Group #21424884 — subgroups

Subgroups — #21424884

Order 7552.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16837450	C₂	1	2	yes	maximal
2	C₂×#14273663	C₂	1	1	yes	maximal, characteristic
3	#16837357	C₂	1	1	yes	maximal, characteristic
4	#16837173	C₂	1	2	yes	maximal
5	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×#11745664	C₂²	1	1	yes	characteristic
8	#14273686	C₂²	1	1	yes	characteristic
9	#14273663	C₄	1	4	yes	complemented
10	#14273662	—	2	4	no	—
11	#14273637	—	2	1	no	—
12	C₈×Dic₅₉	—	2	1	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	#108×C₅₉	—	2	1	no	—
16	#108×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	M₁₆×C₅₉	—	2	2	no	—
19	M₁₆×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	Q₈	1	2	yes	—
25	C₂²⋊Dic₅₉	Q₈	1	2	yes	—
26	C₄⋊Dic₅₉	—	2	1	no	—
27	C₄×Dic₅₉	—	2	1	no	—
28	#11745664	—	2	2	no	—
29	#11745664	—	2	1	no	—
30	#11745664	C₂×C₄	1	4	yes	—
31	#11745664	D₄	1	2	yes	—
32	C₂×C₅₉⋊C₈	—	2	1	no	—
33	C₂×C₅₉⋊C₈	—	2	2	no	—
34	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
40	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
41	C₂×Dic₅₉	—	2	2	no	—
42	C₈×C₅₉	—	4	2	no	—
43	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
45	C₅₉⋊C₈	—	2	4	no	—
46	C₅₉⋊C₈	—	2	2	no	—
47	C₂²×C₅₉	—	4	1	no	—
48	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
53	C₄×C₅₉	#121	1	1	yes	characteristic
54	C₄×C₅₉	#125	1	1	yes	characteristic
55	Dic₅₉	—	4	2	no	—
56	#1953	—	59	1	no	maximal, Sylow
57	C₂×C₅₉	—	2	1	no	—
58	C₂×C₅₉	—	2	1	no	—
59	C₂×C₅₉	—	2	1	no	—
60	C₂×C₅₉	#389	1	1	yes	characteristic
61	#430	—	59	2	no	—
62	C₂×#108	—	59	1	no	—
63	C₂×#108	D₅₉	1	1	yes	characteristic
64	#405	—	59	1	no	—
65	#344	—	59	2	no	—
66	C₅₉	#1953	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	59	2	no	—
68	C₂×M₁₆	—	59	1	no	—
69	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
70	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
71	#117	—	59	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	59	4	no	—
75	#108	—	118	4	no	—
76	#97	—	118	1	no	—
77	C₄×C₈	—	118	1	no	—
78	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	59	4	no	—
82	M₁₆	—	59	2	no	—
83	M₁₆	—	118	1	no	—
84	M₁₆	—	118	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	118	2	no	—
87	C₂×C₈	—	118	1	no	—
88	C₂×C₈	Dic₁₁₈	1	2	yes	—
89	C₂×C₈	C₄×D₅₉	1	2	yes	—
90	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
92	C₄⋊C₄	—	118	1	no	—
93	C₂²⋊C₄	—	59	2	no	—
94	C₄²	—	118	1	no	—
95	C₂³	Q₈×D₅₉	1	1	yes	characteristic
96	C₂×C₄	—	118	2	no	—
97	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
99	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
100	C₂×C₄	#11745687	1	1	yes	characteristic
101	C₂×C₄	#11745689	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	118	4	no	—
105	C₈	—	118	2	no	—
106	C₈	Dic₅₉⋊C₄	1	2	yes	—
107	C₈	Dic₅₉⋊C₄	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#14273692	1	1	yes	characteristic
110	C₂²	#14273703	1	1	yes	characteristic
111	C₂²	#14273726	1	1	yes	characteristic
112	C₄	—	236	2	no	—
113	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
114	C₄	#14273689	1	1	yes	characteristic
115	C₄	#14273698	1	1	yes	characteristic
116	C₄	#14273718	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#16837298	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837450

C₂

yes

maximal

C₂×#14273663

C₂

yes

maximal, characteristic

#16837357

C₂

yes

maximal, characteristic

#16837173

C₂

yes

maximal

C₂×#108×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

characteristic

#14273663

C₄

yes

complemented

#14273662

—

#14273637

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

—

C₂²⋊Dic₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

#11745664

C₂×C₄

yes

—

#11745664

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×Q₈

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1953

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

#405

—

#344

—

C₅₉

#1953

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

—

—

—

—

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

C₄²

—

118

—

C₂³

Q₈×D₅₉

yes

characteristic

C₂×C₄

—

118

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

D₄×D₅₉

yes

characteristic

100

C₂×C₄

#11745687

yes

characteristic

101

C₂×C₄

#11745689

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

103

C₈

—

104

C₈

—

118

—

105

C₈

—

118

—

106

C₈

Dic₅₉⋊C₄

yes

—

107

C₈

Dic₅₉⋊C₄

yes

—

108

C₂²

—

109

C₂²

#14273692

yes

characteristic

110

C₂²

#14273703

yes

characteristic

111

C₂²

#14273726

yes

characteristic

112

C₄

—

236

—

113

C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

114

C₄

#14273689

yes

characteristic

115

C₄

#14273698

yes

characteristic

116

C₄

#14273718

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#16837298

yes

characteristic

	subgroup	G/H
center	C₄	#14273698
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂×C₅₉	#389
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#1953
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#14273698

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #14273663	C₄	semidirect
66. C₅₉	#1953	semidirect

H ≅ G/N

type

9. #14273663

C₄

semidirect

66. C₅₉

#1953

semidirect

Subgroups — #21424884

Conjugacy classes of subgroups

Complements