Group #44558554 — subgroups

Subgroups — #44558554

Order 32128.

380 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 115 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	#30631628	C₂	1	3	yes	maximal, complemented
2	#30631625	C₂	1	3	yes	maximal, complemented
3	#30631616	C₂	1	3	yes	maximal
4	#30631606	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631598	C₂	1	1	yes	maximal, characteristic
6	#30631558	C₂	1	3	yes	maximal
7	#395×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₅₁⋊C₄	C₂²	1	3	yes	—
10	C₂×Dic₂₅₁⋊C₄	C₂²	1	3	yes	—
11	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	3	yes	—
12	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	3	yes	—
13	C₂₅₁⋊#95	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	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
20	#95×C₂₅₁	C₂²	1	3	yes	—
21	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
24	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
25	C₂×C₄×D₂₅₁	—	4	3	no	—
26	Dic₂₅₁⋊C₄	—	4	3	no	—
27	Dic₂₅₁⋊C₄	—	4	3	no	—
28	(C₂×D₂₅₁)⋊C₄	—	2	6	no	—
29	(C₂×D₂₅₁)⋊C₄	—	4	3	no	—
30	C₂×Q₈×C₂₅₁	—	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	C₄⋊C₄×C₂₅₁	—	4	3	no	—
35	C₂²×D₂₅₁	—	2	3	no	—
36	C₂²×D₂₅₁	—	2	3	no	—
37	C₂²×D₂₅₁	#41	1	2	yes	—
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	3	no	—
43	C₂×Dic₂₅₁	—	2	6	no	—
44	C₄×D₂₅₁	—	4	4	no	—
45	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
46	Q₈×C₂₅₁	—	4	4	no	—
47	C₂×C₄×C₂₅₁	—	2	3	no	—
48	C₂×C₄×C₂₅₁	—	2	3	no	—
49	C₂×C₄×C₂₅₁	—	2	3	no	—
50	C₂×C₄×C₂₅₁	—	2	3	no	—
51	C₂×C₄×C₂₅₁	—	2	3	no	—
52	C₂×C₄×C₂₅₁	—	2	3	no	—
53	C₂×C₄×C₂₅₁	—	2	3	no	—
54	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
55	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
56	C₂²×C₂₅₁	2₋¹⁺⁴	1	3	yes	—
57	C₂×D₂₅₁	—	2	6	no	—
58	C₂×D₂₅₁	—	4	1	no	—
59	C₂×D₂₅₁	—	4	3	no	—
60	C₄×C₂₅₁	—	4	3	no	—
61	C₄×C₂₅₁	—	4	3	no	—
62	C₄×C₂₅₁	—	4	1	no	—
63	Dic₂₅₁	—	4	3	no	—
64	Dic₂₅₁	—	4	1	no	—
65	Dic₂₅₁	—	4	3	no	—
66	C₂×C₂₅₁	#539	1	3	yes	—
67	C₂×C₂₅₁	H(3,𝔽₄)	1	1	yes	characteristic
68	C₂×C₂₅₁	#563	1	3	yes	—
69	D₂₅₁	—	4	2	no	—
70	C₂₅₁	#2835	1	1	yes	characteristic, complemented, Sylow
71	#2835	—	251	1	no	maximal, Sylow
72	#400	—	251	3	no	—
73	#400	—	251	3	no	—
74	#397	—	251	3	no	—
75	#395	—	251	1	no	—
76	#395	D₂₅₁	1	1	yes	characteristic, complemented
77	#393	—	251	1	no	—
78	#385	—	251	3	no	—
79	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
80	C₂³×C₄	—	251	1	no	—
81	C₂×C₄⋊C₄	—	251	3	no	—
82	C₂×C₄⋊C₄	—	251	3	no	—
83	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
84	C₂×C₂²⋊C₄	—	251	3	no	—
85	C₂×C₂²⋊C₄	—	251	3	no	—
86	#95	—	251	3	no	—
87	#95	—	251	3	no	—
88	#95	—	251	3	no	—
89	#95	—	251	3	no	—
90	#95	—	251	3	no	—
91	#95	C₂×D₂₅₁	1	3	yes	—
92	C₂⁴	—	251	1	no	—
93	C₂×Q₈	—	4	3	no	—
94	C₂²×C₄	—	1004	3	no	—
95	C₂²×C₄	—	251	3	no	—
96	C₂²×C₄	—	251	3	no	—
97	C₂²×C₄	—	251	1	no	—
98	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
100	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
101	C₄⋊C₄	—	4	3	no	—
102	C₄⋊C₄	—	1004	3	no	—
103	C₄⋊C₄	—	1004	3	no	—
104	C₂²⋊C₄	—	1004	3	no	—
105	C₂²⋊C₄	—	502	6	no	—
106	C₂³	—	502	3	no	—
107	C₂³	—	502	3	no	—
108	C₂³	—	251	2	no	—
109	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	1004	4	no	—
119	C₂×C₄	—	502	3	no	—
120	C₂×C₄	—	502	3	no	—
121	C₂×C₄	—	502	3	no	—
122	C₂×C₄	—	502	3	no	—
123	C₂×C₄	—	502	3	no	—
124	C₂×C₄	—	502	6	no	—
125	C₂²	—	1004	1	no	—
126	C₂²	—	1004	3	no	—
127	C₂²	—	502	6	no	—
128	C₂²	#41×D₂₅₁	1	1	yes	characteristic
129	C₂²	#22548760	1	3	yes	—
130	C₂²	#22548763	1	3	yes	—
131	C₄	—	4	3	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	3	no	—
134	C₄	—	1004	3	no	—
135	C₄	—	1004	1	no	—
136	C₄	—	1004	3	no	—
137	C₂	—	1004	2	no	—
138	C₂	#30632283	1	3	yes	—
139	C₂	#30632430	1	1	yes	characteristic
140	C₂	#30632436	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30631628

C₂

yes

maximal, complemented

#30631625

C₂

yes

maximal, complemented

#30631616

C₂

yes

maximal

#30631606

C₂

yes

maximal, characteristic, complemented

#30631598

C₂

yes

maximal, characteristic

#30631558

C₂

yes

maximal

#395×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₅₁

C₂²

yes

characteristic

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂²×Q₈×C₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

#95×C₂₅₁

C₂²

yes

—

C₂³×D₂₅₁

C₂³

yes

characteristic

C₂²×Dic₂₅₁

C₂³

yes

characteristic

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₂×C₄×D₂₅₁

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

C₂×Q₈×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

#41

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₄×D₂₅₁

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂²×C₂₅₁

C₂×#41

yes

characteristic

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#539

yes

—

C₂×C₂₅₁

H(3,𝔽₄)

yes

characteristic

C₂×C₂₅₁

#563

yes

—

D₂₅₁

—

C₂₅₁

#2835

yes

characteristic, complemented, Sylow

#2835

—

251

maximal, Sylow

#400

—

251

—

#400

—

251

—

#397

—

251

—

#395

—

251

—

#395

D₂₅₁

yes

characteristic, complemented

#393

—

251

—

#385

—

251

—

C₂²×Q₈

C₂×D₂₅₁

yes

characteristic

C₂³×C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

—

C₂⁴

—

251

—

C₂×Q₈

—

C₂²×C₄

—

1004

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

—

100

C₂²×C₄

C₂²×D₂₅₁

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

1004

—

103

C₄⋊C₄

—

1004

—

104

C₂²⋊C₄

—

1004

—

105

C₂²⋊C₄

—

502

—

106

C₂³

—

502

—

107

C₂³

—

502

—

108

C₂³

—

251

—

109

C₂³

C₂³×D₂₅₁

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

1004

—

119

C₂×C₄

—

502

—

120

C₂×C₄

—

502

—

121

C₂×C₄

—

502

—

122

C₂×C₄

—

502

—

123

C₂×C₄

—

502

—

124

C₂×C₄

—

502

—

125

C₂²

—

1004

—

126

C₂²

—

1004

—

127

C₂²

—

502

—

128

C₂²

#41×D₂₅₁

yes

characteristic

129

C₂²

#22548760

yes

—

130

C₂²

#22548763

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

1004

—

135

C₄

—

1004

—

136

C₄

—

1004

—

137

C₂

—

1004

—

138

C₂

#30632283

yes

—

139

C₂

#30632430

yes

characteristic

140

C₂

#30632436

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#395×C₂₅₁	C₂
prime radical	C₂³×D₂₅₁	C₂³
hypercenter	#395	D₂₅₁
hypocenter	C₂₅₁	#2835
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#395×C₂₅₁

C₂

prime radical

C₂³×D₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
1. #30631628	C₂	semidirect
2. #30631625	C₂	semidirect
4. #30631606	C₂	semidirect
7. #395×C₂₅₁	C₂	semidirect
70. C₂₅₁	#2835	semidirect
76. #395	D₂₅₁	semidirect

H ≅ G/N

type

1. #30631628

C₂

semidirect

2. #30631625

C₂

semidirect

4. #30631606

C₂

semidirect

7. #395×C₂₅₁

C₂

semidirect

70. C₂₅₁

#2835

semidirect

76. #395

D₂₅₁

semidirect

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

subgroup	index
#30631628	2
#30631625	2
#30631616	2
#30631606	2
#30631598	2
#30631558	2
#395×C₂₅₁	2
#2835	251

subgroup

index

#395×C₂₅₁

251

Subgroups — #44558554

Conjugacy classes of subgroups

Complements

Maximal subgroups