Group #44558710 — subgroups

Subgroups — #44558710

Order 32128.

924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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	C₂×#22548712	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548689	C₂	1	6	yes	maximal, complemented
3	#30631589	C₂	1	6	yes	maximal, complemented
4	#30631557	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₀₄	C₂²	1	6	yes	—
7	C₂²×C₂²⋊D₂₅₁	C₂²	1	4	yes	—
8	#22548712	—	2	4	no	—
9	#22548689	—	2	24	no	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	6	yes	—
11	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	12	yes	—
12	C₂³×C₄×C₂₅₁	C₂²	1	3	yes	—
13	C₂×C₄²×C₂₅₁	—	2	4	no	—
14	C₂×C₄²×C₂₅₁	C₂²	1	3	yes	—
15	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂³×D₂₅₁	C₂³	1	4	yes	—
17	C₂×D₁₀₀₄	—	2	12	no	—
18	C₂×D₁₀₀₄	—	2	12	no	—
19	C₂×D₁₀₀₄	—	2	12	no	—
20	C₂×C₂²⋊D₂₅₁	—	2	24	no	—
21	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
22	C₄⋊Dic₂₅₁	—	2	12	no	—
23	(C₂×D₂₅₁)⋊C₄	—	2	24	no	—
24	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₅₁	—	2	6	no	—
26	C₂²×C₄×C₂₅₁	—	2	6	no	—
27	C₂²×C₄×C₂₅₁	D₄	1	12	yes	complemented
28	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
29	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
30	C₄²×C₂₅₁	—	2	6	no	—
31	C₄²×C₂₅₁	D₄	1	4	yes	complemented
32	C₂²×D₂₅₁	—	2	12	no	—
33	C₂²×D₂₅₁	—	2	4	no	—
34	C₂²×D₂₅₁	—	2	12	no	—
35	D₁₀₀₄	—	4	24	no	—
36	C₂²⋊D₂₅₁	—	4	16	no	—
37	C₂×Dic₂₅₁	—	2	12	no	—
38	C₂³×C₂₅₁	—	2	1	no	—
39	C₂³×C₂₅₁	—	2	3	no	—
40	C₂³×C₂₅₁	C₂×D₄	1	6	yes	—
41	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₂₅₁	—	2	3	no	—
43	C₂×C₄×C₂₅₁	—	2	3	no	—
44	C₂×C₄×C₂₅₁	—	2	24	no	—
45	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
46	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
47	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₂₅₁	#41	1	6	yes	—
49	C₂²×C₂₅₁	—	2	6	no	—
50	C₂²×C₂₅₁	—	2	6	no	—
51	C₂²×C₂₅₁	#127	1	4	yes	complemented
52	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
53	C₂²×C₂₅₁	C₂²×D₄	1	3	yes	—
54	C₂²×C₂₅₁	C₂×#41	1	3	yes	—
55	C₂×D₂₅₁	—	4	12	no	—
56	C₂×D₂₅₁	—	4	4	no	—
57	C₂×D₂₅₁	—	4	12	no	—
58	C₄×C₂₅₁	—	2	6	no	—
59	C₄×C₂₅₁	#121	1	12	yes	complemented
60	Dic₂₅₁	—	4	4	no	—
61	C₂×C₂₅₁	—	2	4	no	—
62	C₂×C₂₅₁	C₂×#121	1	3	yes	—
63	C₂×C₂₅₁	C₂×#127	1	1	yes	characteristic
64	C₂×C₂₅₁	#532	1	3	yes	—
65	D₂₅₁	—	8	4	no	—
66	C₂₅₁	#2856	1	1	yes	characteristic, complemented, Sylow
67	#2856	—	251	1	no	maximal, Sylow
68	C₂×#127	—	251	1	no	—
69	C₂×#121	—	251	6	no	—
70	C₂²×C₄²	D₂₅₁	1	1	yes	characteristic, complemented
71	#390	—	251	6	no	—
72	#384	—	251	1	no	—
73	C₂²×D₄	—	251	4	no	—
74	C₂²×D₄	—	251	6	no	—
75	C₂³×C₄	C₂×D₂₅₁	1	3	yes	—
76	#127	—	502	4	no	—
77	#121	—	502	24	no	—
78	C₂×C₄⋊C₄	—	251	6	no	—
79	C₂×C₂²⋊C₄	—	251	12	no	—
80	C₂×C₄²	—	2	4	no	—
81	C₂×C₄²	C₂×D₂₅₁	1	3	yes	—
82	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
83	C₂⁴	—	251	4	no	—
84	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
85	C₂×D₄	—	502	24	no	—
86	C₂×D₄	—	502	12	no	—
87	C₂×D₄	—	502	12	no	—
88	C₂×D₄	—	502	12	no	—
89	C₂²×C₄	—	2	6	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	251	4	no	—
92	C₂²×C₄	D₁₀₀₄	1	12	yes	complemented
93	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
94	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
95	C₄⋊C₄	—	502	12	no	—
96	C₂²⋊C₄	—	502	24	no	—
97	C₄²	—	2	6	no	—
98	C₄²	C₂²⋊D₂₅₁	1	4	yes	complemented
99	C₂³	—	2	1	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	502	4	no	—
102	C₂³	—	502	12	no	—
103	C₂³	—	502	12	no	—
104	C₂³	C₂×D₁₀₀₄	1	6	yes	—
105	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
106	D₄	—	1004	16	no	—
107	D₄	—	1004	24	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	3	no	—
110	C₂×C₄	—	2	24	no	—
111	C₂×C₄	—	502	12	no	—
112	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	6	yes	—
113	C₂×C₄	C₂×D₁₀₀₄	1	6	yes	—
114	C₂×C₄	C₂×D₁₀₀₄	1	6	yes	—
115	C₂×C₄	#16860810	1	6	yes	—
116	C₂²	—	2	6	no	—
117	C₂²	—	2	6	no	—
118	C₂²	—	1004	4	no	—
119	C₂²	—	1004	12	no	—
120	C₂²	—	1004	12	no	—
121	C₂²	#22548712	1	4	yes	complemented
122	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
123	C₂²	C₂²×D₁₀₀₄	1	3	yes	—
124	C₂²	C₂×#16860810	1	3	yes	—
125	C₄	—	2	6	no	—
126	C₄	—	1004	4	no	—
127	C₄	#22548689	1	12	yes	complemented
128	C₂	—	2	4	no	—
129	C₂	—	2008	4	no	—
130	C₂	C₂×#22548689	1	3	yes	—
131	C₂	C₂×#22548712	1	1	yes	characteristic
132	C₂	#30632237	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548712

C₂

yes

maximal, characteristic, complemented

C₂×#22548689

C₂

yes

maximal, complemented

#30631589

C₂

yes

maximal, complemented

#30631557

C₂

yes

maximal, characteristic, complemented

C₂²×C₄²×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₀₀₄

C₂²

yes

—

C₂²×C₂²⋊D₂₅₁

C₂²

yes

—

#22548712

—

#22548689

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂³×C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂³×D₂₅₁

C₂³

yes

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×C₂²⋊D₂₅₁

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

C₂⁴×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

complemented

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

complemented

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

D₁₀₀₄

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂×D₄

yes

—

C₂³×C₂₅₁

C₂⁴

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

—

C₂×C₄×C₂₅₁

#41

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#127

yes

complemented

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₂²×C₂₅₁

C₂²×D₄

yes

—

C₂²×C₂₅₁

C₂×#41

yes

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#121

yes

complemented

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×#121

yes

—

C₂×C₂₅₁

C₂×#127

yes

characteristic

C₂×C₂₅₁

#532

yes

—

D₂₅₁

—

C₂₅₁

#2856

yes

characteristic, complemented, Sylow

#2856

—

251

maximal, Sylow

C₂×#127

—

251

—

C₂×#121

—

251

—

C₂²×C₄²

D₂₅₁

yes

characteristic, complemented

#390

—

251

—

#384

—

251

—

C₂²×D₄

—

251

—

C₂²×D₄

—

251

—

C₂³×C₄

C₂×D₂₅₁

yes

—

#127

—

502

—

#121

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₂⁴

—

251

—

C₂⁴

C₂²×D₂₅₁

yes

characteristic

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

251

—

C₂²×C₄

D₁₀₀₄

yes

complemented

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

502

—

C₂²⋊C₄

—

502

—

C₄²

—

C₄²

C₂²⋊D₂₅₁

yes

complemented

C₂³

—

100

C₂³

—

101

C₂³

—

502

—

102

C₂³

—

502

—

103

C₂³

—

502

—

104

C₂³

C₂×D₁₀₀₄

yes

—

105

C₂³

C₂³×D₂₅₁

yes

characteristic

106

D₄

—

1004

—

107

D₄

—

1004

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

502

—

112

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

113

C₂×C₄

C₂×D₁₀₀₄

yes

—

114

C₂×C₄

C₂×D₁₀₀₄

yes

—

115

C₂×C₄

#16860810

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

1004

—

119

C₂²

—

1004

—

120

C₂²

—

1004

—

121

C₂²

#22548712

yes

complemented

122

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

123

C₂²

C₂²×D₁₀₀₄

yes

—

124

C₂²

C₂×#16860810

yes

—

125

C₄

—

126

C₄

—

1004

—

127

C₄

#22548689

yes

complemented

128

C₂

—

129

C₂

—

2008

—

130

C₂

C₂×#22548689

yes

—

131

C₂

C₂×#22548712

yes

characteristic

132

C₂

#30632237

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	C₂²×C₄²×C₂₅₁	C₂
hypercenter	C₂²×C₄²	D₂₅₁
hypocenter	C₂₅₁	#2856
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

C₂²×C₄²×C₂₅₁

C₂

hypercenter

C₂²×C₄²

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
1. C₂×#22548712	C₂	semidirect
2. C₂×#22548689	C₂	semidirect
3. #30631589	C₂	semidirect
4. #30631557	C₂	semidirect
5. C₂²×C₄²×C₂₅₁	C₂	semidirect
27. C₂²×C₄×C₂₅₁	D₄	semidirect
31. C₄²×C₂₅₁	D₄	semidirect
51. C₂²×C₂₅₁	#127	semidirect
59. C₄×C₂₅₁	#121	semidirect
66. C₂₅₁	#2856	semidirect
70. C₂²×C₄²	D₂₅₁	semidirect
92. C₂²×C₄	D₁₀₀₄	semidirect
98. C₄²	C₂²⋊D₂₅₁	semidirect
121. C₂²	#22548712	semidirect
127. C₄	#22548689	semidirect

H ≅ G/N

type

1. C₂×#22548712

C₂

semidirect

2. C₂×#22548689

C₂

semidirect

3. #30631589

C₂

semidirect

4. #30631557

C₂

semidirect

5. C₂²×C₄²×C₂₅₁

C₂

semidirect

27. C₂²×C₄×C₂₅₁

D₄

semidirect

31. C₄²×C₂₅₁

D₄

semidirect

51. C₂²×C₂₅₁

#127

semidirect

59. C₄×C₂₅₁

#121

semidirect

66. C₂₅₁

#2856

semidirect

70. C₂²×C₄²

D₂₅₁

semidirect

92. C₂²×C₄

D₁₀₀₄

semidirect

98. C₄²

C₂²⋊D₂₅₁

semidirect

121. C₂²

#22548712

semidirect

127. C₄

#22548689

semidirect

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

subgroup	index
C₂×#22548712	2
C₂×#22548689	2
#30631589	2
#30631557	2
C₂²×C₄²×C₂₅₁	2
#2856	251

subgroup

index

C₂²×C₄²×C₂₅₁

251

Subgroups — #44558710

Conjugacy classes of subgroups

Complements

Maximal subgroups