Group #21424229 — subgroups

Subgroups — #21424229

Order 7552.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14273638	C₂	1	1	yes	maximal, characteristic
2	#16837272	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	4	yes	maximal
4	C₅₉×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₉	—	2	2	no	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	#14273638	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	#95×C₅₉	—	2	2	no	—
12	C₂²×C₄×C₅₉	—	2	2	no	—
13	C₂²×C₄×C₅₉	D₄	1	2	yes	—
14	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
15	C₂²×Dic₅₉	—	2	2	no	—
16	C₄⋊C₄×C₅₉	—	2	4	no	—
17	C₄²×C₅₉	C₂×C₄	1	2	yes	—
18	C₄²×C₅₉	Q₈	1	2	yes	—
19	C₄⋊Dic₅₉	—	2	4	no	—
20	C₄⋊Dic₅₉	—	4	4	no	—
21	C₂×C₅₉⋊C₈	—	2	2	no	—
22	C₂×C₅₉⋊C₈	—	2	2	no	—
23	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
24	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₅₉	—	2	2	no	—
26	C₂×C₄×C₅₉	—	2	2	no	—
27	C₂×C₄×C₅₉	—	4	2	no	—
28	C₂×C₄×C₅₉	—	4	2	no	—
29	C₂×C₄×C₅₉	—	4	2	no	—
30	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅₉	#41	1	2	yes	—
34	C₂×C₄×C₅₉	#41	1	2	yes	—
35	C₂×Dic₅₉	—	4	2	no	—
36	C₂×Dic₅₉	—	4	4	no	—
37	C₅₉⋊C₈	—	2	4	no	—
38	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₅₉	#124	1	1	yes	characteristic
40	C₂²×C₅₉	#124	1	1	yes	characteristic
41	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₅₉	#137	1	2	yes	—
43	C₄×C₅₉	—	2	4	no	—
44	C₄×C₅₉	—	8	2	no	—
45	C₄×C₅₉	#117	1	2	yes	—
46	C₄×C₅₉	#125	1	2	yes	—
47	Dic₅₉	—	8	2	no	—
48	#1836	—	59	1	no	maximal, Sylow
49	C₂×C₅₉	#382	1	1	yes	characteristic
50	C₂×C₅₉	#428	1	2	yes	—
51	C₂×C₅₉	#489	1	2	yes	—
52	C₂×C₅₉	#490	1	1	yes	characteristic
53	C₂×C₅₉	#491	1	1	yes	characteristic
54	C₂×#97	—	59	1	no	—
55	#384	—	59	1	no	—
56	#382	D₅₉	1	1	yes	characteristic
57	#340	—	59	4	no	—
58	C₅₉	#1836	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	59	2	no	—
60	C₂×C₄⋊C₄	—	59	2	no	—
61	C₂×C₄⋊C₄	—	118	2	no	—
62	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
63	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
64	#97	—	59	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	118	2	no	—
68	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
69	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
70	C₂×C₈	—	59	4	no	—
71	C₂×C₈	—	118	2	no	—
72	C₂×C₈	—	118	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	118	4	no	—
75	C₄⋊C₄	—	236	4	no	—
76	C₄²	Dic₁₁₈	1	2	yes	—
77	C₄²	C₄×D₅₉	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	236	2	no	—
85	C₂×C₄	—	236	4	no	—
86	C₂×C₄	Dic₅₉⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
88	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
89	C₂×C₄	#11745687	1	2	yes	—
90	C₂×C₄	#11745689	1	2	yes	—
91	C₈	—	118	4	no	—
92	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
93	C₂²	#14273714	1	1	yes	characteristic
94	C₂²	#14273716	1	1	yes	characteristic
95	C₂²	#14273752	1	2	yes	—
96	C₂²	#14273757	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	472	2	no	—
100	C₄	#14273685	1	2	yes	—
101	C₄	#14273718	1	2	yes	—
102	C₂	#16837267	1	1	yes	characteristic
103	C₂	#16837445	1	2	yes	—
104	C₂	#16837761	1	2	yes	—
105	C₂	#16837768	1	1	yes	characteristic
106	C₂	#16837772	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273638

C₂

yes

maximal, characteristic

#16837272

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

C₅₉×#382

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273638

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#117

yes

—

C₄×C₅₉

#125

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#489

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×#97

—

#384

—

#382

D₅₉

yes

characteristic

#340

—

C₅₉

#1836

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#97

—

#95

—

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

—

118

—

C₄⋊C₄

—

236

—

C₄²

Dic₁₁₈

yes

—

C₄²

C₄×D₅₉

yes

—

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

Dic₅₉⋊C₄

yes

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

#11745687

yes

—

C₂×C₄

#11745689

yes

—

C₈

—

118

—

C₂²

C₂×Dic₅₉⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#14273757

yes

—

C₄

—

C₄

—

C₄

—

472

—

100

C₄

#14273685

yes

—

101

C₄

#14273718

yes

—

102

C₂

#16837267

yes

characteristic

103

C₂

#16837445

yes

—

104

C₂

#16837761

yes

—

105

C₂

#16837768

yes

characteristic

106

C₂

#16837772

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#382	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#382	D₅₉
hypocenter	C₅₉	#1836
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#382

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #14273638	C₄	semidirect
58. C₅₉	#1836	semidirect

H ≅ G/N

type

8. #14273638

C₄

semidirect

58. C₅₉

#1836

semidirect

Subgroups — #21424229

Conjugacy classes of subgroups

Complements