Group #28509201 — subgroups

Subgroups — #28509201

Order 15104.

178 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16837235	C₂	1	1	yes	maximal, characteristic
2	#368×D₅₉	C₂	1	2	yes	maximal
3	#21422300	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422299	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
7	#108×D₅₉	—	2	2	no	—
8	#16837235	C₄	1	4	yes	—
9	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
10	#16837232	C₂²	1	2	yes	—
11	#16837231	—	2	2	no	—
12	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#368	C₂²	1	2	yes	—
14	M₁₆×D₅₉	—	4	2	no	—
15	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
16	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
17	C₁₆×D₅₉	C₂×C₄	1	2	yes	—
18	C₁₆×D₅₉	Q₈	1	2	yes	—
19	#14273663	—	2	2	no	—
20	#14273662	—	4	2	no	—
21	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
23	#108×C₅₉	—	2	2	no	—
24	C₃₂×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
26	M₁₆×C₅₉	—	4	2	no	—
27	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
28	#11745664	—	4	2	no	—
29	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
30	#11745659	—	8	2	no	—
31	C₈×D₅₉	—	8	2	no	—
32	C₈×D₅₉	C₄⋊C₄	1	2	yes	—
33	C₈×D₅₉	Q₁₆	1	2	yes	—
34	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	C₅₉⋊C₁₆	C₄⋊C₄	1	2	yes	—
37	C₂²×D₅₉	Q₃₂	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
39	C₂×Dic₅₉	Q₃₂	1	1	yes	characteristic
40	C₄×D₅₉	#107	1	2	yes	—
41	C₄×D₅₉	D₁₆	1	2	yes	—
42	C₈×C₅₉	—	8	2	no	—
43	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
45	C₅₉⋊C₈	—	8	2	no	—
46	C₅₉⋊C₈	#107	1	2	yes	—
47	#13636	—	59	1	no	maximal, Sylow
48	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
49	C₂×D₅₉	—	4	1	no	—
50	C₂×D₅₉	#366	1	2	yes	—
51	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
52	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
53	Dic₅₉	#366	1	2	yes	—
54	C₂×M₆₄	—	59	1	no	—
55	C₂×#368	—	59	2	no	—
56	#1425	—	59	1	no	—
57	#1425	—	59	2	no	—
58	#1425	D₅₉	1	1	yes	characteristic, complemented
59	C₂×C₅₉	—	2	1	no	—
60	C₂×C₅₉	C₂×#366	1	1	yes	characteristic
61	D₅₉	—	2	2	no	—
62	C₂²×C₁₆	—	59	1	no	—
63	C₂×#108	—	118	2	no	—
64	M₆₄	—	59	4	no	—
65	C₂×C₃₂	—	59	1	no	—
66	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
67	#368	—	59	2	no	—
68	#368	—	118	2	no	—
69	#368	C₂×D₅₉	1	2	yes	—
70	C₅₉	#13636	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	236	2	no	—
72	C₂²×C₈	—	59	1	no	—
73	C₂×C₁₆	—	59	1	no	—
74	C₂×C₁₆	—	59	2	no	—
75	C₂×C₁₆	—	59	2	no	—
76	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
77	#108	—	2	2	no	—
78	#108	—	118	2	no	—
79	#108	—	236	2	no	—
80	C₃₂	—	59	2	no	—
81	C₃₂	C₄×D₅₉	1	2	yes	—
82	C₂²×C₄	—	59	1	no	—
83	M₁₆	—	4	2	no	—
84	M₁₆	—	472	2	no	—
85	M₁₆	—	236	2	no	—
86	C₂×C₈	—	59	1	no	—
87	C₂×C₈	—	59	2	no	—
88	C₂×C₈	—	59	2	no	—
89	C₂×C₈	—	472	2	no	—
90	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
91	C₁₆	—	59	2	no	—
92	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
93	C₁₆	Q₈×D₅₉	1	1	yes	characteristic
94	C₂³	—	59	1	no	—
95	C₂×C₄	—	59	1	no	—
96	C₂×C₄	—	59	2	no	—
97	C₂×C₄	—	59	2	no	—
98	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
99	C₈	—	8	2	no	—
100	C₈	—	59	2	no	—
101	C₈	—	472	2	no	—
102	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
103	C₈	Q₁₆×D₅₉	1	1	yes	characteristic
104	C₂²	—	59	2	no	—
105	C₂²	—	236	1	no	—
106	C₂²	Q₃₂×D₅₉	1	1	yes	characteristic
107	C₄	—	59	2	no	—
108	C₄	#107×D₅₉	1	1	yes	characteristic
109	C₄	D₁₆×D₅₉	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	118	2	no	—
112	C₂	#366×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837235

C₂

yes

maximal, characteristic

#368×D₅₉

C₂

yes

maximal

#21422300

C₂

yes

maximal, characteristic, complemented

#21422299

C₂

yes

maximal, complemented

C₅₉×#1425

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₅₉

C₂²

yes

characteristic

#108×D₅₉

—

#16837235

C₄

yes

—

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

#16837232

C₂²

yes

—

#16837231

—

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×#368

C₂²

yes

—

M₁₆×D₅₉

—

C₂×C₈×D₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₁₆×D₅₉

C₂×C₄

yes

—

C₁₆×D₅₉

Q₈

yes

—

#14273663

—

#14273662

—

C₅₉⋊C₃₂

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#108×C₅₉

—

C₃₂×C₅₉

C₂×C₄

yes

—

C₂×C₄×D₅₉

D₈

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

D₈

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

C₄⋊C₄

yes

—

C₈×D₅₉

Q₁₆

yes

—

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₅₉⋊C₁₆

C₄⋊C₄

yes

—

C₂²×D₅₉

Q₃₂

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

Q₃₂

yes

characteristic

C₄×D₅₉

#107

yes

—

C₄×D₅₉

D₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

#107

yes

—

#13636

—

maximal, Sylow

C₂²×C₅₉

C₂×Q₃₂

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

#366

yes

—

C₄×C₅₉

C₂×#107

yes

characteristic

C₄×C₅₉

C₂×D₁₆

yes

characteristic

Dic₅₉

#366

yes

—

C₂×M₆₄

—

—

—

—

D₅₉

yes

characteristic, complemented

C₂×C₅₉

—

C₂×C₅₉

C₂×#366

yes

characteristic

D₅₉

—

C₂²×C₁₆

—

C₂×#108

—

118

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

#368

—

#368

—

118

—

#368

C₂×D₅₉

yes

—

C₅₉

#13636

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

236

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#108

—

#108

—

118

—

#108

—

236

—

C₃₂

—

C₃₂

C₄×D₅₉

yes

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

472

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

472

—

C₂×C₈

D₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

C₂×C₄×D₅₉

yes

characteristic

C₁₆

Q₈×D₅₉

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×D₅₉

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

472

—

102

C₈

C₄⋊C₄×D₅₉

yes

characteristic

103

C₈

Q₁₆×D₅₉

yes

characteristic

104

C₂²

—

105

C₂²

—

236

—

106

C₂²

Q₃₂×D₅₉

yes

characteristic

107

C₄

—

108

C₄

#107×D₅₉

yes

characteristic

109

C₄

D₁₆×D₅₉

yes

characteristic

110

C₂

—

111

C₂

—

118

—

112

C₂

#366×D₅₉

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₅₉
commutator	C₁₆×C₅₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1425	C₂
socle	C₂×C₅₉	C₂×#366
prime radical	C₂²×D₅₉	Q₃₂
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#1425	D₅₉
hypocenter	C₅₉	#13636
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₄

D₁₆×D₅₉

commutator

C₁₆×C₅₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#1425

C₂

socle

C₂×C₅₉

C₂×#366

prime radical

C₂²×D₅₉

Q₃₂

cosocle

C₂×C₁₆×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

N	H ≅ G/N	type
3. #21422300	C₂	semidirect
4. #21422299	C₂	semidirect
5. C₅₉×#1425	C₂	semidirect
58. #1425	D₅₉	semidirect
70. C₅₉	#13636	semidirect

H ≅ G/N

type

3. #21422300

C₂

semidirect

4. #21422299

C₂

semidirect

5. C₅₉×#1425

C₂

semidirect

58. #1425

D₅₉

semidirect

70. C₅₉

#13636

semidirect

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

subgroup	index
C₂×#16837235	2
#368×D₅₉	2
#21422300	2
#21422299	2
C₅₉×#1425	2
#13636	59

subgroup

index

#368×D₅₉

C₅₉×#1425

Subgroups — #28509201

Conjugacy classes of subgroups

Complements

Maximal subgroups