Group #44550456 — subgroups

Subgroups — #44550456

Order 32128.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#30631547	C₂	1	1	yes	maximal, characteristic
3	#30631450	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₅₁	—	2	1	no	—
6	C₂×C₄⋊Dic₂₅₁	—	2	2	no	—
7	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
8	C₂×C₄⋊Dic₂₅₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₂₅₁	—	2	1	no	—
12	C₂₅₁⋊#95	—	2	1	no	—
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₂₅₁	—	2	2	no	—
16	C₂²×Dic₂₅₁	—	2	4	no	—
17	C₂²×Dic₂₅₁	—	2	2	no	—
18	C₂²×Dic₂₅₁	—	2	2	no	—
19	C₂²×Dic₂₅₁	—	4	4	no	—
20	C₄⋊Dic₂₅₁	—	2	4	no	—
21	C₄⋊Dic₂₅₁	—	2	2	no	—
22	C₄⋊Dic₂₅₁	—	2	4	no	—
23	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₂₅₁	—	4	2	no	—
25	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₁	—	2	4	no	—
27	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
29	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₅₁	—	2	2	no	—
32	C₂×C₈×C₂₅₁	—	2	4	no	—
33	C₂×Dic₂₅₁	—	2	4	no	—
34	C₂×Dic₂₅₁	—	4	8	no	—
35	C₂×Dic₂₅₁	—	4	4	no	—
36	C₂×Dic₂₅₁	—	4	2	no	—
37	C₂×Dic₂₅₁	—	4	2	no	—
38	C₂×Dic₂₅₁	—	4	2	no	—
39	C₂³×C₂₅₁	—	2	2	no	—
40	C₂³×C₂₅₁	—	2	2	no	—
41	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₂₅₁	SD₁₆	1	2	yes	—
43	C₂³×C₂₅₁	Q₁₆	1	2	yes	—
44	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₁	—	2	2	no	—
46	C₂×C₄×C₂₅₁	—	2	4	no	—
47	C₂×C₄×C₂₅₁	—	2	4	no	—
48	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₁	#41	1	4	yes	—
52	C₈×C₂₅₁	—	4	2	no	—
53	C₂²×C₂₅₁	—	2	2	no	—
54	C₂²×C₂₅₁	—	2	2	no	—
55	C₂²×C₂₅₁	—	2	4	no	—
56	C₂²×C₂₅₁	—	2	4	no	—
57	C₂²×C₂₅₁	#103	1	4	yes	—
58	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₅₁	#123	1	2	yes	—
60	C₂²×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₂₅₁	—	2	2	no	—
64	C₄×C₂₅₁	#117	1	2	yes	—
65	C₄×C₂₅₁	#123	1	2	yes	—
66	Dic₂₅₁	—	4	4	no	—
67	Dic₂₅₁	—	8	2	no	—
68	C₂×C₂₅₁	—	2	4	no	—
69	C₂×C₂₅₁	#381	1	1	yes	characteristic
70	C₂×C₂₅₁	C₂×#103	1	1	yes	characteristic
71	C₂×C₂₅₁	#418	1	1	yes	characteristic
72	C₂×C₂₅₁	#481	1	2	yes	—
73	C₂×C₂₅₁	#484	1	2	yes	—
74	C₂₅₁	#1824	1	1	yes	characteristic, complemented, Sylow
75	#1824	—	251	1	no	maximal, Sylow
76	C₂²×C₄⋊C₄	—	251	1	no	—
77	C₂×C₂²⋊C₈	D₂₅₁	1	1	yes	characteristic
78	#380	—	251	1	no	—
79	#340	—	251	4	no	—
80	C₂³×C₄	—	502	1	no	—
81	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
83	C₂×C₄⋊C₄	—	502	2	no	—
84	C₂×C₄⋊C₄	—	502	1	no	—
85	C₂×C₄⋊C₄	—	251	2	no	—
86	C₂×C₄⋊C₄	—	251	4	no	—
87	C₂×C₄⋊C₄	—	251	2	no	—
88	C₂×C₂²⋊C₄	—	502	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	#95	—	502	1	no	—
91	C₂⁴	D₁₀₀₄	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	1004	4	no	—
94	C₂²×C₄	—	502	2	no	—
95	C₂²×C₄	—	502	4	no	—
96	C₂²×C₄	—	502	2	no	—
97	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
99	C₂²×C₄	D₁₀₀₄	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₄⋊C₄	—	502	4	no	—
104	C₄⋊C₄	—	502	4	no	—
105	C₄⋊C₄	—	502	2	no	—
106	C₄⋊C₄	—	251	4	no	—
107	C₂²⋊C₄	—	1004	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
111	C₂³	#16860799	1	2	yes	—
112	C₂³	Dic₁₀₀₄	1	2	yes	—
113	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	1004	2	no	—
118	C₂×C₄	—	1004	2	no	—
119	C₂×C₄	—	1004	2	no	—
120	C₂×C₄	—	1004	8	no	—
121	C₂×C₄	—	1004	4	no	—
122	C₂×C₄	—	502	4	no	—
123	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
126	C₂×C₄	#16860809	1	4	yes	—
127	C₈	—	4	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#22548640	1	4	yes	—
133	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
134	C₂²	#22548699	1	2	yes	—
135	C₂²	C₂×#16860799	1	1	yes	characteristic
136	C₂²	C₂×Dic₁₀₀₄	1	1	yes	characteristic
137	C₂²	#22548743	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	2008	2	no	—
140	C₄	—	1004	4	no	—
141	C₄	#22548671	1	2	yes	—
142	C₄	#22548696	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#30631548	1	1	yes	characteristic
145	C₂	C₂×#22548640	1	1	yes	characteristic
146	C₂	#30631693	1	1	yes	characteristic
147	C₂	#30632006	1	2	yes	—
148	C₂	#30632023	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₂₅₁

C₂

yes

maximal, characteristic

#30631547

C₂

yes

maximal, characteristic

#30631450

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₂₅₁

C₂

yes

maximal, characteristic

C₂³×Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₂²⋊Dic₂₅₁

—

C₂₅₁⋊#95

—

C₂³×C₄×C₂₅₁

C₂²

yes

characteristic

C₂²×C₈×C₂₅₁

C₂²

yes

—

C₂²⋊C₈×C₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

C₂²⋊Dic₂₅₁

—

C₂⁴×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

SD₁₆

yes

—

C₂³×C₂₅₁

Q₁₆

yes

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#103

yes

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

#123

yes

—

C₂²×C₂₅₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₅₁

C₂×Q₁₆

yes

characteristic

C₂²×C₂₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#117

yes

—

C₄×C₂₅₁

#123

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

—

C₂×C₂₅₁

#484

yes

—

C₂₅₁

#1824

yes

characteristic, complemented, Sylow

#1824

—

251

maximal, Sylow

C₂²×C₄⋊C₄

—

251

—

C₂×C₂²⋊C₈

D₂₅₁

yes

characteristic

#380

—

251

—

#340

—

251

—

C₂³×C₄

—

502

—

C₂³×C₄

C₂×D₂₅₁

yes

characteristic

C₂²×C₈

C₂×D₂₅₁

yes

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

502

—

C₂²⋊C₈

—

#95

—

502

—

C₂⁴

D₁₀₀₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

1004

—

C₂²×C₄

—

502

—

C₂²×C₄

—

502

—

C₂²×C₄

—

502

—

C₂²×C₄

C₄×D₂₅₁

yes

—

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

C₂²×C₄

D₁₀₀₄

yes

characteristic

100

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

502

—

104

C₄⋊C₄

—

502

—

105

C₄⋊C₄

—

502

—

106

C₄⋊C₄

—

251

—

107

C₂²⋊C₄

—

1004

—

108

C₂³

—

109

C₂³

—

110

C₂³

(C₂×D₂₅₁)⋊C₄

yes

—

111

C₂³

#16860799

yes

—

112

C₂³

Dic₁₀₀₄

yes

—

113

C₂³

C₂×D₁₀₀₄

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

1004

—

118

C₂×C₄

—

1004

—

119

C₂×C₄

—

1004

—

120

C₂×C₄

—

1004

—

121

C₂×C₄

—

1004

—

122

C₂×C₄

—

502

—

123

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

124

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

126

C₂×C₄

#16860809

yes

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#22548640

yes

—

133

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

134

C₂²

#22548699

yes

—

135

C₂²

C₂×#16860799

yes

characteristic

136

C₂²

C₂×Dic₁₀₀₄

yes

characteristic

137

C₂²

#22548743

yes

—

138

C₄

—

139

C₄

—

2008

—

140

C₄

—

1004

—

141

C₄

#22548671

yes

—

142

C₄

#22548696

yes

—

143

C₂

—

144

C₂

#30631548

yes

characteristic

145

C₂

C₂×#22548640

yes

characteristic

146

C₂

#30631693

yes

characteristic

147

C₂

#30632006

yes

—

148

C₂

#30632023

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₀₄
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×C₂²⋊C₈×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂×D₄
prime radical	C₂⁴×C₂₅₁	D₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₅₁
hypocenter	C₂₅₁	#1824
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₀₄

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₂×C₂²⋊C₈×C₂₅₁

C₂

socle

C₂³×C₂₅₁

C₂×D₄

prime radical

C₂⁴×C₂₅₁

D₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₅₁

hypocenter

C₂₅₁

#1824

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #30631450	C₂	semidirect
8. C₂×C₄⋊Dic₂₅₁	C₄	semidirect
74. C₂₅₁	#1824	semidirect

H ≅ G/N

type

3. #30631450

C₂

semidirect

8. C₂×C₄⋊Dic₂₅₁

C₄

semidirect

74. C₂₅₁

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₂₅₁	2
#30631547	2
#30631450	2
C₂×C₂²⋊C₈×C₂₅₁	2
#1824	251

subgroup

index

C₂²×C₄⋊Dic₂₅₁

#30631547

#30631450

C₂×C₂²⋊C₈×C₂₅₁

#1824

251

Subgroups — #44550456

Conjugacy classes of subgroups

Complements

Maximal subgroups