Group #28475798 — subgroups

Subgroups — #28475798

Order 15104.

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

normal subgroups only

This group has 200 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	#21422855	C₂	1	2	yes	maximal
2	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
3	#16837426	—	2	2	no	—
4	#16837155	—	2	2	no	—
5	#16837155	C₄	1	4	yes	—
6	C₂×M₃₂×C₅₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×C₄⋊C₁₆	—	2	2	no	—
9	C₂×#11745664	—	4	2	no	—
10	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
11	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
15	M₃₂×C₅₉	—	4	2	no	—
16	C₂×C₁₆×C₅₉	—	2	2	no	—
17	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₉	D₄	1	2	yes	—
19	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₂×C₈×C₅₉	—	2	1	no	—
23	C₂×C₈×C₅₉	—	4	1	no	—
24	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
27	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
28	C₄²×C₅₉	D₈	1	1	yes	characteristic
29	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
30	#11745664	—	8	4	no	—
31	C₂×C₅₉⋊C₈	—	4	4	no	—
32	C₁₆×C₅₉	—	4	2	no	—
33	C₂³×C₅₉	#106	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₉	#102	1	2	yes	—
40	C₂×C₄×C₅₉	#103	1	2	yes	—
41	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
42	C₈×C₅₉	—	2	2	no	—
43	C₈×C₅₉	—	2	1	no	—
44	C₈×C₅₉	#100	1	1	yes	characteristic
45	C₈×C₅₉	#101	1	1	yes	characteristic
46	C₅₉⋊C₈	—	8	4	no	—
47	#7245	—	59	1	no	maximal, Sylow
48	C₂²×C₅₉	—	2	1	no	—
49	C₂²×C₅₉	—	2	2	no	—
50	C₂²×C₅₉	#340	1	1	yes	characteristic
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	—	2	2	no	—
53	C₄×C₅₉	#338	1	1	yes	characteristic
54	C₄×C₅₉	#343	1	1	yes	characteristic
55	C₄×C₅₉	#361	1	2	yes	—
56	C₄×C₅₉	#362	1	2	yes	—
57	#2150	D₅₉	1	1	yes	characteristic
58	#1567	—	59	2	no	—
59	C₂×C₅₉	—	4	1	no	—
60	C₂×C₅₉	#1292	1	1	yes	characteristic
61	C₂×C₅₉	#1387	1	2	yes	—
62	C₂×M₃₂	Dic₅₉	1	2	yes	—
63	#423	—	118	2	no	—
64	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	59	4	no	—
67	#335	—	118	2	no	—
68	C₅₉	#7245	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	236	2	no	—
70	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
71	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
72	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
73	M₃₂	—	4	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₄⋊C₈	—	118	4	no	—
76	C₄⋊C₈	—	236	2	no	—
77	C₄×C₈	C₄×D₅₉	1	2	yes	—
78	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
79	C₂²×C₄	#11745660	1	2	yes	—
80	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
81	M₁₆	—	472	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	236	4	no	—
86	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
87	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
88	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
89	C₄²	D₄₇₂	1	1	yes	characteristic
90	C₄²	Dic₂₃₆	1	1	yes	characteristic
91	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
92	C₁₆	—	4	2	no	—
93	C₂³	#14273659	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
99	C₂×C₄	#14273649	1	2	yes	—
100	C₂×C₄	#14273652	1	2	yes	—
101	C₂×C₄	#14273661	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	1	no	—
104	C₈	—	472	4	no	—
105	C₈	#14273644	1	1	yes	characteristic
106	C₈	#14273646	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#16837165	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	C₅₉⋊#338	1	1	yes	characteristic
113	C₄	#16837171	1	1	yes	characteristic
114	C₄	#16837216	1	2	yes	—
115	C₄	#16837219	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	#21421971	1	1	yes	characteristic
118	C₂	#21422205	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422855

C₂

yes

maximal

C₅₉×#2150

C₂

yes

maximal, characteristic

#16837426

—

#16837155

—

#16837155

C₄

yes

—

C₂×M₃₂×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

Q₁₆

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₂³×C₅₉

#106

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#103

yes

—

C₂×C₄×C₅₉

#107

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#7245

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#340

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#362

yes

—

#2150

D₅₉

yes

characteristic

#1567

—

C₂×C₅₉

—

C₂×C₅₉

#1292

yes

characteristic

C₂×C₅₉

#1387

yes

—

C₂×M₃₂

Dic₅₉

yes

—

#423

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

118

—

C₅₉

#7245

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

236

—

C₂²×C₈

Dic₁₁₈

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

C₄×C₈

C₄×D₅₉

yes

—

C₄×C₈

C₂²⋊D₅₉

yes

—

C₂²×C₄

#11745660

yes

—

C₂²×C₄

C₄⋊Dic₅₉

yes

characteristic

M₁₆

—

472

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₄×Dic₅₉

yes

characteristic

C₂×C₈

Dic₅₉⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

C₄²

D₄₇₂

yes

characteristic

C₄²

Dic₂₃₆

yes

characteristic

C₄²

(C₂×D₅₉)⋊C₄

yes

—

C₁₆

—

C₂³

#14273659

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₅₉⋊#95

yes

characteristic

C₂×C₄

#14273649

yes

—

100

C₂×C₄

#14273652

yes

—

101

C₂×C₄

#14273661

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

472

—

105

C₈

#14273644

yes

characteristic

106

C₈

#14273646

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

#16837165

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₅₉⋊#338

yes

characteristic

113

C₄

#16837171

yes

characteristic

114

C₄

#16837216

yes

—

115

C₄

#16837219

yes

—

116

C₂

—

117

C₂

#21421971

yes

characteristic

118

C₂

#21422205

yes

—

	subgroup	G/H
center	C₂²	#16837165
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#106
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#7245
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

#16837165

commutator

C₂×C₈×C₅₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₅₉×#2150

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
68. C₅₉	#7245	semidirect

H ≅ G/N

type

68. C₅₉

#7245

semidirect

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

subgroup	index
#21422855	2
C₅₉×#2150	2
#7245	59

subgroup

index

#21422855

C₅₉×#2150

#7245

Subgroups — #28475798

Conjugacy classes of subgroups

Complements

Maximal subgroups