Group #21424146 — subgroups

Subgroups — #21424146

Order 7552.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#14273663	C₂	1	2	yes	maximal, complemented
2	#16837386	C₂	1	2	yes	maximal
3	C₅₉⋊#338	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	C₂²	1	4	yes	—
6	#14273663	—	2	4	no	—
7	#14273646	—	2	2	no	—
8	#14273644	—	2	2	no	—
9	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
10	#131×C₅₉	—	2	4	no	—
11	C₂×M₁₆×C₅₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	C₄	1	2	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	#41×C₅₉	—	2	2	no	—
16	#41×C₅₉	—	2	2	no	—
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂×D₄×C₅₉	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₅₉	—	2	2	no	—
23	M₁₆×C₅₉	—	2	4	no	—
24	C₂×C₈×C₅₉	—	2	4	no	—
25	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	D₄	1	4	yes	—
29	#11745664	—	2	4	no	—
30	#11745664	—	2	4	no	—
31	C₂×C₅₉⋊C₈	—	2	4	no	—
32	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
33	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	Q₈×C₅₉	—	2	2	no	—
35	D₄×C₅₉	—	2	4	no	—
36	D₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	—	2	2	no	—
39	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₅₉	#41	1	2	yes	—
44	C₈×C₅₉	—	2	2	no	—
45	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₅₉⋊C₈	—	4	4	no	—
48	C₂²×C₅₉	—	2	2	no	—
49	C₂²×C₅₉	—	4	1	no	—
50	C₂²×C₅₉	—	4	2	no	—
51	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₅₉	—	2	2	no	—
54	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₅₉	#117	1	1	yes	characteristic
56	C₄×C₅₉	#121	1	2	yes	—
57	#1813	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	2	1	no	—
59	C₂×C₅₉	—	2	2	no	—
60	C₂×C₅₉	—	4	2	no	—
61	C₂×C₅₉	#380	1	1	yes	characteristic
62	C₂×#131	D₅₉	1	1	yes	characteristic
63	C₂×#108	—	59	2	no	—
64	#413	—	59	2	no	—
65	#338	—	59	2	no	—
66	C₅₉	#1813	1	1	yes	characteristic, complemented, Sylow
67	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
68	#131	—	2	4	no	—
69	C₂×M₁₆	—	59	4	no	—
70	C₂×M₁₆	Dic₅₉	1	2	yes	—
71	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
72	C₂²×C₈	Dic₅₉	1	2	yes	—
73	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
74	#108	—	118	4	no	—
75	#101	—	118	2	no	—
76	#100	—	118	2	no	—
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
80	C₂×D₄	Dic₁₁₈	1	2	yes	—
81	C₂×D₄	D₂₃₆	1	1	yes	characteristic
82	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	118	4	no	—
87	M₁₆	—	118	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	118	4	no	—
90	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
91	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₅₉	1	4	yes	—
94	C₂³	C₄⋊Dic₅₉	1	2	yes	—
95	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
102	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
103	C₂×C₄	#11745685	1	2	yes	—
104	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	236	4	no	—
108	C₈	C₂²⋊Dic₅₉	1	2	yes	—
109	C₈	C₂²⋊Dic₅₉	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
114	C₂²	#14273706	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
117	C₄	#14273686	1	1	yes	characteristic
118	C₄	#14273701	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#16837262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273663

C₂

yes

maximal, complemented

#16837386

C₂

yes

maximal

C₅₉⋊#338

C₂

yes

maximal, complemented

C₂×#131×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

#14273663

—

#14273646

—

#14273644

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

C₂×M₁₆×C₅₉

C₄

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₄

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#1813

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#380

yes

characteristic

C₂×#131

D₅₉

yes

—

—

—

C₅₉

#1813

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₅₉

yes

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

Dic₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#108

—

118

—

#101

—

118

—

#100

—

118

—

#41

—

#41

—

C₂×Q₈

D₂₃₆

yes

characteristic

C₂×D₄

Dic₁₁₈

yes

—

C₂×D₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

—

C₂³

C₄⋊Dic₅₉

yes

—

C₂³

C₂×Dic₁₁₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₅₉

yes

—

102

C₂×C₄

C₂×D₂₃₆

yes

characteristic

103

C₂×C₄

#11745685

yes

—

104

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

106

C₈

—

107

C₈

—

236

—

108

C₈

C₂²⋊Dic₅₉

yes

—

109

C₈

C₂²⋊Dic₅₉

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

114

C₂²

#14273706

yes

—

115

C₄

—

116

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

117

C₄

#14273686

yes

characteristic

118

C₄

#14273701

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#16837262

yes

characteristic

	subgroup	G/H
center	C₄	#14273686
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#131×C₅₉	C₂
socle	C₂×C₅₉	#380
hypercenter	C₂×#131	D₅₉
hypocenter	C₅₉	#1813
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#14273686

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#131×C₅₉

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#14273663	C₂	semidirect
3. C₅₉⋊#338	C₂	semidirect
66. C₅₉	#1813	semidirect

H ≅ G/N

type

1. C₂×#14273663

C₂

semidirect

3. C₅₉⋊#338

C₂

semidirect

66. C₅₉

#1813

semidirect

Subgroups — #21424146

Conjugacy classes of subgroups

Complements