Group #12015720 — subgroups

Subgroups — #12015720

Order 1280.

342 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₅×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂×#64683	C₂	1	4	yes	maximal
3	#10600536	C₂	1	2	yes	maximal
4	C₂×C₅×#97	C₄	1	2	yes	—
5	C₂×C₅×#97	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#376	C₄	1	1	yes	characteristic
9	C₂×#3944	C₂²	1	4	yes	—
10	#64683	—	2	8	no	—
11	C₄⋊#672	—	2	4	no	—
12	#12901	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₅×#97	—	2	4	no	—
19	C₅×#97	C₈	1	4	yes	—
20	C₄×C₅×C₈	D₄	1	2	yes	—
21	C₄×C₅×C₈	D₄	1	4	yes	—
22	C₄×C₅×C₈	Q₈	1	4	yes	—
23	C₄×C₅×C₈	Q₈	1	2	yes	—
24	C₅×#95	—	2	2	no	—
25	#3944	—	2	8	no	—
26	C₂×#672	—	2	4	no	—
27	C₂×#364	—	5	4	no	—
28	#2150	—	5	2	no	—
29	#1726	D₅	1	1	yes	characteristic
30	C₂²×C₄×C₅	—	2	2	no	—
31	C₂²×C₄×C₅	M₁₆	1	2	yes	—
32	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₅×C₈	—	2	2	no	—
34	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
35	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
36	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
37	C₂×C₅×C₈	M₁₆	1	4	yes	—
38	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
39	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
41	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₅	—	2	2	no	—
43	C₄²×C₅	C₄⋊C₄	1	4	yes	—
44	C₄²×C₅	C₄⋊C₄	1	2	yes	—
45	C₄²×C₅	C₂×C₈	1	2	yes	—
46	#672	—	4	4	no	—
47	C₂×M₃₂	—	5	4	no	—
48	C₂×#97	Dic₅	1	2	yes	—
49	C₂×#97	Dic₅	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
51	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
52	#376	Dic₅	1	1	yes	characteristic
53	#364	—	10	8	no	—
54	C₄⋊C₁₆	—	10	4	no	—
55	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	—	4	2	no	—
58	C₂×C₄×C₅	—	4	4	no	—
59	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
60	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
64	C₅×C₈	—	2	2	no	—
65	C₅×C₈	C₄⋊C₈	1	8	yes	—
66	C₅×C₈	#127	1	1	yes	characteristic
67	C₅×C₈	C₄⋊Q₈	1	2	yes	—
68	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₅	1	2	yes	—
71	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
72	C₂×C₄²	C₂×Dic₅	1	2	yes	—
73	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
74	M₃₂	—	10	8	no	—
75	C₂×C₁₆	—	10	4	no	—
76	#97	—	2	4	no	—
77	#97	#168	1	4	yes	—
78	C₄×C₈	Dic₁₀	1	4	yes	—
79	C₄×C₈	Dic₁₀	1	2	yes	—
80	C₄×C₈	D₂₀	1	4	yes	—
81	C₄×C₈	D₂₀	1	2	yes	—
82	#95	—	2	2	no	—
83	C₂²×C₅	—	2	2	no	—
84	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₅	#423	1	2	yes	—
86	C₄×C₅	—	2	4	no	—
87	C₄×C₅	—	4	4	no	—
88	C₄×C₅	#384	1	1	yes	characteristic
89	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
90	C₄×C₅	#423	1	1	yes	characteristic
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	#681	1	2	yes	—
93	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂×#168	1	4	yes	—
96	C₂×C₈	#681	1	4	yes	—
97	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
98	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
99	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
100	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
101	C₂×C₈	C₂×D₂₀	1	2	yes	—
102	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
103	C₄²	—	2	2	no	—
104	C₄²	C₂×#168	1	2	yes	—
105	C₄²	C₄⋊Dic₅	1	4	yes	—
106	C₄²	C₄⋊Dic₅	1	2	yes	—
107	C₁₆	—	20	4	no	—
108	C₂×C₅	—	2	2	no	—
109	C₂×C₅	#1842	1	1	yes	characteristic
110	C₂×C₅	#2153	1	2	yes	—
111	C₂³	C₂×#681	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	C₄⋊#168	1	4	yes	—
116	C₂×C₄	C₂²×#168	1	1	yes	characteristic
117	C₂×C₄	C₂×#681	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	C₄⋊#168	1	8	yes	—
122	C₈	#4015	1	2	yes	—
123	C₈	#4015	1	1	yes	characteristic
124	C₈	#4020	1	1	yes	characteristic
125	C₅	#12901	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
128	C₂²	#65213	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄⋊#168	1	2	yes	—
132	C₄	#65213	1	1	yes	characteristic
133	C₄	#65224	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#10600187	1	1	yes	characteristic
136	C₂	#10600678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1726

C₂

yes

maximal, characteristic

C₂×#64683

C₂

yes

maximal

#10600536

C₂

yes

maximal

C₂×C₅×#97

C₄

yes

—

C₂×C₅×#97

C₄

yes

characteristic

C₂×C₄×C₅×C₈

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#376

C₄

yes

characteristic

C₂×#3944

C₂²

yes

—

—

—

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₅×#97

—

C₅×#97

C₈

yes

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

C₄×C₅×C₈

Q₈

yes

—

C₅×#95

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂×C₈

yes

—

C₂×C₅×C₈

M₁₆

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×Q₈

yes

—

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₄²×C₅

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

C₂×C₈

yes

—

#672

—

C₂×M₃₂

—

C₂×#97

Dic₅

yes

—

C₂×#97

Dic₅

yes

characteristic

C₂×C₄×C₈

C₂×D₅

yes

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#376

Dic₅

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₂³×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₅×C₈

—

C₅×C₈

C₄⋊C₈

yes

—

C₅×C₈

#127

yes

characteristic

C₅×C₈

C₄⋊Q₈

yes

—

C₅×C₈

C₄⋊Q₈

yes

characteristic

C₂²×C₈

C₂×Dic₅

yes

characteristic

C₂²×C₈

C₂×Dic₅

yes

—

C₂²×C₈

C₂²×D₅

yes

characteristic

C₂×C₄²

C₂×Dic₅

yes

—

C₂×C₄²

C₂×Dic₅

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

#168

yes

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

D₂₀

yes

—

C₄×C₈

D₂₀

yes

—

#95

—

C₂²×C₅

—

C₂²×C₅

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅

#423

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#384

yes

characteristic

C₄×C₅

C₂×C₄⋊C₈

yes

—

C₄×C₅

#423

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#681

yes

—

C₂²×C₄

C₂²×Dic₅

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×#168

yes

—

C₂×C₈

#681

yes

—

C₂×C₈

C₄⋊Dic₅

yes

—

C₂×C₈

C₄⋊Dic₅

yes

—

C₂×C₈

C₂×Dic₁₀

yes

characteristic

100

C₂×C₈

C₂×Dic₁₀

yes

—

101

C₂×C₈

C₂×D₂₀

yes

—

102

C₂×C₈

C₂×D₂₀

yes

characteristic

103

C₄²

—

104

C₄²

C₂×#168

yes

—

105

C₄²

C₄⋊Dic₅

yes

—

106

C₄²

C₄⋊Dic₅

yes

—

107

C₁₆

—

108

C₂×C₅

—

109

C₂×C₅

#1842

yes

characteristic

110

C₂×C₅

#2153

yes

—

111

C₂³

C₂×#681

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₄⋊#168

yes

—

116

C₂×C₄

C₂²×#168

yes

characteristic

117

C₂×C₄

C₂×#681

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₅

yes

—

119

C₂×C₄

C₂×C₄⋊Dic₅

yes

characteristic

120

C₈

—

121

C₈

C₄⋊#168

yes

—

122

C₈

#4015

yes

—

123

C₈

#4015

yes

characteristic

124

C₈

#4020

yes

characteristic

125

C₅

#12901

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

C₂×C₄⋊#168

yes

characteristic

128

C₂²

#65213

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄⋊#168

yes

—

132

C₄

#65213

yes

characteristic

133

C₄

#65224

yes

characteristic

134

C₂

—

135

C₂

#10600187

yes

characteristic

136

C₂

#10600678

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1726	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1726	D₅
hypocenter	C₅	#12901
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₂₀

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₅×#1726

C₂

socle

C₂²×C₅

C₂×C₄⋊C₈

prime radical

C₂³×C₅

C₂×M₁₆

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
125. C₅	#12901	semidirect

H ≅ G/N

type

125. C₅

#12901

semidirect

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

subgroup	index
C₅×#1726	2
C₂×#64683	2
#10600536	2
#12901	5

subgroup

index

Subgroups — #12015720

Conjugacy classes of subgroups

Complements

Maximal subgroups