Group #21425898 — subgroups

Subgroups — #21425898

Order 7552.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₅₉)⋊C₁₆	C₂	1	2	yes	maximal
4	#16837177	C₂	1	2	yes	maximal, complemented
5	C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₉	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
8	C₄²×D₅₉	C₄	1	1	yes	characteristic
9	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
11	C₈×Dic₅₉	C₄	1	1	yes	characteristic
12	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅₉	C₈	1	1	yes	characteristic
16	C₂×C₄×D₅₉	C₈	1	1	yes	characteristic
17	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
20	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₅₉	C₈	1	1	yes	characteristic
22	C₄×Dic₅₉	C₈	1	1	yes	characteristic
23	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₈×D₅₉	—	2	1	no	—
27	C₈×D₅₉	—	2	1	no	—
28	C₈×D₅₉	—	2	2	no	—
29	C₁₆×C₅₉	—	2	2	no	—
30	C₅₉⋊C₁₆	—	2	2	no	—
31	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
38	C₄×D₅₉	—	2	2	no	—
39	C₄×D₅₉	—	2	2	no	—
40	C₄×D₅₉	—	2	1	no	—
41	C₄×D₅₉	—	2	1	no	—
42	C₈×C₅₉	—	2	1	no	—
43	C₈×C₅₉	#41	1	1	yes	characteristic
44	C₈×C₅₉	#41	1	1	yes	characteristic
45	C₅₉⋊C₈	—	2	1	no	—
46	C₅₉⋊C₈	M₁₆	1	2	yes	—
47	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
48	C₂×D₅₉	—	2	1	no	—
49	C₂×D₅₉	—	2	1	no	—
50	C₂×D₅₉	M₃₂	1	2	yes	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	M₃₂	1	2	yes	—
53	C₄×C₅₉	#117	1	1	yes	characteristic
54	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
55	Dic₅₉	—	2	1	no	—
56	Dic₅₉	—	2	1	no	—
57	Dic₅₉	—	2	1	no	—
58	#2163	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	#431	1	1	yes	characteristic
60	C₂×C₅₉	C₂×M₃₂	1	1	yes	characteristic
61	C₂×C₅₉	C₂×M₃₂	1	1	yes	characteristic
62	D₅₉	—	2	2	no	—
63	C₂×C₄×C₈	—	59	1	no	—
64	C₄⋊C₁₆	—	59	1	no	—
65	C₄⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
66	C₂²⋊C₁₆	—	59	2	no	—
67	#346	—	59	2	no	—
68	C₅₉	#2163	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	59	1	no	—
70	C₂²×C₈	—	59	1	no	—
71	C₂×C₄²	—	59	1	no	—
72	C₂×C₁₆	—	59	2	no	—
73	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
74	C₄×C₈	—	59	1	no	—
75	C₄×C₈	—	59	1	no	—
76	C₄×C₈	—	59	1	no	—
77	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂²×C₄	—	59	1	no	—
79	C₂²×C₄	—	59	1	no	—
80	C₂²×C₄	—	59	1	no	—
81	C₂×C₈	—	59	1	no	—
82	C₂×C₈	—	59	1	no	—
83	C₂×C₈	—	118	2	no	—
84	C₂×C₈	—	118	1	no	—
85	C₂×C₈	—	118	1	no	—
86	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
88	C₄²	—	59	1	no	—
89	C₄²	—	59	1	no	—
90	C₄²	—	59	1	no	—
91	C₄²	C₄×D₅₉	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₁₆	—	118	2	no	—
94	C₂³	—	59	1	no	—
95	C₂×C₄	—	59	1	no	—
96	C₂×C₄	—	59	1	no	—
97	C₂×C₄	—	59	1	no	—
98	C₂×C₄	—	118	2	no	—
99	C₂×C₄	—	118	2	no	—
100	C₂×C₄	—	118	1	no	—
101	C₂×C₄	—	118	1	no	—
102	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
103	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	59	2	no	—
107	C₈	—	118	1	no	—
108	C₈	#11745687	1	1	yes	characteristic
109	C₈	#11745689	1	1	yes	characteristic
110	C₂²	—	59	2	no	—
111	C₂²	—	118	1	no	—
112	C₂²	—	118	1	no	—
113	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	118	1	no	—
116	C₄	—	118	1	no	—
117	C₄	—	118	1	no	—
118	C₄	#14273666	1	2	yes	—
119	C₄	#14273685	1	1	yes	characteristic
120	C₄	M₁₆×D₅₉	1	1	yes	characteristic
121	C₂	—	118	2	no	—
122	C₂	#16837454	1	1	yes	characteristic
123	C₂	M₃₂×D₅₉	1	1	yes	characteristic
124	C₂	C₂×#14273666	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₅₉

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal

#16837177

C₂

yes

maximal, complemented

C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₉

C₄

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

C₄²×D₅₉

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

—

C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₈×Dic₅₉

C₄

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₈

yes

characteristic

C₂×C₄×D₅₉

C₈

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

C₈

yes

characteristic

C₄×Dic₅₉

C₈

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₈×D₅₉

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

C₅₉⋊C₈

M₁₆

yes

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

M₃₂

yes

—

C₄×C₅₉

—

C₄×C₅₉

M₃₂

yes

—

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×M₃₂

yes

characteristic

C₂×C₅₉

C₂×M₃₂

yes

characteristic

D₅₉

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₅₉

yes

characteristic, complemented

C₂²⋊C₁₆

—

#346

—

C₅₉

#2163

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

100

C₂×C₄

—

118

—

101

C₂×C₄

—

118

—

102

C₂×C₄

C₈×D₅₉

yes

characteristic

103

C₂×C₄

C₈×D₅₉

yes

characteristic

104

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

118

—

108

C₈

#11745687

yes

characteristic

109

C₈

#11745689

yes

characteristic

110

C₂²

—

111

C₂²

—

118

—

112

C₂²

—

118

—

113

C₂²

C₂×C₈×D₅₉

yes

characteristic

114

C₄

—

115

C₄

—

118

—

116

C₄

—

118

—

117

C₄

—

118

—

118

C₄

#14273666

yes

—

119

C₄

#14273685

yes

characteristic

120

C₄

M₁₆×D₅₉

yes

characteristic

121

C₂

—

118

—

122

C₂

#16837454

yes

characteristic

123

C₂

M₃₂×D₅₉

yes

characteristic

124

C₂

C₂×#14273666

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₈
Fitting	C₅₉×C₄⋊C₁₆	C₂
hypercenter	C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#2163
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₈

Fitting

C₅₉×C₄⋊C₁₆

C₂

hypercenter

C₄⋊C₁₆

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
4. #16837177	C₂	semidirect
5. C₅₉×C₄⋊C₁₆	C₂	semidirect
65. C₄⋊C₁₆	D₅₉	semidirect
68. C₅₉	#2163	semidirect

H ≅ G/N

type

2. C₄⋊(C₅₉⋊C₁₆)

C₂

semidirect

4. #16837177

C₂

semidirect

5. C₅₉×C₄⋊C₁₆

C₂

semidirect

65. C₄⋊C₁₆

D₅₉

semidirect

68. C₅₉

#2163

semidirect

Subgroups — #21425898

Conjugacy classes of subgroups

Complements