Group #13014167 — subgroups

Subgroups — #13014167

Order 1280.

152 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 38 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	#10600585	C₂	1	1	yes	maximal, characteristic
2	#10598691	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
4	C₂×#3951	—	2	2	no	—
5	#65213	C₄	1	1	yes	characteristic
6	#65213	C₂²	1	1	yes	characteristic
7	#64912	—	2	2	no	—
8	#7373	—	5	1	no	maximal, Sylow
9	C₂²×C₅×C₈	—	2	1	no	—
10	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
11	C₂×#681	—	2	1	no	—
12	C₂×#681	D₄	1	2	yes	—
13	#3998	—	4	2	no	—
14	C₄×C₅×C₈	—	2	1	no	—
15	C₄×C₅×C₈	C₈	1	2	yes	—
16	#3951	—	4	4	no	—
17	C₄⋊#168	—	4	1	no	—
18	C₄⋊#168	—	4	1	no	—
19	#1848	—	5	1	no	—
20	#1327	—	5	2	no	—
21	C₂²×C₄×C₅	—	2	1	no	—
22	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₅×C₈	—	2	2	no	—
24	C₂×C₅×C₈	—	2	2	no	—
25	C₂×C₅×C₈	—	2	2	no	—
26	C₄²×C₅	—	2	1	no	—
27	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
28	C₄²×C₅	M₁₆	1	1	yes	characteristic
29	#681	—	4	2	no	—
30	#681	—	4	2	no	—
31	C₂×#168	—	2	2	no	—
32	C₂×#168	—	4	1	no	—
33	#674	—	8	2	no	—
34	C₂×#108	—	10	2	no	—
35	#423	—	5	1	no	—
36	#423	—	5	1	no	—
37	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
38	#349	—	10	2	no	—
39	C₂³×C₅	#101	1	1	yes	characteristic
40	C₂×C₄×C₅	—	2	1	no	—
41	C₂×C₄×C₅	—	2	1	no	—
42	C₂×C₄×C₅	—	2	1	no	—
43	C₂×C₄×C₅	—	2	1	no	—
44	C₂×C₄×C₅	—	4	2	no	—
45	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅	#99	1	1	yes	characteristic
47	C₅×C₈	—	2	4	no	—
48	#168	—	4	2	no	—
49	#168	—	8	1	no	—
50	C₂×M₁₆	—	5	2	no	—
51	C₂×M₁₆	—	10	1	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
54	M₃₂	—	20	2	no	—
55	#108	—	20	4	no	—
56	C₄⋊C₈	—	20	1	no	—
57	C₄⋊C₈	—	20	1	no	—
58	C₄×C₈	—	2	1	no	—
59	C₄×C₈	#170	1	2	yes	—
60	C₂²×C₅	—	2	1	no	—
61	C₂²×C₅	—	4	1	no	—
62	C₂²×C₅	#324	1	1	yes	characteristic
63	C₄×C₅	—	2	2	no	—
64	C₄×C₅	—	2	1	no	—
65	C₄×C₅	—	4	1	no	—
66	C₄×C₅	#353	1	1	yes	characteristic
67	C₄×C₅	#356	1	1	yes	characteristic
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
70	M₁₆	—	20	2	no	—
71	M₁₆	—	20	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	10	2	no	—
76	C₂×C₈	—	20	1	no	—
77	C₄²	—	2	1	no	—
78	C₄²	C₂×#170	1	1	yes	characteristic
79	C₄²	#704	1	1	yes	characteristic
80	C₁₆	—	40	2	no	—
81	C₂×C₅	—	2	1	no	—
82	C₂×C₅	—	4	1	no	—
83	C₂×C₅	#1325	1	1	yes	characteristic
84	C₂³	#4005	1	1	yes	characteristic
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#3999	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊#170	1	1	yes	characteristic
92	C₈	—	2	4	no	—
93	C₈	—	20	2	no	—
94	C₈	—	40	1	no	—
95	C₅	#7373	1	1	yes	characteristic, complemented, Sylow
96	C₂²	—	2	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	#64913	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	1	no	—
102	C₄	#64853	1	1	yes	characteristic
103	C₄	#64855	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	#10598689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10600585

C₂

yes

maximal, characteristic

#10598691

C₂

yes

maximal

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×#3951

—

#65213

C₄

yes

characteristic

#65213

C₂²

yes

—

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×#681

—

C₂×#681

D₄

yes

—

#3998

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₈

yes

—

—

—

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

M₁₆

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

#349

—

C₂³×C₅

#101

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

#99

yes

characteristic

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

M₃₂

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

#170

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#324

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₂²×C₄

—

C₂²×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3999

yes

characteristic

C₂×C₄

C₂²⋊#170

yes

characteristic

C₈

—

C₈

—

C₈

—

C₅

#7373

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#64913

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#64853

yes

characteristic

103

C₄

#64855

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#10598689

yes

characteristic

	subgroup	G/H
center	C₄	#64853
commutator	C₄²×C₅	C₂×C₈
Frattini	C₂×C₄²	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₈	C₄
socle	C₂×C₅	#1325
prime radical	C₂³×C₅	#101
cosocle	#65213	C₂²
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#7373
LCS	G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄

#64853

commutator

C₄²×C₅

C₂×C₈

Frattini

C₂×C₄²

C₂×AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅×C₈

C₄

C₂×C₅

C₂³×C₅

C₂²

C₂×C₄×C₈

C₅

G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
95. C₅	#7373	semidirect

H ≅ G/N

type

95. C₅

#7373

semidirect

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

subgroup	index
#10600585	2
#10598691	2
#7373	5

subgroup

index

#10600585

#10598691

#7373

Subgroups — #13014167

Conjugacy classes of subgroups

Complements

Maximal subgroups