Group #11935630 — subgroups

Subgroups — C₅×#12910 #11935630

Order 1280.

306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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×#368	C₂	1	4	yes	maximal
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
3	C₅×#1848	C₂	1	2	yes	maximal
4	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
5	C₂×C₅×#108	—	2	2	no	—
6	C₂×C₅×#108	C₂²	1	4	yes	—
7	C₅×#423	—	2	2	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#368	—	2	8	no	—
10	C₅×#346	C₄	1	4	yes	—
11	#12910	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂×C₅×M₁₆	—	2	4	no	—
13	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
16	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₅×C₁₆	D₄	1	4	yes	—
18	C₂×C₅×C₁₆	Q₈	1	4	yes	—
19	C₅×#108	—	2	8	no	—
20	C₅×#108	—	4	4	no	—
21	C₅×C₄⋊C₈	—	4	4	no	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	Q₈	1	2	yes	—
24	C₂×#368	C₂×C₅	1	4	yes	—
25	C₂×#346	C₂×C₅	1	1	yes	characteristic
26	#1848	C₂×C₅	1	2	yes	—
27	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
29	C₅×M₁₆	—	4	4	no	—
30	C₅×M₁₆	—	4	4	no	—
31	C₂×C₅×C₈	—	4	4	no	—
32	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
33	C₂×C₅×C₈	D₈	1	2	yes	—
34	C₂×C₅×C₈	Q₁₆	1	2	yes	—
35	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
37	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
38	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₅	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅	D₈	1	2	yes	—
41	C₅×C₁₆	C₄⋊C₄	1	8	yes	—
42	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	C₂²×C₅	1	4	yes	—
45	#423	—	2	2	no	—
46	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
47	#368	—	2	8	no	—
48	#346	C₄×C₅	1	4	yes	—
49	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅	—	4	2	no	—
51	C₂×C₄×C₅	#107	1	4	yes	—
52	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅	#127	1	1	yes	characteristic
54	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
58	C₅×C₈	—	8	4	no	—
59	C₅×C₈	#107	1	4	yes	—
60	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
61	C₅×C₈	C₄⋊Q₈	1	2	yes	—
62	C₂×M₁₆	—	2	4	no	—
63	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
64	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
65	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
66	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
67	C₂×C₁₆	C₅×D₄	1	4	yes	—
68	C₂×C₁₆	C₅×Q₈	1	4	yes	—
69	#108	—	2	8	no	—
70	#108	—	4	4	no	—
71	C₄⋊C₈	—	4	4	no	—
72	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
73	C₄×C₈	C₅×Q₈	1	2	yes	—
74	C₂²×C₅	—	2	2	no	—
75	C₂²×C₅	C₂×#107	1	1	yes	characteristic
76	C₂²×C₅	#494	1	1	yes	characteristic
77	C₂²×C₅	#500	1	1	yes	characteristic
78	C₄×C₅	—	2	4	no	—
79	C₄×C₅	#384	1	1	yes	characteristic
80	C₄×C₅	C₂×#107	1	1	yes	characteristic
81	C₄×C₅	#493	1	1	yes	characteristic
82	C₄×C₅	#500	1	1	yes	characteristic
83	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
84	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
85	M₁₆	—	4	4	no	—
86	M₁₆	—	4	4	no	—
87	C₂×C₈	—	4	4	no	—
88	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
89	C₂×C₈	C₅×D₈	1	2	yes	—
90	C₂×C₈	C₅×Q₁₆	1	2	yes	—
91	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
92	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
93	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
94	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
95	C₄²	C₅×C₄⋊C₄	1	2	yes	—
96	C₄²	C₅×D₈	1	2	yes	—
97	C₁₆	C₅×C₄⋊C₄	1	8	yes	—
98	C₂×C₅	—	2	1	no	—
99	C₂×C₅	—	2	1	no	—
100	C₂×C₅	#1845	1	1	yes	characteristic
101	C₂×C₅	#2161	1	2	yes	—
102	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₅×#107	1	4	yes	—
105	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄	C₅×#127	1	1	yes	characteristic
107	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
108	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
109	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
110	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
111	C₈	—	8	4	no	—
112	C₈	C₅×#107	1	4	yes	—
113	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
114	C₈	C₅×C₄⋊Q₈	1	2	yes	—
115	C₅	#12910	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²	—	2	2	no	—
117	C₂²	C₂×C₅×#107	1	1	yes	characteristic
118	C₂²	C₅×#494	1	1	yes	characteristic
119	C₂²	C₅×#500	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	C₅×#384	1	1	yes	characteristic
122	C₄	C₂×C₅×#107	1	1	yes	characteristic
123	C₄	C₅×#493	1	1	yes	characteristic
124	C₄	C₅×#500	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₅×#1845	1	1	yes	characteristic
128	C₂	C₅×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#368

C₂

yes

maximal

C₂×C₅×#346

C₂

yes

maximal, characteristic

C₅×#1848

C₂

yes

maximal

C₂²×C₅×C₁₆

C₂²

yes

—

C₂×C₅×#108

—

C₂×C₅×#108

C₂²

yes

—

C₅×#423

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#368

—

C₅×#346

C₄

yes

—

#12910

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₂×C₅×C₁₆

D₄

yes

—

C₂×C₅×C₁₆

Q₈

yes

—

C₅×#108

—

C₅×#108

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

C₂×#368

C₂×C₅

yes

—

C₂×#346

C₂×C₅

yes

characteristic

#1848

C₂×C₅

yes

—

C₂²×C₄×C₅

Q₁₆

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

Q₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×Q₈

yes

—

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

D₈

yes

—

C₅×C₁₆

C₄⋊C₄

yes

—

C₂²×C₁₆

C₂²×C₅

yes

—

C₂×#108

—

C₂×#108

C₂²×C₅

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₅

yes

characteristic

#368

—

#346

C₄×C₅

yes

—

C₂³×C₅

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

#107

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×Q₁₆

yes

characteristic

C₅×C₈

—

C₅×C₈

#107

yes

—

C₅×C₈

C₂×C₄⋊C₄

yes

—

C₅×C₈

C₄⋊Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

C₅×D₄

yes

characteristic

C₂²×C₈

C₂³×C₅

yes

characteristic

C₂×C₄²

C₅×D₄

yes

characteristic

C₂×C₁₆

C₂×C₄×C₅

yes

—

C₂×C₁₆

C₅×D₄

yes

—

C₂×C₁₆

C₅×Q₈

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅

yes

—

C₄×C₈

C₅×Q₈

yes

—

C₂²×C₅

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

C₂²×C₄

C₅×Q₁₆

yes

—

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₅×C₄⋊C₄

yes

—

C₂×C₈

C₅×D₈

yes

—

C₂×C₈

C₅×Q₁₆

yes

—

C₂×C₈

C₂²×C₄×C₅

yes

characteristic

C₂×C₈

C₂×C₅×D₄

yes

—

C₂×C₈

C₂×C₅×Q₈

yes

—

C₂×C₈

C₂×C₅×Q₈

yes

characteristic

C₄²

C₅×C₄⋊C₄

yes

—

C₄²

C₅×D₈

yes

—

C₁₆

C₅×C₄⋊C₄

yes

—

C₂×C₅

—

C₂×C₅

—

100

C₂×C₅

#1845

yes

characteristic

101

C₂×C₅

#2161

yes

—

102

C₂³

C₂×C₅×Q₁₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

C₅×#107

yes

—

105

C₂×C₄

C₂×C₅×C₄⋊C₄

yes

characteristic

106

C₂×C₄

C₅×#127

yes

characteristic

107

C₂×C₄

C₅×C₄⋊Q₈

yes

characteristic

108

C₂×C₄

C₂×C₅×D₈

yes

characteristic

109

C₂×C₄

C₂×C₅×D₈

yes

characteristic

110

C₂×C₄

C₂×C₅×Q₁₆

yes

characteristic

111

C₈

—

112

C₈

C₅×#107

yes

—

113

C₈

C₂×C₅×C₄⋊C₄

yes

—

114

C₈

C₅×C₄⋊Q₈

yes

—

115

C₅

#12910

yes

characteristic, direct product factor, Sylow

116

C₂²

—

117

C₂²

C₂×C₅×#107

yes

characteristic

118

C₂²

C₅×#494

yes

characteristic

119

C₂²

C₅×#500

yes

characteristic

120

C₄

—

121

C₄

C₅×#384

yes

characteristic

122

C₄

C₂×C₅×#107

yes

characteristic

123

C₄

C₅×#493

yes

characteristic

124

C₄

C₅×#500

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₅×#1845

yes

characteristic

128

C₂

C₅×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

subgroup

G/H

center

C₂×C₄×C₅

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₅

Frattini, cosocle

C₂²×C₈

C₂³×C₅

socle

C₂²×C₅

C₂×#107

prime radical

C₂³×C₅

C₂×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

N	H ≅ G/N	type
11. #12910	C₅	direct (N × H)
115. C₅	#12910	direct (N × H)

H ≅ G/N

type

11. #12910

C₅

direct (N × H)

115. C₅

#12910

direct (N × H)

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

subgroup	index
C₂×C₅×#368	2
C₂×C₅×#346	2
C₅×#1848	2
#12910	5

subgroup

index

C₂×C₅×#368

C₂×C₅×#346

C₅×#1848

#12910

Subgroups — C₅×#12910 #11935630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#12910 #11935630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12910 #11935630