Group #44551301 — subgroups

Subgroups — #44551301

Order 32128.

292 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#30631441	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₀₂	C₄	1	1	yes	characteristic
6	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₅₀₂	—	2	2	no	—
8	C₄×Dic₅₀₂	—	2	2	no	—
9	C₂×C₄⋊Dic₂₅₁	C₄	1	1	yes	characteristic
10	C₂×Dic₂₅₁⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
12	C₄⋊(C₂₅₁⋊C₈)	—	2	2	no	—
13	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₅₁	—	2	2	no	—
16	C₂×Dic₅₀₂	—	2	2	no	—
17	C₂×Dic₅₀₂	—	2	2	no	—
18	C₂×Dic₅₀₂	C₈	1	4	yes	complemented
19	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
21	C₂×C₂₅₁⋊C₈	—	2	2	no	—
22	C₂×C₂₅₁⋊C₈	—	2	4	no	—
23	C₄⋊Dic₂₅₁	—	2	2	no	—
24	Dic₂₅₁⋊C₄	—	2	4	no	—
25	C₄×Dic₂₅₁	—	2	2	no	—
26	C₄×Dic₂₅₁	—	2	2	no	—
27	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₅₁	—	2	2	no	—
31	C₂×C₈×C₂₅₁	—	2	2	no	—
32	C₂×C₈×C₂₅₁	Q₈	1	4	yes	complemented
33	C₄²×C₂₅₁	D₄	1	2	yes	—
34	C₄²×C₂₅₁	D₄	1	2	yes	—
35	Dic₅₀₂	—	2	8	no	—
36	C₂×Dic₂₅₁	—	2	2	no	—
37	C₂×Dic₂₅₁	—	2	2	no	—
38	C₂×Dic₂₅₁	—	2	2	no	—
39	C₂×Dic₂₅₁	—	2	4	no	—
40	C₂×Dic₂₅₁	C₂×C₈	1	4	yes	—
41	C₂₅₁⋊C₈	—	4	2	no	—
42	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₅₁	—	2	2	no	—
44	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₅₁	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₂₅₁	M₁₆	1	2	yes	—
48	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₅₁	#41	1	2	yes	—
52	C₈×C₂₅₁	—	2	4	no	—
53	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
55	C₂²×C₂₅₁	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₂₅₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₂₅₁	#131	1	2	yes	—
58	C₄×C₂₅₁	—	2	2	no	—
59	C₄×C₂₅₁	C₂²⋊C₈	1	4	yes	—
60	C₄×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₅₁	#124	1	1	yes	characteristic
62	C₄×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₂₅₁	—	2	4	no	—
64	Dic₂₅₁	—	4	2	no	—
65	C₂×C₂₅₁	#391	1	1	yes	characteristic
66	C₂×C₂₅₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₂₅₁	#408	1	1	yes	characteristic
68	C₂×C₂₅₁	C₈×Q₈	1	2	yes	—
69	C₂×C₂₅₁	#446	1	2	yes	—
70	C₂₅₁	#1984	1	1	yes	characteristic, complemented, Sylow
71	#1984	—	251	1	no	maximal, Sylow
72	C₂×C₄×Q₈	—	251	1	no	—
73	C₂×C₄⋊C₈	—	251	1	no	—
74	C₂×C₄×C₈	D₂₅₁	1	1	yes	characteristic
75	#336	—	251	4	no	—
76	C₂²×Q₈	—	251	1	no	—
77	C₂²×C₈	—	251	2	no	—
78	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
79	C₄×Q₈	—	502	2	no	—
80	C₄×Q₈	—	502	2	no	—
81	C₂×C₄⋊C₄	—	251	2	no	—
82	C₂×C₄⋊C₄	—	251	1	no	—
83	C₂×C₄²	—	251	2	no	—
84	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
85	C₄⋊C₈	—	502	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₂×Q₈	—	502	2	no	—
88	C₂×Q₈	—	502	2	no	—
89	C₂×Q₈	—	251	4	no	—
90	C₂²×C₄	—	251	2	no	—
91	C₂²×C₄	—	251	2	no	—
92	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	502	2	no	—
98	C₂×C₈	—	502	4	no	—
99	C₂×C₈	Dic₅₀₂	1	4	yes	complemented
100	C₄⋊C₄	—	502	4	no	—
101	C₄⋊C₄	—	502	2	no	—
102	C₄²	—	502	2	no	—
103	C₄²	—	502	2	no	—
104	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
105	C₄²	D₁₀₀₄	1	2	yes	—
106	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
107	Q₈	—	502	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	502	2	no	—
110	C₂×C₄	—	502	4	no	—
111	C₂×C₄	—	502	2	no	—
112	C₂×C₄	—	502	2	no	—
113	C₂×C₄	—	251	4	no	—
114	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
115	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
116	C₂×C₄	C₈×D₂₅₁	1	2	yes	—
117	C₂×C₄	#16860793	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
121	C₂×C₄	#16860810	1	2	yes	—
122	C₈	—	2	4	no	—
123	C₈	—	1004	2	no	—
124	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
125	C₂²	C₄×Dic₅₀₂	1	2	yes	—
126	C₂²	C₂×C₈×D₂₅₁	1	1	yes	characteristic
127	C₂²	C₂×#16860793	1	1	yes	characteristic
128	C₂²	#22548721	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	1004	2	no	—
131	C₄	—	502	4	no	—
132	C₄	(C₂×D₂₅₁)⋊C₈	1	4	yes	—
133	C₄	#22548694	1	2	yes	—
134	C₄	#22548703	1	1	yes	characteristic
135	C₄	#22548715	1	1	yes	characteristic
136	C₂	#30631593	1	1	yes	characteristic
137	C₂	C₂×(C₂×D₂₅₁)⋊C₈	1	1	yes	characteristic
138	C₂	#30631653	1	1	yes	characteristic
139	C₂	C₈×Dic₅₀₂	1	2	yes	—
140	C₂	#30631817	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₀₂

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₅₁⋊C₈)

C₂

yes

maximal, characteristic

#30631441

C₂

yes

maximal

C₂×C₄×C₈×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₄

yes

characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₄×Dic₅₀₂

—

C₄×Dic₅₀₂

—

C₂×C₄⋊Dic₂₅₁

C₄

yes

characteristic

C₂×Dic₂₅₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

C₄⋊(C₂₅₁⋊C₈)

—

C₂²×C₈×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

C₈

yes

complemented

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

Q₈

yes

complemented

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

D₄

yes

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂×C₈

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂×C₈

yes

—

C₂×C₄×C₂₅₁

M₁₆

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₄×Q₈

yes

—

C₂²×C₂₅₁

C₂²×C₈

yes

characteristic

C₂²×C₂₅₁

C₂×M₁₆

yes

characteristic

C₂²×C₂₅₁

#131

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂²⋊C₈

yes

—

C₄×C₂₅₁

C₂²⋊Q₈

yes

—

C₄×C₂₅₁

#124

yes

characteristic

C₄×C₂₅₁

C₄⋊Q₈

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#391

yes

characteristic

C₂×C₂₅₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₅₁

#408

yes

characteristic

C₂×C₂₅₁

C₈×Q₈

yes

—

C₂×C₂₅₁

#446

yes

—

C₂₅₁

#1984

yes

characteristic, complemented, Sylow

#1984

—

251

maximal, Sylow

C₂×C₄×Q₈

—

251

—

C₂×C₄⋊C₈

—

251

—

C₂×C₄×C₈

D₂₅₁

yes

characteristic

#336

—

251

—

C₂²×Q₈

—

251

—

C₂²×C₈

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

—

C₄×Q₈

—

502

—

C₄×Q₈

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

502

—

C₄×C₈

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₄×D₂₅₁

yes

characteristic

C₂²×C₄

C₄×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

502

—

C₂×C₈

—

502

—

C₂×C₈

Dic₅₀₂

yes

complemented

100

C₄⋊C₄

—

502

—

101

C₄⋊C₄

—

502

—

102

C₄²

—

502

—

103

C₄²

—

502

—

104

C₄²

C₂²⋊D₂₅₁

yes

—

105

C₄²

D₁₀₀₄

yes

—

106

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

107

Q₈

—

502

—

108

C₂×C₄

—

109

C₂×C₄

—

502

—

110

C₂×C₄

—

502

—

111

C₂×C₄

—

502

—

112

C₂×C₄

—

502

—

113

C₂×C₄

—

251

—

114

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

115

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

116

C₂×C₄

C₈×D₂₅₁

yes

—

117

C₂×C₄

#16860793

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

119

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

120

C₂×C₄

C₂×Dic₅₀₂

yes

—

121

C₂×C₄

#16860810

yes

—

122

C₈

—

123

C₈

—

1004

—

124

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

125

C₂²

C₄×Dic₅₀₂

yes

—

126

C₂²

C₂×C₈×D₂₅₁

yes

characteristic

127

C₂²

C₂×#16860793

yes

characteristic

128

C₂²

#22548721

yes

—

129

C₄

—

130

C₄

—

1004

—

131

C₄

—

502

—

132

C₄

(C₂×D₂₅₁)⋊C₈

yes

—

133

C₄

#22548694

yes

—

134

C₄

#22548703

yes

characteristic

135

C₄

#22548715

yes

characteristic

136

C₂

#30631593

yes

characteristic

137

C₂

C₂×(C₂×D₂₅₁)⋊C₈

yes

characteristic

138

C₂

#30631653

yes

characteristic

139

C₂

C₈×Dic₅₀₂

yes

—

140

C₂

#30631817

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂²×C₈
Fitting	C₂×C₄×C₈×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₁
hypocenter	C₂₅₁	#1984
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₅₁

commutator

C₂²×C₂₅₁

C₂²×C₈

Fitting

C₂×C₄×C₈×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₂×C₄×C₈

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
18. C₂×Dic₅₀₂	C₈	semidirect
32. C₂×C₈×C₂₅₁	Q₈	semidirect
70. C₂₅₁	#1984	semidirect
99. C₂×C₈	Dic₅₀₂	semidirect

H ≅ G/N

type

18. C₂×Dic₅₀₂

C₈

semidirect

32. C₂×C₈×C₂₅₁

Q₈

semidirect

70. C₂₅₁

#1984

semidirect

99. C₂×C₈

Dic₅₀₂

semidirect

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

subgroup	index
C₂×C₄×Dic₅₀₂	2
C₂×C₄⋊(C₂₅₁⋊C₈)	2
#30631441	2
C₂×C₄×C₈×C₂₅₁	2
#1984	251

subgroup

index

C₂×C₄×Dic₅₀₂

C₂×C₄⋊(C₂₅₁⋊C₈)

#30631441

C₂×C₄×C₈×C₂₅₁

#1984

251

Subgroups — #44551301

Conjugacy classes of subgroups

Complements

Maximal subgroups