Group #44558402 — subgroups

Subgroups — #44558402

Order 32128.

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

normal subgroups only

This group has 444 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	C₂×#22548694	C₂	1	2	yes	maximal
2	#30631627	C₂	1	4	yes	maximal, complemented
3	#30631612	C₂	1	4	yes	maximal
4	#30631608	C₂	1	4	yes	maximal, complemented
5	#381×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
7	#22548694	—	2	8	no	—
8	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
10	C₂×Dic₂₅₁⋊C₄	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₂₅₁	C₂²	1	4	yes	—
12	C₂₅₁⋊#95	C₂²	1	8	yes	—
13	C₂₅₁⋊#95	C₂²	1	4	yes	—
14	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
16	#95×C₂₅₁	C₂²	1	4	yes	—
17	C₂×Dic₅₀₂	—	2	4	no	—
18	C₂×Dic₅₀₂	—	4	4	no	—
19	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
20	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
21	C₄⋊Dic₂₅₁	—	2	4	no	—
22	C₄⋊Dic₂₅₁	—	4	4	no	—
23	Dic₂₅₁⋊C₄	—	2	8	no	—
24	C₂²⋊Dic₂₅₁	—	2	8	no	—
25	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₁	—	4	2	no	—
27	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
28	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
30	C₂²⋊C₄×C₂₅₁	—	4	2	no	—
31	Dic₅₀₂	—	4	8	no	—
32	C₂×Dic₂₅₁	—	2	4	no	—
33	C₂×Dic₂₅₁	—	2	4	no	—
34	C₂×Dic₂₅₁	—	2	8	no	—
35	C₂×Dic₂₅₁	—	2	8	no	—
36	C₂³×C₂₅₁	—	2	2	no	—
37	C₂³×C₂₅₁	—	2	4	no	—
38	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
39	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	—	2	4	no	—
41	C₂×C₄×C₂₅₁	—	2	4	no	—
42	C₂×C₄×C₂₅₁	—	2	8	no	—
43	C₂×C₄×C₂₅₁	—	4	4	no	—
44	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
45	C₂²×C₂₅₁	—	2	2	no	—
46	C₂²×C₂₅₁	—	2	4	no	—
47	C₂²×C₂₅₁	—	4	4	no	—
48	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
50	C₂²×C₂₅₁	2₋¹⁺⁴	1	4	yes	—
51	C₄×C₂₅₁	—	2	4	no	—
52	C₄×C₂₅₁	—	4	4	no	—
53	Dic₂₅₁	—	4	4	no	—
54	Dic₂₅₁	—	4	4	no	—
55	C₂×C₂₅₁	—	4	2	no	—
56	C₂×C₂₅₁	#534	1	1	yes	characteristic
57	C₂×C₂₅₁	#536	1	2	yes	—
58	C₂×C₂₅₁	#563	1	4	yes	—
59	C₂₅₁	#2817	1	1	yes	characteristic, complemented, Sylow
60	#2817	—	251	1	no	maximal, Sylow
61	C₂×C₂²⋊Q₈	—	251	2	no	—
62	#400	—	251	4	no	—
63	#397	—	251	4	no	—
64	#395	—	251	4	no	—
65	#381	D₂₅₁	1	1	yes	characteristic
66	C₂²×Q₈	—	251	2	no	—
67	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
68	C₂²⋊Q₈	—	502	8	no	—
69	C₂×C₄⋊C₄	—	251	4	no	—
70	C₂×C₄⋊C₄	—	251	4	no	—
71	C₂×C₄⋊C₄	—	251	2	no	—
72	C₂×C₂²⋊C₄	—	251	4	no	—
73	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
74	#95	—	251	8	no	—
75	#95	—	251	4	no	—
76	#95	C₂×D₂₅₁	1	4	yes	—
77	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
78	C₂×Q₈	—	1004	4	no	—
79	C₂×Q₈	—	502	4	no	—
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	—	251	4	no	—
82	C₂²×C₄	—	251	4	no	—
83	C₂²×C₄	D₁₀₀₄	1	4	yes	—
84	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
85	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
86	C₄⋊C₄	—	1004	4	no	—
87	C₄⋊C₄	—	502	8	no	—
88	C₄⋊C₄	—	502	4	no	—
89	C₂²⋊C₄	—	4	2	no	—
90	C₂²⋊C₄	—	502	8	no	—
91	C₂³	—	2	2	no	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×D₁₀₀₄	1	2	yes	—
94	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
95	Q₈	—	1004	8	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	502	4	no	—
101	C₂×C₄	—	502	4	no	—
102	C₂×C₄	—	502	8	no	—
103	C₂×C₄	—	502	8	no	—
104	C₂×C₄	C₂×D₁₀₀₄	1	4	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	4	4	no	—
108	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
109	C₂²	#22548760	1	2	yes	—
110	C₂²	#22548762	1	4	yes	—
111	C₄	—	2	4	no	—
112	C₄	—	4	4	no	—
113	C₄	—	1004	4	no	—
114	C₄	—	1004	4	no	—
115	C₂	—	4	2	no	—
116	C₂	#30632249	1	1	yes	characteristic
117	C₂	#30632263	1	2	yes	—
118	C₂	#30632440	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548694

C₂

yes

maximal

#30631627

C₂

yes

maximal, complemented

#30631612

C₂

yes

maximal

#30631608

C₂

yes

maximal, complemented

#381×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

—

#22548694

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₂₅₁

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂³×C₄×C₂₅₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₅₁

C₂²

yes

—

#95×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

C₂⁴×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²⋊C₄×C₂₅₁

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂×D₄

yes

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#534

yes

characteristic

C₂×C₂₅₁

#536

yes

—

C₂×C₂₅₁

#563

yes

—

C₂₅₁

#2817

yes

characteristic, complemented, Sylow

#2817

—

251

maximal, Sylow

C₂×C₂²⋊Q₈

—

251

—

#400

—

251

—

#397

—

251

—

#395

—

251

—

#381

D₂₅₁

yes

characteristic

C₂²×Q₈

—

251

—

C₂³×C₄

C₂×D₂₅₁

yes

characteristic

C₂²⋊Q₈

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

C₂×D₂₅₁

yes

—

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

—

C₂⁴

C₂²×D₂₅₁

yes

characteristic

C₂×Q₈

—

1004

—

C₂×Q₈

—

502

—

C₂²×C₄

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

D₁₀₀₄

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₂²⋊C₄

—

C₂²⋊C₄

—

502

—

C₂³

—

C₂³

—

C₂³

C₂×D₁₀₀₄

yes

—

C₂³

C₂³×D₂₅₁

yes

characteristic

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

502

—

101

C₂×C₄

—

502

—

102

C₂×C₄

—

502

—

103

C₂×C₄

—

502

—

104

C₂×C₄

C₂×D₁₀₀₄

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²×D₁₀₀₄

yes

characteristic

109

C₂²

#22548760

yes

—

110

C₂²

#22548762

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

1004

—

114

C₄

—

1004

—

115

C₂

—

116

C₂

#30632249

yes

characteristic

117

C₂

#30632263

yes

—

118

C₂

#30632440

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#381×C₂₅₁	C₂
prime radical	C₂⁴×C₂₅₁	C₂³
hypercenter	#381	D₂₅₁
hypocenter	C₂₅₁	#2817
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #381

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#381×C₂₅₁

C₂

prime radical

C₂⁴×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #381

N	H ≅ G/N	type
2. #30631627	C₂	semidirect
4. #30631608	C₂	semidirect
59. C₂₅₁	#2817	semidirect

H ≅ G/N

type

2. #30631627

C₂

semidirect

4. #30631608

C₂

semidirect

59. C₂₅₁

#2817

semidirect

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

subgroup	index
C₂×#22548694	2
#30631627	2
#30631612	2
#30631608	2
#381×C₂₅₁	2
#2817	251

subgroup

index

#381×C₂₅₁

251

Subgroups — #44558402

Conjugacy classes of subgroups

Complements

Maximal subgroups