Group #28509324 — subgroups

Subgroups — #28509324

Order 15104.

158 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₁₆×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₅₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	Dic₅₉⋊C₃₂	C₂	1	2	yes	maximal
4	#21422229	C₂	1	2	yes	maximal
5	C₄×C₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic
7	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
8	C₄⋊(C₅₉⋊C₁₆)	C₄	1	1	yes	characteristic
9	Dic₅₉⋊C₁₆	C₄	1	2	yes	—
10	C₁₆×Dic₅₉	C₂²	1	2	yes	—
11	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
15	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
16	C₄⋊(C₅₉⋊C₈)	C₈	1	1	yes	characteristic
17	Dic₅₉⋊C₈	C₈	1	2	yes	—
18	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
19	C₅₉⋊C₃₂	—	2	2	no	—
20	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₃₂×C₅₉	—	2	1	no	—
24	C₃₂×C₅₉	Q₈	1	2	yes	complemented
25	C₂×Dic₁₁₈	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
29	C₄⋊Dic₅₉	C₁₆	1	1	yes	characteristic
30	Dic₅₉⋊C₄	C₁₆	1	2	yes	—
31	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
32	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
33	C₁₆×C₅₉	—	2	1	no	—
34	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₁₆×C₅₉	#41	1	1	yes	characteristic
36	C₅₉⋊C₁₆	—	2	2	no	—
37	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₅₉	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₅₉	C₂×C₁₆	1	2	yes	—
42	Dic₁₁₈	—	2	2	no	—
43	C₈×C₅₉	—	2	1	no	—
44	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₅₉	#131	1	1	yes	characteristic
46	C₅₉⋊C₈	—	2	2	no	—
47	#13660	—	59	1	no	maximal, Sylow
48	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	M₆₄	1	2	yes	—
51	C₄×C₅₉	C₈×Q₈	1	1	yes	characteristic
52	C₄×C₅₉	#504	1	1	yes	characteristic
53	Dic₅₉	—	2	2	no	—
54	Dic₅₉	—	2	2	no	—
55	Q₈×C₁₆	—	59	1	no	—
56	C₄⋊C₃₂	—	59	2	no	—
57	C₄⋊C₃₂	—	59	1	no	—
58	#1397	—	59	2	no	—
59	C₄×C₃₂	D₅₉	1	1	yes	characteristic
60	C₂×C₅₉	Q₈×C₁₆	1	1	yes	characteristic
61	C₂×C₅₉	C₂×M₆₄	1	1	yes	characteristic
62	C₂×C₅₉	#2258	1	1	yes	characteristic
63	C₈×Q₈	—	59	1	no	—
64	C₂×C₃₂	—	59	2	no	—
65	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
66	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
67	C₄⋊C₁₆	—	59	2	no	—
68	C₄⋊C₁₆	—	59	1	no	—
69	C₄×C₁₆	—	59	2	no	—
70	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
71	C₅₉	#13660	1	1	yes	characteristic, complemented, Sylow
72	C₄×Q₈	—	59	1	no	—
73	C₂×C₁₆	—	59	2	no	—
74	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
76	C₄⋊C₈	—	59	2	no	—
77	C₄⋊C₈	—	59	1	no	—
78	C₄×C₈	—	59	2	no	—
79	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
80	C₃₂	—	2	1	no	—
81	C₃₂	—	118	2	no	—
82	C₃₂	Dic₁₁₈	1	2	yes	complemented
83	C₂×Q₈	—	59	1	no	—
84	C₂×C₈	—	59	2	no	—
85	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
87	C₄⋊C₄	—	59	2	no	—
88	C₄⋊C₄	—	59	1	no	—
89	C₄²	—	59	2	no	—
90	C₄²	C₈×D₅₉	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	118	2	no	—
93	C₁₆	C₂×Dic₁₁₈	1	1	yes	characteristic
94	C₁₆	#11745685	1	1	yes	characteristic
95	Q₈	—	118	2	no	—
96	C₂×C₄	—	59	2	no	—
97	C₂×C₄	—	59	2	no	—
98	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	118	2	no	—
103	C₈	C₄×Dic₁₁₈	1	1	yes	characteristic
104	C₈	#14273733	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	118	2	no	—
108	C₄	—	118	2	no	—
109	C₄	#16837235	1	2	yes	—
110	C₄	C₈×Dic₁₁₈	1	1	yes	characteristic
111	C₄	#16837818	1	1	yes	characteristic
112	C₂	C₁₆×Dic₁₁₈	1	1	yes	characteristic
113	C₂	C₂×#16837235	1	1	yes	characteristic
114	C₂	#21426389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₁₈

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₃₂)

C₂

yes

maximal, characteristic

Dic₅₉⋊C₃₂

C₂

yes

maximal

#21422229

C₂

yes

maximal

C₄×C₃₂×C₅₉

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

C₂²

yes

—

C₄⋊(C₅₉⋊C₁₆)

C₄

yes

characteristic

Dic₅₉⋊C₁₆

C₄

yes

—

C₁₆×Dic₅₉

C₂²

yes

—

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₈

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

C₈

yes

characteristic

Dic₅₉⋊C₈

C₈

yes

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₅₉⋊C₃₂

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₃₂×C₅₉

—

C₃₂×C₅₉

Q₈

yes

complemented

C₂×Dic₁₁₈

C₁₆

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄⋊Dic₅₉

C₁₆

yes

characteristic

Dic₅₉⋊C₄

C₁₆

yes

—

C₄×Dic₅₉

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₁₆×C₅₉

#41

yes

characteristic

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

—

C₂×Dic₅₉

C₂×C₁₆

yes

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₅₉⋊C₈

—

#13660

—

maximal, Sylow

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

M₆₄

yes

—

C₄×C₅₉

C₈×Q₈

yes

characteristic

C₄×C₅₉

#504

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

D₅₉

yes

characteristic

C₂×C₅₉

Q₈×C₁₆

yes

characteristic

C₂×C₅₉

C₂×M₆₄

yes

characteristic

C₂×C₅₉

#2258

yes

characteristic

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

C₂×C₃₂

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#13660

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₉

yes

characteristic

C₃₂

—

C₃₂

—

118

—

C₃₂

Dic₁₁₈

yes

complemented

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₅₉

yes

characteristic

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₁₆

C₂×Dic₁₁₈

yes

characteristic

C₁₆

#11745685

yes

characteristic

Q₈

—

118

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₅₉

yes

characteristic

C₂×C₄

C₁₆×D₅₉

yes

characteristic

100

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

101

C₈

—

102

C₈

—

118

—

103

C₈

C₄×Dic₁₁₈

yes

characteristic

104

C₈

#14273733

yes

characteristic

105

C₂²

C₂×C₁₆×D₅₉

yes

characteristic

106

C₄

—

107

C₄

—

118

—

108

C₄

—

118

—

109

C₄

#16837235

yes

—

110

C₄

C₈×Dic₁₁₈

yes

characteristic

111

C₄

#16837818

yes

characteristic

112

C₂

C₁₆×Dic₁₁₈

yes

characteristic

113

C₂

C₂×#16837235

yes

characteristic

114

C₂

#21426389

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator, socle, prime radical	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₄×C₃₂×C₅₉	C₂
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₄×C₃₂	D₅₉
hypocenter	C₅₉	#13660
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator, socle, prime radical

C₂²×C₅₉

C₂²×C₁₆

Fitting

C₄×C₃₂×C₅₉

C₂

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₄×C₃₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
24. C₃₂×C₅₉	Q₈	semidirect
71. C₅₉	#13660	semidirect
82. C₃₂	Dic₁₁₈	semidirect

H ≅ G/N

type

24. C₃₂×C₅₉

Q₈

semidirect

71. C₅₉

#13660

semidirect

82. C₃₂

Dic₁₁₈

semidirect

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

subgroup	index
C₁₆×Dic₁₁₈	2
C₄⋊(C₅₉⋊C₃₂)	2
Dic₅₉⋊C₃₂	2
#21422229	2
C₄×C₃₂×C₅₉	2
#13660	59

subgroup

index

C₁₆×Dic₁₁₈

C₄⋊(C₅₉⋊C₃₂)

Dic₅₉⋊C₃₂

#21422229

C₄×C₃₂×C₅₉

#13660

Subgroups — #28509324

Conjugacy classes of subgroups

Complements

Maximal subgroups