Group #21425088 — subgroups

Subgroups — #21425088

Order 7552.

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

normal subgroups only

This group has 228 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	C₂×#14273719	C₂	1	1	yes	maximal, characteristic
2	C₄×#11745664	C₂	1	1	yes	maximal, characteristic
3	#16837170	C₂	1	4	yes	maximal
4	C₅₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	C₂²	1	2	yes	—
6	#14273719	—	2	1	no	—
7	#14273719	—	2	1	no	—
8	#14273719	—	2	2	no	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	C₂×Dic₅₉⋊C₄	C₄	1	4	yes	—
11	#14273638	—	2	1	no	—
12	C₄×C₅₉⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	C₂×C₄	1	4	yes	—
19	M₁₆×C₅₉	—	2	4	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₄²×C₅₉	D₄	1	1	yes	characteristic
22	C₄²×C₅₉	D₄	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	2	yes	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	Dic₅₉⋊C₄	—	2	4	no	—
27	Dic₅₉⋊C₄	—	2	4	no	—
28	#11745664	—	2	2	no	—
29	#11745664	Q₈	1	4	yes	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	4	2	no	—
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₅₉	#41	1	2	yes	—
39	C₂×Dic₅₉	—	2	4	no	—
40	C₂×Dic₅₉	—	4	4	no	—
41	C₈×C₅₉	—	4	2	no	—
42	C₅₉⋊C₈	—	2	4	no	—
43	C₂²×C₅₉	—	2	2	no	—
44	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
46	C₄×C₅₉	—	2	2	no	—
47	C₄×C₅₉	—	2	2	no	—
48	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₅₉	#124	1	1	yes	characteristic
50	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₅₉	—	4	4	no	—
52	#1998	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	2	2	no	—
54	C₂×C₅₉	#391	1	1	yes	characteristic
55	C₂×C₅₉	#413	1	2	yes	—
56	C₂×#125	—	59	1	no	—
57	#423	D₅₉	1	1	yes	characteristic
58	C₄×M₁₆	—	59	1	no	—
59	#343	—	59	4	no	—
60	C₅₉	#1998	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	59	2	no	—
62	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
63	#125	—	118	1	no	—
64	#125	—	118	1	no	—
65	#125	—	118	2	no	—
66	C₂×C₄⋊C₄	—	59	4	no	—
67	C₂×C₄⋊C₄	—	59	2	no	—
68	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	118	1	no	—
71	C₄×C₈	—	118	1	no	—
72	C₂²×C₄	—	59	4	no	—
73	C₂²×C₄	C₄×D₅₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	59	4	no	—
77	M₁₆	—	118	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	118	2	no	—
80	C₄⋊C₄	—	118	4	no	—
81	C₄⋊C₄	—	118	4	no	—
82	C₄⋊C₄	—	118	2	no	—
83	C₄⋊C₄	—	118	2	no	—
84	C₄²	D₂₃₆	1	1	yes	characteristic
85	C₄²	D₂₃₆	1	1	yes	characteristic
86	C₄²	C₂²⋊D₅₉	1	2	yes	—
87	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	118	4	no	—
90	C₂×C₄	—	236	4	no	—
91	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
94	C₂×C₄	#11745687	1	2	yes	—
95	C₂×C₄	Q₈×D₅₉	1	2	yes	—
96	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
97	C₈	—	4	2	no	—
98	C₈	—	118	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
101	C₂²	#14273692	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	236	4	no	—
105	C₄	#14273707	1	2	yes	—
106	C₄	#14273714	1	1	yes	characteristic
107	C₄	#14273725	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#16837307	1	1	yes	characteristic
110	C₂	#16837384	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273719

C₂

yes

maximal, characteristic

C₄×#11745664

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal

C₅₉×#423

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

#14273719

—

#14273719

—

#14273719

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

C₄

yes

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

#11745664

—

#11745664

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

#1998

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#391

yes

characteristic

C₂×C₅₉

#413

yes

—

C₂×#125

—

#423

D₅₉

yes

characteristic

C₄×M₁₆

—

#343

—

C₅₉

#1998

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

#125

—

118

—

#125

—

118

—

#125

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

D₂₃₆

yes

characteristic

C₄²

D₂₃₆

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

—

C₂³

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745687

yes

—

C₂×C₄

Q₈×D₅₉

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

118

—

C₂²

—

100

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

101

C₂²

#14273692

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

236

—

105

C₄

#14273707

yes

—

106

C₄

#14273714

yes

characteristic

107

C₄

#14273725

yes

characteristic

108

C₂

—

109

C₂

#16837307

yes

characteristic

110

C₂

#16837384

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂³×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#423	D₅₉
hypocenter	C₅₉	#1998
LCS	G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₅₉)⋊C₄

commutator

C₂³×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#423

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
60. C₅₉	#1998	semidirect

H ≅ G/N

type

60. C₅₉

#1998

semidirect

Subgroups — #21425088

Conjugacy classes of subgroups

Complements