Group #44551288 — subgroups

Subgroups — #44551288

Order 32128.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22548715	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#22548638	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
6	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
7	#22548715	—	2	2	no	—
8	#22548715	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
10	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
11	C₄⋊(C₂₅₁⋊C₈)	—	2	2	no	—
12	#22548638	—	2	8	no	—
13	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₁	—	2	1	no	—
15	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
16	#95×C₂₅₁	—	2	1	no	—
17	C₂×Dic₅₀₂	—	2	2	no	—
18	C₂×Dic₅₀₂	—	2	4	no	—
19	C₂×Dic₅₀₂	—	2	4	no	—
20	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	—
21	C₂²×Dic₂₅₁	—	2	2	no	—
22	C₂×C₂₅₁⋊C₈	—	2	4	no	—
23	C₂×C₂₅₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₂₅₁	—	2	4	no	—
25	C₄⋊Dic₂₅₁	—	2	4	no	—
26	C₂²×C₄×C₂₅₁	—	2	2	no	—
27	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₅₁	—	2	2	no	—
31	C₄⋊C₄×C₂₅₁	D₄	1	4	yes	—
32	C₄²×C₂₅₁	—	4	2	no	—
33	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
34	C₄²×C₂₅₁	D₄	1	2	yes	—
35	Dic₅₀₂	—	2	8	no	—
36	Dic₅₀₂	—	4	2	no	—
37	Dic₅₀₂	—	4	2	no	—
38	C₂×Dic₂₅₁	—	2	4	no	—
39	C₂×Dic₂₅₁	—	4	4	no	—
40	C₂₅₁⋊C₈	—	4	2	no	—
41	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₁	—	2	2	no	—
43	C₂×C₄×C₂₅₁	—	2	4	no	—
44	C₂×C₄×C₂₅₁	—	4	2	no	—
45	C₂×C₄×C₂₅₁	—	4	2	no	—
46	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₅₁	SD₁₆	1	2	yes	—
49	C₂×C₄×C₂₅₁	Q₁₆	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	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₅₁	#121	1	2	yes	—
56	C₂²×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₂₅₁	#137	1	2	yes	—
59	C₄×C₂₅₁	—	2	2	no	—
60	C₄×C₂₅₁	—	4	4	no	—
61	C₄×C₂₅₁	#103	1	4	yes	complemented
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	C₂×C₂₅₁	#390	1	1	yes	characteristic
67	C₂×C₂₅₁	C₂×#103	1	1	yes	characteristic
68	C₂×C₂₅₁	#419	1	1	yes	characteristic
69	C₂×C₂₅₁	#461	1	2	yes	—
70	C₂×C₂₅₁	#462	1	2	yes	—
71	C₂₅₁	#1981	1	1	yes	characteristic, complemented, Sylow
72	#1981	—	251	1	no	maximal, Sylow
73	C₂×C₄⋊Q₈	—	251	1	no	—
74	C₂×C₄⋊C₈	—	251	1	no	—
75	C₂×#103	—	251	4	no	—
76	C₄×C₄⋊C₄	D₂₅₁	1	1	yes	characteristic
77	C₂²×Q₈	—	251	2	no	—
78	C₂²×C₈	—	251	2	no	—
79	C₄⋊Q₈	—	502	2	no	—
80	C₄⋊Q₈	—	251	4	no	—
81	C₂×C₄⋊C₄	—	502	1	no	—
82	C₂×C₄⋊C₄	—	502	1	no	—
83	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
86	C₄⋊C₈	—	502	2	no	—
87	#103	—	502	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	502	4	no	—
90	C₂×Q₈	—	502	4	no	—
91	C₂×Q₈	—	502	2	no	—
92	C₂×Q₈	—	251	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	502	2	no	—
95	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
98	C₂×C₈	—	502	4	no	—
99	C₂×C₈	—	502	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	502	4	no	—
102	C₄⋊C₄	—	502	4	no	—
103	C₄⋊C₄	D₁₀₀₄	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₂₅₁	1	2	yes	—
106	C₄²	D₁₀₀₄	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
108	Q₈	—	1004	2	no	—
109	Q₈	—	1004	2	no	—
110	Q₈	—	502	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	1004	4	no	—
116	C₂×C₄	—	502	4	no	—
117	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
119	C₂×C₄	#16860798	1	2	yes	—
120	C₂×C₄	#16860800	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
123	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
124	C₂×C₄	#16860810	1	2	yes	—
125	C₈	—	1004	2	no	—
126	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
127	C₂²	#22548689	1	2	yes	—
128	C₂²	C₂×#16860798	1	1	yes	characteristic
129	C₂²	C₂×#16860800	1	1	yes	characteristic
130	C₂²	#22548745	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	1004	4	no	—
134	C₄	#22548638	1	4	yes	complemented
135	C₄	C₄×D₁₀₀₄	1	2	yes	—
136	C₄	#22548703	1	1	yes	characteristic
137	C₄	#22548712	1	1	yes	characteristic
138	C₂	#30631589	1	1	yes	characteristic
139	C₂	C₂×#22548638	1	1	yes	characteristic
140	C₂	#30631695	1	1	yes	characteristic
141	C₂	#30631892	1	2	yes	—
142	C₂	#30631897	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548715

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₅₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#22548638

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

#22548715

—

#22548715

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

—

C₄⋊(C₂₅₁⋊C₈)

—

#22548638

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#95×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂²×Dic₂₅₁

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

D₄

yes

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

SD₁₆

yes

—

C₂×C₄×C₂₅₁

Q₁₆

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

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

#121

yes

—

C₂²×C₂₅₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₅₁

C₂×Q₁₆

yes

characteristic

C₂²×C₂₅₁

#137

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#103

yes

complemented

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

—

C₂×C₂₅₁

#462

yes

—

C₂₅₁

#1981

yes

characteristic, complemented, Sylow

#1981

—

251

maximal, Sylow

C₂×C₄⋊Q₈

—

251

—

C₂×C₄⋊C₈

—

251

—

C₂×#103

—

251

—

C₄×C₄⋊C₄

D₂₅₁

yes

characteristic

C₂²×Q₈

—

251

—

C₂²×C₈

—

251

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

251

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

502

—

#103

—

502

—

#95

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂²×C₄

—

C₂²×C₄

—

502

—

C₂²×C₄

C₂²⋊D₂₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

502

—

C₂×C₈

—

502

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

502

—

102

C₄⋊C₄

—

502

—

103

C₄⋊C₄

D₁₀₀₄

yes

—

104

C₄²

—

105

C₄²

C₄×D₂₅₁

yes

—

106

C₄²

D₁₀₀₄

yes

—

107

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

108

Q₈

—

1004

—

109

Q₈

—

1004

—

110

Q₈

—

502

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

1004

—

116

C₂×C₄

—

502

—

117

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

119

C₂×C₄

#16860798

yes

—

120

C₂×C₄

#16860800

yes

—

121

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

122

C₂×C₄

C₂×D₁₀₀₄

yes

—

123

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

124

C₂×C₄

#16860810

yes

—

125

C₈

—

1004

—

126

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

127

C₂²

#22548689

yes

—

128

C₂²

C₂×#16860798

yes

characteristic

129

C₂²

C₂×#16860800

yes

characteristic

130

C₂²

#22548745

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

1004

—

134

C₄

#22548638

yes

complemented

135

C₄

C₄×D₁₀₀₄

yes

—

136

C₄

#22548703

yes

characteristic

137

C₄

#22548712

yes

characteristic

138

C₂

#30631589

yes

characteristic

139

C₂

C₂×#22548638

yes

characteristic

140

C₂

#30631695

yes

characteristic

141

C₂

#30631892

yes

—

142

C₂

#30631897

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₄×C₄⋊C₄×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂×D₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₅₁
hypocenter	C₂₅₁	#1981
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₄×C₄⋊C₄×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂×D₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₅₁

hypocenter

C₂₅₁

#1981

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #22548715	C₄	semidirect
61. C₄×C₂₅₁	#103	semidirect
71. C₂₅₁	#1981	semidirect
134. C₄	#22548638	semidirect

H ≅ G/N

type

8. #22548715

C₄

semidirect

61. C₄×C₂₅₁

#103

semidirect

71. C₂₅₁

#1981

semidirect

134. C₄

#22548638

semidirect

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

subgroup	index
C₂×#22548715	2
C₂×C₄⋊(C₂₅₁⋊C₈)	2
C₂×#22548638	2
C₄×C₄⋊C₄×C₂₅₁	2
#1981	251

subgroup

index

C₂×#22548715

C₂×C₄⋊(C₂₅₁⋊C₈)

C₂×#22548638

C₄×C₄⋊C₄×C₂₅₁

#1981

251

Subgroups — #44551288

Conjugacy classes of subgroups

Complements

Maximal subgroups