Group #44549872 — subgroups

Subgroups — #44549872

Order 32128.

146 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 55 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	#30632085	C₂	1	1	yes	maximal, characteristic
2	#30631983	C₂	1	1	yes	maximal, characteristic
3	#30631898	C₂	1	1	yes	maximal, characteristic
4	#30631817	C₂	1	1	yes	maximal, characteristic
5	#30631435	C₂	1	1	yes	maximal, characteristic
6	#30631434	C₂	1	1	yes	maximal, characteristic
7	#335×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₀₄	—	2	1	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548715	C₂²	1	1	yes	characteristic
11	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
12	#22548649	—	2	1	no	—
13	#22548647	—	2	1	no	—
14	#22548647	—	2	1	no	—
15	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
16	Dic₂₅₁⋊C₈	—	2	1	no	—
17	#22548640	—	2	1	no	—
18	#22548640	—	2	1	no	—
19	#22548625	—	2	1	no	—
20	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₀₂	—	2	1	no	—
24	C₂×Dic₅₀₂	—	2	1	no	—
25	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
26	Dic₁₀₀₄	—	4	2	no	—
27	C₂×C₂₅₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₂₅₁	—	2	1	no	—
29	C₄⋊Dic₂₅₁	—	2	1	no	—
30	C₄⋊Dic₂₅₁	—	2	1	no	—
31	C₄⋊Dic₂₅₁	—	2	1	no	—
32	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
33	Dic₂₅₁⋊C₄	—	2	1	no	—
34	C₄×Dic₂₅₁	—	2	1	no	—
35	C₂×C₈×C₂₅₁	—	2	1	no	—
36	C₂×C₈×C₂₅₁	—	2	1	no	—
37	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
38	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
39	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
40	Dic₅₀₂	—	2	1	no	—
41	Dic₅₀₂	—	2	1	no	—
42	Dic₅₀₂	—	4	1	no	—
43	Dic₅₀₂	—	4	1	no	—
44	Dic₅₀₂	—	8	1	no	—
45	C₂×Dic₂₅₁	—	2	1	no	—
46	C₂×Dic₂₅₁	—	2	1	no	—
47	C₂×Dic₂₅₁	—	4	1	no	—
48	C₂×Dic₂₅₁	—	4	1	no	—
49	C₂₅₁⋊C₈	—	4	1	no	—
50	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
53	C₈×C₂₅₁	—	2	1	no	—
54	C₈×C₂₅₁	—	4	1	no	—
55	C₈×C₂₅₁	—	4	1	no	—
56	C₈×C₂₅₁	Q₁₆	1	2	yes	complemented
57	C₂²×C₂₅₁	#121	1	1	yes	characteristic
58	C₄×C₂₅₁	—	2	1	no	—
59	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₂₅₁	#137	1	1	yes	characteristic
62	C₄×C₂₅₁	#137	1	1	yes	characteristic
63	Dic₂₅₁	—	4	1	no	—
64	Dic₂₅₁	—	4	1	no	—
65	Dic₂₅₁	—	8	1	no	—
66	Dic₂₅₁	—	8	1	no	—
67	C₂×C₂₅₁	#462	1	1	yes	characteristic
68	C₂×C₂₅₁	#468	1	1	yes	characteristic
69	C₂×C₂₅₁	#471	1	1	yes	characteristic
70	C₂₅₁	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	251	1	no	maximal, Sylow
72	#498	—	251	1	no	—
73	#477	—	251	1	no	—
74	#462	—	251	1	no	—
75	#446	—	251	1	no	—
76	#335	D₂₅₁	1	1	yes	characteristic
77	#333	—	251	1	no	—
78	#332	—	251	1	no	—
79	C₂×Q₁₆	—	502	1	no	—
80	C₄⋊Q₈	—	251	1	no	—
81	C₄⋊Q₈	—	251	1	no	—
82	C₄×Q₈	—	251	1	no	—
83	#107	—	502	1	no	—
84	#106	—	502	1	no	—
85	#106	—	502	1	no	—
86	C₄⋊C₈	—	502	1	no	—
87	C₄⋊C₈	—	251	1	no	—
88	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
90	#103	—	502	1	no	—
91	#103	—	502	1	no	—
92	#97	—	502	1	no	—
93	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
94	C₂×Q₈	—	502	1	no	—
95	C₂×Q₈	—	502	1	no	—
96	C₂×Q₈	—	251	1	no	—
97	Q₁₆	—	1004	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	502	1	no	—
101	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
102	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
103	C₄⋊C₄	—	502	1	no	—
104	C₄⋊C₄	—	502	1	no	—
105	C₄⋊C₄	—	502	1	no	—
106	C₄⋊C₄	—	502	1	no	—
107	C₄⋊C₄	—	502	1	no	—
108	C₄⋊C₄	—	251	1	no	—
109	C₄²	—	502	1	no	—
110	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
111	Q₈	—	2008	1	no	—
112	Q₈	—	1004	1	no	—
113	Q₈	—	1004	1	no	—
114	Q₈	—	502	1	no	—
115	Q₈	—	502	1	no	—
116	C₂×C₄	—	1004	1	no	—
117	C₂×C₄	—	1004	1	no	—
118	C₂×C₄	—	502	1	no	—
119	C₂×C₄	—	502	1	no	—
120	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
121	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
122	C₂×C₄	#16860811	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	1004	1	no	—
127	C₈	Dic₁₀₀₄	1	2	yes	complemented
128	C₂²	#22548690	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2008	1	no	—
131	C₄	—	2008	1	no	—
132	C₄	—	1004	1	no	—
133	C₄	—	1004	1	no	—
134	C₄	C₂×Dic₁₀₀₄	1	1	yes	characteristic
135	C₄	#22548739	1	1	yes	characteristic
136	C₄	#22548747	1	1	yes	characteristic
137	C₄	#22548748	1	1	yes	characteristic
138	C₂	#30631898	1	1	yes	characteristic
139	C₂	#30631934	1	1	yes	characteristic
140	C₂	#30631951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632085

C₂

yes

maximal, characteristic

#30631983

C₂

yes

maximal, characteristic

#30631898

C₂

yes

maximal, characteristic

#30631817

C₂

yes

maximal, characteristic

#30631435

C₂

yes

maximal, characteristic

#30631434

C₂

yes

maximal, characteristic

#335×C₂₅₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₄

—

#22548715

C₂²

yes

characteristic

#22548715

C₂²

yes

characteristic

C₄×Dic₅₀₂

C₂²

yes

—

—

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

Dic₂₅₁⋊C₈

—

#22548640

—

#22548640

—

#22548625

—

C₄⋊C₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

D₄

yes

characteristic

Dic₁₀₀₄

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

D₄

yes

characteristic

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

D₄

yes

characteristic

C₂×C₈×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₈×C₂₅₁

Q₁₆

yes

complemented

C₂²×C₂₅₁

#121

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×Q₁₆

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1692

—

251

maximal, Sylow

#498

—

251

—

#477

—

251

—

#462

—

251

—

#446

—

251

—

#335

D₂₅₁

yes

characteristic

#333

—

251

—

#332

—

251

—

C₂×Q₁₆

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

—

251

—

C₄×Q₈

—

251

—

#107

—

502

—

#106

—

502

—

#106

—

502

—

C₄⋊C₈

—

502

—

C₄⋊C₈

—

251

—

C₄⋊C₈

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

C₂×D₂₅₁

yes

characteristic

#103

—

502

—

#103

—

502

—

#97

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

Q₁₆

—

1004

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

502

—

101

C₂×C₈

D₁₀₀₄

yes

characteristic

102

C₂×C₈

D₁₀₀₄

yes

characteristic

103

C₄⋊C₄

—

502

—

104

C₄⋊C₄

—

502

—

105

C₄⋊C₄

—

502

—

106

C₄⋊C₄

—

502

—

107

C₄⋊C₄

—

502

—

108

C₄⋊C₄

—

251

—

109

C₄²

—

502

—

110

C₄²

C₂²×D₂₅₁

yes

characteristic

111

Q₈

—

2008

—

112

Q₈

—

1004

—

113

Q₈

—

1004

—

114

Q₈

—

502

—

115

Q₈

—

502

—

116

C₂×C₄

—

1004

—

117

C₂×C₄

—

1004

—

118

C₂×C₄

—

502

—

119

C₂×C₄

—

502

—

120

C₂×C₄

D₄×D₂₅₁

yes

characteristic

121

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

122

C₂×C₄

#16860811

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

1004

—

127

C₈

Dic₁₀₀₄

yes

complemented

128

C₂²

#22548690

yes

characteristic

129

C₄

—

130

C₄

—

2008

—

131

C₄

—

2008

—

132

C₄

—

1004

—

133

C₄

—

1004

—

134

C₄

C₂×Dic₁₀₀₄

yes

characteristic

135

C₄

#22548739

yes

characteristic

136

C₄

#22548747

yes

characteristic

137

C₄

#22548748

yes

characteristic

138

C₂

#30631898

yes

characteristic

139

C₂

#30631934

yes

characteristic

140

C₂

#30631951

yes

characteristic

	subgroup	G/H
center	C₂²	#22548690
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#335×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#121
hypercenter	#335	D₂₅₁
hypocenter	C₂₅₁	#1692
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#22548690

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#335×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
56. C₈×C₂₅₁	Q₁₆	semidirect
70. C₂₅₁	#1692	semidirect
127. C₈	Dic₁₀₀₄	semidirect

H ≅ G/N

type

56. C₈×C₂₅₁

Q₁₆

semidirect

70. C₂₅₁

#1692

semidirect

127. C₈

Dic₁₀₀₄

semidirect

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

subgroup	index
#30632085	2
#30631983	2
#30631898	2
#30631817	2
#30631435	2
#30631434	2
#335×C₂₅₁	2
#1692	251

subgroup

index

#335×C₂₅₁

251

Subgroups — #44549872

Conjugacy classes of subgroups

Complements

Maximal subgroups