Group #44563007 — subgroups

Subgroups — #44563007

Order 32128.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#30632416	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632039	C₂	1	3	yes	maximal, complemented
3	#30631969	C₂	1	3	yes	maximal, complemented
4	#30631892	C₂	1	3	yes	maximal
5	Q₈×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#16860798	C₂	1	3	yes	maximal
7	D₄×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16860798	C₂²	1	3	yes	—
9	#22548715	C₂²	1	3	yes	—
10	#22548706	—	2	3	no	—
11	Q₈×Dic₂₅₁	—	2	1	no	—
12	C₄×Dic₅₀₂	C₂²	1	3	yes	—
13	#22548646	C₂²	1	3	yes	—
14	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	3	yes	—
15	#22548638	C₂²	1	6	yes	—
16	#22548638	C₂²	1	3	yes	—
17	#22548636	C₂²	1	1	yes	characteristic
18	C₄×C₂₅₁⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₂₅₁	—	2	1	no	—
20	C₄⋊Q₈×C₂₅₁	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₂₅₁	—	2	3	no	—
22	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₅₁	C₂²	1	3	yes	—
24	C₂×Dic₅₀₂	C₂³	1	3	yes	—
25	#16860798	—	2	6	no	—
26	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
27	C₂×C₂₅₁⋊C₈	C₂³	1	3	yes	—
28	C₄⋊Dic₂₅₁	—	2	3	no	—
29	C₄⋊Dic₂₅₁	—	2	3	no	—
30	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
31	Dic₂₅₁⋊C₄	—	2	3	no	—
32	C₄×Dic₂₅₁	—	2	3	no	—
33	C₂×Q₈×C₂₅₁	—	2	3	no	—
34	C₂×Q₈×C₂₅₁	—	4	4	no	—
35	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
36	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₅₁	—	2	3	no	—
38	C₄⋊C₄×C₂₅₁	—	2	3	no	—
39	C₄⋊C₄×C₂₅₁	D₄	1	3	yes	—
40	C₄⋊C₄×C₂₅₁	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₂₅₁	—	2	3	no	—
42	C₄²×C₂₅₁	C₂³	1	3	yes	—
43	Dic₅₀₂	—	2	3	no	—
44	Dic₅₀₂	—	2	3	no	—
45	C₂×Dic₂₅₁	—	2	1	no	—
46	C₂×Dic₂₅₁	—	2	3	no	—
47	C₂₅₁⋊C₈	—	2	3	no	—
48	C₂₅₁⋊C₈	—	2	1	no	—
49	C₂³×C₂₅₁	—	2	1	no	—
50	Q₈×C₂₅₁	—	4	6	no	—
51	Q₈×C₂₅₁	SD₁₆	1	4	yes	complemented
52	D₄×C₂₅₁	—	2	1	no	—
53	D₄×C₂₅₁	#41	1	2	yes	—
54	C₂×C₄×C₂₅₁	—	2	3	no	—
55	C₂×C₄×C₂₅₁	—	4	6	no	—
56	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
58	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
59	C₂²×C₂₅₁	—	2	2	no	—
60	C₂²×C₂₅₁	—	4	1	no	—
61	C₂²×C₂₅₁	—	4	1	no	—
62	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₂₅₁	—	2	3	no	—
64	C₄×C₂₅₁	—	4	3	no	—
65	C₄×C₂₅₁	C₂×SD₁₆	1	6	yes	—
66	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
67	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₂₅₁	—	4	3	no	—
69	Dic₂₅₁	—	4	1	no	—
70	C₂×C₂₅₁	—	4	2	no	—
71	C₂×C₂₅₁	#548	1	1	yes	characteristic
72	C₂×C₂₅₁	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₂₅₁	#578	1	1	yes	characteristic
74	C₂₅₁	#3359	1	1	yes	characteristic, complemented, Sylow
75	#3359	—	251	1	no	maximal, Sylow
76	#557	—	251	1	no	—
77	D₄×Q₈	D₂₅₁	1	1	yes	characteristic
78	#487	—	251	3	no	—
79	#475	—	251	3	no	—
80	#461	—	251	3	no	—
81	C₈×Q₈	—	251	1	no	—
82	C₄×SD₁₆	—	251	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	251	3	no	—
85	C₄⋊Q₈	—	251	3	no	—
86	C₄⋊Q₈	C₂×D₂₅₁	1	3	yes	—
87	#125	—	502	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	502	1	no	—
90	C₄×Q₈	—	251	3	no	—
91	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
92	C₄×D₄	C₂×D₂₅₁	1	3	yes	—
93	#106	—	251	3	no	—
94	C₄⋊C₈	—	251	3	no	—
95	#103	—	251	3	no	—
96	#103	—	251	6	no	—
97	#102	—	251	1	no	—
98	C₄×C₈	—	251	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	251	3	no	—
102	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	502	6	no	—
106	C₂×C₈	—	251	1	no	—
107	C₂×C₈	—	251	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	502	3	no	—
110	C₄⋊C₄	—	502	3	no	—
111	C₄⋊C₄	—	502	3	no	—
112	C₄⋊C₄	—	251	1	no	—
113	C₄⋊C₄	C₂²⋊D₂₅₁	1	3	yes	—
114	C₄⋊C₄	C₂²×D₂₅₁	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	502	3	no	—
117	C₄²	C₂²×D₂₅₁	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	502	3	no	—
121	Q₈	—	502	3	no	—
122	Q₈	#16860798	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#16860811	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	502	1	no	—
128	C₂×C₄	—	502	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	3	yes	—
131	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
132	C₈	—	502	3	no	—
133	C₈	—	502	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	1004	1	no	—
141	C₄	—	1004	3	no	—
142	C₄	C₂×#16860798	1	6	yes	—
143	C₄	C₂×#16860811	1	1	yes	characteristic
144	C₄	#22548763	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#30632348	1	1	yes	characteristic
147	C₂	C₂²×#16860798	1	1	yes	characteristic
148	C₂	#30632502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632416

C₂

yes

maximal, characteristic, complemented

#30632039

C₂

yes

maximal, complemented

#30631969

C₂

yes

maximal, complemented

#30631892

C₂

yes

maximal

Q₈×C₂₅₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#16860798

C₂

yes

maximal

D₄×Q₈×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860798

C₂²

yes

—

#22548715

C₂²

yes

—

#22548706

—

Q₈×Dic₂₅₁

—

C₄×Dic₅₀₂

C₂²

yes

—

#22548646

C₂²

yes

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

#22548638

C₂²

yes

—

#22548638

C₂²

yes

—

#22548636

C₂²

yes

characteristic

C₄×C₂₅₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

—

C₂²⋊Q₈×C₂₅₁

—

C₄×Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×D₄×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

C₂³

yes

—

#16860798

—

C₂×C₂₅₁⋊C₈

C₂³

yes

characteristic

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂³

yes

characteristic

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

characteristic

C₂×D₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄⋊C₄×C₂₅₁

C₂³

yes

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

SD₁₆

yes

complemented

D₄×C₂₅₁

—

D₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×SD₁₆

yes

—

C₄×C₂₅₁

C₂×#41

yes

characteristic

C₄×C₂₅₁

2₋¹⁺⁴

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#548

yes

characteristic

C₂×C₂₅₁

C₂²×SD₁₆

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#3359

—

251

maximal, Sylow

#557

—

251

—

D₄×Q₈

D₂₅₁

yes

characteristic

#487

—

251

—

#475

—

251

—

#461

—

251

—

C₈×Q₈

—

251

—

C₄×SD₁₆

—

251

—

C₂²×Q₈

—

C₂×SD₁₆

—

251

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

—

#125

—

502

—

C₂²⋊Q₈

—

C₄×Q₈

—

502

—

C₄×Q₈

—

251

—

C₄×Q₈

C₂×D₂₅₁

yes

characteristic

C₄×D₄

C₂×D₂₅₁

yes

—

#106

—

251

—

C₄⋊C₈

—

251

—

#103

—

251

—

#103

—

251

—

#102

—

251

—

C₄×C₈

—

251

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

251

—

102

C₂×Q₈

C₂²⋊D₂₅₁

yes

characteristic

103

C₂×D₄

C₂²×D₂₅₁

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

502

—

106

C₂×C₈

—

251

—

107

C₂×C₈

—

251

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

502

—

110

C₄⋊C₄

—

502

—

111

C₄⋊C₄

—

502

—

112

C₄⋊C₄

—

251

—

113

C₄⋊C₄

C₂²⋊D₂₅₁

yes

—

114

C₄⋊C₄

C₂²×D₂₅₁

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

502

—

117

C₄²

C₂²×D₂₅₁

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

502

—

121

Q₈

—

502

—

122

Q₈

#16860798

yes

complemented

123

D₄

—

124

D₄

#16860811

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

502

—

128

C₂×C₄

—

502

—

129

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

130

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

131

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

132

C₈

—

502

—

133

C₈

—

502

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

1004

—

141

C₄

—

1004

—

142

C₄

C₂×#16860798

yes

—

143

C₄

C₂×#16860811

yes

characteristic

144

C₄

#22548763

yes

characteristic

145

C₂

—

146

C₂

#30632348

yes

characteristic

147

C₂

C₂²×#16860798

yes

characteristic

148

C₂

#30632502

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	D₄×Q₈×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×D₄×C₂₅₁	C₂³
hypercenter	D₄×Q₈	D₂₅₁
hypocenter	C₂₅₁	#3359
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

D₄×Q₈×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

prime radical

C₂×D₄×C₂₅₁

C₂³

hypercenter

D₄×Q₈

D₂₅₁

hypocenter

C₂₅₁

#3359

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #30632416	C₂	semidirect
2. #30632039	C₂	semidirect
3. #30631969	C₂	semidirect
5. Q₈×C₂₅₁⋊C₈	C₂	semidirect
51. Q₈×C₂₅₁	SD₁₆	semidirect
74. C₂₅₁	#3359	semidirect
122. Q₈	#16860798	semidirect

H ≅ G/N

type

1. #30632416

C₂

semidirect

2. #30632039

C₂

semidirect

3. #30631969

C₂

semidirect

5. Q₈×C₂₅₁⋊C₈

C₂

semidirect

51. Q₈×C₂₅₁

SD₁₆

semidirect

74. C₂₅₁

#3359

semidirect

122. Q₈

#16860798

semidirect

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

subgroup	index
#30632416	2
#30632039	2
#30631969	2
#30631892	2
Q₈×C₂₅₁⋊C₈	2
C₄×#16860798	2
D₄×Q₈×C₂₅₁	2
#3359	251

subgroup

index

Q₈×C₂₅₁⋊C₈

D₄×Q₈×C₂₅₁

251

Subgroups — #44563007

Conjugacy classes of subgroups

Complements

Maximal subgroups