Group #12014716 — subgroups

Subgroups — #12014716

Order 1280.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#64726	C₂	1	2	yes	maximal
3	C₄×#3951	C₂	1	2	yes	maximal
4	#10598299	C₂	1	2	yes	maximal
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3951	C₂²	1	4	yes	—
9	C₄×#681	—	2	2	no	—
10	C₅×#346	C₄	1	4	yes	—
11	#64772	—	2	2	no	—
12	#64726	C₄	1	8	yes	complemented
13	#12676	—	5	1	no	maximal, Sylow
14	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
17	C₂×#681	—	2	4	no	—
18	C₂×C₅×C₁₆	—	2	2	no	—
19	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
20	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	Q₈	1	2	yes	—
24	#3951	—	2	4	no	—
25	#3951	C₂×C₄	1	8	yes	—
26	#3935	—	4	2	no	—
27	C₄×#168	—	4	2	no	—
28	C₂×#368	—	5	2	no	—
29	C₂×#346	D₅	1	1	yes	characteristic
30	C₄×#108	—	5	2	no	—
31	#1381	—	5	2	no	—
32	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
33	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
34	C₂×C₅×C₈	—	2	2	no	—
35	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
37	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₅×C₈	#41	1	2	yes	—
40	C₄²×C₅	C₄⋊C₄	1	2	yes	—
41	C₄²×C₅	D₈	1	2	yes	—
42	#681	—	2	8	no	—
43	#681	—	4	4	no	—
44	C₂×#168	—	4	4	no	—
45	C₅×C₁₆	—	2	2	no	—
46	C₅×C₁₆	C₄²	1	4	yes	—
47	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
48	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
49	C₂×#108	—	5	4	no	—
50	C₄×M₁₆	—	10	2	no	—
51	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
52	#368	—	5	8	no	—
53	#346	Dic₅	1	4	yes	—
54	#341	—	10	2	no	—
55	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅	—	4	2	no	—
57	C₂×C₄×C₅	#107	1	4	yes	—
58	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅	#135	1	2	yes	—
62	C₅×C₈	—	2	2	no	—
63	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
64	C₅×C₈	#117	1	1	yes	characteristic
65	C₅×C₈	C₄×Q₈	1	2	yes	—
66	#168	—	4	8	no	—
67	C₂×M₁₆	—	10	4	no	—
68	C₂²×C₈	D₂₀	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
70	C₂×C₄²	D₂₀	1	1	yes	characteristic
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₄×D₅	1	4	yes	—
73	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
74	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
75	#108	—	5	8	no	—
76	#108	—	10	4	no	—
77	#97	—	20	2	no	—
78	C₄×C₈	—	20	2	no	—
79	C₄×C₈	Dic₁₀	1	2	yes	—
80	C₄×C₈	C₂×Dic₅	1	2	yes	—
81	C₂²×C₅	—	2	2	no	—
82	C₂²×C₅	C₂×#107	1	1	yes	characteristic
83	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
84	C₄×C₅	—	2	4	no	—
85	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅	#427	1	1	yes	characteristic
87	C₄×C₅	C₄×D₈	1	2	yes	—
88	C₂²×C₄	Dic₂₀	1	2	yes	—
89	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
90	M₁₆	—	10	8	no	—
91	M₁₆	—	20	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	20	4	no	—
94	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
95	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
96	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
97	C₂×C₈	#709	1	2	yes	—
98	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
99	C₄²	D₄₀	1	2	yes	—
100	C₄²	C₄⋊Dic₅	1	2	yes	—
101	C₁₆	—	2	2	no	—
102	C₁₆	C₄×Dic₅	1	4	yes	—
103	C₂×C₅	—	2	2	no	—
104	C₂×C₅	C₄×#107	1	1	yes	characteristic
105	C₂×C₅	#2161	1	2	yes	—
106	C₂³	C₂×Dic₂₀	1	1	yes	characteristic
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	#3950	1	4	yes	—
109	C₂×C₄	C₄×D₂₀	1	2	yes	—
110	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
111	C₂×C₄	#4050	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	20	8	no	—
115	C₈	C₄×Dic₁₀	1	2	yes	—
116	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
117	C₈	#4072	1	1	yes	characteristic
118	C₅	#12676	1	1	yes	characteristic, complemented, Sylow
119	C₂²	—	2	2	no	—
120	C₂²	C₄×Dic₂₀	1	2	yes	—
121	C₂²	C₂×#3950	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×D₄₀	1	2	yes	—
124	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
125	C₄	#65395	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₄×#3950	1	1	yes	characteristic
128	C₂	#10602222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#346

C₂

yes

maximal, characteristic

C₂×#64726

C₂

yes

maximal

C₄×#3951

C₂

yes

maximal

#10598299

C₂

yes

maximal

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3951

C₂²

yes

—

C₄×#681

—

C₅×#346

C₄

yes

—

#64772

—

#64726

C₄

yes

complemented

#12676

—

maximal, Sylow

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×#681

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

#3951

—

#3951

C₂×C₄

yes

—

—

—

—

D₅

yes

characteristic

C₄×#108

—

#1381

—

C₂²×C₄×C₅

Q₁₆

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

D₈

yes

—

#681

—

#681

—

C₂×#168

—

C₅×C₁₆

—

C₅×C₁₆

C₄²

yes

—

C₂²×C₁₆

C₂×D₅

yes

characteristic

C₂²×C₁₆

C₂×D₅

yes

characteristic

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#368

—

#346

Dic₅

yes

—

#341

—

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

C₄×D₄

yes

—

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₅×C₈

—

C₅×C₈

C₂×C₄²

yes

characteristic

C₅×C₈

#117

yes

characteristic

C₅×C₈

C₄×Q₈

yes

—

#168

—

C₂×M₁₆

—

C₂²×C₈

D₂₀

yes

characteristic

C₂²×C₈

C₂²×D₅

yes

characteristic

C₂×C₄²

D₂₀

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅

yes

—

C₂×C₁₆

C₂×Dic₅

yes

—

C₂×C₁₆

C₂×Dic₅

yes

—

#108

—

#108

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

C₂×Dic₅

yes

—

C₂²×C₅

—

C₂²×C₅

C₂×#107

yes

characteristic

C₂²×C₅

C₄×Q₁₆

yes

—

C₄×C₅

—

C₄×C₅

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅

#427

yes

characteristic

C₄×C₅

C₄×D₈

yes

—

C₂²×C₄

Dic₂₀

yes

—

C₂²×C₄

C₂×D₂₀

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₅

yes

—

C₂×C₈

C₂×Dic₁₀

yes

characteristic

C₂×C₈

C₂×C₄×D₅

yes

—

C₂×C₈

#709

yes

—

C₂×C₈

C₂²×Dic₅

yes

characteristic

C₄²

D₄₀

yes

—

100

C₄²

C₄⋊Dic₅

yes

—

101

C₁₆

—

102

C₁₆

C₄×Dic₅

yes

—

103

C₂×C₅

—

104

C₂×C₅

C₄×#107

yes

characteristic

105

C₂×C₅

#2161

yes

—

106

C₂³

C₂×Dic₂₀

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

#3950

yes

—

109

C₂×C₄

C₄×D₂₀

yes

—

110

C₂×C₄

C₂×D₄₀

yes

characteristic

111

C₂×C₄

#4050

yes

—

112

C₂×C₄

C₂×C₄⋊Dic₅

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

C₄×Dic₁₀

yes

—

116

C₈

C₂×C₄×Dic₅

yes

characteristic

117

C₈

#4072

yes

characteristic

118

C₅

#12676

yes

characteristic, complemented, Sylow

119

C₂²

—

120

C₂²

C₄×Dic₂₀

yes

—

121

C₂²

C₂×#3950

yes

characteristic

122

C₄

—

123

C₄

C₄×D₄₀

yes

—

124

C₄

C₄×C₄⋊Dic₅

yes

characteristic

125

C₄

#65395

yes

characteristic

126

C₂

—

127

C₂

C₄×#3950

yes

characteristic

128

C₂

#10602222

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#346	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12676
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₀

commutator

C₅×C₈

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₂×C₅×#346

C₂

socle

C₂²×C₅

C₂×#107

prime radical

C₂³×C₅

C₂×Q₁₆

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
12. #64726	C₄	semidirect
118. C₅	#12676	semidirect

H ≅ G/N

type

12. #64726

C₄

semidirect

118. C₅

#12676

semidirect

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

subgroup	index
C₂×C₅×#346	2
C₂×#64726	2
C₄×#3951	2
#10598299	2
#12676	5

subgroup

index

C₂×C₅×#346

Subgroups — #12014716

Conjugacy classes of subgroups

Complements

Maximal subgroups