Group #28505072 — subgroups

Subgroups — #28505072

Order 15104.

310 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21424260	C₂	1	2	yes	maximal
2	#21422188	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273661	C₂²	1	4	yes	—
5	C₂×#14273659	—	2	2	no	—
6	#16837268	—	2	2	no	—
7	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₅₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₉	—	2	4	no	—
11	#14273661	—	2	8	no	—
12	#14273659	—	4	4	no	—
13	C₅₉⋊#95	—	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	D₈	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
23	C₂²×Dic₅₉	—	4	4	no	—
24	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
26	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈×C₅₉	#41	1	4	yes	—
29	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
30	C₄²×C₅₉	Q₁₆	1	2	yes	—
31	C₄⋊Dic₅₉	—	4	4	no	—
32	C₄⋊Dic₅₉	—	4	4	no	—
33	C₁₆×C₅₉	—	2	4	no	—
34	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	4	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×SD₁₆	1	2	yes	—
40	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×Dic₅₉	—	8	4	no	—
42	C₂×Dic₅₉	—	8	8	no	—
43	C₈×C₅₉	#106	1	4	yes	—
44	C₈×C₅₉	#117	1	2	yes	—
45	C₈×C₅₉	#125	1	2	yes	—
46	#12889	—	59	1	no	maximal, Sylow
47	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
48	C₂²×C₅₉	#486	1	2	yes	—
49	C₂²×C₅₉	#508	1	4	yes	—
50	C₄×C₅₉	—	2	4	no	—
51	C₄×C₅₉	#382	1	1	yes	characteristic
52	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
53	C₄×C₅₉	#487	1	2	yes	—
54	Dic₅₉	—	16	4	no	—
55	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
56	#1846	—	59	2	no	—
57	#1380	—	59	4	no	—
58	C₂×C₅₉	#1834	1	1	yes	characteristic
59	C₂×C₅₉	#2158	1	2	yes	—
60	C₂×C₅₉	#2242	1	4	yes	—
61	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
62	C₂×#107	—	59	4	no	—
63	C₂×#106	—	118	2	no	—
64	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
65	#382	—	118	2	no	—
66	C₄×C₁₆	Dic₅₉	1	4	yes	complemented
67	C₅₉	#12889	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	118	4	no	—
71	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
74	#107	—	118	8	no	—
75	#106	—	236	4	no	—
76	C₄×C₈	Dic₁₁₈	1	2	yes	—
77	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
78	#95	—	236	4	no	—
79	C₂²×C₄	—	236	4	no	—
80	C₂²×C₄	D₄₇₂	1	2	yes	—
81	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
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	C₄⋊C₄	—	236	4	no	—
88	C₄⋊C₄	—	236	4	no	—
89	C₄²	Dic₂₃₆	1	2	yes	—
90	C₄²	C₄⋊Dic₅₉	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	472	4	no	—
95	C₂×C₄	—	472	8	no	—
96	C₂×C₄	#14273661	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
98	C₂×C₄	#14273715	1	2	yes	—
99	C₂×C₄	C₂×#11745660	1	2	yes	—
100	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
101	C₈	#14273659	1	4	yes	—
102	C₈	#14273686	1	2	yes	—
103	C₈	#14273719	1	2	yes	—
104	C₂²	C₂×#14273661	1	1	yes	characteristic
105	C₂²	#16837749	1	2	yes	—
106	C₂²	#16837834	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	944	4	no	—
109	C₄	#16837268	1	1	yes	characteristic
110	C₄	C₂×#14273659	1	1	yes	characteristic
111	C₄	#16837753	1	2	yes	—
112	C₂	#21424224	1	1	yes	characteristic
113	C₂	#21425880	1	2	yes	—
114	C₂	#21426328	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424260

C₂

yes

maximal

#21422188

C₂

yes

maximal

C₂×C₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273661

C₂²

yes

—

C₂×#14273659

—

#16837268

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273659

—

C₅₉⋊#95

—

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

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

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

Q₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×D₈

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×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#106

yes

—

C₈×C₅₉

#117

yes

—

C₈×C₅₉

#125

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#508

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

C₂×C₄×C₁₆

D₅₉

yes

characteristic

#1846

—

#1380

—

C₂×C₅₉

#1834

yes

characteristic

C₂×C₅₉

#2158

yes

—

C₂×C₅₉

#2242

yes

—

C₂²×C₁₆

C₂×D₅₉

yes

—

C₂×#107

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

—

118

—

C₄×C₁₆

Dic₅₉

yes

complemented

C₅₉

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₃₆

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

#107

—

118

—

#106

—

236

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

#95

—

236

—

C₂²×C₄

—

236

—

C₂²×C₄

D₄₇₂

yes

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

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

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

Dic₂₃₆

yes

—

C₄²

C₄⋊Dic₅₉

yes

—

C₁₆

—

C₂³

C₂×D₄₇₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

472

—

C₂×C₄

—

472

—

C₂×C₄

#14273661

yes

—

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

#14273715

yes

—

C₂×C₄

C₂×#11745660

yes

—

100

C₂×C₄

C₂×Dic₂₃₆

yes

characteristic

101

C₈

#14273659

yes

—

102

C₈

#14273686

yes

—

103

C₈

#14273719

yes

—

104

C₂²

C₂×#14273661

yes

characteristic

105

C₂²

#16837749

yes

—

106

C₂²

#16837834

yes

—

107

C₄

—

108

C₄

—

944

—

109

C₄

#16837268

yes

characteristic

110

C₄

C₂×#14273659

yes

characteristic

111

C₄

#16837753

yes

—

112

C₂

#21424224

yes

characteristic

113

C₂

#21425880

yes

—

114

C₂

#21426328

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#12889
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₄×C₁₆×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₅₉

hypocenter

C₅₉

#12889

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
9. C₄×C₁₆×C₅₉	C₄	semidirect
66. C₄×C₁₆	Dic₅₉	semidirect
67. C₅₉	#12889	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₅₉

C₄

semidirect

66. C₄×C₁₆

Dic₅₉

semidirect

67. C₅₉

#12889

semidirect

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

subgroup	index
#21424260	2
#21422188	2
C₂×C₄×C₁₆×C₅₉	2
#12889	59

subgroup

index

#21424260

#21422188

C₂×C₄×C₁₆×C₅₉

#12889

Subgroups — #28505072

Conjugacy classes of subgroups

Complements

Maximal subgroups