Group #44559436 — subgroups

Subgroups — #44559436

Order 32128.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#30631759	C₂	1	8	yes	maximal, complemented
2	#30631744	C₂	1	2	yes	maximal, complemented
3	#30631654	C₂	1	4	yes	maximal, complemented
4	#532×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16860794	C₂²	1	4	yes	—
6	C₂²×C₂₅₁⋊C₈	C₂²	1	4	yes	—
7	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	—
8	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	—
9	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	8	yes	—
10	#22548626	C₂²	1	4	yes	—
11	C₂×#41×C₂₅₁	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₅₁	C₄	1	1	yes	characteristic
13	#127×C₂₅₁	C₄	1	1	yes	characteristic
14	#124×C₂₅₁	C₄	1	2	yes	—
15	#121×C₂₅₁	C₄	1	4	yes	—
16	C₄×D₄×C₂₅₁	C₂²	1	4	yes	—
17	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
18	#16860794	—	4	4	no	—
19	C₂×C₂₅₁⋊C₈	—	4	4	no	—
20	C₂×C₂₅₁⋊C₈	C₂³	1	8	yes	—
21	#41×C₂₅₁	—	4	2	no	—
22	#41×C₂₅₁	—	4	2	no	—
23	C₂×Q₈×C₂₅₁	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	—
26	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
30	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	8	yes	—
32	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₅₁	C₂³	1	2	yes	—
35	C₂₅₁⋊C₈	—	2	8	no	—
36	C₂³×C₂₅₁	C₂²×C₄	1	4	yes	—
37	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
38	Q₈×C₂₅₁	—	4	2	no	—
39	D₄×C₂₅₁	—	4	4	no	—
40	D₄×C₂₅₁	—	4	2	no	—
41	D₄×C₂₅₁	—	4	4	no	—
42	C₂×C₄×C₂₅₁	—	2	2	no	—
43	C₂×C₄×C₂₅₁	—	2	4	no	—
44	C₂×C₄×C₂₅₁	—	2	4	no	—
45	C₂×C₄×C₂₅₁	—	4	2	no	—
46	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
51	C₂²×C₂₅₁	—	2	1	no	—
52	C₂²×C₂₅₁	—	2	4	no	—
53	C₂²×C₂₅₁	—	2	2	no	—
54	C₂²×C₂₅₁	—	2	8	no	—
55	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₂₅₁	—	2	1	no	—
57	C₄×C₂₅₁	—	2	4	no	—
58	C₄×C₂₅₁	—	2	2	no	—
59	C₄×C₂₅₁	—	2	2	no	—
60	C₄×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₂₅₁	—	4	1	no	—
62	C₂×C₂₅₁	—	4	4	no	—
63	C₂×C₂₅₁	#518	1	1	yes	characteristic
64	C₂×C₂₅₁	#568	1	2	yes	—
65	C₂₅₁	#2980	1	1	yes	characteristic, complemented, Sylow
66	#2980	—	251	1	no	maximal, Sylow
67	#532	D₂₅₁	1	1	yes	characteristic
68	#435	—	251	8	no	—
69	#432	—	251	2	no	—
70	#408	—	251	4	no	—
71	C₂×#41	C₂×D₂₅₁	1	2	yes	—
72	C₂×M₁₆	—	251	4	no	—
73	C₂²×C₈	—	251	4	no	—
74	C₄⋊Q₈	Dic₂₅₁	1	1	yes	characteristic
75	#127	Dic₂₅₁	1	1	yes	characteristic
76	#124	Dic₂₅₁	1	2	yes	—
77	#121	Dic₂₅₁	1	4	yes	—
78	C₄×D₄	C₂×D₂₅₁	1	4	yes	—
79	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
80	C₄⋊C₈	—	251	4	no	—
81	C₂²⋊C₈	—	251	4	no	—
82	C₂²⋊C₈	—	251	8	no	—
83	#97	—	251	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₂₅₁	1	2	yes	—
87	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	—
88	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	—
89	C₂×D₄	C₂×Dic₂₅₁	1	2	yes	—
90	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
93	M₁₆	—	1004	4	no	—
94	C₂×C₈	—	1004	4	no	—
95	C₂×C₈	—	251	8	no	—
96	C₄⋊C₄	C₂×Dic₂₅₁	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₂₅₁	1	8	yes	—
98	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
99	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
100	C₄²	C₂²×D₂₅₁	1	2	yes	—
101	C₂³	C₂²×Dic₂₅₁	1	4	yes	—
102	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
113	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
114	C₂×C₄	C₂²×Dic₂₅₁	1	4	yes	—
115	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
116	C₈	—	502	8	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	4	no	—
121	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	4	no	—
126	C₄	#22548760	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#30632162	1	1	yes	characteristic
130	C₂	#30632454	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30631759

C₂

yes

maximal, complemented

#30631744

C₂

yes

maximal, complemented

#30631654

C₂

yes

maximal, complemented

#532×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₅₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₅₁⋊C₈)

C₂²

yes

—

#22548626

C₂²

yes

—

C₂×#41×C₂₅₁

C₂²

yes

—

C₄⋊Q₈×C₂₅₁

C₄

yes

characteristic

#127×C₂₅₁

C₄

yes

characteristic

#124×C₂₅₁

C₄

yes

—

#121×C₂₅₁

C₄

yes

—

C₄×D₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

#41×C₂₅₁

—

#41×C₂₅₁

—

C₂×Q₈×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂³×C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

H(5,𝔽₂)

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#518

yes

characteristic

C₂×C₂₅₁

#568

yes

—

C₂₅₁

#2980

yes

characteristic, complemented, Sylow

#2980

—

251

maximal, Sylow

#532

D₂₅₁

yes

characteristic

#435

—

251

—

#432

—

251

—

#408

—

251

—

C₂×#41

C₂×D₂₅₁

yes

—

C₂×M₁₆

—

251

—

C₂²×C₈

—

251

—

C₄⋊Q₈

Dic₂₅₁

yes

characteristic

#127

Dic₂₅₁

yes

characteristic

#124

Dic₂₅₁

yes

—

#121

Dic₂₅₁

yes

—

C₄×D₄

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

251

—

C₂²⋊C₈

—

251

—

C₂²⋊C₈

—

251

—

#97

—

251

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₂₅₁

yes

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂²×C₄

C₂×Dic₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

—

M₁₆

—

1004

—

C₂×C₈

—

1004

—

C₂×C₈

—

251

—

C₄⋊C₄

C₂×Dic₂₅₁

yes

—

C₂²⋊C₄

C₂×Dic₂₅₁

yes

—

C₄²

C₂×Dic₂₅₁

yes

characteristic

C₄²

C₂×Dic₂₅₁

yes

characteristic

100

C₄²

C₂²×D₂₅₁

yes

—

101

C₂³

C₂²×Dic₂₅₁

yes

—

102

C₂³

C₂²×Dic₂₅₁

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

112

C₂×C₄

C₂²×Dic₂₅₁

yes

—

113

C₂×C₄

C₂²×Dic₂₅₁

yes

—

114

C₂×C₄

C₂²×Dic₂₅₁

yes

—

115

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

116

C₈

—

502

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₂₅₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#22548760

yes

—

127

C₂

—

128

C₂

—

129

C₂

#30632162

yes

characteristic

130

C₂

#30632454

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting, prime radical	#532×C₂₅₁	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#532	D₂₅₁
hypocenter	C₂₅₁	#2980
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₅₁

commutator, socle

C₂²×C₂₅₁

C₂³×C₄

Fitting, prime radical

#532×C₂₅₁

C₂

cosocle

C₂×C₄×C₂₅₁

C₂⁴

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #30631759	C₂	semidirect
2. #30631744	C₂	semidirect
3. #30631654	C₂	semidirect
65. C₂₅₁	#2980	semidirect

H ≅ G/N

type

1. #30631759

C₂

semidirect

2. #30631744

C₂

semidirect

3. #30631654

C₂

semidirect

65. C₂₅₁

#2980

semidirect

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

subgroup	index
#30631759	2
#30631744	2
#30631654	2
#532×C₂₅₁	2
#2980	251

subgroup

index

#532×C₂₅₁

251

Subgroups — #44559436

Conjugacy classes of subgroups

Complements

Maximal subgroups