Group #21424138 — subgroups

Subgroups — #21424138

Order 7552.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16837366	C₂	1	2	yes	maximal
3	#16837167	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	C₂²	1	4	yes	—
6	#14273663	—	2	4	no	—
7	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₅₉	—	2	4	no	—
10	C₂×M₁₆×C₅₉	C₄	1	4	yes	—
11	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂⁴×C₅₉	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	—	2	2	no	—
15	C₂²×C₄×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₉	D₄	1	2	yes	—
18	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	M₁₆×C₅₉	—	2	8	no	—
21	C₂×C₈×C₅₉	—	2	2	no	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₉	D₄	1	4	yes	—
25	#11745664	—	2	4	no	—
26	#11745664	—	2	4	no	—
27	C₂×C₅₉⋊C₈	—	2	4	no	—
28	C₂³×C₅₉	—	2	2	no	—
29	C₂³×C₅₉	—	4	2	no	—
30	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	2	no	—
35	C₂×C₄×C₅₉	—	2	4	no	—
36	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₅₉	#41	1	2	yes	—
41	C₈×C₅₉	—	2	2	no	—
42	C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
43	C₅₉⋊C₈	—	4	4	no	—
44	C₂²×C₅₉	—	2	2	no	—
45	C₂²×C₅₉	—	2	2	no	—
46	C₂²×C₅₉	—	2	4	no	—
47	C₂²×C₅₉	—	4	4	no	—
48	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₅₉	—	2	2	no	—
51	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₅₉	#117	1	1	yes	characteristic
53	C₄×C₅₉	#121	1	2	yes	—
54	#1811	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	—	2	2	no	—
56	C₂×C₅₉	—	4	2	no	—
57	C₂×C₅₉	#380	1	1	yes	characteristic
58	C₂×C₅₉	#430	1	2	yes	—
59	C₂²×M₁₆	D₅₉	1	1	yes	characteristic
60	C₂×#108	—	59	2	no	—
61	#407	—	59	2	no	—
62	#341	—	59	2	no	—
63	C₅₉	#1811	1	1	yes	characteristic, complemented, Sylow
64	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	—	59	4	no	—
67	C₂×M₁₆	Dic₅₉	1	4	yes	—
68	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
70	#108	—	118	4	no	—
71	C₂²⋊C₈	—	118	4	no	—
72	C₂⁴	Dic₁₁₈	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
76	C₂²×C₄	D₂₃₆	1	2	yes	—
77	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
78	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	118	4	no	—
81	M₁₆	—	118	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	118	4	no	—
84	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
85	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
86	C₂×C₈	C₂²⋊D₅₉	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₅₉	1	2	yes	—
90	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
96	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂×C₄	#11745685	1	2	yes	—
98	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	236	4	no	—
102	C₈	C₂²⋊Dic₅₉	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
108	C₂²	#14273706	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
111	C₄	#14273686	1	1	yes	characteristic
112	C₄	#14273701	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#16837262	1	1	yes	characteristic
116	C₂	#16837451	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273663

C₂

yes

maximal, complemented

#16837366

C₂

yes

maximal

#16837167

C₂

yes

maximal, complemented

C₂²×M₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

#14273663

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

C₄

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×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

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

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

—

#1811

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#380

yes

characteristic

C₂×C₅₉

#430

yes

—

C₂²×M₁₆

D₅₉

yes

—

—

—

C₅₉

#1811

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

#108

—

118

—

C₂²⋊C₈

—

118

—

C₂⁴

Dic₁₁₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₅₉

yes

—

C₂³

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745685

yes

—

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

100

C₈

—

101

C₈

—

236

—

102

C₈

C₂²⋊Dic₅₉

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

108

C₂²

#14273706

yes

—

109

C₄

—

110

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

111

C₄

#14273686

yes

characteristic

112

C₄

#14273701

yes

—

113

C₂

—

114

C₂

—

115

C₂

#16837262

yes

characteristic

116

C₂

#16837451

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂²×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	C₂²×M₁₆	D₅₉
hypocenter	C₅₉	#1811
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂²×M₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

hypercenter

C₂²×M₁₆

D₅₉

hypocenter

C₅₉

#1811

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#14273663	C₂	semidirect
3. #16837167	C₂	semidirect
63. C₅₉	#1811	semidirect

H ≅ G/N

type

1. C₂×#14273663

C₂

semidirect

3. #16837167

C₂

semidirect

63. C₅₉

#1811

semidirect

Subgroups — #21424138

Conjugacy classes of subgroups

Complements