Group #21424850 — subgroups

Subgroups — #21424850

Order 7552.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	#16837429	C₂	1	2	yes	maximal
3	C₂×#14273637	C₂	1	1	yes	maximal, characteristic
4	#16837161	C₂	1	2	yes	maximal
5	C₅₉×#427	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
9	#14273663	—	2	2	no	—
10	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
11	#14273637	—	2	2	no	—
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	#117×C₅₉	C₂²	1	2	yes	—
14	#107×C₅₉	C₄	1	2	yes	complemented
15	#106×C₅₉	C₄	1	2	yes	complemented
16	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	2	1	no	—
18	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	D₄	1	2	yes	—
21	C₂×C₈×C₅₉	Q₈	1	2	yes	—
22	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
23	C₂²⋊C₄×C₅₉	—	2	2	no	—
24	C₄²×C₅₉	—	2	2	no	—
25	C₄×Dic₅₉	—	2	1	no	—
26	C₄×Dic₅₉	Q₈	1	2	yes	—
27	#11745664	—	2	2	no	—
28	#11745664	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	1	no	—
30	C₂×C₅₉⋊C₈	—	2	1	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
33	C₂³×C₅₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	2	4	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	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
41	C₂×Dic₅₉	—	2	2	no	—
42	C₂×Dic₅₉	—	4	2	no	—
43	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
45	C₅₉⋊C₈	—	2	2	no	—
46	C₅₉⋊C₈	—	4	2	no	—
47	C₂²×C₅₉	—	2	2	no	—
48	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₅₉	#125	1	1	yes	characteristic
51	C₄×C₅₉	—	4	4	no	—
52	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₅₉	#121	1	1	yes	characteristic
55	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₅₉	—	4	2	no	—
57	#1946	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	2	1	no	—
59	C₂×C₅₉	—	2	1	no	—
60	C₂×C₅₉	#389	1	1	yes	characteristic
61	C₂×C₅₉	#444	1	2	yes	—
62	C₂×#108	—	59	1	no	—
63	#427	D₅₉	1	1	yes	characteristic
64	#424	—	59	2	no	—
65	C₂×#97	—	59	1	no	—
66	#339	—	59	2	no	—
67	C₅₉	#1946	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	59	2	no	—
69	C₂²×C₈	—	59	1	no	—
70	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
71	#117	C₂×D₅₉	1	2	yes	—
72	C₂×C₄²	—	59	1	no	—
73	#108	—	118	2	no	—
74	#107	Dic₅₉	1	2	yes	complemented
75	#106	Dic₅₉	1	2	yes	complemented
76	C₄⋊C₈	—	118	4	no	—
77	#97	—	118	2	no	—
78	C₂²×C₄	—	118	1	no	—
79	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
80	M₁₆	—	118	2	no	—
81	M₁₆	—	118	2	no	—
82	C₂×C₈	—	59	2	no	—
83	C₂×C₈	—	118	1	no	—
84	C₂×C₈	—	118	1	no	—
85	C₂×C₈	—	118	2	no	—
86	C₂×C₈	Dic₁₁₈	1	2	yes	—
87	C₂×C₈	D₂₃₆	1	2	yes	—
88	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	59	2	no	—
94	C₄²	—	118	1	no	—
95	C₂³	#11745687	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	118	2	no	—
98	C₂×C₄	—	236	2	no	—
99	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
100	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
101	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
103	C₂×C₄	#11745689	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
105	C₈	—	118	2	no	—
106	C₈	—	236	2	no	—
107	C₈	C₄⋊Dic₅₉	1	2	yes	—
108	C₈	C₄⋊Dic₅₉	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₅₉	1	1	yes	characteristic
111	C₂²	#14273707	1	1	yes	characteristic
112	C₂²	#14273718	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	236	2	no	—
115	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
116	C₄	Q₈×Dic₅₉	1	1	yes	characteristic
117	C₄	#14273702	1	1	yes	characteristic
118	C₄	#14273726	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#16837301	1	1	yes	characteristic
122	C₂	#16837524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273663

C₂

yes

maximal, characteristic

#16837429

C₂

yes

maximal

C₂×#14273637

C₂

yes

maximal, characteristic

#16837161

C₂

yes

maximal

C₅₉×#427

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

#14273637

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

—

#107×C₅₉

C₄

yes

complemented

#106×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

Q₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

—

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

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

#125

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

#1946

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#389

yes

characteristic

C₂×C₅₉

#444

yes

—

C₂×#108

—

#427

D₅₉

yes

characteristic

#424

—

C₂×#97

—

#339

—

C₅₉

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

C₂×D₅₉

yes

—

C₂×C₄²

—

#108

—

118

—

#107

Dic₅₉

yes

complemented

#106

Dic₅₉

yes

complemented

C₄⋊C₈

—

118

—

#97

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

118

—

C₂³

#11745687

yes

characteristic

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

100

C₂×C₄

C₂×D₂₃₆

yes

characteristic

101

C₂×C₄

D₄×D₅₉

yes

characteristic

102

C₂×C₄

Q₈×D₅₉

yes

characteristic

103

C₂×C₄

#11745689

yes

characteristic

104

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

105

C₈

—

118

—

106

C₈

—

236

—

107

C₈

C₄⋊Dic₅₉

yes

—

108

C₈

C₄⋊Dic₅₉

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₅₉

yes

characteristic

111

C₂²

#14273707

yes

characteristic

112

C₂²

#14273718

yes

characteristic

113

C₄

—

114

C₄

—

236

—

115

C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

116

C₄

Q₈×Dic₅₉

yes

characteristic

117

C₄

#14273702

yes

characteristic

118

C₄

#14273726

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#16837301

yes

characteristic

122

C₂

#16837524

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#427	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	#427	D₅₉
hypocenter	C₅₉	#1946
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#427

C₂

socle

C₂²×C₅₉

C₄×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₅₉	C₄	semidirect
15. #106×C₅₉	C₄	semidirect
67. C₅₉	#1946	semidirect
74. #107	Dic₅₉	semidirect
75. #106	Dic₅₉	semidirect

H ≅ G/N

type

14. #107×C₅₉

C₄

semidirect

15. #106×C₅₉

C₄

semidirect

67. C₅₉

#1946

semidirect

74. #107

Dic₅₉

semidirect

75. #106

Dic₅₉

semidirect

Subgroups — #21424850

Conjugacy classes of subgroups

Complements