Group #21422882 — subgroups

Subgroups — #21422882

Order 7552.

160 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16837455	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837154	C₂	1	2	yes	maximal, complemented
4	#16837152	C₂	1	2	yes	maximal, complemented
5	C₅₉×#432	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
11	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₅₉	—	2	1	no	—
16	#97×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₉	—	2	1	no	—
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	Q₈	1	1	yes	characteristic
25	C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	—	4	2	no	—
28	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
29	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	—	2	2	no	—
32	C₂×C₄×C₅₉	—	2	1	no	—
33	C₂×C₄×C₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
40	C₈×C₅₉	—	4	1	no	—
41	C₈×C₅₉	—	4	1	no	—
42	C₅₉⋊C₈	—	2	4	no	—
43	C₅₉⋊C₈	—	4	2	no	—
44	C₂²×C₅₉	—	2	1	no	—
45	C₂²×C₅₉	—	2	1	no	—
46	C₂²×C₅₉	#108	1	2	yes	—
47	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	—	2	1	no	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	#131	1	2	yes	—
53	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₅₉	#137	1	1	yes	characteristic
55	#1573	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	2	no	—
57	C₂×C₅₉	#424	1	1	yes	characteristic
58	C₂×C₅₉	#428	1	1	yes	characteristic
59	C₂×C₅₉	C₂×#108	1	1	yes	characteristic
60	#432	D₅₉	1	1	yes	characteristic
61	#431	—	59	1	no	—
62	C₂×C₄⋊C₈	—	59	1	no	—
63	#335	—	59	2	no	—
64	#334	—	59	2	no	—
65	C₅₉	#1573	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	—	118	1	no	—
67	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
68	C₄⋊C₈	—	59	2	no	—
69	C₄⋊C₈	—	59	2	no	—
70	C₄⋊C₈	—	118	1	no	—
71	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
72	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
73	C₂²⋊C₈	—	2	1	no	—
74	C₂²⋊C₈	—	59	2	no	—
75	#97	—	2	1	no	—
76	C₄×C₈	—	59	2	no	—
77	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	59	4	no	—
83	C₂×C₈	—	118	2	no	—
84	C₂×C₈	—	236	2	no	—
85	C₄²	Dic₁₁₈	1	1	yes	characteristic
86	C₄²	C₄×D₅₉	1	1	yes	characteristic
87	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
88	C₄²	C₂²×D₅₉	1	1	yes	characteristic
89	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
100	C₈	—	4	1	no	—
101	C₈	—	4	1	no	—
102	C₈	—	118	4	no	—
103	C₈	—	236	2	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#14273662	1	2	yes	—
107	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#14273735	1	2	yes	—
113	C₄	#14273752	1	1	yes	characteristic
114	C₄	#14273757	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#16837430	1	1	yes	characteristic
117	C₂	#16837445	1	1	yes	characteristic
118	C₂	C₂×#14273662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837455

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837154

C₂

yes

maximal, complemented

#16837152

C₂

yes

maximal, complemented

C₅₉×#432

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₄

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#108

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#431

—

C₂×C₄⋊C₈

—

#335

—

#334

—

C₅₉

#1573

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

118

—

103

C₈

—

236

—

104

C₂²

—

105

C₂²

—

106

C₂²

#14273662

yes

—

107

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#14273735

yes

—

113

C₄

#14273752

yes

characteristic

114

C₄

#14273757

yes

characteristic

115

C₂

—

116

C₂

#16837430

yes

characteristic

117

C₂

#16837445

yes

characteristic

118

C₂

C₂×#14273662

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#432	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#432	D₅₉
hypocenter	C₅₉	#1573
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#432

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #16837154	C₂	semidirect
4. #16837152	C₂	semidirect
65. C₅₉	#1573	semidirect

H ≅ G/N

type

3. #16837154

C₂

semidirect

4. #16837152

C₂

semidirect

65. C₅₉

#1573

semidirect

Subgroups — #21422882

Conjugacy classes of subgroups

Complements