Group #44559151 — subgroups

Subgroups — C₂×#30631793 #44559151

Order 32128.

480 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×Dic₁₀₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548680	C₂	1	2	yes	maximal
3	#30631793	C₂	1	8	yes	maximal, direct product factor
4	C₂×#22548638	C₂	1	2	yes	maximal
5	C₂×#22548625	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₂₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
8	C₂×Dic₁₀₀₄	C₄	1	8	yes	complemented
9	C₂×Dic₁₀₀₄	C₂²	1	4	yes	—
10	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	#22548680	—	2	4	no	—
12	#22548680	C₂²	1	8	yes	—
13	C₂×Dic₂₅₁⋊C₄	—	2	2	no	—
14	C₂×C₄×Dic₂₅₁	—	2	2	no	—
15	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
16	#22548638	C₂²	1	8	yes	—
17	#22548625	C₂²	1	4	yes	—
18	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
20	#106×C₂₅₁	C₂²	1	4	yes	—
21	C₂×Dic₅₀₂	—	2	2	no	—
22	C₂×Dic₅₀₂	—	2	4	no	—
23	C₂×Dic₅₀₂	C₂×C₄	1	8	yes	—
24	C₂×Dic₅₀₂	C₂³	1	4	yes	—
25	C₂²×Dic₂₅₁	—	2	2	no	—
26	C₂²×Dic₂₅₁	D₄	1	2	yes	—
27	Dic₁₀₀₄	C₂×C₄	1	16	yes	complemented
28	C₂×C₂₅₁⋊C₈	—	2	2	no	—
29	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
30	Dic₂₅₁⋊C₄	—	2	8	no	—
31	C₄×Dic₂₅₁	—	2	8	no	—
32	C₄×Dic₂₅₁	D₄	1	2	yes	—
33	C₄×Dic₂₅₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₅₁	—	2	2	no	—
35	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
38	C₄⋊C₄×C₂₅₁	—	2	2	no	—
39	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
40	Dic₅₀₂	—	2	8	no	—
41	Dic₅₀₂	—	2	4	no	—
42	Dic₅₀₂	C₂²×C₄	1	8	yes	—
43	C₂×Dic₂₅₁	—	2	8	no	—
44	C₂×Dic₂₅₁	—	2	4	no	—
45	C₂×Dic₂₅₁	—	2	4	no	—
46	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
47	C₂₅₁⋊C₈	—	2	2	no	—
48	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₅₁	—	2	4	no	—
50	C₂×C₄×C₂₅₁	—	4	4	no	—
51	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₁	#41	1	2	yes	—
54	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
55	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
57	C₂²×C₂₅₁	#137	1	4	yes	—
58	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₂₅₁	—	4	4	no	—
60	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
61	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
63	Dic₂₅₁	—	2	8	no	—
64	Dic₂₅₁	—	2	4	no	—
65	C₂×C₂₅₁	#441	1	4	yes	complemented
66	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
67	C₂×C₂₅₁	C₂×#137	1	2	yes	—
68	C₂₅₁	C₂×#441	1	1	yes	characteristic, complemented, Sylow
69	C₂×#441	—	251	1	no	maximal, Sylow
70	C₂²×Q₁₆	—	251	1	no	—
71	C₂×C₄×Q₈	—	251	2	no	—
72	#441	—	251	8	no	—
73	C₂×#106	D₂₅₁	1	1	yes	characteristic
74	C₂×#103	—	251	2	no	—
75	C₂×#97	—	251	1	no	—
76	C₂²×Q₈	—	251	2	no	—
77	C₂×Q₁₆	—	251	8	no	—
78	C₂×Q₁₆	—	251	4	no	—
79	C₂²×C₈	—	251	1	no	—
80	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
81	C₄×Q₈	—	502	4	no	—
82	C₄×Q₈	—	251	8	no	—
83	C₂×C₄⋊C₄	—	502	2	no	—
84	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
85	C₂×C₄²	—	502	2	no	—
86	C₂×C₄²	—	251	1	no	—
87	#106	C₂×D₂₅₁	1	4	yes	—
88	#103	—	251	8	no	—
89	#97	—	251	4	no	—
90	C₂×Q₈	—	502	2	no	—
91	C₂×Q₈	—	502	4	no	—
92	C₂×Q₈	—	251	8	no	—
93	C₂×Q₈	—	251	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	502	2	no	—
96	C₂²×C₄	—	251	2	no	—
97	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
98	Q₁₆	—	251	16	no	—
99	C₂×C₈	—	502	2	no	—
100	C₂×C₈	—	251	2	no	—
101	C₂×C₈	C₄×D₂₅₁	1	4	yes	—
102	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	502	8	no	—
105	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
106	C₄²	—	502	8	no	—
107	C₄²	—	251	2	no	—
108	C₄²	—	251	2	no	—
109	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
110	Q₈	—	502	4	no	—
111	Q₈	—	502	8	no	—
112	Q₈	—	251	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	502	4	no	—
116	C₂×C₄	—	502	4	no	—
117	C₂×C₄	—	502	8	no	—
118	C₂×C₄	—	251	4	no	—
119	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
120	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
121	C₂×C₄	#16860811	1	2	yes	—
122	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
123	C₈	—	502	2	no	—
124	C₈	C₂×C₄×D₂₅₁	1	4	yes	—
125	C₂²	#22548677	1	2	yes	—
126	C₂²	#22548744	1	4	yes	—
127	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	502	4	no	—
130	C₄	—	502	8	no	—
131	C₄	#22548677	1	2	yes	—
132	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
133	C₄	C₂×#16860811	1	1	yes	characteristic
134	C₂	#30631793	1	4	yes	direct product factor
135	C₂	C₂×#22548677	1	1	yes	characteristic
136	C₂	C₂×#22548744	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₀₀₄

C₂

yes

maximal, characteristic

C₂×#22548680

C₂

yes

maximal

#30631793

C₂

yes

maximal, direct product factor

C₂×#22548638

C₂

yes

maximal

C₂×#22548625

C₂

yes

maximal, characteristic

C₂×#106×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

—

C₂×Dic₁₀₀₄

C₄

yes

complemented

C₂×Dic₁₀₀₄

C₂²

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

#22548680

—

#22548680

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

—

C₂×C₄×Dic₂₅₁

—

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

#22548638

C₂²

yes

—

#22548625

C₂²

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

#106×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂×Dic₅₀₂

C₂³

yes

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

D₄

yes

—

Dic₁₀₀₄

C₂×C₄

yes

complemented

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

—

C₄×Dic₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

C₂³

yes

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

C₂²×C₄

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂×D₄

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

C₂²×C₄

yes

—

C₂²×C₂₅₁

C₄×D₄

yes

—

C₂²×C₂₅₁

#137

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

C₂³×C₄

yes

characteristic

C₄×C₂₅₁

C₂×#41

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#441

yes

complemented

C₂×C₂₅₁

C₂×C₄×D₄

yes

characteristic

C₂×C₂₅₁

C₂×#137

yes

—

C₂₅₁

C₂×#441

yes

characteristic, complemented, Sylow

C₂×#441

—

251

maximal, Sylow

C₂²×Q₁₆

—

251

—

C₂×C₄×Q₈

—

251

—

#441

—

251

—

C₂×#106

D₂₅₁

yes

characteristic

C₂×#103

—

251

—

C₂×#97

—

251

—

C₂²×Q₈

—

251

—

C₂×Q₁₆

—

251

—

C₂×Q₁₆

—

251

—

C₂²×C₈

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

C₄×Q₈

—

502

—

C₄×Q₈

—

251

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

—

502

—

C₂×C₄²

—

251

—

#106

C₂×D₂₅₁

yes

—

#103

—

251

—

#97

—

251

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂×Q₈

—

251

—

C₂²×C₄

—

C₂²×C₄

—

502

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

Q₁₆

—

251

—

C₂×C₈

—

502

—

100

C₂×C₈

—

251

—

101

C₂×C₈

C₄×D₂₅₁

yes

—

102

C₂×C₈

C₂²×D₂₅₁

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

502

—

105

C₄⋊C₄

C₂²×D₂₅₁

yes

—

106

C₄²

—

502

—

107

C₄²

—

251

—

108

C₄²

—

251

—

109

C₂³

D₄×D₂₅₁

yes

characteristic

110

Q₈

—

502

—

111

Q₈

—

502

—

112

Q₈

—

251

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

502

—

116

C₂×C₄

—

502

—

117

C₂×C₄

—

502

—

118

C₂×C₄

—

251

—

119

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

120

C₂×C₄

D₄×D₂₅₁

yes

characteristic

121

C₂×C₄

#16860811

yes

—

122

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

123

C₈

—

502

—

124

C₈

C₂×C₄×D₂₅₁

yes

—

125

C₂²

#22548677

yes

—

126

C₂²

#22548744

yes

—

127

C₂²

C₂×D₄×D₂₅₁

yes

characteristic

128

C₄

—

129

C₄

—

502

—

130

C₄

—

502

—

131

C₄

#22548677

yes

—

132

C₄

C₂²×C₄×D₂₅₁

yes

characteristic

133

C₄

C₂×#16860811

yes

characteristic

134

C₂

#30631793

yes

direct product factor

135

C₂

C₂×#22548677

yes

characteristic

136

C₂

C₂×#22548744

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#106×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂×D₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#106	D₂₅₁
hypocenter	C₂₅₁	C₂×#441
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₂₅₁

commutator

C₄×C₂₅₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

C₂×#106×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂×D₄

cosocle

C₂×C₄×C₂₅₁

C₂⁴

D₂₅₁

C₂₅₁

G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #30631793	C₂	direct (N × H)
8. C₂×Dic₁₀₀₄	C₄	semidirect
27. Dic₁₀₀₄	C₂×C₄	semidirect
65. C₂×C₂₅₁	#441	semidirect
68. C₂₅₁	C₂×#441	semidirect
134. C₂	#30631793	direct (N × H)

H ≅ G/N

type

3. #30631793

C₂

direct (N × H)

8. C₂×Dic₁₀₀₄

C₄

semidirect

27. Dic₁₀₀₄

C₂×C₄

semidirect

65. C₂×C₂₅₁

#441

semidirect

68. C₂₅₁

C₂×#441

semidirect

134. C₂

#30631793

direct (N × H)

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

subgroup	index
C₂²×Dic₁₀₀₄	2
C₂×#22548680	2
#30631793	2
C₂×#22548638	2
C₂×#22548625	2
C₂×#106×C₂₅₁	2
C₂×#441	251

subgroup

index

C₂²×Dic₁₀₀₄

C₂×#106×C₂₅₁

251

Subgroups — C₂×#30631793 #44559151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#30631793 #44559151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30631793 #44559151