Group #21422981 — subgroups

Subgroups — #21422981

Order 7552.

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

normal subgroups only

This group has 180 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	#16837480	C₂	1	2	yes	maximal, complemented
2	#16837155	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	—	2	2	no	—
7	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
9	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₅₉	C₄	1	1	yes	characteristic
12	C₄×D₄×C₅₉	C₂²	1	2	yes	—
13	#107×C₅₉	C₄	1	1	yes	characteristic
14	#102×C₅₉	C₄	1	2	yes	—
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₉	—	2	2	no	—
18	D₈×C₅₉	—	2	2	no	—
19	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²⋊C₄×C₅₉	—	2	2	no	—
23	C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	#11745664	—	4	4	no	—
25	C₂×C₅₉⋊C₈	—	2	1	no	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂³×C₅₉	—	2	2	no	—
30	D₄×C₅₉	—	2	2	no	—
31	D₄×C₅₉	—	2	2	no	—
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	—	4	2	no	—
34	C₂×C₄×C₅₉	—	4	2	no	—
35	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
38	C₈×C₅₉	—	2	1	no	—
39	C₈×C₅₉	M₁₆	1	2	yes	complemented
40	C₅₉⋊C₈	—	2	2	no	—
41	C₅₉⋊C₈	—	2	1	no	—
42	C₅₉⋊C₈	—	4	2	no	—
43	C₅₉⋊C₈	D₈	1	2	yes	complemented
44	C₂²×C₅₉	—	4	2	no	—
45	C₂²×C₅₉	—	4	2	no	—
46	C₂²×C₅₉	—	4	2	no	—
47	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	—	4	2	no	—
50	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
51	C₄×C₅₉	#131	1	1	yes	characteristic
52	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₄×C₅₉	#135	1	1	yes	characteristic
54	#1589	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	—	8	2	no	—
56	C₂×C₅₉	#436	1	1	yes	characteristic
57	C₂×C₅₉	C₄×D₈	1	1	yes	characteristic
58	C₂×C₅₉	#443	1	1	yes	characteristic
59	C₄×D₈	D₅₉	1	1	yes	characteristic
60	#436	—	59	2	no	—
61	#335	—	59	1	no	—
62	D₄⋊C₈	—	59	2	no	—
63	C₈²	—	59	1	no	—
64	C₅₉	#1589	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	Dic₅₉	1	1	yes	characteristic
66	C₂×M₁₆	—	118	2	no	—
67	C₄×D₄	C₂×D₅₉	1	2	yes	—
68	#107	Dic₅₉	1	1	yes	characteristic
69	C₄⋊C₈	—	59	2	no	—
70	#102	Dic₅₉	1	2	yes	—
71	C₂²⋊C₈	—	118	2	no	—
72	C₄×C₈	—	59	1	no	—
73	C₄×C₈	—	59	1	no	—
74	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
75	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
76	C₂²×C₄	—	2	2	no	—
77	D₈	—	2	2	no	—
78	M₁₆	—	236	4	no	—
79	C₂×C₈	—	59	1	no	—
80	C₂×C₈	—	59	1	no	—
81	C₂×C₈	—	118	1	no	—
82	C₂×C₈	—	118	2	no	—
83	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
85	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₄²	C₂²×D₅₉	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
95	C₂×C₄	#11745687	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	59	2	no	—
99	C₈	—	118	2	no	—
100	C₈	—	118	1	no	—
101	C₈	—	236	2	no	—
102	C₈	#11745664	1	2	yes	complemented
103	C₂²	—	4	2	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	D₄×Dic₅₉	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	C₂×#11745664	1	1	yes	characteristic
110	C₄	#14273734	1	1	yes	characteristic
111	C₄	D₈×D₅₉	1	1	yes	characteristic
112	C₄	#14273746	1	1	yes	characteristic
113	C₂	—	8	2	no	—
114	C₂	#16837480	1	1	yes	characteristic
115	C₂	D₈×Dic₅₉	1	1	yes	characteristic
116	C₂	#16837519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837480

C₂

yes

maximal, complemented

#16837155

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, complemented

C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₄

yes

characteristic

C₄×D₄×C₅₉

C₂²

yes

—

#107×C₅₉

C₄

yes

characteristic

#102×C₅₉

C₄

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

D₈

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

#436

yes

characteristic

C₂×C₅₉

C₄×D₈

yes

characteristic

C₂×C₅₉

#443

yes

characteristic

C₄×D₈

D₅₉

yes

characteristic

#436

—

#335

—

D₄⋊C₈

—

C₈²

—

C₅₉

#1589

yes

characteristic, complemented, Sylow

C₂×D₈

Dic₅₉

yes

characteristic

C₂×M₁₆

—

118

—

C₄×D₄

C₂×D₅₉

yes

—

#107

Dic₅₉

yes

characteristic

C₄⋊C₈

—

#102

Dic₅₉

yes

—

C₂²⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×D₄

C₂×Dic₅₉

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745687

yes

characteristic

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

C₈

—

C₈

—

118

—

100

C₈

—

118

—

101

C₈

—

236

—

102

C₈

#11745664

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

D₄×Dic₅₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₂×#11745664

yes

characteristic

110

C₄

#14273734

yes

characteristic

111

C₄

D₈×D₅₉

yes

characteristic

112

C₄

#14273746

yes

characteristic

113

C₂

—

114

C₂

#16837480

yes

characteristic

115

C₂

D₈×Dic₅₉

yes

characteristic

116

C₂

#16837519

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₄×D₈×C₅₉	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	C₄×D₈	D₅₉
hypocenter	C₅₉	#1589
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₄×D₈×C₅₉

C₂

socle

C₂²×C₅₉

C₄×D₄

hypercenter

C₄×D₈

D₅₉

hypocenter

C₅₉

#1589

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #16837480	C₂	semidirect
2. #16837155	C₂	semidirect
3. D₄⋊(C₅₉⋊C₈)	C₂	semidirect
4. C₈×C₅₉⋊C₈	C₂	semidirect
39. C₈×C₅₉	M₁₆	semidirect
43. C₅₉⋊C₈	D₈	semidirect
64. C₅₉	#1589	semidirect
102. C₈	#11745664	semidirect

H ≅ G/N

type

1. #16837480

C₂

semidirect

2. #16837155

C₂

semidirect

3. D₄⋊(C₅₉⋊C₈)

C₂

semidirect

4. C₈×C₅₉⋊C₈

C₂

semidirect

39. C₈×C₅₉

M₁₆

semidirect

43. C₅₉⋊C₈

D₈

semidirect

64. C₅₉

#1589

semidirect

102. C₈

#11745664

semidirect

Subgroups — #21422981

Conjugacy classes of subgroups

Complements