Group #44558632 — subgroups

Subgroups — #44558632

Order 32128.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#30631629	C₂	1	3	yes	maximal
2	#30631621	C₂	1	3	yes	maximal
3	#30631619	C₂	1	3	yes	maximal
4	#30631608	C₂	1	3	yes	maximal
5	#30631557	C₂	1	1	yes	maximal, characteristic
6	#30631555	C₂	1	1	yes	maximal, characteristic
7	#383×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
12	C₂×Dic₂₅₁⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₂₅₁⋊C₄	C₂²	1	3	yes	—
14	C₂₅₁⋊#95	C₂²	1	3	yes	—
15	C₂₅₁⋊#95	C₂²	1	3	yes	—
16	C₂₅₁⋊#95	C₂²	1	3	yes	—
17	C₂₅₁⋊#95	C₂²	1	3	yes	—
18	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
19	#95×C₂₅₁	C₂²	1	3	yes	—
20	#95×C₂₅₁	C₂²	1	3	yes	—
21	C₂×Dic₅₀₂	—	4	3	no	—
22	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
25	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
26	C₄⋊Dic₂₅₁	—	4	3	no	—
27	C₄⋊Dic₂₅₁	—	4	3	no	—
28	C₄⋊Dic₂₅₁	—	4	3	no	—
29	Dic₂₅₁⋊C₄	—	4	3	no	—
30	Dic₂₅₁⋊C₄	—	4	3	no	—
31	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
33	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
34	C₄²×C₂₅₁	Q₈	1	4	yes	complemented
35	Dic₅₀₂	—	4	4	no	—
36	C₂×Dic₂₅₁	—	2	3	no	—
37	C₂×Dic₂₅₁	—	2	3	no	—
38	C₂×Dic₂₅₁	—	2	3	no	—
39	C₂×Dic₂₅₁	—	2	3	no	—
40	C₂×Dic₂₅₁	—	2	3	no	—
41	C₂×Dic₂₅₁	—	2	3	no	—
42	C₂×Dic₂₅₁	—	2	6	no	—
43	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₂₅₁	—	2	3	no	—
45	C₂×C₄×C₂₅₁	—	2	3	no	—
46	C₂×C₄×C₂₅₁	—	2	3	no	—
47	C₂×C₄×C₂₅₁	—	2	3	no	—
48	C₂×C₄×C₂₅₁	—	2	6	no	—
49	C₂×C₄×C₂₅₁	C₂×Q₈	1	6	yes	—
50	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
52	C₂²×C₂₅₁	2₋¹⁺⁴	1	3	yes	—
53	C₄×C₂₅₁	—	2	6	no	—
54	C₄×C₂₅₁	—	4	1	no	—
55	C₄×C₂₅₁	—	4	3	no	—
56	Dic₂₅₁	—	4	3	no	—
57	Dic₂₅₁	—	4	1	no	—
58	Dic₂₅₁	—	4	1	no	—
59	Dic₂₅₁	—	4	3	no	—
60	C₂×C₂₅₁	#544	1	3	yes	—
61	C₂×C₂₅₁	#560	1	1	yes	characteristic
62	C₂×C₂₅₁	#563	1	3	yes	—
63	C₂₅₁	#2843	1	1	yes	characteristic, complemented, Sylow
64	#2843	—	251	1	no	maximal, Sylow
65	#400	—	251	3	no	—
66	#398	—	251	3	no	—
67	#398	—	251	3	no	—
68	#395	—	251	3	no	—
69	#384	—	251	1	no	—
70	#383	—	251	1	no	—
71	#383	D₂₅₁	1	1	yes	characteristic
72	C₂²×Q₈	—	251	1	no	—
73	C₂×C₄⋊C₄	—	251	3	no	—
74	C₂×C₄⋊C₄	—	251	3	no	—
75	C₂×C₄⋊C₄	—	251	3	no	—
76	C₂×C₄⋊C₄	—	251	3	no	—
77	C₂×C₄⋊C₄	—	251	3	no	—
78	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
79	#95	—	251	3	no	—
80	#95	—	251	3	no	—
81	#95	—	251	3	no	—
82	#95	—	251	3	no	—
83	#95	C₂×D₂₅₁	1	3	yes	—
84	#95	C₂×D₂₅₁	1	3	yes	—
85	C₂×Q₈	—	1004	3	no	—
86	C₂²×C₄	—	251	3	no	—
87	C₂²×C₄	—	251	1	no	—
88	C₂²×C₄	—	251	1	no	—
89	C₂²×C₄	—	251	3	no	—
90	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
91	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
93	C₄⋊C₄	—	1004	3	no	—
94	C₄⋊C₄	—	1004	3	no	—
95	C₄⋊C₄	—	1004	3	no	—
96	C₄⋊C₄	—	1004	3	no	—
97	C₄⋊C₄	—	1004	3	no	—
98	C₄²	Dic₅₀₂	1	4	yes	complemented
99	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
100	Q₈	—	1004	4	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	502	3	no	—
107	C₂×C₄	—	502	3	no	—
108	C₂×C₄	—	502	3	no	—
109	C₂×C₄	—	502	3	no	—
110	C₂×C₄	—	502	3	no	—
111	C₂×C₄	—	502	3	no	—
112	C₂×C₄	—	502	6	no	—
113	C₂×C₄	C₂×Dic₅₀₂	1	6	yes	—
114	C₂²	C₂²×Dic₅₀₂	1	1	yes	characteristic
115	C₂²	#22548761	1	3	yes	—
116	C₂²	#22548762	1	3	yes	—
117	C₄	—	2	6	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	3	no	—
120	C₄	—	1004	1	no	—
121	C₄	—	1004	1	no	—
122	C₄	—	1004	3	no	—
123	C₄	—	1004	3	no	—
124	C₂	#30632315	1	3	yes	—
125	C₂	#30632428	1	1	yes	characteristic
126	C₂	#30632439	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30631629

C₂

yes

maximal

#30631621

C₂

yes

maximal

#30631619

C₂

yes

maximal

#30631608

C₂

yes

maximal

#30631557

C₂

yes

maximal, characteristic

#30631555

C₂

yes

maximal, characteristic

#383×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#95×C₂₅₁

C₂²

yes

—

#95×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

C₂³

yes

characteristic

C₂²×Dic₂₅₁

C₂³

yes

characteristic

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

Q₈

yes

complemented

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂²×C₂₅₁

C₂²×Q₈

yes

characteristic

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#544

yes

—

C₂×C₂₅₁

#560

yes

characteristic

C₂×C₂₅₁

#563

yes

—

C₂₅₁

#2843

yes

characteristic, complemented, Sylow

#2843

—

251

maximal, Sylow

#400

—

251

—

#398

—

251

—

#398

—

251

—

#395

—

251

—

#384

—

251

—

#383

—

251

—

#383

D₂₅₁

yes

characteristic

C₂²×Q₈

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

—

#95

C₂×D₂₅₁

yes

—

C₂×Q₈

—

1004

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄²

Dic₅₀₂

yes

complemented

C₂³

C₂³×D₂₅₁

yes

characteristic

100

Q₈

—

1004

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

502

—

107

C₂×C₄

—

502

—

108

C₂×C₄

—

502

—

109

C₂×C₄

—

502

—

110

C₂×C₄

—

502

—

111

C₂×C₄

—

502

—

112

C₂×C₄

—

502

—

113

C₂×C₄

C₂×Dic₅₀₂

yes

—

114

C₂²

C₂²×Dic₅₀₂

yes

characteristic

115

C₂²

#22548761

yes

—

116

C₂²

#22548762

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

1004

—

121

C₄

—

1004

—

122

C₄

—

1004

—

123

C₄

—

1004

—

124

C₂

#30632315

yes

—

125

C₂

#30632428

yes

characteristic

126

C₂

#30632439

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#383×C₂₅₁	C₂
hypercenter	#383	D₂₅₁
hypocenter	C₂₅₁	#2843
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, prime radical, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#383×C₂₅₁

C₂

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
34. C₄²×C₂₅₁	Q₈	semidirect
63. C₂₅₁	#2843	semidirect
98. C₄²	Dic₅₀₂	semidirect

H ≅ G/N

type

34. C₄²×C₂₅₁

Q₈

semidirect

63. C₂₅₁

#2843

semidirect

98. C₄²

Dic₅₀₂

semidirect

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

subgroup	index
#30631629	2
#30631621	2
#30631619	2
#30631608	2
#30631557	2
#30631555	2
#383×C₂₅₁	2
#2843	251

subgroup

index

#383×C₂₅₁

251

Subgroups — #44558632

Conjugacy classes of subgroups

Complements

Maximal subgroups