Group #44548989 — subgroups

Subgroups — #44548989

Order 32128.

248 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 77 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	#30631744	C₂	1	2	yes	maximal, complemented
2	#30631427	C₂	1	4	yes	maximal, complemented
3	C₂×#124×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₂₅₁⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₂₅₁⋊C₈)	—	2	2	no	—
6	#22548626	C₂²	1	4	yes	complemented
7	C₂²×Q₈×C₂₅₁	C₄	1	1	yes	characteristic
8	C₂²×D₄×C₂₅₁	C₄	1	1	yes	characteristic
9	#124×C₂₅₁	—	2	2	no	—
10	#124×C₂₅₁	C₄	1	2	yes	—
11	#124×C₂₅₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
13	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₂₅₁⋊C₈	—	2	4	no	—
15	C₂⁴×C₂₅₁	—	2	1	no	—
16	C₂×Q₈×C₂₅₁	—	2	1	no	—
17	C₂×Q₈×C₂₅₁	—	4	2	no	—
18	C₂×Q₈×C₂₅₁	C₂×C₄	1	2	yes	—
19	C₂×D₄×C₂₅₁	—	2	1	no	—
20	C₂×D₄×C₂₅₁	—	4	2	no	—
21	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₅₁	—	2	1	no	—
23	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
25	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
26	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
27	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₅₁	D₄	1	2	yes	—
29	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₂₅₁⋊C₈	—	4	4	no	—
31	C₂³×C₂₅₁	—	2	2	no	—
32	C₂³×C₂₅₁	—	4	4	no	—
33	C₂³×C₂₅₁	—	4	2	no	—
34	C₂³×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₂₅₁	—	4	2	no	—
36	Q₈×C₂₅₁	—	4	2	no	—
37	D₄×C₂₅₁	—	4	2	no	—
38	D₄×C₂₅₁	—	4	2	no	—
39	C₂×C₄×C₂₅₁	—	2	2	no	—
40	C₂×C₄×C₂₅₁	—	2	2	no	—
41	C₂×C₄×C₂₅₁	—	4	2	no	—
42	C₂×C₄×C₂₅₁	—	4	2	no	—
43	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
47	C₂²×C₂₅₁	—	2	2	no	—
48	C₂²×C₂₅₁	—	4	2	no	—
49	C₂²×C₂₅₁	—	4	4	no	—
50	C₂²×C₂₅₁	—	8	4	no	—
51	C₂²×C₂₅₁	#100	1	2	yes	—
52	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₅₁	—	2	1	no	—
54	C₄×C₂₅₁	—	2	1	no	—
55	C₄×C₂₅₁	—	2	2	no	—
56	C₄×C₂₅₁	—	2	2	no	—
57	C₄×C₂₅₁	—	4	2	no	—
58	C₂×C₂₅₁	—	2	2	no	—
59	C₂×C₂₅₁	—	8	2	no	—
60	C₂×C₂₅₁	C₂×#100	1	1	yes	characteristic
61	C₂×C₂₅₁	#421	1	2	yes	—
62	C₂₅₁	#1533	1	1	yes	characteristic, complemented, Sylow
63	#1533	—	251	1	no	maximal, Sylow
64	C₂×#124	D₂₅₁	1	1	yes	characteristic
65	#432	—	251	2	no	—
66	#329	—	251	4	no	—
67	C₂²×Q₈	Dic₂₅₁	1	1	yes	characteristic
68	C₂²×D₄	Dic₂₅₁	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#124	Dic₂₅₁	1	2	yes	—
71	#124	C₂×D₂₅₁	1	2	yes	—
72	C₂×C₂²⋊C₄	—	2	2	no	—
73	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
74	C₄⋊C₈	—	502	2	no	—
75	C₂²⋊C₈	—	502	2	no	—
76	#97	—	251	4	no	—
77	C₂⁴	—	2	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂×Dic₂₅₁	1	2	yes	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	C₂×Dic₂₅₁	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
87	C₂×C₈	—	502	4	no	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
91	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
92	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	—	4	4	no	—
95	C₂³	—	4	2	no	—
96	C₂³	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
109	C₈	—	1004	4	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	4	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	8	4	no	—
114	C₂²	#22548632	1	2	yes	—
115	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₂	—	2	2	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#22548632	1	1	yes	characteristic
124	C₂	#30631704	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30631744

C₂

yes

maximal, complemented

#30631427

C₂

yes

maximal, complemented

C₂×#124×C₂₅₁

C₂

yes

maximal, characteristic

C₄⋊(C₂₅₁⋊C₈)

—

C₂²⋊(C₂₅₁⋊C₈)

—

#22548626

C₂²

yes

complemented

C₂²×Q₈×C₂₅₁

C₄

yes

characteristic

C₂²×D₄×C₂₅₁

C₄

yes

characteristic

#124×C₂₅₁

—

#124×C₂₅₁

C₄

yes

—

#124×C₂₅₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂×C₂₅₁⋊C₈

—

C₂⁴×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

characteristic

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#100

yes

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×#100

yes

characteristic

C₂×C₂₅₁

#421

yes

—

C₂₅₁

#1533

yes

characteristic, complemented, Sylow

#1533

—

251

maximal, Sylow

C₂×#124

D₂₅₁

yes

characteristic

#432

—

251

—

#329

—

251

—

C₂²×Q₈

Dic₂₅₁

yes

characteristic

C₂²×D₄

Dic₂₅₁

yes

characteristic

#124

—

#124

Dic₂₅₁

yes

—

#124

C₂×D₂₅₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

502

—

C₂²⋊C₈

—

502

—

#97

—

251

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₅₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

C₂×C₈

—

502

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₂₅₁

yes

characteristic

C₄²

C₂²⋊D₂₅₁

yes

—

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₂₅₁

yes

characteristic

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₂₅₁

yes

—

106

C₂×C₄

C₂²⋊Dic₂₅₁

yes

characteristic

107

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

109

C₈

—

1004

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#22548632

yes

—

115

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#22548632

yes

characteristic

124

C₂

#30631704

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₅₁
Fitting	C₂×#124×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₅₁	C₄
cosocle	C₄²×C₂₅₁	C₂³
hypercenter	C₂×#124	D₂₅₁
hypocenter	C₂₅₁	#1533
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₅₁

Fitting

C₂×#124×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₂₅₁

C₄

cosocle

C₄²×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #30631744	C₂	semidirect
2. #30631427	C₂	semidirect
6. #22548626	C₂²	semidirect
62. C₂₅₁	#1533	semidirect

H ≅ G/N

type

1. #30631744

C₂

semidirect

2. #30631427

C₂

semidirect

6. #22548626

C₂²

semidirect

62. C₂₅₁

#1533

semidirect

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

subgroup	index
#30631744	2
#30631427	2
C₂×#124×C₂₅₁	2
#1533	251

subgroup

index

#30631744

#30631427

C₂×#124×C₂₅₁

#1533

251

Subgroups — #44548989

Conjugacy classes of subgroups

Complements

Maximal subgroups