Group #11931530 — subgroups

Subgroups — C₅×#8810 #11931530

Order 1280.

402 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 222 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×#1842	C₂	1	2	yes	maximal
2	C₅×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊C₈	—	2	4	no	—
5	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	C₅×#336	C₄	1	4	yes	—
9	C₅×#335	—	2	8	no	—
10	#8810	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₅×C₈	—	2	4	no	—
12	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅	D₄	1	2	yes	—
15	C₂×C₄²×C₅	Q₈	1	2	yes	—
16	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
17	C₅×C₄⋊C₈	—	2	8	no	—
18	C₅×C₄⋊C₈	—	4	8	no	—
19	C₄×C₅×C₈	D₄	1	4	yes	—
20	C₄×C₅×C₈	Q₈	1	4	yes	—
21	#1842	C₂×C₅	1	2	yes	—
22	#1744	C₂×C₅	1	1	yes	characteristic
23	C₂×#335	C₂×C₅	1	4	yes	—
24	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
25	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₅×C₈	—	2	8	no	—
29	C₂×C₅×C₈	—	4	4	no	—
30	C₂×C₅×C₈	—	4	8	no	—
31	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
32	C₄²×C₅	—	2	4	no	—
33	C₄²×C₅	—	4	2	no	—
34	C₄²×C₅	D₈	1	4	yes	—
35	C₄²×C₅	Q₁₆	1	4	yes	—
36	C₄²×C₅	C₂×D₄	1	2	yes	—
37	C₄²×C₅	C₂×Q₈	1	2	yes	—
38	C₂×C₄⋊C₈	—	2	4	no	—
39	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
40	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
41	C₄³	C₂²×C₅	1	1	yes	characteristic
42	#336	C₄×C₅	1	4	yes	—
43	#335	—	2	8	no	—
44	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅	—	2	4	no	—
46	C₂×C₄×C₅	—	2	4	no	—
47	C₂×C₄×C₅	—	2	2	no	—
48	C₂×C₄×C₅	#107	1	8	yes	—
49	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₅	#127	1	1	yes	characteristic
51	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅	#131	1	4	yes	—
54	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
55	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
56	C₅×C₈	—	2	8	no	—
57	C₅×C₈	—	8	4	no	—
58	C₂²×C₈	—	2	4	no	—
59	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
60	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
61	C₂×C₄²	C₅×D₄	1	2	yes	—
62	C₂×C₄²	C₅×Q₈	1	2	yes	—
63	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
64	C₄⋊C₈	—	2	8	no	—
65	C₄⋊C₈	—	4	8	no	—
66	C₄×C₈	C₅×D₄	1	4	yes	—
67	C₄×C₈	C₅×Q₈	1	4	yes	—
68	C₂²×C₅	#384	1	1	yes	characteristic
69	C₂²×C₅	#424	1	2	yes	—
70	C₂²×C₅	C₂×#107	1	2	yes	—
71	C₂²×C₅	#430	1	2	yes	—
72	C₄×C₅	—	2	8	no	—
73	C₄×C₅	—	2	2	no	—
74	C₄×C₅	#424	1	4	yes	—
75	C₄×C₅	#493	1	1	yes	characteristic
76	C₄×C₅	#494	1	1	yes	characteristic
77	C₄×C₅	#500	1	2	yes	—
78	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
79	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	—	4	4	no	—
84	C₂×C₈	—	4	8	no	—
85	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
86	C₄²	—	2	4	no	—
87	C₄²	—	4	2	no	—
88	C₄²	C₅×D₈	1	4	yes	—
89	C₄²	C₅×Q₁₆	1	4	yes	—
90	C₄²	C₂×C₅×D₄	1	2	yes	—
91	C₄²	C₂×C₅×Q₈	1	2	yes	—
92	C₂×C₅	#1571	1	4	yes	—
93	C₂×C₅	#1843	1	1	yes	characteristic
94	C₂×C₅	#1845	1	1	yes	characteristic
95	C₂×C₅	#1849	1	1	yes	characteristic
96	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	C₅×#107	1	8	yes	—
101	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
102	C₂×C₄	C₅×#127	1	1	yes	characteristic
103	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
104	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄	C₅×#131	1	4	yes	—
106	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
107	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
108	C₈	—	2	8	no	—
109	C₈	—	8	4	no	—
110	C₅	#8810	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²	C₅×#384	1	1	yes	characteristic
112	C₂²	C₅×#424	1	2	yes	—
113	C₂²	C₂×C₅×#107	1	2	yes	—
114	C₂²	C₅×#430	1	2	yes	—
115	C₄	—	2	8	no	—
116	C₄	—	2	2	no	—
117	C₄	C₅×#424	1	4	yes	—
118	C₄	C₅×#493	1	1	yes	characteristic
119	C₄	C₅×#494	1	1	yes	characteristic
120	C₄	C₅×#500	1	2	yes	—
121	C₂	C₅×#1571	1	4	yes	—
122	C₂	C₅×#1843	1	1	yes	characteristic
123	C₂	C₅×#1845	1	1	yes	characteristic
124	C₂	C₅×#1849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1842

C₂

yes

maximal

C₅×#1744

C₂

yes

maximal, characteristic

C₂×C₅×#335

C₂

yes

maximal

C₂×C₅×C₄⋊C₈

—

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

C₄³×C₅

C₂²

yes

characteristic

C₅×#336

C₄

yes

—

C₅×#335

—

#8810

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×C₈

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

Q₈

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

#1842

C₂×C₅

yes

—

#1744

C₂×C₅

yes

characteristic

C₂×#335

C₂×C₅

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

D₈

yes

—

C₄²×C₅

Q₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₅

yes

—

C₂×C₄×C₈

C₂²×C₅

yes

—

C₄³

C₂²×C₅

yes

characteristic

#336

C₄×C₅

yes

—

#335

—

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#107

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₅×C₈

—

C₅×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₅

yes

—

C₂×C₄²

C₂×C₄×C₅

yes

—

C₂×C₄²

C₅×D₄

yes

—

C₂×C₄²

C₅×Q₈

yes

—

C₂×C₄²

C₂³×C₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₅×D₄

yes

—

C₄×C₈

C₅×Q₈

yes

—

C₂²×C₅

#384

yes

characteristic

C₂²×C₅

#424

yes

—

C₂²×C₅

C₂×#107

yes

—

C₂²×C₅

#430

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#424

yes

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

C₂²×C₄

C₅×C₄⋊C₄

yes

—

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

C₂²×C₄

C₂×C₅×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₅×C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₅×D₈

yes

—

C₄²

C₅×Q₁₆

yes

—

C₄²

C₂×C₅×D₄

yes

—

C₄²

C₂×C₅×Q₈

yes

—

C₂×C₅

#1571

yes

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂³

C₂×C₅×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₅×#107

yes

—

101

C₂×C₄

C₂×C₅×C₄⋊C₄

yes

—

102

C₂×C₄

C₅×#127

yes

characteristic

103

C₂×C₄

C₅×C₄⋊Q₈

yes

—

104

C₂×C₄

C₅×C₄⋊Q₈

yes

characteristic

105

C₂×C₄

C₅×#131

yes

—

106

C₂×C₄

C₂×C₅×D₈

yes

—

107

C₂×C₄

C₂×C₅×Q₁₆

yes

—

108

C₈

—

109

C₈

—

110

C₅

#8810

yes

characteristic, direct product factor, Sylow

111

C₂²

C₅×#384

yes

characteristic

112

C₂²

C₅×#424

yes

—

113

C₂²

C₂×C₅×#107

yes

—

114

C₂²

C₅×#430

yes

—

115

C₄

—

116

C₄

—

117

C₄

C₅×#424

yes

—

118

C₄

C₅×#493

yes

characteristic

119

C₄

C₅×#494

yes

characteristic

120

C₄

C₅×#500

yes

—

121

C₂

C₅×#1571

yes

—

122

C₂

C₅×#1843

yes

characteristic

123

C₂

C₅×#1845

yes

characteristic

124

C₂

C₅×#1849

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

subgroup

G/H

center

C₂²×C₄×C₅

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₅

Frattini, cosocle

C₂×C₄²

C₂³×C₅

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

N	H ≅ G/N	type
10. #8810	C₅	direct (N × H)
110. C₅	#8810	direct (N × H)

H ≅ G/N

type

10. #8810

C₅

direct (N × H)

110. C₅

#8810

direct (N × H)

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

subgroup	index
C₅×#1842	2
C₅×#1744	2
C₂×C₅×#335	2
#8810	5

subgroup

index

C₅×#1842

C₅×#1744

C₂×C₅×#335

#8810

Subgroups — C₅×#8810 #11931530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#8810 #11931530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8810 #11931530