Group #44560897 — subgroups

Subgroups — #44560897

Order 32128.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#30632315	C₂	1	2	yes	maximal
2	#30632095	C₂	1	4	yes	maximal, complemented
3	#30632087	C₂	1	2	yes	maximal, complemented
4	#30632084	C₂	1	2	yes	maximal, complemented
5	#30631730	C₂	1	2	yes	maximal
6	C₂×#22548647	C₂	1	1	yes	maximal, characteristic
7	#30631642	C₂	1	1	yes	maximal, characteristic
8	#427×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#16860794	C₂²	1	1	yes	characteristic
10	#22548715	—	2	2	no	—
11	#22548714	C₂²	1	4	yes	—
12	#22548707	—	2	2	no	—
13	#22548706	C₂²	1	4	yes	—
14	#22548692	—	2	4	no	—
15	#22548674	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
17	#22548648	C₂²	1	4	yes	—
18	#22548647	C₂²	1	2	yes	—
19	#22548647	C₂²	1	2	yes	—
20	#22548646	C₂²	1	4	yes	—
21	#22548625	C₂²	1	2	yes	—
22	C₈×Dic₂₅₁	C₂²	1	2	yes	—
23	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
24	#117×C₂₅₁	C₂²	1	2	yes	—
25	#107×C₂₅₁	C₂²	1	2	yes	—
26	#106×C₂₅₁	C₂²	1	1	yes	characteristic
27	#106×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₅₀₂	—	2	4	no	—
29	C₂²×Dic₂₅₁	—	2	2	no	—
30	#16860794	Q₈	1	4	yes	complemented
31	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₂₅₁	—	2	4	no	—
33	C₄⋊Dic₂₅₁	—	2	2	no	—
34	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
35	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
36	Dic₂₅₁⋊C₄	—	2	4	no	—
37	Dic₂₅₁⋊C₄	—	2	4	no	—
38	C₂²⋊Dic₂₅₁	D₄	1	2	yes	—
39	C₄×Dic₂₅₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₂₅₁	Q₈	1	4	yes	complemented
42	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
47	C₄²×C₂₅₁	—	2	2	no	—
48	Dic₅₀₂	—	8	4	no	—
49	C₂×Dic₂₅₁	—	2	4	no	—
50	C₂×Dic₂₅₁	—	8	2	no	—
51	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
52	C₂₅₁⋊C₈	C₂×Q₈	1	4	yes	—
53	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₁	—	2	2	no	—
55	C₂×C₄×C₂₅₁	—	2	2	no	—
56	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
60	C₈×C₂₅₁	C₂×Q₈	1	2	yes	—
61	C₈×C₂₅₁	C₂×Q₈	1	2	yes	—
62	C₂²×C₂₅₁	—	2	2	no	—
63	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₂₅₁	—	4	2	no	—
66	C₄×C₂₅₁	—	4	2	no	—
67	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
68	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
70	Dic₂₅₁	—	2	4	no	—
71	Dic₂₅₁	—	4	4	no	—
72	C₂×C₂₅₁	—	2	2	no	—
73	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₂₅₁	#577	1	2	yes	—
75	C₂₅₁	#3163	1	1	yes	characteristic, complemented, Sylow
76	#3163	—	251	1	no	maximal, Sylow
77	#544	—	251	2	no	—
78	#501	—	251	4	no	—
79	#499	—	251	2	no	—
80	#498	—	251	2	no	—
81	#428	—	251	2	no	—
82	#427	D₂₅₁	1	1	yes	characteristic
83	C₂×#106	—	251	1	no	—
84	#405	—	251	1	no	—
85	C₂×M₁₆	—	251	1	no	—
86	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
87	C₄⋊Q₈	—	502	2	no	—
88	C₄⋊Q₈	—	251	4	no	—
89	#125	—	502	2	no	—
90	#125	—	251	4	no	—
91	C₂²⋊Q₈	—	502	4	no	—
92	#117	—	251	1	no	—
93	#117	C₂×D₂₅₁	1	2	yes	—
94	C₂×C₄⋊C₄	—	251	2	no	—
95	#107	—	251	4	no	—
96	#107	C₂×D₂₅₁	1	2	yes	—
97	#106	—	251	2	no	—
98	#106	—	251	2	no	—
99	#106	—	251	4	no	—
100	#106	C₂×D₂₅₁	1	1	yes	characteristic
101	#106	C₂×D₂₅₁	1	1	yes	characteristic
102	#97	—	251	2	no	—
103	C₄×C₈	—	251	2	no	—
104	C₂×Q₈	—	502	4	no	—
105	C₂²×C₄	—	502	2	no	—
106	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
107	M₁₆	—	251	4	no	—
108	C₂×C₈	—	251	2	no	—
109	C₂×C₈	Dic₅₀₂	1	4	yes	complemented
110	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
112	C₄⋊C₄	—	502	4	no	—
113	C₄⋊C₄	—	502	4	no	—
114	C₄⋊C₄	—	502	4	no	—
115	C₄⋊C₄	—	502	2	no	—
116	C₄⋊C₄	—	251	4	no	—
117	C₄⋊C₄	—	251	2	no	—
118	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
119	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	251	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	251	2	no	—
124	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
125	Q₈	—	2008	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2008	2	no	—
129	C₂×C₄	—	502	4	no	—
130	C₂×C₄	—	251	4	no	—
131	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
133	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
134	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
135	C₈	—	251	4	no	—
136	C₈	C₂×Dic₅₀₂	1	2	yes	—
137	C₈	C₂×Dic₅₀₂	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#22548714	1	2	yes	—
140	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	1004	4	no	—
144	C₄	—	502	4	no	—
145	C₄	#22548714	1	2	yes	—
146	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
147	C₄	C₂²×Dic₅₀₂	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#22548714	1	1	yes	characteristic
150	C₂	#30632494	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632315

C₂

yes

maximal

#30632095

C₂

yes

maximal, complemented

#30632087

C₂

yes

maximal, complemented

#30632084

C₂

yes

maximal, complemented

#30631730

C₂

yes

maximal

C₂×#22548647

C₂

yes

maximal, characteristic

#30631642

C₂

yes

maximal, characteristic

#427×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

characteristic

#22548715

—

#22548714

C₂²

yes

—

#22548707

—

#22548706

C₂²

yes

—

#22548692

—

#22548674

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

#22548648

C₂²

yes

—

#22548647

C₂²

yes

—

#22548647

C₂²

yes

—

#22548646

C₂²

yes

—

#22548625

C₂²

yes

—

C₈×Dic₂₅₁

C₂²

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

#117×C₂₅₁

C₂²

yes

—

#107×C₂₅₁

C₂²

yes

—

#106×C₂₅₁

C₂²

yes

characteristic

#106×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

—

#16860794

Q₈

yes

complemented

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₄⋊Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

D₄

yes

—

C₄×Dic₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

Q₈

yes

complemented

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

C₂³

yes

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂×D₄

yes

—

C₂₅₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

C₂×Q₈

yes

—

C₈×C₂₅₁

C₂×Q₈

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₄⋊Q₈

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄⋊Q₈

yes

—

C₄×C₂₅₁

C₂²×Q₈

yes

characteristic

C₄×C₂₅₁

C₂²×Q₈

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₅₁

#577

yes

—

C₂₅₁

#3163

yes

characteristic, complemented, Sylow

#3163

—

251

maximal, Sylow

#544

—

251

—

#501

—

251

—

#499

—

251

—

#498

—

251

—

#428

—

251

—

#427

D₂₅₁

yes

characteristic

C₂×#106

—

251

—

#405

—

251

—

C₂×M₁₆

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

251

—

#125

—

502

—

#125

—

251

—

C₂²⋊Q₈

—

502

—

#117

—

251

—

#117

C₂×D₂₅₁

yes

—

C₂×C₄⋊C₄

—

251

—

#107

—

251

—

#107

C₂×D₂₅₁

yes

—

#106

—

251

—

#106

—

251

—

#106

—

251

—

100

#106

C₂×D₂₅₁

yes

characteristic

101

#106

C₂×D₂₅₁

yes

characteristic

102

#97

—

251

—

103

C₄×C₈

—

251

—

104

C₂×Q₈

—

502

—

105

C₂²×C₄

—

502

—

106

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

107

M₁₆

—

251

—

108

C₂×C₈

—

251

—

109

C₂×C₈

Dic₅₀₂

yes

complemented

110

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

111

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

112

C₄⋊C₄

—

502

—

113

C₄⋊C₄

—

502

—

114

C₄⋊C₄

—

502

—

115

C₄⋊C₄

—

502

—

116

C₄⋊C₄

—

251

—

117

C₄⋊C₄

—

251

—

118

C₄⋊C₄

C₂²×D₂₅₁

yes

—

119

C₄⋊C₄

C₂²×D₂₅₁

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

251

—

122

C₄²

—

123

C₄²

—

251

—

124

C₂³

D₄×D₂₅₁

yes

characteristic

125

Q₈

—

2008

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

2008

—

129

C₂×C₄

—

502

—

130

C₂×C₄

—

251

—

131

C₂×C₄

D₄×D₂₅₁

yes

characteristic

132

C₂×C₄

Q₈×D₂₅₁

yes

—

133

C₂×C₄

C₂×Dic₅₀₂

yes

—

134

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

135

C₈

—

251

—

136

C₈

C₂×Dic₅₀₂

yes

—

137

C₈

C₂×Dic₅₀₂

yes

—

138

C₂²

—

139

C₂²

#22548714

yes

—

140

C₂²

C₂×D₄×D₂₅₁

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

1004

—

144

C₄

—

502

—

145

C₄

#22548714

yes

—

146

C₄

C₂×Q₈×D₂₅₁

yes

characteristic

147

C₄

C₂²×Dic₅₀₂

yes

characteristic

148

C₂

—

149

C₂

C₂×#22548714

yes

characteristic

150

C₂

#30632494

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#427×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂³×C₂₅₁	C₂×D₄
hypercenter	#427	D₂₅₁
hypocenter	C₂₅₁	#3163
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

#427×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

prime radical

C₂³×C₂₅₁

C₂×D₄

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #30632095	C₂	semidirect
3. #30632087	C₂	semidirect
4. #30632084	C₂	semidirect
30. #16860794	Q₈	semidirect
41. C₂×C₈×C₂₅₁	Q₈	semidirect
75. C₂₅₁	#3163	semidirect
109. C₂×C₈	Dic₅₀₂	semidirect

H ≅ G/N

type

2. #30632095

C₂

semidirect

3. #30632087

C₂

semidirect

4. #30632084

C₂

semidirect

30. #16860794

Q₈

semidirect

41. C₂×C₈×C₂₅₁

Q₈

semidirect

75. C₂₅₁

#3163

semidirect

109. C₂×C₈

Dic₅₀₂

semidirect

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

subgroup	index
#30632315	2
#30632095	2
#30632087	2
#30632084	2
#30631730	2
C₂×#22548647	2
#30631642	2
#427×C₂₅₁	2
#3163	251

subgroup

index

#427×C₂₅₁

251

Subgroups — #44560897

Conjugacy classes of subgroups

Complements

Maximal subgroups