Group #44550002 — subgroups

Subgroups — #44550002

Order 32128.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#30632063	C₂	1	1	yes	maximal, characteristic
2	#30632039	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632035	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632034	C₂	1	1	yes	maximal, characteristic, complemented
5	#30632033	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631409	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860795	—	2	2	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548712	C₂²	1	1	yes	characteristic
11	#22548638	—	2	2	no	—
12	#22548637	—	2	2	no	—
13	#22548636	—	2	2	no	—
14	#22548635	—	2	2	no	—
15	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₂₅₁	—	2	2	no	—
18	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	#127×C₂₅₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₀₂	—	2	1	no	—
22	C₂×D₁₀₀₄	—	2	1	no	—
23	C₂×D₁₀₀₄	—	2	1	no	—
24	C₂×D₁₀₀₄	—	2	1	no	—
25	#16860795	—	4	4	no	—
26	C₂×C₂₅₁⋊C₈	D₄	1	2	yes	—
27	C₂×C₂₅₁⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₂₅₁	—	2	1	no	—
29	C₄⋊Dic₂₅₁	—	2	1	no	—
30	C₂×Q₈×C₂₅₁	—	2	1	no	—
31	C₂×D₄×C₂₅₁	—	2	1	no	—
32	C₂×D₄×C₂₅₁	—	2	1	no	—
33	C₂×D₄×C₂₅₁	—	2	1	no	—
34	SD₁₆×C₂₅₁	—	4	4	no	—
35	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
36	C₄⋊C₄×C₂₅₁	—	2	1	no	—
37	C₄⋊C₄×C₂₅₁	—	2	1	no	—
38	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
39	C₂²×D₂₅₁	—	4	1	no	—
40	Dic₅₀₂	—	4	2	no	—
41	D₁₀₀₄	—	4	2	no	—
42	D₁₀₀₄	—	4	2	no	—
43	D₁₀₀₄	—	4	2	no	—
44	C₂×Dic₂₅₁	—	4	1	no	—
45	C₂₅₁⋊C₈	—	2	2	no	—
46	C₂₅₁⋊C₈	—	2	2	no	—
47	C₂³×C₂₅₁	—	4	1	no	—
48	Q₈×C₂₅₁	—	4	2	no	—
49	D₄×C₂₅₁	—	4	2	no	—
50	D₄×C₂₅₁	—	4	2	no	—
51	D₄×C₂₅₁	—	4	2	no	—
52	C₂×C₄×C₂₅₁	—	4	1	no	—
53	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₂₅₁	—	2	2	no	—
57	C₂²×C₂₅₁	—	8	1	no	—
58	C₂²×C₂₅₁	—	8	1	no	—
59	C₂²×C₂₅₁	—	8	1	no	—
60	C₂²×C₂₅₁	#127	1	1	yes	characteristic
61	C₂×D₂₅₁	—	8	1	no	—
62	C₂×D₂₅₁	—	8	1	no	—
63	C₂×D₂₅₁	—	8	1	no	—
64	C₄×C₂₅₁	—	8	1	no	—
65	C₄×C₂₅₁	#135	1	2	yes	—
66	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₂₅₁	—	8	1	no	—
69	C₂×C₂₅₁	—	16	1	no	—
70	C₂×C₂₅₁	#495	1	1	yes	characteristic
71	C₂×C₂₅₁	#496	1	1	yes	characteristic
72	C₂×C₂₅₁	#496	1	1	yes	characteristic
73	D₂₅₁	—	16	1	no	—
74	C₂₅₁	#1717	1	1	yes	characteristic, complemented, Sylow
75	#1717	—	251	1	no	maximal, Sylow
76	#493	—	251	1	no	—
77	#492	D₂₅₁	1	1	yes	characteristic, complemented
78	#487	—	251	1	no	—
79	#486	—	251	1	no	—
80	#486	—	251	1	no	—
81	#486	—	251	1	no	—
82	#322	—	251	1	no	—
83	C₂×SD₁₆	—	2	2	no	—
84	C₂×D₈	—	502	2	no	—
85	C₄⋊Q₈	—	251	1	no	—
86	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
87	#127	—	251	1	no	—
88	#127	C₂×D₂₅₁	1	1	yes	characteristic
89	#103	—	502	2	no	—
90	#102	—	502	2	no	—
91	#102	—	502	2	no	—
92	#102	—	502	2	no	—
93	C₄×C₈	—	251	1	no	—
94	C₄×C₈	—	251	1	no	—
95	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	502	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	502	1	no	—
102	C₂×D₄	—	502	1	no	—
103	C₂×D₄	—	502	1	no	—
104	SD₁₆	—	4	4	no	—
105	D₈	—	1004	4	no	—
106	C₂×C₈	—	251	2	no	—
107	C₂×C₈	—	251	2	no	—
108	C₂×C₈	C₂²⋊D₂₅₁	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	502	1	no	—
112	C₄⋊C₄	—	502	1	no	—
113	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	1004	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	1004	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	1004	2	no	—
122	D₄	—	1004	2	no	—
123	D₄	—	1004	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	1004	1	no	—
126	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	502	2	no	—
131	C₈	—	502	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	2008	1	no	—
136	C₂²	—	2008	1	no	—
137	C₂²	—	2008	1	no	—
138	C₂²	#22548711	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	2008	1	no	—
141	C₄	#22548736	1	2	yes	—
142	C₄	#22548741	1	2	yes	—
143	C₄	#22548742	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	4016	1	no	—
146	C₂	#30632068	1	1	yes	characteristic
147	C₂	#30632073	1	1	yes	characteristic
148	C₂	#30632074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632063

C₂

yes

maximal, characteristic

#30632039

C₂

yes

maximal, characteristic, complemented

#30632035

C₂

yes

maximal, characteristic, complemented

#30632034

C₂

yes

maximal, characteristic, complemented

#30632033

C₂

yes

maximal, characteristic, complemented

#30631409

C₂

yes

maximal, characteristic, complemented

#492×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860795

—

#22548715

C₂²

yes

characteristic

#22548712

C₂²

yes

—

—

—

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

#127×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

#16860795

—

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

SD₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

D₄

yes

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#127

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#135

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

D₂₅₁

—

C₂₅₁

#1717

yes

characteristic, complemented, Sylow

#1717

—

251

maximal, Sylow

#493

—

251

—

#492

D₂₅₁

yes

characteristic, complemented

#487

—

251

—

#486

—

251

—

#486

—

251

—

#486

—

251

—

#322

—

251

—

C₂×SD₁₆

—

C₂×D₈

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

#127

—

251

—

#127

C₂×D₂₅₁

yes

characteristic

#103

—

502

—

#102

—

502

—

#102

—

502

—

#102

—

502

—

C₄×C₈

—

251

—

C₄×C₈

—

251

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

502

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

502

—

102

C₂×D₄

—

502

—

103

C₂×D₄

—

502

—

104

SD₁₆

—

105

D₈

—

1004

—

106

C₂×C₈

—

251

—

107

C₂×C₈

—

251

—

108

C₂×C₈

C₂²⋊D₂₅₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

502

—

112

C₄⋊C₄

—

502

—

113

C₄²

C₂²×D₂₅₁

yes

characteristic

114

C₂³

—

115

C₂³

—

1004

—

116

Q₈

—

117

Q₈

—

1004

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

1004

—

122

D₄

—

1004

—

123

D₄

—

1004

—

124

C₂×C₄

—

125

C₂×C₄

—

1004

—

126

C₂×C₄

D₄×D₂₅₁

yes

characteristic

127

C₂×C₄

D₄×D₂₅₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

129

C₈

—

130

C₈

—

502

—

131

C₈

—

502

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

2008

—

136

C₂²

—

2008

—

137

C₂²

—

2008

—

138

C₂²

#22548711

yes

characteristic

139

C₄

—

140

C₄

—

2008

—

141

C₄

#22548736

yes

—

142

C₄

#22548741

yes

—

143

C₄

#22548742

yes

—

144

C₂

—

145

C₂

—

4016

—

146

C₂

#30632068

yes

characteristic

147

C₂

#30632073

yes

characteristic

148

C₂

#30632074

yes

characteristic

	subgroup	G/H
center	C₂²	#22548711
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#492×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#127
prime radical	#30632063	C₂
hypercenter	#492	D₂₅₁
hypocenter	C₂₅₁	#1717
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22548711

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#492×C₂₅₁

C₂

C₂²×C₂₅₁

C₂

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #30632039	C₂	semidirect
3. #30632035	C₂	semidirect
4. #30632034	C₂	semidirect
5. #30632033	C₂	semidirect
6. #30631409	C₂	semidirect
7. #492×C₂₅₁	C₂	semidirect
74. C₂₅₁	#1717	semidirect
77. #492	D₂₅₁	semidirect

H ≅ G/N

type

2. #30632039

C₂

semidirect

3. #30632035

C₂

semidirect

4. #30632034

C₂

semidirect

5. #30632033

C₂

semidirect

6. #30631409

C₂

semidirect

7. #492×C₂₅₁

C₂

semidirect

74. C₂₅₁

#1717

semidirect

77. #492

D₂₅₁

semidirect

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

subgroup	index
#30632063	2
#30632039	2
#30632035	2
#30632034	2
#30632033	2
#30631409	2
#492×C₂₅₁	2
#1717	251

subgroup

index

#492×C₂₅₁

251

Subgroups — #44550002

Conjugacy classes of subgroups

Complements

Maximal subgroups