Group #28509662 — subgroups

Subgroups — #28509662

Order 15104.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21426371	C₂	1	1	yes	maximal, characteristic
2	#21426349	C₂	1	1	yes	maximal, characteristic
3	#21422278	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₇₂	—	2	1	no	—
6	C₂×Dic₄₇₂	C₂²	1	2	yes	—
7	#16837807	—	2	1	no	—
8	#16837782	—	2	1	no	—
9	#16837228	—	2	1	no	—
10	#16837228	C₂²	1	2	yes	—
11	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
12	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₃₆	—	2	2	no	—
14	C₂×Dic₂₃₆	—	4	1	no	—
15	#14273727	—	4	1	no	—
16	#14273727	—	4	1	no	—
17	Dic₄₇₂	—	2	4	no	—
18	Dic₄₇₂	—	4	2	no	—
19	#14273661	—	2	2	no	—
20	#14273661	—	4	1	no	—
21	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₃₂×C₅₉	—	2	2	no	—
25	C₂×Dic₁₁₈	—	4	2	no	—
26	C₂×Dic₁₁₈	—	8	1	no	—
27	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
28	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₅₉	D₈	1	1	yes	characteristic
30	C₄⋊Dic₅₉	—	4	2	no	—
31	C₄⋊Dic₅₉	—	8	1	no	—
32	C₄⋊Dic₅₉	—	8	1	no	—
33	C₄⋊Dic₅₉	—	8	1	no	—
34	Dic₂₃₆	—	4	2	no	—
35	Dic₂₃₆	—	4	2	no	—
36	Dic₂₃₆	—	8	2	no	—
37	C₁₆×C₅₉	SD₁₆	1	2	yes	—
38	C₁₆×C₅₉	#41	1	2	yes	—
39	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₅₉	—	8	2	no	—
43	C₂×Dic₅₉	—	8	2	no	—
44	Dic₁₁₈	—	8	2	no	—
45	Dic₁₁₈	—	8	2	no	—
46	Dic₁₁₈	—	16	2	no	—
47	C₈×C₅₉	SD₃₂	1	2	yes	—
48	C₈×C₅₉	#124	1	1	yes	characteristic
49	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
50	#13720	—	59	1	no	maximal, Sylow
51	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
52	C₄×C₅₉	SD₆₄	1	2	yes	—
53	C₄×C₅₉	Q₆₄	1	2	yes	—
54	C₄×C₅₉	#486	1	1	yes	characteristic
55	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
56	Dic₅₉	—	16	2	no	—
57	Dic₅₉	—	16	2	no	—
58	#2252	—	59	1	no	—
59	#2247	—	59	1	no	—
60	#1416	—	59	4	no	—
61	C₄×C₃₂	D₅₉	1	1	yes	characteristic
62	C₂×C₅₉	#2240	1	1	yes	characteristic
63	C₂×C₅₉	C₂×SD₆₄	1	1	yes	characteristic
64	C₂×C₅₉	C₂×Q₆₄	1	1	yes	characteristic
65	C₂×Q₃₂	—	59	2	no	—
66	C₂×Q₃₂	—	118	1	no	—
67	#500	—	118	1	no	—
68	#494	—	118	1	no	—
69	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
70	#366	—	59	2	no	—
71	#366	—	118	1	no	—
72	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
73	C₅₉	#13720	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₁₆	—	118	2	no	—
75	C₂×Q₁₆	—	236	1	no	—
76	C₄⋊Q₈	—	236	1	no	—
77	C₄⋊Q₈	—	236	1	no	—
78	Q₃₂	—	118	4	no	—
79	Q₃₂	—	236	2	no	—
80	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
82	#107	—	118	2	no	—
83	#107	—	236	1	no	—
84	C₄×C₈	D₂₃₆	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	472	1	no	—
87	C₂×Q₈	—	236	2	no	—
88	Q₁₆	—	472	2	no	—
89	Q₁₆	—	236	2	no	—
90	Q₁₆	—	236	2	no	—
91	C₂×C₈	D₄₇₂	1	1	yes	characteristic
92	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
93	C₄⋊C₄	—	472	1	no	—
94	C₄⋊C₄	—	472	1	no	—
95	C₄⋊C₄	—	472	1	no	—
96	C₄⋊C₄	—	236	2	no	—
97	C₄²	D₄₇₂	1	1	yes	characteristic
98	C₁₆	#11745660	1	2	yes	—
99	C₁₆	#11745685	1	2	yes	—
100	Q₈	—	944	2	no	—
101	Q₈	—	472	2	no	—
102	Q₈	—	472	2	no	—
103	C₂×C₄	—	472	2	no	—
104	C₂×C₄	—	472	2	no	—
105	C₂×C₄	D₉₄₄	1	1	yes	characteristic
106	C₂×C₄	D₉₄₄	1	1	yes	characteristic
107	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
108	C₈	#14273672	1	2	yes	—
109	C₈	#14273715	1	1	yes	characteristic
110	C₈	C₂×#11745660	1	1	yes	characteristic
111	C₂²	C₂×D₉₄₄	1	1	yes	characteristic
112	C₄	—	944	2	no	—
113	C₄	—	944	2	no	—
114	C₄	#16837241	1	2	yes	—
115	C₄	Dic₉₄₄	1	2	yes	—
116	C₄	#16837749	1	1	yes	characteristic
117	C₄	C₂×#14273672	1	1	yes	characteristic
118	C₂	#21426320	1	1	yes	characteristic
119	C₂	C₂×#16837241	1	1	yes	characteristic
120	C₂	C₂×Dic₉₄₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic

#21426349

C₂

yes

maximal, characteristic

#21422278

C₂

yes

maximal

C₄×C₃₂×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

—

C₂×Dic₄₇₂

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₅₉

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

#14273727

—

#14273727

—

Dic₄₇₂

—

Dic₄₇₂

—

#14273661

—

#14273661

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₃₂×C₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

SD₁₆

yes

—

C₁₆×C₅₉

#41

yes

—

C₂×C₄×C₅₉

D₁₆

yes

characteristic

C₂×C₄×C₅₉

D₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

SD₃₂

yes

—

C₈×C₅₉

#124

yes

characteristic

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

#13720

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

SD₆₄

yes

—

C₄×C₅₉

Q₆₄

yes

—

C₄×C₅₉

#486

yes

characteristic

C₄×C₅₉

C₂×SD₃₂

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2252

—

#2247

—

#1416

—

C₄×C₃₂

D₅₉

yes

characteristic

C₂×C₅₉

#2240

yes

characteristic

C₂×C₅₉

C₂×SD₆₄

yes

characteristic

C₂×C₅₉

C₂×Q₆₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

118

—

#500

—

118

—

#494

—

118

—

C₂×C₃₂

C₂×D₅₉

yes

—

#366

—

#366

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#13720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

236

—

C₄⋊Q₈

—

236

—

C₄⋊Q₈

—

236

—

Q₃₂

—

118

—

Q₃₂

—

236

—

C₂×C₁₆

D₂₃₆

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#107

—

118

—

#107

—

236

—

C₄×C₈

D₂₃₆

yes

characteristic

C₃₂

—

C₂×Q₈

—

472

—

C₂×Q₈

—

236

—

Q₁₆

—

472

—

Q₁₆

—

236

—

Q₁₆

—

236

—

C₂×C₈

D₄₇₂

yes

characteristic

C₂×C₈

C₂×D₂₃₆

yes

characteristic

C₄⋊C₄

—

472

—

C₄⋊C₄

—

472

—

C₄⋊C₄

—

472

—

C₄⋊C₄

—

236

—

C₄²

D₄₇₂

yes

characteristic

C₁₆

#11745660

yes

—

C₁₆

#11745685

yes

—

100

Q₈

—

944

—

101

Q₈

—

472

—

102

Q₈

—

472

—

103

C₂×C₄

—

472

—

104

C₂×C₄

—

472

—

105

C₂×C₄

D₉₄₄

yes

characteristic

106

C₂×C₄

D₉₄₄

yes

characteristic

107

C₂×C₄

C₂×D₄₇₂

yes

characteristic

108

C₈

#14273672

yes

—

109

C₈

#14273715

yes

characteristic

110

C₈

C₂×#11745660

yes

characteristic

111

C₂²

C₂×D₉₄₄

yes

characteristic

112

C₄

—

944

—

113

C₄

—

944

—

114

C₄

#16837241

yes

—

115

C₄

Dic₉₄₄

yes

—

116

C₄

#16837749

yes

characteristic

117

C₄

C₂×#14273672

yes

characteristic

118

C₂

#21426320

yes

characteristic

119

C₂

C₂×#16837241

yes

characteristic

120

C₂

C₂×Dic₉₄₄

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₄₄
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₄×C₃₂×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₅₉
hypocenter	C₅₉	#13720
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₉₄₄

commutator, cosocle

C₂×C₁₆×C₅₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₄×C₃₂×C₅₉

C₂

socle, prime radical

C₂²×C₅₉

C₂×D₁₆

hypercenter

C₄×C₃₂

D₅₉

hypocenter

C₅₉

#13720

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
73. C₅₉	#13720	semidirect

H ≅ G/N

type

73. C₅₉

#13720

semidirect

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

subgroup	index
#21426371	2
#21426349	2
#21422278	2
C₄×C₃₂×C₅₉	2
#13720	59

subgroup

index

C₄×C₃₂×C₅₉

Subgroups — #28509662

Conjugacy classes of subgroups

Complements

Maximal subgroups