Group #44559017 — subgroups

Subgroups — #44559017

Order 32128.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#16860810	C₂	1	1	yes	maximal, characteristic
2	#30631746	C₂	1	4	yes	maximal, complemented
3	#30631742	C₂	1	4	yes	maximal, complemented
4	#30631715	C₂	1	2	yes	maximal, complemented
5	#30631655	C₂	1	2	yes	maximal
6	C₂×#22548626	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860810	C₂²	1	1	yes	characteristic
9	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₅₀₂	C₄	1	2	yes	—
11	C₄×D₁₀₀₄	C₄	1	2	yes	—
12	C₄×C₂²⋊D₂₅₁	C₄	1	4	yes	—
13	#22548674	C₂²	1	1	yes	characteristic
14	#22548672	C₂²	1	2	yes	—
15	C₄²×D₂₅₁	C₂²	1	2	yes	—
16	Dic₂₅₁⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₂₅₁)⋊C₈	C₂²	1	8	yes	—
18	#22548626	C₂²	1	2	yes	—
19	#22548626	C₂²	1	2	yes	—
20	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
22	#97×C₂₅₁	C₂²	1	2	yes	—
23	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
24	#16860810	—	2	4	no	—
25	C₂×Dic₅₀₂	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
30	C₂×C₂₅₁⋊C₈	—	2	4	no	—
31	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
33	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
35	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
36	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
37	C₄×Dic₂₅₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₂₅₁	—	2	4	no	—
41	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
42	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
43	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
44	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
46	Dic₅₀₂	—	2	2	no	—
47	D₁₀₀₄	—	2	2	no	—
48	C₂²⋊D₂₅₁	—	2	4	no	—
49	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
50	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
51	C₄×D₂₅₁	—	2	4	no	—
52	C₄×D₂₅₁	—	4	4	no	—
53	C₂₅₁⋊C₈	—	2	4	no	—
54	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₁	—	4	2	no	—
56	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₅₁	#41	1	4	yes	—
60	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₂₅₁	—	2	4	no	—
62	C₂²×C₂₅₁	—	2	2	no	—
63	C₂²×C₂₅₁	#117	1	2	yes	—
64	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₂₅₁	—	2	2	no	—
66	C₂×D₂₅₁	—	2	4	no	—
67	C₄×C₂₅₁	—	2	2	no	—
68	C₄×C₂₅₁	—	2	2	no	—
69	C₄×C₂₅₁	#117	1	2	yes	—
70	C₄×C₂₅₁	C₂×#41	1	2	yes	—
71	Dic₂₅₁	—	2	2	no	—
72	Dic₂₅₁	—	2	4	no	—
73	C₂×C₂₅₁	—	2	2	no	—
74	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
75	C₂×C₂₅₁	#568	1	2	yes	—
76	D₂₅₁	—	4	2	no	—
77	C₂₅₁	#2923	1	1	yes	characteristic, complemented, Sylow
78	#2923	—	251	1	no	maximal, Sylow
79	C₄×#41	—	251	1	no	—
80	#433	—	251	4	no	—
81	#432	—	251	4	no	—
82	#424	—	251	2	no	—
83	#408	—	251	2	no	—
84	C₄×M₁₆	D₂₅₁	1	1	yes	characteristic, complemented
85	C₂×#97	—	251	1	no	—
86	C₂×#41	—	251	1	no	—
87	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
88	C₂²×C₈	—	251	2	no	—
89	C₄×Q₈	—	251	2	no	—
90	C₄×D₄	—	251	4	no	—
91	C₄×D₄	—	251	2	no	—
92	#117	—	251	1	no	—
93	#117	—	251	2	no	—
94	C₂×C₄²	—	251	2	no	—
95	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
96	C₄⋊C₈	—	251	8	no	—
97	C₂²⋊C₈	—	251	8	no	—
98	#97	—	251	2	no	—
99	#97	—	251	2	no	—
100	#97	C₂×D₂₅₁	1	2	yes	—
101	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
102	#41	—	502	4	no	—
103	C₂×Q₈	—	251	1	no	—
104	C₂×D₄	—	251	2	no	—
105	C₂×D₄	—	251	1	no	—
106	C₂²×C₄	—	251	4	no	—
107	C₂²×C₄	—	251	2	no	—
108	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	502	4	no	—
112	C₂×C₈	—	251	4	no	—
113	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
114	C₄⋊C₄	—	251	4	no	—
115	C₄⋊C₄	—	251	2	no	—
116	C₂²⋊C₄	—	251	2	no	—
117	C₂²⋊C₄	—	251	4	no	—
118	C₄²	—	251	4	no	—
119	C₄²	—	251	2	no	—
120	C₄²	C₄×D₂₅₁	1	2	yes	—
121	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
122	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
123	C₂³	—	251	2	no	—
124	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
125	Q₈	—	502	2	no	—
126	D₄	—	502	4	no	—
127	D₄	—	502	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	1004	4	no	—
130	C₂×C₄	—	502	4	no	—
131	C₂×C₄	—	251	4	no	—
132	C₂×C₄	—	251	2	no	—
133	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
136	C₂×C₄	#16860810	1	4	yes	—
137	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	502	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	502	4	no	—
142	C₂²	—	502	2	no	—
143	C₂²	#22548672	1	2	yes	—
144	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	502	4	no	—
148	C₄	—	502	2	no	—
149	C₄	#22548672	1	2	yes	—
150	C₄	C₂×#16860810	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	1004	2	no	—
153	C₂	C₂×#22548672	1	1	yes	characteristic
154	C₂	#30632453	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#16860810

C₂

yes

maximal, characteristic

#30631746

C₂

yes

maximal, complemented

#30631742

C₂

yes

maximal, complemented

#30631715

C₂

yes

maximal, complemented

#30631655

C₂

yes

maximal

C₂×#22548626

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860810

C₂²

yes

characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₄×Dic₅₀₂

C₄

yes

—

C₄×D₁₀₀₄

C₄

yes

—

C₄×C₂²⋊D₂₅₁

C₄

yes

—

#22548674

C₂²

yes

characteristic

#22548672

C₂²

yes

—

C₄²×D₂₅₁

C₂²

yes

—

Dic₂₅₁⋊C₈

C₂²

yes

—

(C₂×D₂₅₁)⋊C₈

C₂²

yes

—

#22548626

C₂²

yes

—

#22548626

C₂²

yes

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#97×C₂₅₁

C₂²

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

—

#16860810

—

C₂×Dic₅₀₂

C₂×C₄

yes

characteristic

C₂×D₁₀₀₄

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₂₅₁

C₂×C₄

yes

—

C₂×C₄×D₂₅₁

C₂×C₄

yes

—

C₂×C₄×D₂₅₁

C₂³

yes

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

Dic₂₅₁⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₂₅₁

C₂×C₄

yes

—

(C₂×D₂₅₁)⋊C₄

C₂×C₄

yes

—

C₄×Dic₂₅₁

C₂×C₄

yes

—

C₄×Dic₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

C₂²×C₄

yes

—

Dic₅₀₂

—

D₁₀₀₄

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

C₂²×C₄

yes

—

C₂×Dic₂₅₁

C₂²×C₄

yes

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#117

yes

—

C₂²×C₂₅₁

C₂³×C₄

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#117

yes

—

C₄×C₂₅₁

C₂×#41

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×#117

yes

characteristic

C₂×C₂₅₁

#568

yes

—

D₂₅₁

—

C₂₅₁

#2923

yes

characteristic, complemented, Sylow

#2923

—

251

maximal, Sylow

C₄×#41

—

251

—

#433

—

251

—

#432

—

251

—

#424

—

251

—

#408

—

251

—

C₄×M₁₆

D₂₅₁

yes

characteristic, complemented

C₂×#97

—

251

—

C₂×#41

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₄×Q₈

—

251

—

C₄×D₄

—

251

—

C₄×D₄

—

251

—

#117

—

251

—

#117

—

251

—

C₂×C₄²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

251

—

C₂²⋊C₈

—

251

—

#97

—

251

—

#97

—

251

—

100

#97

C₂×D₂₅₁

yes

—

101

C₄×C₈

C₂×D₂₅₁

yes

—

102

#41

—

502

—

103

C₂×Q₈

—

251

—

104

C₂×D₄

—

251

—

105

C₂×D₄

—

251

—

106

C₂²×C₄

—

251

—

107

C₂²×C₄

—

251

—

108

C₂²×C₄

C₄×D₂₅₁

yes

—

109

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

502

—

112

C₂×C₈

—

251

—

113

C₂×C₈

C₂²×D₂₅₁

yes

—

114

C₄⋊C₄

—

251

—

115

C₄⋊C₄

—

251

—

116

C₂²⋊C₄

—

251

—

117

C₂²⋊C₄

—

251

—

118

C₄²

—

251

—

119

C₄²

—

251

—

120

C₄²

C₄×D₂₅₁

yes

—

121

C₄²

C₂²×D₂₅₁

yes

characteristic

122

C₄²

C₂²×D₂₅₁

yes

characteristic

123

C₂³

—

251

—

124

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

125

Q₈

—

502

—

126

D₄

—

502

—

127

D₄

—

502

—

128

C₂×C₄

—

129

C₂×C₄

—

1004

—

130

C₂×C₄

—

502

—

131

C₂×C₄

—

251

—

132

C₂×C₄

—

251

—

133

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

134

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

135

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

136

C₂×C₄

#16860810

yes

—

137

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

138

C₈

—

139

C₈

—

502

—

140

C₂²

—

141

C₂²

—

502

—

142

C₂²

—

502

—

143

C₂²

#22548672

yes

—

144

C₂²

C₂²×C₄×D₂₅₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

502

—

148

C₄

—

502

—

149

C₄

#22548672

yes

—

150

C₄

C₂×#16860810

yes

—

151

C₂

—

152

C₂

—

1004

—

153

C₂

C₂×#22548672

yes

characteristic

154

C₂

#30632453

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₄×M₁₆×C₂₅₁	C₂
prime radical	C₂×#16860810	C₂²
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₄×M₁₆	D₂₅₁
hypocenter	C₂₅₁	#2923
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₅₁

commutator, socle

C₂²×C₂₅₁

C₂³×C₄

Fitting

C₄×M₁₆×C₂₅₁

C₂

prime radical

C₂×#16860810

C₂²

cosocle

C₂×C₄×C₂₅₁

C₂⁴

hypercenter

C₄×M₁₆

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #30631746	C₂	semidirect
3. #30631742	C₂	semidirect
4. #30631715	C₂	semidirect
6. C₂×#22548626	C₂	semidirect
7. C₄×M₁₆×C₂₅₁	C₂	semidirect
77. C₂₅₁	#2923	semidirect
84. C₄×M₁₆	D₂₅₁	semidirect

H ≅ G/N

type

2. #30631746

C₂

semidirect

3. #30631742

C₂

semidirect

4. #30631715

C₂

semidirect

6. C₂×#22548626

C₂

semidirect

7. C₄×M₁₆×C₂₅₁

C₂

semidirect

77. C₂₅₁

#2923

semidirect

84. C₄×M₁₆

D₂₅₁

semidirect

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

subgroup	index
C₄×#16860810	2
#30631746	2
#30631742	2
#30631715	2
#30631655	2
C₂×#22548626	2
C₄×M₁₆×C₂₅₁	2
#2923	251

subgroup

index

C₄×M₁₆×C₂₅₁

251

Subgroups — #44559017

Conjugacy classes of subgroups

Complements

Maximal subgroups