Group #21424316 — subgroups

Subgroups — #21424316

Order 7552.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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, complemented
2	#16837386	C₂	1	1	yes	maximal, characteristic
3	#16837369	C₂	1	1	yes	maximal, characteristic
4	#16837166	C₂	1	2	yes	maximal, complemented
5	C₅₉×#568	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	2	yes	—
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14273662	—	2	4	no	—
10	#14273646	C₄	1	2	yes	—
11	#14273644	C₄	1	2	yes	—
12	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
13	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
14	#131×C₅₉	—	2	4	no	—
15	C₂×M₁₆×C₅₉	—	2	2	no	—
16	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
17	#41×C₅₉	—	2	2	no	—
18	#41×C₅₉	—	2	2	no	—
19	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₅₉	Q₈	1	2	yes	—
22	C₂²×C₄×C₅₉	D₄	1	2	yes	—
23	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₅₉	—	2	4	no	—
25	M₁₆×C₅₉	—	2	2	no	—
26	M₁₆×C₅₉	D₄	1	4	yes	—
27	C₂×C₈×C₅₉	—	2	4	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	#11745664	—	2	2	no	—
30	#11745664	—	2	2	no	—
31	#11745664	C₂×C₄	1	4	yes	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	—	2	1	no	—
34	C₂×C₅₉⋊C₈	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	2	no	—
36	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₅₉	—	2	2	no	—
39	D₄×C₅₉	—	2	4	no	—
40	D₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	#41	1	2	yes	—
48	C₈×C₅₉	—	2	4	no	—
49	C₅₉⋊C₈	—	2	4	no	—
50	C₅₉⋊C₈	—	4	1	no	—
51	C₅₉⋊C₈	—	4	1	no	—
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	—	4	1	no	—
54	C₂²×C₅₉	—	4	2	no	—
55	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₅₉	—	2	2	no	—
58	C₄×C₅₉	C₄×D₄	1	2	yes	—
59	C₄×C₅₉	#120	1	1	yes	characteristic
60	C₄×C₅₉	#123	1	1	yes	characteristic
61	#1858	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	—	4	2	no	—
65	C₂×C₅₉	#385	1	1	yes	characteristic
66	#568	D₅₉	1	1	yes	characteristic
67	#430	—	59	2	no	—
68	#413	—	59	1	no	—
69	#408	—	59	1	no	—
70	#341	—	59	2	no	—
71	C₅₉	#1858	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	59	2	no	—
76	C₂×M₁₆	—	59	1	no	—
77	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
78	C₂²×C₈	—	59	1	no	—
79	#108	—	118	4	no	—
80	#101	—	59	2	no	—
81	#100	—	59	2	no	—
82	C₂²⋊C₈	—	118	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
86	C₂×D₄	Dic₁₁₈	1	2	yes	—
87	C₂×D₄	C₄×D₅₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	59	4	no	—
93	M₁₆	—	118	2	no	—
94	M₁₆	—	118	2	no	—
95	M₁₆	C₂²⋊D₅₉	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	118	2	no	—
99	C₂×C₈	—	118	2	no	—
100	C₂×C₈	—	118	1	no	—
101	C₂×C₈	—	118	1	no	—
102	C₂³	Dic₅₉⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
111	C₂×C₄	#11745685	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
114	C₈	—	2	4	no	—
115	C₈	—	118	4	no	—
116	C₈	—	236	1	no	—
117	C₈	—	236	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
122	C₂²	#14273706	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
125	C₄	#14273696	1	1	yes	characteristic
126	C₄	#14273710	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#16837275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837450

C₂

yes

maximal, complemented

#16837386

C₂

yes

maximal, characteristic

#16837369

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal, complemented

C₅₉×#568

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273662

—

#14273646

C₄

yes

—

#14273644

C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

#11745664

—

#11745664

—

#11745664

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

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

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#385

yes

characteristic

#568

D₅₉

yes

—

—

—

—

C₅₉

#1858

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

#108

—

118

—

#101

—

#100

—

C₂²⋊C₈

—

118

—

#41

—

#41

—

C₂×Q₈

C₄×D₅₉

yes

characteristic

C₂×D₄

Dic₁₁₈

yes

—

C₂×D₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

C₂²⋊D₅₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

100

C₂×C₈

—

118

—

101

C₂×C₈

—

118

—

102

C₂³

Dic₅₉⋊C₄

yes

—

103

C₂³

C₂×Dic₁₁₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₅₉⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

111

C₂×C₄

#11745685

yes

—

112

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

113

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

114

C₈

—

115

C₈

—

118

—

116

C₈

—

236

—

117

C₈

—

236

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

122

C₂²

#14273706

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₅₉

yes

—

125

C₄

#14273696

yes

characteristic

126

C₄

#14273710

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16837275

yes

characteristic

	subgroup	G/H
center	C₄	#14273696
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#568	C₂
socle	C₂×C₅₉	#385
hypercenter	#568	D₅₉
hypocenter	C₅₉	#1858
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#14273696

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#568

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #16837450	C₂	semidirect
4. #16837166	C₂	semidirect
71. C₅₉	#1858	semidirect

H ≅ G/N

type

1. #16837450

C₂

semidirect

4. #16837166

C₂

semidirect

71. C₅₉

#1858

semidirect

Subgroups — #21424316

Conjugacy classes of subgroups

Complements