Group #12013972 — subgroups

Subgroups — #12013972

Order 1280.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	#10602103	C₂	1	2	yes	maximal
3	#10602089	C₂	1	2	yes	maximal
4	#10599839	C₂	1	1	yes	maximal, characteristic
5	#10597989	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Q₃₂	—	2	2	no	—
7	C₅×#494	C₂²	1	2	yes	—
8	#65382	—	2	2	no	—
9	#65368	—	2	2	no	—
10	#64971	C₂²	1	2	yes	—
11	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	#64784	—	2	4	no	—
13	#64725	—	2	2	no	—
14	C₄×#672	C₂²	1	1	yes	characteristic
15	C₈×#168	C₂²	1	1	yes	characteristic
16	#12507	—	5	1	no	maximal, Sylow
17	C₂×C₅×Q₁₆	—	2	2	no	—
18	C₂×C₅×Q₁₆	—	2	2	no	—
19	C₅×C₄⋊Q₈	—	2	2	no	—
20	#4015	—	2	2	no	—
21	C₅×Q₃₂	—	4	4	no	—
22	C₂×C₅×C₁₆	D₄	1	2	yes	—
23	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
24	#3950	—	2	2	no	—
25	#3950	—	2	2	no	—
26	C₂×#672	D₄	1	2	yes	—
27	#3942	—	4	4	no	—
28	#3940	—	4	4	no	—
29	C₄×#168	D₄	1	2	yes	—
30	#2254	—	5	1	no	—
31	#2247	D₅	1	1	yes	characteristic
32	#2241	—	5	2	no	—
33	#1707	—	5	2	no	—
34	#1311	—	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	2	no	—
42	C₅×C₄⋊C₄	—	4	2	no	—
43	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₅	—	4	2	no	—
45	C₄⋊Dic₅	—	4	2	no	—
46	C₂×#168	D₈	1	4	yes	—
47	C₅×C₁₆	SD₁₆	1	4	yes	—
48	#672	—	2	2	no	—
49	C₂×Q₃₂	—	2	2	no	—
50	#500	—	5	2	no	—
51	#498	—	10	2	no	—
52	#494	C₂×D₅	1	2	yes	—
53	#487	—	10	2	no	—
54	#367	—	10	2	no	—
55	#358	—	10	4	no	—
56	C₄×C₁₆	—	5	1	no	—
57	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
58	C₈²	—	5	1	no	—
59	C₅×Q₈	—	8	2	no	—
60	C₅×Q₈	—	8	2	no	—
61	C₂×C₄×C₅	—	8	2	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	4	no	—
66	C₅×C₈	C₂×SD₁₆	1	2	yes	—
67	C₅×C₈	#135	1	2	yes	—
68	#168	—	2	4	no	—
69	C₂×Q₁₆	—	2	2	no	—
70	C₂×Q₁₆	—	2	2	no	—
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	—	10	2	no	—
73	Q₃₂	—	4	4	no	—
74	C₂×C₁₆	—	5	2	no	—
75	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
76	#107	—	10	2	no	—
77	#107	—	10	2	no	—
78	#106	—	20	4	no	—
79	#103	—	20	4	no	—
80	C₄×C₈	—	5	2	no	—
81	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
82	C₂²×C₅	#493	1	1	yes	characteristic
83	C₄×C₅	—	16	2	no	—
84	C₄×C₅	#492	1	1	yes	characteristic
85	C₄×C₅	#495	1	1	yes	characteristic
86	C₄×C₅	#510	1	4	yes	—
87	Dic₅	—	16	2	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂×Q₈	—	20	2	no	—
90	Q₁₆	—	4	4	no	—
91	Q₁₆	—	4	4	no	—
92	C₂×C₈	—	5	4	no	—
93	C₂×C₈	D₄×D₅	1	1	yes	characteristic
94	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	20	2	no	—
98	C₄⋊C₄	—	20	2	no	—
99	C₄²	D₄×D₅	1	1	yes	characteristic
100	C₁₆	—	10	2	no	—
101	C₁₆	#687	1	4	yes	—
102	C₂×C₅	#1706	1	1	yes	characteristic
103	C₂×C₅	#2251	1	2	yes	—
104	Q₈	—	8	2	no	—
105	Q₈	—	8	2	no	—
106	Q₈	—	40	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	40	2	no	—
109	C₂×C₄	D₅×D₈	1	2	yes	—
110	C₂×C₄	#4086	1	1	yes	characteristic
111	C₈	—	10	4	no	—
112	C₈	#4058	1	2	yes	—
113	C₈	C₂×#687	1	2	yes	—
114	C₅	#12507	1	1	yes	characteristic, complemented, Sylow
115	C₂²	#65440	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	80	2	no	—
118	C₄	#65207	1	4	yes	—
119	C₄	#65365	1	1	yes	characteristic
120	C₄	#65444	1	1	yes	characteristic
121	C₂	#10602067	1	1	yes	characteristic
122	C₂	#10602294	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2247

C₂

yes

maximal, characteristic

#10602103

C₂

yes

maximal

#10602089

C₂

yes

maximal

#10599839

C₂

yes

maximal, characteristic

#10597989

C₂

yes

maximal, characteristic

C₂×C₅×Q₃₂

—

C₅×#494

C₂²

yes

—

#65382

—

#65368

—

#64971

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12507

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

#4015

—

C₅×Q₃₂

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

D₄

yes

—

#3942

—

#3940

—

C₄×#168

D₄

yes

—

#2254

—

#2247

D₅

yes

—

—

—

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

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₈

yes

—

C₅×C₁₆

SD₁₆

yes

—

#672

—

C₂×Q₃₂

—

#500

—

#498

—

#494

C₂×D₅

yes

—

#487

—

#367

—

#358

—

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

—

C₅×C₈

C₂×SD₁₆

yes

—

C₅×C₈

#135

yes

—

#168

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₅

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×D₅

yes

characteristic

100

C₁₆

—

101

C₁₆

#687

yes

—

102

C₂×C₅

#1706

yes

characteristic

103

C₂×C₅

#2251

yes

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

D₅×D₈

yes

—

110

C₂×C₄

#4086

yes

characteristic

111

C₈

—

112

C₈

#4058

yes

—

113

C₈

C₂×#687

yes

—

114

C₅

#12507

yes

characteristic, complemented, Sylow

115

C₂²

#65440

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#65207

yes

—

119

C₄

#65365

yes

characteristic

120

C₄

#65444

yes

characteristic

121

C₂

#10602067

yes

characteristic

122

C₂

#10602294

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	#12507
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
114. C₅	#12507	semidirect

H ≅ G/N

type

114. C₅

#12507

semidirect

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

subgroup	index
C₅×#2247	2
#10602103	2
#10602089	2
#10599839	2
#10597989	2
#12507	5

subgroup

index

Subgroups — #12013972

Conjugacy classes of subgroups

Complements

Maximal subgroups