C₂×C₈×C₂₅₁	—	2	2	no	—
33	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₂₅₁	D₄	1	2	yes	—
36	C₂²×D₂₅₁	—	2	2	no	—
37	Dic₅₀₂	—	2	4	no	—
38	D₁₀₀₄	—	2	4	no	—
39	D₁₀₀₄	—	4	4	no	—
40	C₂²⋊D₂₅₁	—	4	4	no	—
41	C₂×Dic₂₅₁	—	2	2	no	—
42	C₂×Dic₂₅₁	—	2	4	no	—
43	C₂×Dic₂₅₁	—	4	4	no	—
44	C₄×D₂₅₁	—	4	2	no	—
45	C₄×D₂₅₁	—	4	2	no	—
46	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₁	—	4	2	no	—
48	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₁	#41	1	2	yes	—
54	C₈×C₂₅₁	—	4	2	no	—
55	C₂²×C₂₅₁	—	2	2	no	—
56	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₅₁	#121	1	2	yes	—
58	C₂×D₂₅₁	—	4	2	no	—
59	C₂×D₂₅₁	—	4	4	no	—
60	C₄×C₂₅₁	—	2	2	no	—
61	C₄×C₂₅₁	—	2	2	no	—
62	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
63	C₄×C₂₅₁	#124	1	1	yes	characteristic
64	C₄×C₂₅₁	#127	1	1	yes	characteristic
65	Dic₂₅₁	—	4	4	no	—
66	Dic₂₅₁	—	4	2	no	—
67	C₂×C₂₅₁	—	2	2	no	—
68	C₂×C₂₅₁	#390	1	1	yes	characteristic
69	C₂×C₂₅₁	#421	1	2	yes	—
70	D₂₅₁	—	8	2	no	—
71	C₂₅₁	#1963	1	1	yes	characteristic, complemented, Sylow
72	#1963	—	251	1	no	maximal, Sylow
73	#532	—	251	1	no	—
74	#423	D₂₅₁	1	1	yes	characteristic, complemented
75	C₂×#104	—	251	4	no	—
76	#376	—	251	1	no	—
77	C₂×#41	—	251	2	no	—
78	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
79	C₄⋊Q₈	—	251	1	no	—
80	#127	—	251	1	no	—
81	#124	—	251	2	no	—
82	#121	—	502	2	no	—
83	C₄×D₄	—	502	2	no	—
84	C₂×C₄²	—	251	2	no	—
85	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#104	—	502	8	no	—
88	#95	—	502	2	no	—
89	#41	—	502	4	no	—
90	#41	—	502	4	no	—
91	C₂×Q₈	—	251	2	no	—
92	C₂×D₄	—	502	2	no	—
93	C₂×D₄	—	502	2	no	—
94	C₂×D₄	—	251	2	no	—
95	C₂²×C₄	—	502	2	no	—
96	C₂²×C₄	—	502	2	no	—
97	C₂²×C₄	D₁₀₀₄	1	2	yes	—
98	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	502	2	no	—
102	C₂²⋊C₄	—	502	4	no	—
103	C₄²	—	502	2	no	—
104	C₄²	—	251	4	no	—
105	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
106	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
107	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
108	C₂³	—	502	2	no	—
109	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
110	Q₈	—	502	4	no	—
111	D₄	—	1004	4	no	—
112	D₄	—	1004	4	no	—
113	D₄	—	502	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	1004	4	no	—
116	C₂×C₄	—	1004	2	no	—
117	C₂×C₄	—	1004	2	no	—
118	C₂×C₄	—	502	2	no	—
119	C₂×C₄	—	502	4	no	—
120	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
123	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
125	C₂×C₄	#16860811	1	2	yes	—
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	1004	4	no	—
129	C₂²	—	1004	2	no	—
130	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
131	C₂²	#22548690	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	1004	4	no	—
135	C₄	—	1004	2	no	—
136	C₄	#22548677	1	2	yes	—
137	C₄	#22548704	1	1	yes	characteristic
138	C₄	#22548711	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	2008	2	no	—
141	C₂	#30631588	1	1	yes	characteristic
142	C₂	#30631703	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632237

C₂

yes

maximal, characteristic

C₂×#22548641

C₂

yes

maximal, complemented

#30631533

C₂

yes

maximal, characteristic, complemented

#423×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860810

C₂²

yes

—

#22548715

C₄

yes

characteristic, complemented

#22548712

C₄

yes

characteristic, complemented

#22548703

C₄

yes

complemented

#22548689

—

C₄×D₁₀₀₄

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

#22548641

—

C₂₅₁⋊#95

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

—

#16860810

—

#16860810

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

C₂×C₄

yes

—

C₂×C₂²⋊D₂₅₁

—

C₂²×Dic₂₅₁

—

C₂×C₄×D₂₅₁

—

C₄⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

complemented

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

—

C₂²×D₂₅₁

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

#121

yes

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#390

yes

characteristic

C₂×C₂₅₁

#421

yes

—

D₂₅₁

—

C₂₅₁

#1963

yes

characteristic, complemented, Sylow

#1963

—

251

maximal, Sylow

#532

—

251

—

#423

D₂₅₁

yes

characteristic, complemented

C₂×#104

—

251

—

#376

—

251

—

C₂×#41

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₄⋊Q₈

—

251

—

#127

—

251

—

#124

—

251

—

#121

—

502

—

C₄×D₄

—

502

—

C₂×C₄²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

#104

—

502

—

#95

—

502

—

#41

—

502

—

#41

—

502

—

C₂×Q₈

—

251

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

251

—

C₂²×C₄

—

502

—

C₂²×C₄

—

502

—

C₂²×C₄

D₁₀₀₄

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

502

—

102

C₂²⋊C₄

—

502

—

103

C₄²

—

502

—

104

C₄²

—

251

—

105

C₄²

C₄×D₂₅₁

yes

characteristic

106

C₄²

C₄×D₂₅₁

yes

characteristic

107

C₄²

C₂²⋊D₂₅₁

yes

—

108

C₂³

—

502

—

109

C₂³

C₂×D₁₀₀₄

yes

characteristic

110

Q₈

—

502

—

111

D₄

—

1004

—

112

D₄

—

1004

—

113

D₄

—

502

—

114

C₂×C₄

—

115

C₂×C₄

—

1004

—

116

C₂×C₄

—

1004

—

117

C₂×C₄

—

1004

—

118

C₂×C₄

—

502

—

119

C₂×C₄

—

502

—

120

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

123

C₂×C₄

D₄×D₂₅₁

yes

—

124

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

125

C₂×C₄

#16860811

yes

—

126

C₈

—

127

C₂²

—

128

C₂²

—

1004

—

129

C₂²

—

1004

—

130

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

131

C₂²

#22548690

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

1004

—

135

C₄

—

1004

—

136

C₄

#22548677

yes

—

137

C₄

#22548704

yes

characteristic

138

C₄

#22548711

yes

characteristic

139

C₂

—

140

C₂

—

2008

—

141

C₂

#30631588

yes

characteristic

142

C₂

#30631703

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#423×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	#30632237	C₂
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#1963
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#423×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

prime radical

#30632237

C₂

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#22548641	C₂	semidirect
3. #30631533	C₂	semidirect
4. #423×C₂₅₁	C₂	semidirect
6. #22548715	C₄	semidirect
7. #22548712	C₄	semidirect
8. #22548703	C₄	semidirect
28. C₄×Dic₂₅₁	D₄	semidirect
71. C₂₅₁	#1963	semidirect
74. #423	D₂₅₁	semidirect

H ≅ G/N

type

2. C₂×#22548641

C₂

semidirect

3. #30631533

C₂

semidirect

4. #423×C₂₅₁

C₂

semidirect

6. #22548715

C₄

semidirect

7. #22548712

C₄

semidirect

8. #22548703

C₄

semidirect

28. C₄×Dic₂₅₁

D₄

semidirect

71. C₂₅₁

#1963

semidirect

74. #423

D₂₅₁

semidirect

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

subgroup	index
#30632237	2
C₂×#22548641	2
#30631533	2
#423×C₂₅₁	2
#1963	251

subgroup

index

#423×C₂₅₁

251

Subgroups — #44551196

Conjugacy classes of subgroups

Complements

Maximal subgroups