Group #11751172 — subgroups

Subgroups — #11751172

Order 960.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	#91759	C₂	1	1	yes	maximal, characteristic, complemented
3	#91116	C₂	1	4	yes	maximal, complemented
4	#91087	C₂	1	1	yes	maximal, characteristic, complemented
5	#65360	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×D₄	—	2	2	no	—
7	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
8	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₅	C₂²	1	2	yes	—
10	#6712	—	2	2	no	—
11	C₂×C₃⋊#168	C₂²	1	2	yes	—
12	#5071	—	5	1	no	maximal, Hall
13	C₅×#127	D₃	1	1	yes	characteristic
14	#4016	—	3	1	no	—
15	#3964	—	3	4	no	—
16	#3935	—	3	1	no	—
17	C₂³×C₃×C₅	—	2	2	no	—
18	C₃×C₅×D₄	—	2	4	no	—
19	C₃×C₅×D₄	—	4	4	no	—
20	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₅	—	2	2	no	—
23	C₃⋊#168	—	2	2	no	—
24	C₃×#127	D₅	1	1	yes	characteristic
25	#870	—	5	1	no	—
26	#832	—	5	4	no	—
27	#803	—	5	1	no	—
28	C₂×C₅×D₄	—	2	2	no	—
29	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
30	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
31	C₄⋊Dic₅	—	3	2	no	—
32	#683	—	6	2	no	—
33	C₂×#168	—	3	2	no	—
34	#490	—	15	1	no	Sylow
35	C₂²×C₃×C₅	—	4	2	no	—
36	C₂²×C₃×C₅	—	4	4	no	—
37	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
38	C₃×C₄×C₅	—	2	2	no	—
39	C₃×C₄×C₅	#41	1	2	yes	—
40	Dic₁₅	—	4	2	no	—
41	C₂×C₃×D₄	—	2	2	no	—
42	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
43	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
44	C₄⋊Dic₃	—	5	2	no	—
45	Dic₃⋊C₄	—	10	2	no	—
46	C₂×C₃⋊C₈	—	5	2	no	—
47	C₂³×C₅	—	2	2	no	—
48	C₅×D₄	—	2	4	no	—
49	C₅×D₄	—	4	4	no	—
50	C₂×C₄×C₅	#67	1	2	yes	—
51	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
52	C₂×Dic₅	—	6	2	no	—
53	#168	—	6	2	no	—
54	#127	D₁₅	1	1	yes	characteristic
55	#125	—	15	1	no	—
56	#102	—	15	4	no	—
57	#97	—	15	1	no	—
58	C₂×C₃×C₅	—	8	2	no	—
59	C₂×C₃×C₅	#124	1	1	yes	characteristic
60	C₂×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
61	C₂³×C₃	—	2	2	no	—
62	C₃×D₄	—	2	4	no	—
63	C₃×D₄	—	4	4	no	—
64	C₂×C₃×C₄	C₂²⋊D₅	1	2	yes	—
65	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
66	C₂×Dic₃	—	10	2	no	—
67	C₃⋊C₈	—	10	2	no	—
68	C₂²×C₅	—	4	4	no	—
69	C₂²×C₅	—	4	2	no	—
70	C₂²×C₅	C₂×#67	1	1	yes	characteristic
71	C₄×C₅	—	2	2	no	—
72	C₄×C₅	#237	1	2	yes	—
73	Dic₅	—	12	2	no	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	C₂×D₁₅	1	2	yes	—
76	C₂×C₈	—	15	2	no	—
77	C₄⋊C₄	—	15	2	no	—
78	C₄⋊C₄	—	30	2	no	—
79	C₄²	C₂×D₁₅	1	1	yes	characteristic
80	C₃×C₅	#490	1	1	yes	characteristic, complemented, Hall
81	C₂²×C₃	—	4	4	no	—
82	C₂²×C₃	—	4	2	no	—
83	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
84	C₃×C₄	—	2	2	no	—
85	C₃×C₄	#711	1	2	yes	—
86	Dic₃	—	20	2	no	—
87	C₂×C₅	—	8	2	no	—
88	C₂×C₅	#932	1	2	yes	—
89	C₂×C₅	#936	1	1	yes	characteristic
90	C₂³	—	2	2	no	—
91	D₄	—	2	4	no	—
92	D₄	—	4	4	no	—
93	C₂×C₄	—	30	2	no	—
94	C₂×C₄	#1220	1	2	yes	—
95	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
96	C₈	—	30	2	no	—
97	C₂×C₃	—	8	2	no	—
98	C₂×C₃	#4078	1	2	yes	—
99	C₂×C₃	#4082	1	1	yes	characteristic
100	C₅	#5071	1	1	yes	characteristic, complemented, Sylow
101	C₂²	—	4	4	no	—
102	C₂²	—	4	2	no	—
103	C₂²	C₂×#1220	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	60	2	no	—
106	C₄	#6819	1	2	yes	—
107	C₃	#65360	1	1	yes	characteristic, complemented, Sylow
108	C₂	—	8	2	no	—
109	C₂	#91821	1	2	yes	—
110	C₂	#91825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#127

C₂

yes

maximal, characteristic

#91759

C₂

yes

maximal, characteristic, complemented

#91116

C₂

yes

maximal, complemented

#91087

C₂

yes

maximal, characteristic, complemented

#65360

—

maximal, Hall

C₂×C₃×C₅×D₄

—

C₂×C₃×C₅×D₄

C₂²

yes

—

C₃×C₄²×C₅

C₂²

yes

characteristic

C₄⋊Dic₁₅

C₂²

yes

—

#6712

—

C₂×C₃⋊#168

C₂²

yes

—

#5071

—

maximal, Hall

C₅×#127

D₃

yes

—

—

—

C₂³×C₃×C₅

—

C₃×C₅×D₄

—

C₃×C₅×D₄

—

C₂×C₃×C₄×C₅

D₄

yes

—

C₂×C₃×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₁₅

—

C₃⋊#168

—

C₃×#127

D₅

yes

—

—

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₃

yes

—

C₄²×C₅

C₂×D₃

yes

characteristic

C₄⋊Dic₅

—

—

—

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×D₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

#41

yes

—

Dic₁₅

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×D₅

yes

—

C₃×C₄²

C₂×D₅

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

#67

yes

—

C₂×C₄×C₅

C₂²×D₃

yes

characteristic

C₂×Dic₅

—

#168

—

#127

D₁₅

yes

—

—

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#124

yes

characteristic

C₂×C₃×C₅

AGL(1,ℤ/8)

yes

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂²⋊D₅

yes

—

C₂×C₃×C₄

C₂²×D₅

yes

characteristic

C₂×Dic₃

—

C₃⋊C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×#67

yes

characteristic

C₄×C₅

—

C₄×C₅

#237

yes

—

Dic₅

—

C₂×D₄

—

C₂×D₄

C₂×D₁₅

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₁₅

yes

characteristic

C₃×C₅

#490

yes

characteristic, complemented, Hall

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊D₅

yes

characteristic

C₃×C₄

—

C₃×C₄

#711

yes

—

Dic₃

—

C₂×C₅

—

C₂×C₅

#932

yes

—

C₂×C₅

#936

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#1220

yes

—

C₂×C₄

C₂²×D₁₅

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

#4078

yes

—

C₂×C₃

#4082

yes

characteristic

100

C₅

#5071

yes

characteristic, complemented, Sylow

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×#1220

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

#6819

yes

—

107

C₃

#65360

yes

characteristic, complemented, Sylow

108

C₂

—

109

C₂

#91821

yes

—

110

C₂

#91825

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₁₅
Fitting, prime radical	C₃×C₅×#127	C₂
socle	C₂²×C₃×C₅	C₂×D₄
hypercenter	#127	D₁₅
hypocenter	C₃×C₅	#490
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×#1220

commutator, cosocle

C₂×C₃×C₄×C₅

C₂³

Frattini

C₂×C₄

C₂²×D₁₅

Fitting, prime radical

C₃×C₅×#127

C₂

socle

C₂²×C₃×C₅

C₂×D₄

D₁₅

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. #91759	C₂	semidirect
3. #91116	C₂	semidirect
4. #91087	C₂	semidirect
80. C₃×C₅	#490	semidirect
100. C₅	#5071	semidirect
107. C₃	#65360	semidirect

H ≅ G/N

type

2. #91759

C₂

semidirect

3. #91116

C₂

semidirect

4. #91087

C₂

semidirect

80. C₃×C₅

#490

semidirect

100. C₅

#5071

semidirect

107. C₃

#65360

semidirect

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

subgroup	index
C₃×C₅×#127	2
#91759	2
#91116	2
#91087	2
#65360	3
#5071	5

subgroup

index

C₃×C₅×#127

Subgroups — #11751172

Conjugacy classes of subgroups

Complements

Maximal subgroups