Group #28505082 — subgroups

Subgroups — #28505082

Order 15104.

274 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 147 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	#21422190	C₂	1	4	yes	maximal
3	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273663	—	2	2	no	—
5	C₂×#14273663	C₂²	1	4	yes	—
6	#16837426	—	2	2	no	—
7	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#346	C₄	1	4	yes	—
10	C₂×#11745664	—	2	4	no	—
11	#14273663	—	2	8	no	—
12	#14273663	—	4	4	no	—
13	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
14	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	—	2	4	no	—
18	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₅₉	Q₈	1	2	yes	—
21	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
25	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₈×C₅₉	#41	1	4	yes	—
28	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
29	C₄²×C₅₉	D₈	1	2	yes	—
30	#11745664	—	4	4	no	—
31	#11745664	—	4	4	no	—
32	C₂×C₅₉⋊C₈	—	4	4	no	—
33	C₁₆×C₅₉	—	2	4	no	—
34	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	4	2	no	—
36	C₂×C₄×C₅₉	#107	1	4	yes	—
37	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	#124	1	2	yes	—
39	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
41	C₈×C₅₉	#106	1	4	yes	—
42	C₈×C₅₉	#117	1	2	yes	—
43	C₈×C₅₉	#125	1	2	yes	—
44	C₅₉⋊C₈	—	8	4	no	—
45	#12892	—	59	1	no	maximal, Sylow
46	C₂²×C₅₉	—	2	2	no	—
47	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
48	C₂²×C₅₉	#487	1	2	yes	—
49	C₄×C₅₉	—	2	4	no	—
50	C₄×C₅₉	#382	1	1	yes	characteristic
51	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
52	C₄×C₅₉	#486	1	2	yes	—
53	C₂×#346	D₅₉	1	1	yes	characteristic
54	#1848	—	59	2	no	—
55	#1381	—	59	4	no	—
56	C₂×C₅₉	—	2	2	no	—
57	C₂×C₅₉	#1834	1	1	yes	characteristic
58	C₂×C₅₉	#2161	1	2	yes	—
59	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
60	C₂×#108	—	59	4	no	—
61	C₂×#108	—	118	2	no	—
62	#423	—	118	2	no	—
63	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
64	#346	Dic₅₉	1	4	yes	—
65	C₅₉	#12892	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	118	4	no	—
67	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
68	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
69	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
72	#108	—	118	8	no	—
73	#108	—	236	4	no	—
74	C₄⋊C₈	—	236	4	no	—
75	C₄×C₈	Dic₁₁₈	1	2	yes	—
76	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
77	C₂²×C₄	Dic₂₃₆	1	2	yes	—
78	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
79	M₁₆	—	236	4	no	—
80	M₁₆	—	236	4	no	—
81	C₂×C₈	—	236	4	no	—
82	C₂×C₈	#11745660	1	4	yes	—
83	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
84	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
85	C₂×C₈	#11745685	1	4	yes	—
86	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
87	C₄²	D₄₇₂	1	2	yes	—
88	C₄²	C₄⋊Dic₅₉	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#14273661	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
94	C₂×C₄	#14273715	1	2	yes	—
95	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
96	C₂×C₄	C₂×#11745660	1	2	yes	—
97	C₈	—	472	4	no	—
98	C₈	#14273659	1	4	yes	—
99	C₈	#14273686	1	2	yes	—
100	C₈	#14273719	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#14273661	1	1	yes	characteristic
103	C₂²	#16837753	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#16837268	1	1	yes	characteristic
106	C₄	C₂×#14273659	1	1	yes	characteristic
107	C₄	#16837749	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#21424224	1	1	yes	characteristic
110	C₂	#21425891	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal

#21422190

C₂

yes

maximal

C₂×C₅₉×#346

C₂

yes

maximal, characteristic

C₂×#14273663

—

C₂×#14273663

C₂²

yes

—

#16837426

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#346

C₄

yes

—

C₂×#11745664

—

#14273663

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

Q₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

#106

yes

—

C₈×C₅₉

#117

yes

—

C₈×C₅₉

#125

yes

—

C₅₉⋊C₈

—

#12892

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

#487

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×#346

D₅₉

yes

characteristic

#1848

—

#1381

—

C₂×C₅₉

—

C₂×C₅₉

#1834

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

C₂×D₅₉

yes

—

C₂×#108

—

C₂×#108

—

118

—

#423

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#346

Dic₅₉

yes

—

C₅₉

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

D₂₃₆

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

#108

—

118

—

#108

—

236

—

C₄⋊C₈

—

236

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

Dic₂₃₆

yes

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

M₁₆

—

236

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₂×C₈

#11745660

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

C₂×C₈

#11745685

yes

—

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

C₄²

D₄₇₂

yes

—

C₄²

C₄⋊Dic₅₉

yes

—

C₁₆

—

C₂³

C₂×Dic₂₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

#14273661

yes

—

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

#14273715

yes

—

C₂×C₄

C₂×D₄₇₂

yes

characteristic

C₂×C₄

C₂×#11745660

yes

—

C₈

—

472

—

C₈

#14273659

yes

—

C₈

#14273686

yes

—

100

C₈

#14273719

yes

—

101

C₂²

—

102

C₂²

C₂×#14273661

yes

characteristic

103

C₂²

#16837753

yes

—

104

C₄

—

105

C₄

#16837268

yes

characteristic

106

C₄

C₂×#14273659

yes

characteristic

107

C₄

#16837749

yes

—

108

C₂

—

109

C₂

#21424224

yes

characteristic

110

C₂

#21425891

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#12892
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#346

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

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

N	H ≅ G/N	type
65. C₅₉	#12892	semidirect

H ≅ G/N

type

65. C₅₉

#12892

semidirect

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

subgroup	index
#21424268	2
#21422190	2
C₂×C₅₉×#346	2
#12892	59

subgroup

index

#21424268

#21422190

C₂×C₅₉×#346

#12892

Subgroups — #28505082

Conjugacy classes of subgroups

Complements

Maximal subgroups