Group #44550414 — subgroups

Subgroups — #44550414

Order 32128.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#22548723	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548650	C₂	1	2	yes	maximal, complemented
3	#30631669	C₂	1	2	yes	maximal
4	C₂₅₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16860810	C₂²	1	1	yes	characteristic
7	#22548723	—	2	4	no	—
8	C₂×#16860794	C₂²	1	2	yes	—
9	C₂×#16860793	C₄	1	2	yes	—
10	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₅₁	C₄	1	2	yes	—
12	#22548650	—	2	4	no	—
13	#22548633	—	2	2	no	—
14	#22548631	—	2	2	no	—
15	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
17	#16860810	—	2	2	no	—
18	#16860810	—	2	2	no	—
19	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
20	C₂×D₁₀₀₄	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₅₁	Q₈	1	2	yes	—
22	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
23	#16860794	—	2	2	no	—
24	#16860794	—	2	2	no	—
25	#16860793	—	2	4	no	—
26	C₂×C₂₅₁⋊C₈	—	2	2	no	—
27	C₂×C₂₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂₅₁⋊C₈	D₄	1	4	yes	—
30	C₈×D₂₅₁	—	2	4	no	—
31	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₅₁	—	2	2	no	—
33	M₁₆×C₂₅₁	—	2	2	no	—
34	M₁₆×C₂₅₁	—	2	2	no	—
35	C₂×C₈×C₂₅₁	—	2	2	no	—
36	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
37	C₂²×D₂₅₁	C₄⋊C₄	1	2	yes	—
38	Dic₅₀₂	—	2	2	no	—
39	D₁₀₀₄	—	2	2	no	—
40	C₂²⋊D₂₅₁	—	2	4	no	—
41	C₂×Dic₂₅₁	C₄⋊C₄	1	2	yes	—
42	C₄×D₂₅₁	—	2	2	no	—
43	C₄×D₂₅₁	—	2	2	no	—
44	C₂₅₁⋊C₈	—	4	2	no	—
45	C₂₅₁⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₂₅₁⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₁	#41	1	2	yes	—
52	C₈×C₂₅₁	—	2	2	no	—
53	C₈×C₂₅₁	—	4	2	no	—
54	C₂²×C₂₅₁	—	4	1	no	—
55	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₂₅₁	—	2	2	no	—
58	C₂×D₂₅₁	—	4	2	no	—
59	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₅₁	#117	1	1	yes	characteristic
61	C₄×C₂₅₁	#121	1	2	yes	—
62	Dic₂₅₁	—	2	2	no	—
63	C₂×C₂₅₁	—	2	1	no	—
64	C₂×C₂₅₁	—	2	2	no	—
65	C₂×C₂₅₁	#380	1	1	yes	characteristic
66	D₂₅₁	—	4	2	no	—
67	C₂₅₁	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	251	1	no	maximal, Sylow
69	C₂×#131	—	251	1	no	—
70	C₂×#108	—	251	2	no	—
71	#413	—	251	2	no	—
72	#338	—	251	1	no	—
73	#338	D₂₅₁	1	1	yes	characteristic, complemented
74	C₂×#41	—	251	1	no	—
75	#131	—	502	4	no	—
76	C₂×M₁₆	—	251	2	no	—
77	C₂×M₁₆	—	251	2	no	—
78	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
79	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
80	C₂²×C₈	—	251	2	no	—
81	C₂²×C₈	—	251	1	no	—
82	#108	—	502	4	no	—
83	#101	—	502	2	no	—
84	#100	—	502	2	no	—
85	#41	—	502	2	no	—
86	#41	—	502	2	no	—
87	C₂×Q₈	—	251	1	no	—
88	C₂×D₄	—	251	2	no	—
89	C₂×D₄	—	251	1	no	—
90	C₂²×C₄	—	251	2	no	—
91	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	502	4	no	—
96	M₁₆	—	502	2	no	—
97	M₁₆	—	502	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	502	4	no	—
100	C₂×C₈	—	502	2	no	—
101	C₂×C₈	—	251	1	no	—
102	C₂×C₈	—	251	1	no	—
103	C₂×C₈	—	251	4	no	—
104	C₂×C₈	C₄×D₂₅₁	1	2	yes	—
105	C₂³	—	251	2	no	—
106	C₂³	Q₈×D₂₅₁	1	1	yes	characteristic
107	Q₈	—	502	2	no	—
108	D₄	—	502	4	no	—
109	D₄	—	502	2	no	—
110	C₂×C₄	—	502	2	no	—
111	C₂×C₄	—	502	2	no	—
112	C₂×C₄	—	251	2	no	—
113	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
114	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
115	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
116	C₂×C₄	#16860810	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	1004	2	no	—
120	C₈	—	251	2	no	—
121	C₈	—	251	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	1004	2	no	—
124	C₂²	—	502	2	no	—
125	C₂²	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
126	C₂²	#22548693	1	2	yes	—
127	C₄	—	502	2	no	—
128	C₄	C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
129	C₄	#22548672	1	1	yes	characteristic
130	C₄	#22548688	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	1004	2	no	—
134	C₂	#30631546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548723

C₂

yes

maximal, characteristic

C₂×#22548650

C₂

yes

maximal, complemented

#30631669

C₂

yes

maximal

C₂₅₁⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860810

C₂²

yes

characteristic

#22548723

—

C₂×#16860794

C₂²

yes

—

C₂×#16860793

C₄

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₅₁

C₄

yes

—

#22548650

—

#22548633

—

#22548631

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×M₁₆×C₂₅₁

C₂²

yes

characteristic

#16860810

—

#16860810

—

C₂×Dic₅₀₂

D₄

yes

characteristic

C₂×D₁₀₀₄

D₄

yes

characteristic

C₂×C₂²⋊D₂₅₁

Q₈

yes

—

C₂×C₄×D₂₅₁

C₂×C₄

yes

—

#16860794

—

#16860794

—

#16860793

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₈×D₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂²×D₂₅₁

C₄⋊C₄

yes

—

Dic₅₀₂

—

D₁₀₀₄

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

C₄⋊C₄

yes

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

C₂²⋊C₄

yes

—

C₂₅₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₂²⋊Q₈

yes

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

C₂×C₂²⋊C₄

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#380

yes

characteristic

D₂₅₁

—

C₂₅₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

251

maximal, Sylow

C₂×#131

—

251

—

C₂×#108

—

251

—

#413

—

251

—

#338

—

251

—

#338

D₂₅₁

yes

characteristic, complemented

C₂×#41

—

251

—

#131

—

502

—

C₂×M₁₆

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₂×M₁₆

C₂×D₂₅₁

yes

characteristic

C₂²×C₈

—

251

—

C₂²×C₈

—

251

—

#108

—

502

—

#101

—

502

—

#100

—

502

—

#41

—

502

—

#41

—

502

—

C₂×Q₈

—

251

—

C₂×D₄

—

251

—

C₂×D₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

502

—

M₁₆

—

502

—

M₁₆

—

502

—

C₂×C₈

—

C₂×C₈

—

502

—

100

C₂×C₈

—

502

—

101

C₂×C₈

—

251

—

102

C₂×C₈

—

251

—

103

C₂×C₈

—

251

—

104

C₂×C₈

C₄×D₂₅₁

yes

—

105

C₂³

—

251

—

106

C₂³

Q₈×D₂₅₁

yes

characteristic

107

Q₈

—

502

—

108

D₄

—

502

—

109

D₄

—

502

—

110

C₂×C₄

—

502

—

111

C₂×C₄

—

502

—

112

C₂×C₄

—

251

—

113

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

114

C₂×C₄

D₄×D₂₅₁

yes

—

115

C₂×C₄

D₄×D₂₅₁

yes

characteristic

116

C₂×C₄

#16860810

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

1004

—

120

C₈

—

251

—

121

C₈

—

251

—

122

C₂²

—

123

C₂²

—

1004

—

124

C₂²

—

502

—

125

C₂²

C₄⋊C₄×D₂₅₁

yes

characteristic

126

C₂²

#22548693

yes

—

127

C₄

—

502

—

128

C₄

C₂²⋊C₄×D₂₅₁

yes

characteristic

129

C₄

#22548672

yes

characteristic

130

C₄

#22548688

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

1004

—

134

C₂

#30631546

yes

characteristic

	subgroup	G/H
center	C₄	#22548672
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#338×C₂₅₁	C₂
socle	C₂×C₂₅₁	#380
prime radical	C₂×#16860810	C₂²
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#338	D₂₅₁
hypocenter	C₂₅₁	#1813
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#22548672

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#338×C₂₅₁

C₂

C₂×C₂₅₁

C₂²

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#22548650	C₂	semidirect
4. C₂₅₁⋊#338	C₂	semidirect
5. #338×C₂₅₁	C₂	semidirect
67. C₂₅₁	#1813	semidirect
73. #338	D₂₅₁	semidirect

H ≅ G/N

type

2. C₂×#22548650

C₂

semidirect

4. C₂₅₁⋊#338

C₂

semidirect

5. #338×C₂₅₁

C₂

semidirect

67. C₂₅₁

#1813

semidirect

73. #338

D₂₅₁

semidirect

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

subgroup	index
C₂×#22548723	2
C₂×#22548650	2
#30631669	2
C₂₅₁⋊#338	2
#338×C₂₅₁	2
#1813	251

subgroup

index

C₂×#22548723

C₂×#22548650

#30631669

C₂₅₁⋊#338

#338×C₂₅₁

#1813

251

Subgroups — #44550414

Conjugacy classes of subgroups

Complements

Maximal subgroups