Group #44552414 — subgroups

Subgroups — #44552414

Order 32128.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₂₀₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548661	C₂	1	1	yes	maximal, characteristic
3	#30632065	C₂	1	1	yes	maximal, characteristic
4	#30631781	C₂	1	1	yes	maximal, characteristic
5	Dic₂₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#30631490	C₂	1	1	yes	maximal, characteristic
7	#358×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
10	C₂×#16860800	—	2	1	no	—
11	#22548713	—	2	1	no	—
12	#22548680	—	2	1	no	—
13	Dic₂₀₀₈	—	2	2	no	—
14	#22548661	—	2	2	no	—
15	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#22548638	—	2	1	no	—
17	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
20	#107×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₀₂	—	2	1	no	—
22	C₂×Dic₅₀₂	—	2	1	no	—
23	Dic₁₀₀₄	—	2	1	no	—
24	Dic₁₀₀₄	—	2	1	no	—
25	Dic₁₀₀₄	—	2	1	no	—
26	Dic₁₀₀₄	D₄	1	2	yes	—
27	#16860800	—	4	2	no	—
28	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
29	Dic₂₅₁⋊C₄	—	4	1	no	—
30	Dic₂₅₁⋊C₄	—	4	1	no	—
31	Dic₂₅₁⋊C₄	—	4	1	no	—
32	C₄×Dic₂₅₁	—	4	1	no	—
33	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
34	C₂₅₁⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₂₅₁	—	2	1	no	—
36	Q₁₆×C₂₅₁	—	2	1	no	—
37	Q₁₆×C₂₅₁	—	2	1	no	—
38	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₂₅₁	—	2	1	no	—
40	C₁₆×C₂₅₁	—	2	1	no	—
41	Dic₅₀₂	—	2	2	no	—
42	Dic₅₀₂	—	4	1	no	—
43	Dic₅₀₂	—	4	1	no	—
44	Dic₅₀₂	—	4	1	no	—
45	C₂×Dic₂₅₁	—	4	1	no	—
46	C₂×Dic₂₅₁	—	4	1	no	—
47	C₂×Dic₂₅₁	D₈	1	1	yes	characteristic
48	C₂×Dic₂₅₁	D₈	1	1	yes	characteristic
49	C₂₅₁⋊C₈	—	2	1	no	—
50	Q₈×C₂₅₁	—	4	1	no	—
51	Q₈×C₂₅₁	—	4	1	no	—
52	C₂×C₄×C₂₅₁	—	4	1	no	—
53	C₂×C₄×C₂₅₁	—	4	1	no	—
54	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₂₅₁	#41	1	1	yes	characteristic
57	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
58	C₄×C₂₅₁	—	8	1	no	—
59	C₄×C₂₅₁	—	8	1	no	—
60	C₄×C₂₅₁	#121	1	1	yes	characteristic
61	C₄×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₂₅₁	—	2	1	no	—
63	Dic₂₅₁	—	4	2	no	—
64	Dic₂₅₁	—	8	1	no	—
65	Dic₂₅₁	Q₃₂	1	2	yes	complemented
66	C₂×C₂₅₁	#459	1	1	yes	characteristic
67	C₂×C₂₅₁	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₂₅₁	#510	1	1	yes	characteristic
69	C₂₅₁	#2209	1	1	yes	characteristic, complemented, Sylow
70	#2209	—	251	1	no	maximal, Sylow
71	C₂×Q₃₂	—	251	1	no	—
72	C₂×Q₃₂	—	251	1	no	—
73	#494	—	251	1	no	—
74	C₄×Q₁₆	—	251	1	no	—
75	C₄⋊C₁₆	—	251	1	no	—
76	#358	—	251	1	no	—
77	#358	D₂₅₁	1	1	yes	characteristic
78	C₂×Q₁₆	—	502	1	no	—
79	C₂×Q₁₆	—	251	1	no	—
80	C₂×Q₁₆	—	251	1	no	—
81	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
82	C₄⋊Q₈	—	502	1	no	—
83	C₄×Q₈	—	502	1	no	—
84	Q₃₂	—	502	2	no	—
85	Q₃₂	—	502	2	no	—
86	C₂×C₁₆	—	251	1	no	—
87	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
88	#107	C₂×D₂₅₁	1	1	yes	characteristic
89	#103	—	502	1	no	—
90	C₄×C₈	—	251	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	502	1	no	—
93	C₂×Q₈	—	502	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	1004	2	no	—
97	Q₁₆	—	502	1	no	—
98	Q₁₆	—	502	1	no	—
99	Q₁₆	—	502	1	no	—
100	Q₁₆	—	251	2	no	—
101	C₂×C₈	—	251	1	no	—
102	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	1004	1	no	—
105	C₄⋊C₄	—	1004	1	no	—
106	C₄⋊C₄	—	1004	1	no	—
107	C₄²	—	1004	1	no	—
108	C₄²	—	251	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	502	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	1004	1	no	—
114	Q₈	—	1004	1	no	—
115	Q₈	—	1004	1	no	—
116	Q₈	—	502	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	1004	1	no	—
120	C₂×C₄	—	1004	1	no	—
121	C₂×C₄	—	251	1	no	—
122	C₂×C₄	—	251	1	no	—
123	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
124	C₈	—	502	1	no	—
125	C₈	D₄×D₂₅₁	1	1	yes	characteristic
126	C₈	#16860810	1	1	yes	characteristic
127	C₂²	D₈×D₂₅₁	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	2008	1	no	—
131	C₄	—	1004	2	no	—
132	C₄	—	502	1	no	—
133	C₄	—	251	2	no	—
134	C₄	#22548688	1	1	yes	characteristic
135	C₄	#22548742	1	1	yes	characteristic
136	C₂	#30631881	1	1	yes	characteristic
137	C₂	Q₃₂×D₂₅₁	1	1	yes	characteristic
138	C₂	#30632126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₀₀₈

C₂

yes

maximal, characteristic

C₂×#22548661

C₂

yes

maximal, characteristic

#30632065

C₂

yes

maximal, characteristic

#30631781

C₂

yes

maximal, characteristic

Dic₂₅₁⋊C₁₆

C₂

yes

maximal, characteristic

#30631490

C₂

yes

maximal, characteristic

#358×C₂₅₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₄

C₂²

yes

characteristic

C₂×Dic₁₀₀₄

C₂²

yes

—

—

—

Dic₂₀₀₈

—

#22548661

—

C₂×C₂₅₁⋊C₁₆

C₂²

yes

characteristic

#22548638

—

C₈×Dic₂₅₁

C₂²

yes

characteristic

C₂×Q₁₆×C₂₅₁

C₂²

yes

characteristic

C₂×C₁₆×C₂₅₁

C₂²

yes

characteristic

#107×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

Dic₁₀₀₄

—

Dic₁₀₀₄

—

Dic₁₀₀₄

—

Dic₁₀₀₄

D₄

yes

—

#16860800

—

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

characteristic

C₂₅₁⋊C₁₆

—

C₂×Q₈×C₂₅₁

—

Q₁₆×C₂₅₁

—

Q₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₁₆×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

D₈

yes

characteristic

C₂×Dic₂₅₁

D₈

yes

characteristic

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

#41

yes

characteristic

C₂²×C₂₅₁

C₂×D₈

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

Q₃₂

yes

complemented

C₂×C₂₅₁

#459

yes

characteristic

C₂×C₂₅₁

C₂×Q₃₂

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#2209

—

251

maximal, Sylow

C₂×Q₃₂

—

251

—

C₂×Q₃₂

—

251

—

#494

—

251

—

C₄×Q₁₆

—

251

—

C₄⋊C₁₆

—

251

—

#358

—

251

—

#358

D₂₅₁

yes

characteristic

C₂×Q₁₆

—

502

—

C₂×Q₁₆

—

251

—

C₂×Q₁₆

—

251

—

C₂×Q₁₆

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

—

502

—

C₄×Q₈

—

502

—

Q₃₂

—

502

—

Q₃₂

—

502

—

C₂×C₁₆

—

251

—

C₂×C₁₆

C₂×D₂₅₁

yes

characteristic

#107

C₂×D₂₅₁

yes

characteristic

#103

—

502

—

C₄×C₈

—

251

—

C₂×Q₈

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

1004

—

Q₁₆

—

502

—

Q₁₆

—

502

—

Q₁₆

—

502

—

100

Q₁₆

—

251

—

101

C₂×C₈

—

251

—

102

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

1004

—

105

C₄⋊C₄

—

1004

—

106

C₄⋊C₄

—

1004

—

107

C₄²

—

1004

—

108

C₄²

—

251

—

109

C₁₆

—

110

C₁₆

—

502

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

1004

—

114

Q₈

—

1004

—

115

Q₈

—

1004

—

116

Q₈

—

502

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

1004

—

120

C₂×C₄

—

1004

—

121

C₂×C₄

—

251

—

122

C₂×C₄

—

251

—

123

C₂×C₄

D₄×D₂₅₁

yes

characteristic

124

C₈

—

502

—

125

C₈

D₄×D₂₅₁

yes

characteristic

126

C₈

#16860810

yes

characteristic

127

C₂²

D₈×D₂₅₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

2008

—

131

C₄

—

1004

—

132

C₄

—

502

—

133

C₄

—

251

—

134

C₄

#22548688

yes

characteristic

135

C₄

#22548742

yes

characteristic

136

C₂

#30631881

yes

characteristic

137

C₂

Q₃₂×D₂₅₁

yes

characteristic

138

C₂

#30632126

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₅₁
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#358×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	#358	D₂₅₁
hypocenter	C₂₅₁	#2209
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₂₅₁

commutator, cosocle

C₂×C₈×C₂₅₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₅₁

Fitting

#358×C₂₅₁

C₂

socle, prime radical

C₂²×C₂₅₁

C₂×D₈

D₂₅₁

C₂₅₁

G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. Dic₂₅₁	Q₃₂	semidirect
69. C₂₅₁	#2209	semidirect

H ≅ G/N

type

65. Dic₂₅₁

Q₃₂

semidirect

69. C₂₅₁

#2209

semidirect

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

subgroup	index
C₂×Dic₂₀₀₈	2
C₂×#22548661	2
#30632065	2
#30631781	2
Dic₂₅₁⋊C₁₆	2
#30631490	2
#358×C₂₅₁	2
#2209	251

subgroup

index

C₂×Dic₂₀₀₈

Dic₂₅₁⋊C₁₆

#358×C₂₅₁

251

Subgroups — #44552414

Conjugacy classes of subgroups

Complements

Maximal subgroups