Group #21423978 — subgroups

Subgroups — #21423978

Order 7552.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16837440	C₂	1	1	yes	maximal, characteristic
2	#16837360	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837163	C₂	1	2	yes	maximal
5	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
7	#14273686	C₂²	1	2	yes	—
8	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
9	#14273661	C₄	1	2	yes	complemented
10	#14273659	C₄	1	2	yes	complemented
11	#14273637	—	2	2	no	—
12	C₈×Dic₅₉	—	2	2	no	—
13	C₈×Dic₅₉	C₄	1	4	yes	complemented
14	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	#108×C₅₉	C₄	1	4	yes	complemented
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	1	no	—
19	M₁₆×C₅₉	—	2	2	no	—
20	M₁₆×C₅₉	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₅₉	—	2	2	no	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂²⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
28	C₄×Dic₅₉	—	2	2	no	—
29	C₄×Dic₅₉	—	2	1	no	—
30	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
35	C₂³×C₅₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
42	C₂×Dic₅₉	—	2	4	no	—
43	C₂×Dic₅₉	—	2	2	no	—
44	C₂×Dic₅₉	—	2	4	no	—
45	C₈×C₅₉	—	2	4	no	—
46	C₈×C₅₉	C₄²	1	2	yes	—
47	C₈×C₅₉	C₄²	1	2	yes	—
48	C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
50	C₂²×C₅₉	—	2	2	no	—
51	C₂²×C₅₉	#117	1	1	yes	characteristic
52	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
57	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
58	Dic₅₉	—	2	4	no	—
59	Dic₅₉	—	4	4	no	—
60	#1772	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	—	2	1	no	—
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₅₉	#443	1	2	yes	—
65	C₂×#108	D₅₉	1	1	yes	characteristic
66	#427	—	59	1	no	—
67	#405	—	59	2	no	—
68	C₂×C₄×C₈	—	59	1	no	—
69	#339	—	59	2	no	—
70	C₅₉	#1772	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
72	C₂²×C₈	—	59	1	no	—
73	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
74	#117	—	59	2	no	—
75	C₂×C₄²	—	59	1	no	—
76	#108	Dic₅₉	1	4	yes	complemented
77	#107	—	59	2	no	—
78	#106	—	59	2	no	—
79	#97	—	118	2	no	—
80	C₄×C₈	—	59	4	no	—
81	C₄×C₈	—	118	2	no	—
82	C₂²×C₄	—	118	1	no	—
83	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	C₂×Dic₅₉	1	4	yes	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	59	2	no	—
88	C₂×C₈	—	59	1	no	—
89	C₂×C₈	—	59	1	no	—
90	C₂×C₈	—	59	2	no	—
91	C₂×C₈	C₄×D₅₉	1	2	yes	—
92	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
93	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
95	C₄⋊C₄	—	59	4	no	—
96	C₂²⋊C₄	—	118	2	no	—
97	C₄²	—	59	2	no	—
98	C₄²	—	118	2	no	—
99	C₄²	—	118	1	no	—
100	C₂³	#11745689	1	1	yes	characteristic
101	C₂×C₄	—	118	4	no	—
102	C₂×C₄	—	118	2	no	—
103	C₂×C₄	—	118	4	no	—
104	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
107	C₂×C₄	#11745687	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	—	59	2	no	—
112	C₈	—	59	2	no	—
113	C₈	C₄×Dic₅₉	1	2	yes	—
114	C₈	C₄×Dic₅₉	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#14273685	1	1	yes	characteristic
117	C₂²	#14273689	1	1	yes	characteristic
118	C₂²	Q₈×Dic₅₉	1	1	yes	characteristic
119	C₄	—	118	4	no	—
120	C₄	—	236	4	no	—
121	C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
123	C₄	D₄×Dic₅₉	1	1	yes	characteristic
124	C₄	#14273692	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
128	C₂	#16837518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837440

C₂

yes

maximal, characteristic

#16837360

C₂

yes

maximal

C₂×C₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837163

C₂

yes

maximal

C₂×#108×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273686

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273661

C₄

yes

complemented

#14273659

C₄

yes

complemented

#14273637

—

C₈×Dic₅₉

—

C₈×Dic₅₉

C₄

yes

complemented

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄²

yes

—

C₈×C₅₉

C₄²

yes

—

C₅₉⋊C₈

C₄⋊C₄

yes

—

C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

#117

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#1772

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₉

#443

yes

—

C₂×#108

D₅₉

yes

characteristic

#427

—

#405

—

C₂×C₄×C₈

—

#339

—

C₅₉

#1772

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

C₂×C₄²

—

#108

Dic₅₉

yes

—

—

—

118

—

C₄×C₈

—

C₄×C₈

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₅₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

118

—

C₄²

—

C₄²

—

118

—

C₄²

—

118

—

100

C₂³

#11745689

yes

characteristic

101

C₂×C₄

—

118

—

102

C₂×C₄

—

118

—

103

C₂×C₄

—

118

—

104

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

105

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

106

C₂×C₄

D₄×D₅₉

yes

characteristic

107

C₂×C₄

#11745687

yes

characteristic

108

C₂×C₄

Q₈×D₅₉

yes

characteristic

109

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₄×Dic₅₉

yes

—

114

C₈

C₄×Dic₅₉

yes

—

115

C₂²

—

116

C₂²

#14273685

yes

characteristic

117

C₂²

#14273689

yes

characteristic

118

C₂²

Q₈×Dic₅₉

yes

characteristic

119

C₄

—

118

—

120

C₄

—

236

—

121

C₄

C₂×C₄×Dic₅₉

yes

characteristic

122

C₄

C₄⋊C₄×D₅₉

yes

characteristic

123

C₄

D₄×Dic₅₉

yes

characteristic

124

C₄

#14273692

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₅₉

yes

characteristic

128

C₂

#16837518

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#1772
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₄×C₅₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

socle

C₂²×C₅₉

C₄×D₄

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #14273661	C₄	semidirect
10. #14273659	C₄	semidirect
13. C₈×Dic₅₉	C₄	semidirect
16. #108×C₅₉	C₄	semidirect
70. C₅₉	#1772	semidirect
76. #108	Dic₅₉	semidirect

H ≅ G/N

type

9. #14273661

C₄

semidirect

10. #14273659

C₄

semidirect

13. C₈×Dic₅₉

C₄

semidirect

16. #108×C₅₉

C₄

semidirect

70. C₅₉

#1772

semidirect

76. #108

Dic₅₉

semidirect

Subgroups — #21423978

Conjugacy classes of subgroups

Complements