Group #12014020 — subgroups

Subgroups — #12014020

Order 1280.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2246	C₂	1	1	yes	maximal, characteristic
2	#10602084	C₂	1	2	yes	maximal, complemented
3	#10602068	C₂	1	2	yes	maximal, complemented
4	#10599859	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₁₆	—	2	2	no	—
7	C₅×#493	C₂²	1	2	yes	—
8	C₂×#3959	—	2	4	no	—
9	#65359	—	2	4	no	—
10	C₂×Dic₄₀	—	2	2	no	—
11	#64988	C₂²	1	2	yes	—
12	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×#672	C₂²	1	1	yes	characteristic
14	C₈×#168	C₂²	1	1	yes	characteristic
15	#12518	—	5	1	no	maximal, Sylow
16	C₂×C₅×D₈	—	2	2	no	—
17	C₂×C₅×D₈	—	2	2	no	—
18	C₅×#127	—	2	2	no	—
19	C₂×Dic₂₀	—	2	2	no	—
20	C₂×Dic₂₀	—	2	2	no	—
21	#4015	—	2	2	no	—
22	C₅×D₁₆	—	4	4	no	—
23	C₂×C₅×C₁₆	D₄	1	2	yes	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	#3964	—	4	8	no	—
26	#3959	—	4	8	no	—
27	C₂×#672	D₄	1	2	yes	—
28	C₄×#168	D₄	1	2	yes	—
29	Dic₄₀	—	4	4	no	—
30	#2248	—	5	2	no	—
31	#2247	—	5	1	no	—
32	#2246	D₅	1	1	yes	characteristic
33	#1709	—	5	2	no	—
34	C₈×C₁₆	—	5	1	no	—
35	C₂×C₅×D₄	—	4	2	no	—
36	C₂×C₅×D₄	—	4	4	no	—
37	C₂×Dic₁₀	—	4	2	no	—
38	C₅×D₈	—	4	4	no	—
39	C₅×D₈	—	4	4	no	—
40	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅	—	4	4	no	—
44	C₂×#168	D₈	1	4	yes	—
45	Dic₂₀	—	4	4	no	—
46	Dic₂₀	—	4	4	no	—
47	C₅×C₁₆	D₈	1	4	yes	—
48	#672	D₈	1	4	yes	—
49	C₂×Q₃₂	—	10	2	no	—
50	C₂×SD₃₂	—	10	4	no	—
51	C₂×D₁₆	—	2	2	no	—
52	#494	—	5	2	no	—
53	#493	C₂×D₅	1	2	yes	—
54	#486	—	10	4	no	—
55	C₄×C₁₆	—	5	1	no	—
56	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
57	C₈²	—	5	1	no	—
58	C₂³×C₅	—	8	2	no	—
59	C₅×D₄	—	8	2	no	—
60	C₅×D₄	—	8	2	no	—
61	C₅×D₄	—	8	8	no	—
62	C₂×C₄×C₅	#127	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
64	C₂×Dic₅	—	8	2	no	—
65	Dic₁₀	—	8	2	no	—
66	Dic₁₀	—	8	2	no	—
67	C₅×C₈	C₂×D₈	1	2	yes	—
68	C₅×C₈	C₂×D₈	1	2	yes	—
69	#168	—	2	4	no	—
70	C₂×Q₁₆	—	10	2	no	—
71	C₂×Q₁₆	—	10	2	no	—
72	C₂×D₈	—	2	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₄⋊Q₈	—	10	2	no	—
75	#127	—	2	2	no	—
76	Q₃₂	—	20	4	no	—
77	SD₃₂	—	20	8	no	—
78	D₁₆	—	4	4	no	—
79	C₂×C₁₆	—	5	2	no	—
80	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
81	#102	—	20	8	no	—
82	C₄×C₈	—	5	2	no	—
83	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
84	C₂²×C₅	—	16	2	no	—
85	C₂²×C₅	—	16	4	no	—
86	C₂²×C₅	#493	1	1	yes	characteristic
87	C₄×C₅	#493	1	1	yes	characteristic
88	C₄×C₅	#493	1	1	yes	characteristic
89	C₄×C₅	#508	1	4	yes	—
90	Dic₅	—	16	2	no	—
91	C₂×Q₈	—	20	2	no	—
92	C₂×D₄	—	4	4	no	—
93	C₂×D₄	—	4	2	no	—
94	Q₁₆	—	20	4	no	—
95	Q₁₆	—	20	4	no	—
96	D₈	—	4	4	no	—
97	D₈	—	4	4	no	—
98	C₂×C₈	—	5	4	no	—
99	C₂×C₈	D₄×D₅	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
101	C₄⋊C₄	—	20	4	no	—
102	C₄²	D₄×D₅	1	1	yes	characteristic
103	C₁₆	—	5	4	no	—
104	C₁₆	#686	1	4	yes	—
105	C₂×C₅	—	32	2	no	—
106	C₂×C₅	#1712	1	1	yes	characteristic
107	C₂×C₅	#2249	1	2	yes	—
108	C₂³	—	8	2	no	—
109	Q₈	—	40	2	no	—
110	Q₈	—	40	2	no	—
111	D₄	—	8	2	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	8	no	—
114	C₂×C₄	—	40	2	no	—
115	C₂×C₄	D₅×D₈	1	2	yes	—
116	C₂×C₄	#4086	1	1	yes	characteristic
117	C₈	—	10	4	no	—
118	C₈	D₅×D₈	1	2	yes	—
119	C₈	C₂×#686	1	2	yes	—
120	C₅	#12518	1	1	yes	characteristic, complemented, Sylow
121	C₂²	—	16	4	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#65440	1	1	yes	characteristic
124	C₄	—	80	2	no	—
125	C₄	#65201	1	4	yes	—
126	C₄	#65362	1	1	yes	characteristic
127	C₄	#65440	1	1	yes	characteristic
128	C₂	—	32	2	no	—
129	C₂	#10602069	1	1	yes	characteristic
130	C₂	#10602258	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2246

C₂

yes

maximal, characteristic

#10602084

C₂

yes

maximal, complemented

#10602068

C₂

yes

maximal, complemented

#10599859

C₂

yes

maximal, characteristic, complemented

C₁₆×#168

C₂

yes

maximal, characteristic, complemented

C₂×C₅×D₁₆

—

C₅×#493

C₂²

yes

—

C₂×#3959

—

#65359

—

C₂×Dic₄₀

—

#64988

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₄×#672

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12518

—

maximal, Sylow

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₅×#127

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

C₅×D₁₆

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

D₄

yes

—

C₄×#168

D₄

yes

—

Dic₄₀

—

#2248

—

#2247

—

#2246

D₅

yes

characteristic

#1709

—

C₈×C₁₆

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×Dic₁₀

—

C₅×D₈

—

C₅×D₈

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₂×#168

D₈

yes

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

D₈

yes

—

#672

D₈

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

—

#493

C₂×D₅

yes

—

#486

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₈²

—

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

C₂×D₈

yes

—

C₅×C₈

C₂×D₈

yes

—

#168

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

D₄×D₅

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

101

C₄⋊C₄

—

102

C₄²

D₄×D₅

yes

characteristic

103

C₁₆

—

104

C₁₆

#686

yes

—

105

C₂×C₅

—

106

C₂×C₅

#1712

yes

characteristic

107

C₂×C₅

#2249

yes

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

D₅×D₈

yes

—

116

C₂×C₄

#4086

yes

characteristic

117

C₈

—

118

C₈

D₅×D₈

yes

—

119

C₈

C₂×#686

yes

—

120

C₅

#12518

yes

characteristic, complemented, Sylow

121

C₂²

—

122

C₂²

—

123

C₂²

#65440

yes

characteristic

124

C₄

—

125

C₄

#65201

yes

—

126

C₄

#65362

yes

characteristic

127

C₄

#65440

yes

characteristic

128

C₂

—

129

C₂

#10602069

yes

characteristic

130

C₂

#10602258

yes

—

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting, prime radical	C₅×#2246	C₂
socle	C₂²×C₅	#493
hypercenter	#2246	D₅
hypocenter	C₅	#12518
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#65440

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

Fitting, prime radical

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
2. #10602084	C₂	semidirect
3. #10602068	C₂	semidirect
4. #10599859	C₂	semidirect
5. C₁₆×#168	C₂	semidirect
120. C₅	#12518	semidirect

H ≅ G/N

type

2. #10602084

C₂

semidirect

3. #10602068

C₂

semidirect

4. #10599859

C₂

semidirect

5. C₁₆×#168

C₂

semidirect

120. C₅

#12518

semidirect

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

subgroup	index
C₅×#2246	2
#10602084	2
#10602068	2
#10599859	2
C₁₆×#168	2
#12518	5

subgroup

index

C₁₆×#168

Subgroups — #12014020

Conjugacy classes of subgroups

Complements

Maximal subgroups