Group #12019787 — subgroups

Subgroups — #12019787

Order 1280.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂×C₅×M₆₄	C₂	1	1	yes	maximal, characteristic
2	#10602192	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3926	C₂	1	1	yes	maximal, characteristic
4	#10598081	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	#65390	C₄	1	1	yes	characteristic
7	C₂×#3944	C₄	1	1	yes	characteristic
8	C₅×M₆₄	—	2	2	no	—
9	C₂×C₅×C₃₂	C₂²	1	2	yes	—
10	#64722	C₄	1	2	yes	—
11	C₁₆×Dic₅	C₂²	1	2	yes	—
12	C₂×#3926	—	2	2	no	—
13	C₂×#3926	C₂²	1	2	yes	—
14	#13628	—	5	1	no	maximal, Sylow
15	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
16	#4072	C₈	1	1	yes	characteristic
17	C₂×#681	C₈	1	1	yes	characteristic
18	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×C₁₆	D₄	1	2	yes	—
20	C₂×C₅×C₁₆	Q₈	1	2	yes	—
21	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
22	#3947	C₈	1	2	yes	—
23	C₈×Dic₅	C₂×C₄	1	2	yes	—
24	#3944	—	2	2	no	—
25	C₂×#672	C₂×C₄	1	2	yes	—
26	C₅×C₃₂	—	2	2	no	—
27	#3926	—	2	2	no	—
28	C₂×M₆₄	D₅	1	1	yes	characteristic
29	C₂²×C₃₂	—	5	1	no	—
30	#2108	—	5	1	no	—
31	C₄⋊C₃₂	—	5	4	no	—
32	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
33	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
34	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₅×C₈	M₁₆	1	2	yes	—
36	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂²⋊Dic₅	C₁₆	1	2	yes	—
38	C₄⋊Dic₅	C₁₆	1	2	yes	—
39	C₄×Dic₅	C₂×C₈	1	2	yes	—
40	#681	—	2	2	no	—
41	C₂×#168	C₂×C₈	1	2	yes	—
42	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
43	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
44	C₅×C₁₆	C₂×Q₈	1	1	yes	characteristic
45	#672	—	2	2	no	—
46	C₂×M₃₂	—	5	1	no	—
47	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
48	#405	—	5	1	no	—
49	M₆₄	—	2	2	no	—
50	C₂×C₃₂	—	5	2	no	—
51	C₂×C₃₂	—	10	2	no	—
52	C₂×C₃₂	C₂×D₅	1	2	yes	—
53	#346	—	5	2	no	—
54	C₄×C₁₆	—	5	2	no	—
55	C₂³×C₅	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅	M₃₂	1	2	yes	—
59	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
60	C₂×Dic₅	C₂×C₁₆	1	4	yes	—
61	C₅×C₈	C₄⋊C₈	1	2	yes	—
62	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
64	#168	—	2	2	no	—
65	C₂×M₁₆	—	5	1	no	—
66	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
67	#117	—	5	1	no	—
68	M₃₂	—	10	2	no	—
69	C₂×C₁₆	—	5	2	no	—
70	C₂×C₁₆	Dic₁₀	1	2	yes	—
71	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
73	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
74	#97	—	5	2	no	—
75	C₄×C₈	—	5	2	no	—
76	C₃₂	—	2	2	no	—
77	C₃₂	—	10	2	no	—
78	C₂²×C₅	—	2	2	no	—
79	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
80	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
81	C₄×C₅	C₄⋊C₁₆	1	2	yes	—
82	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
84	Dic₅	—	2	4	no	—
85	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
86	M₁₆	—	10	2	no	—
87	C₂×C₈	—	5	2	no	—
88	C₂×C₈	C₈×D₅	1	1	yes	characteristic
89	C₂×C₈	#676	1	2	yes	—
90	C₂×C₈	#683	1	2	yes	—
91	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
92	C₄⋊C₄	—	5	2	no	—
93	C₂²⋊C₄	—	5	2	no	—
94	C₄²	—	5	2	no	—
95	C₁₆	—	10	2	no	—
96	C₁₆	#683	1	2	yes	—
97	C₁₆	C₂×Dic₁₀	1	1	yes	characteristic
98	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
99	C₂×C₅	—	2	2	no	—
100	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
101	C₂×C₅	#2258	1	2	yes	—
102	C₂³	D₅×C₁₆	1	1	yes	characteristic
103	C₂×C₄	—	5	4	no	—
104	C₂×C₄	D₅×C₁₆	1	1	yes	characteristic
105	C₂×C₄	#3930	1	2	yes	—
106	C₂×C₄	#3946	1	2	yes	—
107	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
108	C₈	—	10	2	no	—
109	C₈	#3946	1	2	yes	—
110	C₈	C₂×#676	1	1	yes	characteristic
111	C₈	C₂×#683	1	1	yes	characteristic
112	C₅	#13628	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	2	2	no	—
114	C₂²	#64721	1	2	yes	—
115	C₂²	C₂×D₅×C₁₆	1	1	yes	characteristic
116	C₄	—	10	4	no	—
117	C₄	#64721	1	2	yes	—
118	C₄	C₂×#3930	1	1	yes	characteristic
119	C₄	C₂×#3946	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#10600165	1	2	yes	—
122	C₂	C₂×#64721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×M₆₄

C₂

yes

maximal, characteristic

#10602192

C₂

yes

maximal, characteristic

C₂²×#3926

C₂

yes

maximal, characteristic

#10598081

C₂

yes

maximal, complemented

C₂²×C₅×C₁₆

C₂²

yes

characteristic

#65390

C₄

yes

characteristic

C₂×#3944

C₄

yes

characteristic

C₅×M₆₄

—

C₂×C₅×C₃₂

C₂²

yes

—

#64722

C₄

yes

—

C₁₆×Dic₅

C₂²

yes

—

C₂×#3926

—

C₂×#3926

C₂²

yes

—

#13628

—

maximal, Sylow

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

#4072

C₈

yes

characteristic

C₂×#681

C₈

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

—

C₂×C₅×C₁₆

Q₈

yes

—

C₂×C₅×C₁₆

C₂³

yes

characteristic

#3947

C₈

yes

—

C₈×Dic₅

C₂×C₄

yes

—

#3944

—

C₂×#672

C₂×C₄

yes

—

C₅×C₃₂

—

#3926

—

C₂×M₆₄

D₅

yes

characteristic

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

M₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂²⋊Dic₅

C₁₆

yes

—

C₄⋊Dic₅

C₁₆

yes

—

C₄×Dic₅

C₂×C₈

yes

—

#681

—

C₂×#168

C₂×C₈

yes

—

C₅×C₁₆

C₄⋊C₄

yes

—

C₅×C₁₆

C₂×D₄

yes

characteristic

C₅×C₁₆

C₂×Q₈

yes

characteristic

#672

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅

yes

—

#346

—

C₄×C₁₆

—

C₂³×C₅

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅

M₃₂

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×Dic₅

C₂×C₁₆

yes

—

C₅×C₈

C₄⋊C₈

yes

—

C₅×C₈

C₂×C₄⋊C₄

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₅

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₀

yes

—

C₂×C₁₆

C₄×D₅

yes

characteristic

C₂×C₁₆

C₂²⋊D₅

yes

—

C₂×C₁₆

C₂²×D₅

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₅

—

C₂²×C₅

C₄⋊C₁₆

yes

—

C₂²×C₅

C₂²×C₁₆

yes

characteristic

C₄×C₅

C₄⋊C₁₆

yes

—

C₄×C₅

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅

C₂×M₃₂

yes

characteristic

Dic₅

—

C₂²×C₄

C₈×D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₈×D₅

yes

characteristic

C₂×C₈

#676

yes

—

C₂×C₈

#683

yes

—

C₂×C₈

C₂×C₄×D₅

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

#683

yes

—

C₁₆

C₂×Dic₁₀

yes

characteristic

C₁₆

C₂×C₂²⋊D₅

yes

characteristic

C₂×C₅

—

100

C₂×C₅

C₂×C₄⋊C₁₆

yes

characteristic

101

C₂×C₅

#2258

yes

—

102

C₂³

D₅×C₁₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

D₅×C₁₆

yes

characteristic

105

C₂×C₄

#3930

yes

—

106

C₂×C₄

#3946

yes

—

107

C₂×C₄

C₂×C₈×D₅

yes

characteristic

108

C₈

—

109

C₈

#3946

yes

—

110

C₈

C₂×#676

yes

characteristic

111

C₈

C₂×#683

yes

characteristic

112

C₅

#13628

yes

characteristic, complemented, Sylow

113

C₂²

—

114

C₂²

#64721

yes

—

115

C₂²

C₂×D₅×C₁₆

yes

characteristic

116

C₄

—

117

C₄

#64721

yes

—

118

C₄

C₂×#3930

yes

characteristic

119

C₄

C₂×#3946

yes

characteristic

120

C₂

—

121

C₂

#10600165

yes

—

122

C₂

C₂×#64721

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₁₆
Fitting	C₂×C₅×M₆₄	C₂
prime radical	C₂³×C₅	C₂×C₁₆
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	C₂×M₆₄	D₅
hypocenter	C₅	#13628
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅

commutator, socle

C₂²×C₅

C₂²×C₁₆

Fitting

C₂×C₅×M₆₄

C₂

prime radical

C₂³×C₅

C₂×C₁₆

cosocle

C₂×C₅×C₁₆

C₂³

hypercenter

C₂×M₆₄

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
4. #10598081	C₂	semidirect
112. C₅	#13628	semidirect

H ≅ G/N

type

4. #10598081

C₂

semidirect

112. C₅

#13628

semidirect

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

subgroup	index
C₂×C₅×M₆₄	2
#10602192	2
C₂²×#3926	2
#10598081	2
#13628	5

subgroup

index

C₂×C₅×M₆₄

#10602192

C₂²×#3926

#10598081

#13628

Subgroups — #12019787

Conjugacy classes of subgroups

Complements

Maximal subgroups