Group #21425064 — subgroups

Subgroups — #21425064

Order 7552.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14273719	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	4	yes	maximal
4	C₅₉×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
6	#14273719	—	2	2	no	—
7	#14273719	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×Dic₅₉⋊C₄	—	2	2	no	—
10	#14273638	—	2	2	no	—
11	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	—	2	2	no	—
15	C₂²×C₄×C₅₉	D₄	1	2	yes	—
16	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	2	2	no	—
18	C₄⋊C₄×C₅₉	—	2	4	no	—
19	C₄⋊C₄×C₅₉	—	4	4	no	—
20	C₄²×C₅₉	C₂×C₄	1	2	yes	—
21	C₄²×C₅₉	D₄	1	2	yes	—
22	C₄⋊Dic₅₉	—	2	2	no	—
23	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
24	Dic₅₉⋊C₄	—	2	8	no	—
25	C₂×C₅₉⋊C₈	—	2	2	no	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	Q₈	1	4	yes	—
28	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	—	4	2	no	—
32	C₂×C₄×C₅₉	—	4	4	no	—
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₅₉	#41	1	2	yes	—
38	C₂×Dic₅₉	—	2	4	no	—
39	C₂×Dic₅₉	—	4	2	no	—
40	C₂×Dic₅₉	—	4	2	no	—
41	C₅₉⋊C₈	—	2	4	no	—
42	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₅₉	#124	1	1	yes	characteristic
44	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₅₉	#137	1	2	yes	—
47	C₄×C₅₉	—	2	4	no	—
48	C₄×C₅₉	—	8	2	no	—
49	C₄×C₅₉	C₄×Q₈	1	2	yes	—
50	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
51	Dic₅₉	—	4	4	no	—
52	#1992	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	#391	1	1	yes	characteristic
54	C₂×C₅₉	#417	1	2	yes	—
55	C₂×C₅₉	#490	1	1	yes	characteristic
56	C₂×C₅₉	#491	1	1	yes	characteristic
57	C₂×C₅₉	#501	1	2	yes	—
58	C₂×#125	—	59	1	no	—
59	C₂×#97	—	59	1	no	—
60	#384	D₅₉	1	1	yes	characteristic
61	#340	—	59	4	no	—
62	C₅₉	#1992	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	59	2	no	—
64	#125	—	59	4	no	—
65	#125	—	118	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	59	2	no	—
68	C₂×C₄⋊C₄	—	118	2	no	—
69	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
70	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
71	#97	—	118	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	118	2	no	—
74	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
75	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
76	C₂×C₈	—	59	4	no	—
77	C₂×C₈	—	118	2	no	—
78	C₂×C₈	—	118	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	59	4	no	—
82	C₄⋊C₄	—	118	8	no	—
83	C₄⋊C₄	—	118	2	no	—
84	C₄²	C₄×D₅₉	1	2	yes	—
85	C₄²	D₂₃₆	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	118	4	no	—
92	C₂×C₄	—	236	2	no	—
93	C₂×C₄	—	236	2	no	—
94	C₂×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
96	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂×C₄	#11745687	1	2	yes	—
98	C₂×C₄	Q₈×D₅₉	1	2	yes	—
99	C₈	—	118	4	no	—
100	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
101	C₂²	#14273714	1	1	yes	characteristic
102	C₂²	#14273725	1	1	yes	characteristic
103	C₂²	#14273753	1	2	yes	—
104	C₂²	#14273758	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	236	4	no	—
108	C₄	#14273692	1	2	yes	—
109	C₄	#14273707	1	2	yes	—
110	C₂	#16837307	1	1	yes	characteristic
111	C₂	#16837401	1	2	yes	—
112	C₂	#16837766	1	1	yes	characteristic
113	C₂	#16837770	1	1	yes	characteristic
114	C₂	#16837808	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273719

C₂

yes

maximal, characteristic

C₂×#14273638

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

C₅₉×#384

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

#14273719

—

#14273719

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

—

#14273638

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#124

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×#125

—

C₂×#97

—

#384

D₅₉

yes

characteristic

#340

—

C₅₉

#1992

yes

characteristic, complemented, Sylow

C₂²×C₈

—

#125

—

#125

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#97

—

118

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

C₄×D₅₉

yes

—

C₄²

D₂₃₆

yes

—

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745687

yes

—

C₂×C₄

Q₈×D₅₉

yes

—

C₈

—

118

—

100

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

101

C₂²

#14273714

yes

characteristic

102

C₂²

#14273725

yes

characteristic

103

C₂²

#14273753

yes

—

104

C₂²

#14273758

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

236

—

108

C₄

#14273692

yes

—

109

C₄

#14273707

yes

—

110

C₂

#16837307

yes

characteristic

111

C₂

#16837401

yes

—

112

C₂

#16837766

yes

characteristic

113

C₂

#16837770

yes

characteristic

114

C₂

#16837808

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#384	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#384	D₅₉
hypocenter	C₅₉	#1992
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#384

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #14273719	C₄	semidirect
62. C₅₉	#1992	semidirect

H ≅ G/N

type

7. #14273719

C₄

semidirect

62. C₅₉

#1992

semidirect

Subgroups — #21425064

Conjugacy classes of subgroups

Complements