Group #44550247 — subgroups

Subgroups — #44550247

Order 32128.

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

normal subgroups only

This group has 232 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	#30631725	C₂	1	1	yes	maximal, characteristic
2	#30631645	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#30631448	C₂	1	2	yes	maximal
5	C₂×#108×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
7	#22548674	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
9	#22548649	C₄	1	2	yes	complemented
10	#22548647	C₄	1	2	yes	complemented
11	#22548625	—	2	2	no	—
12	C₈×Dic₂₅₁	—	2	2	no	—
13	C₈×Dic₂₅₁	C₄	1	4	yes	complemented
14	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	#108×C₂₅₁	C₄	1	4	yes	complemented
17	C₂²×Dic₂₅₁	—	2	1	no	—
18	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
21	C₂×C₂₅₁⋊C₈	Q₈	1	2	yes	—
22	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
23	C₂²⋊Dic₂₅₁	—	2	2	no	—
24	C₄×Dic₂₅₁	—	2	2	no	—
25	C₄×Dic₂₅₁	—	2	1	no	—
26	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₅₁	—	2	2	no	—
29	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₂₅₁	—	2	2	no	—
31	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₅₁	—	2	4	no	—
36	C₂×Dic₂₅₁	—	2	2	no	—
37	C₂×Dic₂₅₁	—	2	4	no	—
38	C₂₅₁⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₂₅₁⋊C₈	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₅₁	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
47	C₈×C₂₅₁	—	2	4	no	—
48	C₈×C₂₅₁	C₄²	1	2	yes	—
49	C₈×C₂₅₁	C₄²	1	2	yes	—
50	C₂²×C₂₅₁	—	2	2	no	—
51	C₂²×C₂₅₁	#117	1	1	yes	characteristic
52	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₅₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
57	C₄×C₂₅₁	C₄×Q₈	1	1	yes	characteristic
58	Dic₂₅₁	—	2	4	no	—
59	Dic₂₅₁	—	4	4	no	—
60	C₂×C₂₅₁	—	2	1	no	—
61	C₂×C₂₅₁	—	2	1	no	—
62	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₂₅₁	#443	1	2	yes	—
64	C₂₅₁	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	251	1	no	maximal, Sylow
66	C₂×#108	D₂₅₁	1	1	yes	characteristic
67	#427	—	251	1	no	—
68	#405	—	251	2	no	—
69	C₂×C₄×C₈	—	251	1	no	—
70	#339	—	251	2	no	—
71	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
72	C₂²×C₈	—	251	1	no	—
73	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
74	#117	—	251	2	no	—
75	C₂×C₄²	—	251	1	no	—
76	#108	Dic₂₅₁	1	4	yes	complemented
77	#107	—	251	2	no	—
78	#106	—	251	2	no	—
79	#97	—	502	2	no	—
80	C₄×C₈	—	502	2	no	—
81	C₄×C₈	—	251	4	no	—
82	C₂²×C₄	—	502	1	no	—
83	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	C₂×Dic₂₅₁	1	4	yes	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	251	2	no	—
88	C₂×C₈	—	251	1	no	—
89	C₂×C₈	—	251	1	no	—
90	C₂×C₈	—	251	2	no	—
91	C₂×C₈	C₄×D₂₅₁	1	2	yes	—
92	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
93	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
95	C₄⋊C₄	—	251	4	no	—
96	C₂²⋊C₄	—	502	2	no	—
97	C₄²	—	502	2	no	—
98	C₄²	—	502	1	no	—
99	C₄²	—	251	2	no	—
100	C₂³	#16860811	1	1	yes	characteristic
101	C₂×C₄	—	502	4	no	—
102	C₂×C₄	—	502	4	no	—
103	C₂×C₄	—	502	2	no	—
104	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₂₅₁	1	1	yes	characteristic
109	C₂×C₄	#16860809	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	—	251	2	no	—
112	C₈	—	251	2	no	—
113	C₈	C₄×Dic₂₅₁	1	2	yes	—
114	C₈	C₄×Dic₂₅₁	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#22548673	1	1	yes	characteristic
117	C₂²	#22548677	1	1	yes	characteristic
118	C₂²	Q₈×Dic₂₅₁	1	1	yes	characteristic
119	C₄	—	1004	4	no	—
120	C₄	—	502	4	no	—
121	C₄	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
123	C₄	D₄×Dic₂₅₁	1	1	yes	characteristic
124	C₄	#22548680	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₂₅₁	1	1	yes	characteristic
128	C₂	#30631803	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30631725

C₂

yes

maximal, characteristic

#30631645

C₂

yes

maximal

C₂×C₈×Dic₂₅₁

C₂

yes

maximal, characteristic

#30631448

C₂

yes

maximal

C₂×#108×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

#22548674

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

#22548649

C₄

yes

complemented

#22548647

C₄

yes

complemented

#22548625

—

C₈×Dic₂₅₁

—

C₈×Dic₂₅₁

C₄

yes

complemented

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

#108×C₂₅₁

C₄

yes

complemented

C₂²×Dic₂₅₁

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₂×C₂₅₁⋊C₈

Q₈

yes

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

C₄⋊C₄

yes

—

C₂₅₁⋊C₈

C₄⋊C₄

yes

—

C₂³×C₂₅₁

#41

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

C₄²

yes

—

C₈×C₂₅₁

C₄²

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#117

yes

characteristic

C₂²×C₂₅₁

C₄×D₄

yes

characteristic

C₂²×C₂₅₁

C₄×Q₈

yes

characteristic

C₄×C₂₅₁

C₂×C₄²

yes

characteristic

C₄×C₂₅₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₅₁

C₄×D₄

yes

characteristic

C₄×C₂₅₁

C₄×Q₈

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₅₁

#443

yes

—

C₂₅₁

#1772

yes

characteristic, complemented, Sylow

#1772

—

251

maximal, Sylow

C₂×#108

D₂₅₁

yes

characteristic

#427

—

251

—

#405

—

251

—

C₂×C₄×C₈

—

251

—

#339

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

#117

—

251

—

C₂×C₄²

—

251

—

#108

Dic₂₅₁

yes

complemented

#107

—

251

—

#106

—

251

—

#97

—

502

—

C₄×C₈

—

502

—

C₄×C₈

—

251

—

C₂²×C₄

—

502

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₂₅₁

yes

—

C₂×C₈

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

C₄×D₂₅₁

yes

—

C₂×C₈

C₄×D₂₅₁

yes

characteristic

C₂×C₈

C₄×D₂₅₁

yes

characteristic

C₂×C₈

C₂×Dic₂₅₁

yes

—

C₄⋊C₄

—

251

—

C₂²⋊C₄

—

502

—

C₄²

—

502

—

C₄²

—

502

—

C₄²

—

251

—

100

C₂³

#16860811

yes

characteristic

101

C₂×C₄

—

502

—

102

C₂×C₄

—

502

—

103

C₂×C₄

—

502

—

104

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

106

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

107

C₂×C₄

D₄×D₂₅₁

yes

characteristic

108

C₂×C₄

Q₈×D₂₅₁

yes

characteristic

109

C₂×C₄

#16860809

yes

characteristic

110

C₈

—

111

C₈

—

251

—

112

C₈

—

251

—

113

C₈

C₄×Dic₂₅₁

yes

—

114

C₈

C₄×Dic₂₅₁

yes

—

115

C₂²

—

116

C₂²

#22548673

yes

characteristic

117

C₂²

#22548677

yes

characteristic

118

C₂²

Q₈×Dic₂₅₁

yes

characteristic

119

C₄

—

1004

—

120

C₄

—

502

—

121

C₄

C₂×C₄×Dic₂₅₁

yes

characteristic

122

C₄

C₄⋊C₄×D₂₅₁

yes

characteristic

123

C₄

D₄×Dic₂₅₁

yes

characteristic

124

C₄

#22548680

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₂₅₁

yes

characteristic

128

C₂

#30631803

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×#108×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₄×D₄
prime radical	C₂³×C₂₅₁	#41
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#108	D₂₅₁
hypocenter	C₂₅₁	#1772
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₅₁

commutator

C₄×C₂₅₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₂×#108×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₄×D₄

prime radical

C₂³×C₂₅₁

#41

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22548649	C₄	semidirect
10. #22548647	C₄	semidirect
13. C₈×Dic₂₅₁	C₄	semidirect
16. #108×C₂₅₁	C₄	semidirect
64. C₂₅₁	#1772	semidirect
76. #108	Dic₂₅₁	semidirect

H ≅ G/N

type

9. #22548649

C₄

semidirect

10. #22548647

C₄

semidirect

13. C₈×Dic₂₅₁

C₄

semidirect

16. #108×C₂₅₁

C₄

semidirect

64. C₂₅₁

#1772

semidirect

76. #108

Dic₂₅₁

semidirect

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

subgroup	index
#30631725	2
#30631645	2
C₂×C₈×Dic₂₅₁	2
#30631448	2
C₂×#108×C₂₅₁	2
#1772	251

subgroup

index

C₂×C₈×Dic₂₅₁

C₂×#108×C₂₅₁

251

Subgroups — #44550247

Conjugacy classes of subgroups

Complements

Maximal subgroups