Group #28482500 — subgroups

Subgroups — #28482500

Order 15104.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#21424268	C₂	1	2	yes	maximal
2	#21422852	C₂	1	4	yes	maximal
3	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273663	—	2	4	no	—
5	#16837426	C₂²	1	4	yes	—
6	#16837153	—	2	8	no	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#322	C₄	1	4	yes	—
10	C₂×#11745664	—	2	4	no	—
11	#14273663	—	4	8	no	—
12	C₄⋊(C₅₉⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₅₉	D₄	1	2	yes	—
14	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
15	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₉	D₄	1	4	yes	—
19	C₄×C₈×C₅₉	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
22	C₂×C₈×C₅₉	—	4	4	no	—
23	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
24	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
25	C₂×C₈×C₅₉	SD₁₆	1	8	yes	—
26	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
27	C₄²×C₅₉	C₂×D₄	1	2	yes	—
28	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
29	#11745664	—	4	8	no	—
30	C₂×C₅₉⋊C₈	—	4	4	no	—
31	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	2	4	no	—
33	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
38	C₈×C₅₉	—	2	8	no	—
39	C₈×C₅₉	#106	1	8	yes	—
40	C₅₉⋊C₈	—	8	4	no	—
41	#8825	—	59	1	no	maximal, Sylow
42	C₂²×C₅₉	—	2	2	no	—
43	C₂²×C₅₉	#384	1	1	yes	characteristic
44	C₂²×C₅₉	#498	1	2	yes	—
45	C₄×C₅₉	—	2	2	no	—
46	C₄×C₅₉	C₂×#106	1	2	yes	—
47	C₄×C₅₉	#430	1	4	yes	—
48	C₄×C₅₉	#492	1	2	yes	—
49	#1848	—	59	2	no	—
50	#1566	—	59	4	no	—
51	C₂×#322	D₅₉	1	1	yes	characteristic
52	C₂×C₅₉	—	2	2	no	—
53	C₂×C₅₉	#1844	1	1	yes	characteristic
54	C₂×C₅₉	#1849	1	2	yes	—
55	C₂×#108	—	118	4	no	—
56	#423	—	59	4	no	—
57	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
59	#334	—	118	8	no	—
60	#322	Dic₅₉	1	4	yes	—
61	C₅₉	#8825	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	118	4	no	—
63	C₂²×C₈	Dic₁₁₈	1	4	yes	—
64	C₂²×C₈	D₂₃₆	1	2	yes	—
65	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
66	#108	—	236	8	no	—
67	C₄⋊C₈	—	118	8	no	—
68	C₄×C₈	Dic₁₁₈	1	2	yes	—
69	C₄×C₈	D₂₃₆	1	4	yes	—
70	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
71	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
73	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
74	M₁₆	—	236	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	236	4	no	—
77	C₂×C₈	#11745660	1	8	yes	—
78	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
79	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
80	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
81	C₄²	C₂×D₂₃₆	1	2	yes	—
82	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
83	C₂³	#14273727	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
87	C₂×C₄	#14273724	1	1	yes	characteristic
88	C₂×C₄	#14273727	1	2	yes	—
89	C₂×C₄	C₂×#11745660	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	472	4	no	—
92	C₈	#14273659	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#16837272	1	1	yes	characteristic
95	C₂²	#16837800	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#14273659	1	2	yes	—
98	C₄	#16837451	1	4	yes	—
99	C₄	#16837775	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	#21424253	1	1	yes	characteristic
102	C₂	#21424271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal

#21422852

C₂

yes

maximal

C₂×C₅₉×#322

C₂

yes

maximal, characteristic

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837153

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

C₄

yes

—

C₂×#11745664

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

Q₈

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

#106

yes

—

C₅₉⋊C₈

—

#8825

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#384

yes

characteristic

C₂²×C₅₉

#498

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#106

yes

—

C₄×C₅₉

#430

yes

—

C₄×C₅₉

#492

yes

—

#1848

—

#1566

—

C₂×#322

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1844

yes

characteristic

C₂×C₅₉

#1849

yes

—

C₂×#108

—

118

—

#423

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

#334

—

118

—

#322

Dic₅₉

yes

—

C₅₉

#8825

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

Dic₁₁₈

yes

—

C₂²×C₈

D₂₃₆

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#108

—

236

—

C₄⋊C₈

—

118

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

D₂₃₆

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₁₁₈

yes

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

#11745660

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₄²

C₂×Dic₁₁₈

yes

characteristic

C₄²

C₂×D₂₃₆

yes

—

C₄²

C₂²×Dic₅₉

yes

characteristic

C₂³

#14273727

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

—

C₂×C₄

#14273724

yes

characteristic

C₂×C₄

#14273727

yes

—

C₂×C₄

C₂×#11745660

yes

—

C₈

—

C₈

—

472

—

C₈

#14273659

yes

—

C₂²

—

C₂²

#16837272

yes

characteristic

C₂²

#16837800

yes

—

C₄

—

C₄

C₂×#14273659

yes

—

C₄

#16837451

yes

—

C₄

#16837775

yes

—

100

C₂

—

101

C₂

#21424253

yes

characteristic

102

C₂

#21424271

yes

—

	subgroup	G/H
center	C₂×C₄	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle	C₂²×C₅₉	#384
prime radical	C₂³×C₅₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#8825
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#322

C₂

socle

C₂²×C₅₉

#384

prime radical

C₂³×C₅₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
61. C₅₉	#8825	semidirect

H ≅ G/N

type

61. C₅₉

#8825

semidirect

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

subgroup	index
#21424268	2
#21422852	2
C₂×C₅₉×#322	2
#8825	59

subgroup

index

#21424268

#21422852

C₂×C₅₉×#322

#8825

Subgroups — #28482500

Conjugacy classes of subgroups

Complements

Maximal subgroups