Group #44560768 — subgroups

Subgroups — #44560768

Order 32128.

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

normal subgroups only

This group has 392 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	#30632237	C₂	1	1	yes	maximal, characteristic
2	#30632057	C₂	1	2	yes	maximal, complemented
3	#30632053	C₂	1	2	yes	maximal, complemented
4	#30632044	C₂	1	4	yes	maximal, complemented
5	#30631686	C₂	1	4	yes	maximal, complemented
6	C₄×#16860794	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860810	C₂²	1	2	yes	—
9	C₂×#16860794	C₂²	1	2	yes	—
10	#22548715	C₂²	1	1	yes	characteristic
11	#22548712	C₂²	1	1	yes	characteristic
12	#22548703	C₂²	1	2	yes	—
13	#22548689	—	2	2	no	—
14	C₄×D₁₀₀₄	—	2	2	no	—
15	#22548638	C₂²	1	8	yes	—
16	#22548635	C₂²	1	8	yes	—
17	#22548626	C₂²	1	2	yes	—
18	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
19	#125×C₂₅₁	—	2	2	no	—
20	#125×C₂₅₁	C₂²	1	2	yes	—
21	#125×C₂₅₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
25	#16860810	—	2	8	no	—
26	C₂×Dic₅₀₂	C₂³	1	2	yes	—
27	C₂×D₁₀₀₄	—	2	2	no	—
28	C₂×D₁₀₀₄	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
30	C₂×C₄×D₂₅₁	—	2	2	no	—
31	#16860794	—	2	4	no	—
32	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₂₅₁	—	2	2	no	—
34	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₂₅₁	—	2	2	no	—
36	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
37	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₅₁	—	2	4	no	—
39	C₄⋊C₄×C₂₅₁	—	2	4	no	—
40	C₄⋊C₄×C₂₅₁	—	2	2	no	—
41	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
42	C₄²×C₂₅₁	D₄	1	2	yes	—
43	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
44	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×D₂₅₁	—	2	2	no	—
46	Dic₅₀₂	—	2	4	no	—
47	D₁₀₀₄	—	2	4	no	—
48	D₁₀₀₄	—	4	4	no	—
49	C₂²⋊D₂₅₁	—	4	4	no	—
50	C₂×Dic₂₅₁	—	2	2	no	—
51	C₄×D₂₅₁	—	4	4	no	—
52	C₂₅₁⋊C₈	—	2	4	no	—
53	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₁	—	2	4	no	—
55	C₂×C₄×C₂₅₁	—	4	2	no	—
56	C₂×C₄×C₂₅₁	—	4	4	no	—
57	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₅₁	#41	1	4	yes	—
61	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₂₅₁	—	2	2	no	—
63	C₂²×C₂₅₁	#124	1	2	yes	—
64	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₂₅₁	—	4	2	no	—
66	C₂×D₂₅₁	—	4	4	no	—
67	C₄×C₂₅₁	—	2	2	no	—
68	C₄×C₂₅₁	—	2	2	no	—
69	C₄×C₂₅₁	—	4	4	no	—
70	C₄×C₂₅₁	#124	1	2	yes	—
71	C₄×C₂₅₁	C₂×#41	1	2	yes	—
72	Dic₂₅₁	—	4	2	no	—
73	C₂×C₂₅₁	—	2	2	no	—
74	C₂×C₂₅₁	C₂×#124	1	1	yes	characteristic
75	C₂×C₂₅₁	#575	1	2	yes	—
76	D₂₅₁	—	8	2	no	—
77	C₂₅₁	#3142	1	1	yes	characteristic, complemented, Sylow
78	#3142	—	251	1	no	maximal, Sylow
79	#532	—	251	1	no	—
80	C₂×#125	D₂₅₁	1	1	yes	characteristic, complemented
81	#491	—	251	2	no	—
82	#490	—	251	2	no	—
83	#488	—	251	4	no	—
84	#417	—	251	4	no	—
85	C₄×M₁₆	—	251	1	no	—
86	C₂×#41	—	251	2	no	—
87	C₂×M₁₆	—	251	2	no	—
88	C₄⋊Q₈	—	251	1	no	—
89	#127	—	251	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₂₅₁	1	2	yes	—
92	#125	C₂×D₂₅₁	1	2	yes	—
93	#124	—	251	2	no	—
94	#121	—	502	2	no	—
95	C₄×D₄	—	502	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
98	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
99	#103	—	251	8	no	—
100	#102	—	251	8	no	—
101	#97	—	251	2	no	—
102	C₄×C₈	—	251	2	no	—
103	#41	—	502	8	no	—
104	C₂×Q₈	—	251	2	no	—
105	C₂×D₄	—	502	2	no	—
106	C₂×D₄	—	502	2	no	—
107	C₂×D₄	—	251	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	502	2	no	—
110	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
112	M₁₆	—	502	4	no	—
113	C₂×C₈	—	251	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	502	2	no	—
118	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
119	C₂²⋊C₄	—	502	4	no	—
120	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
121	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
122	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
123	C₂³	—	502	2	no	—
124	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
125	Q₈	—	502	4	no	—
126	D₄	—	1004	4	no	—
127	D₄	—	1004	4	no	—
128	D₄	—	502	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	1004	4	no	—
133	C₂×C₄	—	502	2	no	—
134	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
137	C₂×C₄	#16860811	1	4	yes	—
138	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
139	C₈	—	502	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	1004	2	no	—
142	C₂²	—	1004	4	no	—
143	C₂²	#22548704	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	1004	2	no	—
149	C₄	#22548704	1	2	yes	—
150	C₄	C₂×#16860811	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	2008	2	no	—
153	C₂	C₂×#22548704	1	1	yes	characteristic
154	C₂	#30632485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632237

C₂

yes

maximal, characteristic

#30632057

C₂

yes

maximal, complemented

#30632053

C₂

yes

maximal, complemented

#30632044

C₂

yes

maximal, complemented

#30631686

C₂

yes

maximal, complemented

C₄×#16860794

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860810

C₂²

yes

—

C₂×#16860794

C₂²

yes

—

#22548715

C₂²

yes

characteristic

#22548712

C₂²

yes

characteristic

#22548703

C₂²

yes

—

#22548689

—

C₄×D₁₀₀₄

—

#22548638

C₂²

yes

—

#22548635

C₂²

yes

—

#22548626

C₂²

yes

—

C₄×C₂₅₁⋊C₈

C₂²

yes

—

#125×C₂₅₁

—

#125×C₂₅₁

C₂²

yes

—

#125×C₂₅₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#16860810

—

C₂×Dic₅₀₂

C₂³

yes

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

C₂³

yes

—

C₂×C₂²⋊D₂₅₁

—

C₂×C₄×D₂₅₁

—

#16860794

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#124

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#124

yes

—

C₄×C₂₅₁

C₂×#41

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×#124

yes

characteristic

C₂×C₂₅₁

#575

yes

—

D₂₅₁

—

C₂₅₁

#3142

yes

characteristic, complemented, Sylow

#3142

—

251

maximal, Sylow

#532

—

251

—

C₂×#125

D₂₅₁

yes

characteristic, complemented

#491

—

251

—

#490

—

251

—

#488

—

251

—

#417

—

251

—

C₄×M₁₆

—

251

—

C₂×#41

—

251

—

C₂×M₁₆

—

251

—

C₄⋊Q₈

—

251

—

#127

—

251

—

#125

—

#125

C₂×D₂₅₁

yes

—

#125

C₂×D₂₅₁

yes

—

#124

—

251

—

#121

—

502

—

C₄×D₄

—

502

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#103

—

251

—

100

#102

—

251

—

101

#97

—

251

—

102

C₄×C₈

—

251

—

103

#41

—

502

—

104

C₂×Q₈

—

251

—

105

C₂×D₄

—

502

—

106

C₂×D₄

—

502

—

107

C₂×D₄

—

251

—

108

C₂²×C₄

—

109

C₂²×C₄

—

502

—

110

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

111

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

112

M₁₆

—

502

—

113

C₂×C₈

—

251

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

502

—

118

C₄⋊C₄

C₂²×D₂₅₁

yes

—

119

C₂²⋊C₄

—

502

—

120

C₄²

C₂²⋊D₂₅₁

yes

—

121

C₄²

C₂²×D₂₅₁

yes

characteristic

122

C₄²

C₂²×D₂₅₁

yes

characteristic

123

C₂³

—

502

—

124

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

125

Q₈

—

502

—

126

D₄

—

1004

—

127

D₄

—

1004

—

128

D₄

—

502

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

1004

—

133

C₂×C₄

—

502

—

134

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

135

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

136

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

137

C₂×C₄

#16860811

yes

—

138

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

139

C₈

—

502

—

140

C₂²

—

141

C₂²

—

1004

—

142

C₂²

—

1004

—

143

C₂²

#22548704

yes

—

144

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

1004

—

149

C₄

#22548704

yes

—

150

C₄

C₂×#16860811

yes

—

151

C₂

—

152

C₂

—

2008

—

153

C₂

C₂×#22548704

yes

characteristic

154

C₂

#30632485

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#125×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	#30632237	C₂
hypercenter	C₂×#125	D₂₅₁
hypocenter	C₂₅₁	#3142
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

C₂×#125×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

C₂

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #30632057	C₂	semidirect
3. #30632053	C₂	semidirect
4. #30632044	C₂	semidirect
5. #30631686	C₂	semidirect
6. C₄×#16860794	C₂	semidirect
7. C₂×#125×C₂₅₁	C₂	semidirect
77. C₂₅₁	#3142	semidirect
80. C₂×#125	D₂₅₁	semidirect

H ≅ G/N

type

2. #30632057

C₂

semidirect

3. #30632053

C₂

semidirect

4. #30632044

C₂

semidirect

5. #30631686

C₂

semidirect

6. C₄×#16860794

C₂

semidirect

7. C₂×#125×C₂₅₁

C₂

semidirect

77. C₂₅₁

#3142

semidirect

80. C₂×#125

D₂₅₁

semidirect

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

subgroup	index
#30632237	2
#30632057	2
#30632053	2
#30632044	2
#30631686	2
C₄×#16860794	2
C₂×#125×C₂₅₁	2
#3142	251

subgroup

index

C₂×#125×C₂₅₁

251

Subgroups — #44560768

Conjugacy classes of subgroups

Complements

Maximal subgroups