Group #12014013 — subgroups

Subgroups — #12014013

Order 1280.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2247	C₂	1	1	yes	maximal, characteristic
2	#10602105	C₂	1	2	yes	maximal
3	#10602091	C₂	1	2	yes	maximal
4	#10599859	C₂	1	1	yes	maximal, characteristic
5	C₁₆×#168	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Q₃₂	—	2	2	no	—
7	C₅×#494	C₂²	1	2	yes	—
8	C₂×#3961	—	2	4	no	—
9	#65367	—	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	#12516	—	5	1	no	maximal, Sylow
16	C₂×C₅×Q₁₆	—	2	2	no	—
17	C₂×C₅×Q₁₆	—	2	2	no	—
18	C₅×C₄⋊Q₈	—	2	2	no	—
19	C₂×Dic₂₀	—	2	2	no	—
20	C₂×Dic₂₀	—	2	2	no	—
21	#4015	—	2	2	no	—
22	C₅×Q₃₂	—	4	4	no	—
23	C₂×C₅×C₁₆	D₄	1	2	yes	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	#3961	—	4	8	no	—
26	C₂×#672	D₄	1	2	yes	—
27	#3939	—	4	8	no	—
28	C₄×#168	D₄	1	2	yes	—
29	Dic₄₀	—	4	4	no	—
30	#2247	—	5	2	no	—
31	#2247	—	5	1	no	—
32	#2247	D₅	1	1	yes	characteristic
33	#1713	—	5	2	no	—
34	C₈×C₁₆	—	5	1	no	—
35	C₂×C₅×Q₈	—	4	2	no	—
36	C₂×Dic₁₀	—	4	2	no	—
37	C₅×Q₁₆	—	4	4	no	—
38	C₅×Q₁₆	—	4	4	no	—
39	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	C₅×C₄⋊C₄	—	4	4	no	—
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	—	2	2	no	—
50	C₂×Q₃₂	—	10	4	no	—
51	C₂×Q₃₂	—	10	2	no	—
52	#500	—	10	4	no	—
53	#494	—	5	2	no	—
54	#494	C₂×D₅	1	2	yes	—
55	C₄×C₁₆	—	5	1	no	—
56	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
57	C₈²	—	5	1	no	—
58	C₅×Q₈	—	8	2	no	—
59	C₅×Q₈	—	8	2	no	—
60	C₂×C₄×C₅	—	8	2	no	—
61	C₂×C₄×C₅	#127	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
63	C₂×Dic₅	—	8	2	no	—
64	Dic₁₀	—	8	2	no	—
65	Dic₁₀	—	8	2	no	—
66	C₅×C₈	C₂×D₈	1	2	yes	—
67	C₅×C₈	C₂×D₈	1	2	yes	—
68	#168	Q₃₂	1	8	yes	complemented
69	C₂×Q₁₆	—	2	2	no	—
70	C₂×Q₁₆	—	2	2	no	—
71	C₂×Q₁₆	—	10	2	no	—
72	C₂×Q₁₆	—	10	2	no	—
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	—	10	2	no	—
75	Q₃₂	—	4	4	no	—
76	Q₃₂	—	20	8	no	—
77	Q₃₂	—	20	4	no	—
78	C₂×C₁₆	—	5	2	no	—
79	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
80	#107	—	20	8	no	—
81	C₄×C₈	—	5	2	no	—
82	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
83	C₂²×C₅	#493	1	1	yes	characteristic
84	C₄×C₅	—	16	2	no	—
85	C₄×C₅	#493	1	1	yes	characteristic
86	C₄×C₅	#493	1	1	yes	characteristic
87	C₄×C₅	C₂×Q₃₂	1	4	yes	—
88	Dic₅	—	16	2	no	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	20	2	no	—
91	Q₁₆	—	4	4	no	—
92	Q₁₆	—	4	4	no	—
93	Q₁₆	—	20	4	no	—
94	Q₁₆	—	20	4	no	—
95	C₂×C₈	—	5	4	no	—
96	C₂×C₈	D₄×D₅	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	—	20	4	no	—
100	C₄²	D₄×D₅	1	1	yes	characteristic
101	C₁₆	—	5	4	no	—
102	C₁₆	#686	1	4	yes	—
103	C₂×C₅	#1712	1	1	yes	characteristic
104	C₂×C₅	#2247	1	2	yes	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	40	2	no	—
108	Q₈	—	40	2	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	—	40	2	no	—
111	C₂×C₄	D₅×D₈	1	2	yes	—
112	C₂×C₄	#4086	1	1	yes	characteristic
113	C₈	—	5	8	no	—
114	C₈	D₅×D₈	1	2	yes	—
115	C₈	C₂×#686	1	2	yes	—
116	C₅	#12516	1	1	yes	characteristic, complemented, Sylow
117	C₂²	#65440	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	80	2	no	—
120	C₄	D₅×Q₃₂	1	4	yes	—
121	C₄	#65362	1	1	yes	characteristic
122	C₄	#65440	1	1	yes	characteristic
123	C₂	#10602069	1	1	yes	characteristic
124	C₂	#10602292	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2247

C₂

yes

maximal, characteristic

#10602105

C₂

yes

maximal

#10602091

C₂

yes

maximal

#10599859

C₂

yes

maximal, characteristic

C₁₆×#168

C₂

yes

maximal, characteristic

C₂×C₅×Q₃₂

—

C₅×#494

C₂²

yes

—

C₂×#3961

—

#65367

—

C₂×Dic₄₀

—

#64988

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₄×#672

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12516

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

C₅×Q₃₂

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

characteristic

#3961

—

C₂×#672

D₄

yes

—

#3939

—

C₄×#168

D₄

yes

—

Dic₄₀

—

#2247

—

#2247

—

#2247

D₅

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₂×#168

D₈

yes

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

D₈

yes

—

#672

D₈

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

#494

C₂×D₅

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₈²

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

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

Q₃₂

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

—

#107

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

C₂×Q₃₂

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₅

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

D₄×D₅

yes

characteristic

101

C₁₆

—

102

C₁₆

#686

yes

—

103

C₂×C₅

#1712

yes

characteristic

104

C₂×C₅

#2247

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

D₅×D₈

yes

—

112

C₂×C₄

#4086

yes

characteristic

113

C₈

—

114

C₈

D₅×D₈

yes

—

115

C₈

C₂×#686

yes

—

116

C₅

#12516

yes

characteristic, complemented, Sylow

117

C₂²

#65440

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

D₅×Q₃₂

yes

—

121

C₄

#65362

yes

characteristic

122

C₄

#65440

yes

characteristic

123

C₂

#10602069

yes

characteristic

124

C₂

#10602292

yes

—

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2247	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#2247	D₅
hypocenter	C₅	#12516
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#65440

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
68. #168	Q₃₂	semidirect
116. C₅	#12516	semidirect

H ≅ G/N

type

68. #168

Q₃₂

semidirect

116. C₅

#12516

semidirect

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

subgroup	index
C₅×#2247	2
#10602105	2
#10602091	2
#10599859	2
C₁₆×#168	2
#12516	5

subgroup

index

C₁₆×#168

Subgroups — #12014013

Conjugacy classes of subgroups

Complements

Maximal subgroups