Group #21425272 — subgroups

Subgroups — #21425272

Order 7552.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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×#14273641	C₂	1	3	yes	maximal
2	#16837168	C₂	1	3	yes	maximal, complemented
3	C₅₉×#543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₅₉	C₂²	1	3	yes	—
6	#14273641	—	2	6	no	—
7	C₂²⋊Q₈×C₅₉	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	—
9	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
10	C₂³×D₅₉	Q₈	1	1	yes	characteristic, complemented
11	C₂×Q₈×C₅₉	—	2	2	no	—
12	C₂²×C₄×C₅₉	—	2	3	no	—
13	C₂×C₂²⋊D₅₉	—	2	3	no	—
14	C₂×C₂²⋊D₅₉	—	2	3	no	—
15	C₂²×Dic₅₉	—	2	3	no	—
16	C₂²×Dic₅₉	Q₈	1	1	yes	characteristic, complemented
17	C₄⋊C₄×C₅₉	—	2	6	no	—
18	C₂²⋊C₄×C₅₉	—	2	3	no	—
19	C₂²⋊C₄×C₅₉	D₄	1	6	yes	—
20	C₂²⋊Dic₅₉	—	2	3	no	—
21	C₂²⋊Dic₅₉	—	2	3	no	—
22	C₂³×C₅₉	—	8	1	no	—
23	C₂³×C₅₉	C₂×D₄	1	3	yes	—
24	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
25	C₂³×C₅₉	#41	1	3	yes	—
26	C₂²×D₅₉	—	2	3	no	—
27	C₂²×D₅₉	—	8	1	no	—
28	Q₈×C₅₉	—	8	2	no	—
29	C₂×C₄×C₅₉	—	2	6	no	—
30	C₂×C₄×C₅₉	—	8	3	no	—
31	C₂²⋊D₅₉	—	4	1	no	—
32	C₂²⋊D₅₉	—	4	3	no	—
33	C₂×Dic₅₉	—	2	3	no	—
34	C₂×Dic₅₉	—	4	3	no	—
35	C₂×Dic₅₉	—	8	3	no	—
36	C₂²×C₅₉	—	4	3	no	—
37	C₂²×C₅₉	—	4	3	no	—
38	C₂²×C₅₉	—	4	1	no	—
39	C₂²×C₅₉	#120	1	1	yes	characteristic
40	C₂²×C₅₉	C₂²⋊Q₈	1	3	yes	—
41	C₂²×C₅₉	#124	1	3	yes	—
42	C₂×D₅₉	—	4	1	no	—
43	C₂×D₅₉	—	8	3	no	—
44	C₄×C₅₉	—	4	6	no	—
45	Dic₅₉	—	4	1	no	—
46	Dic₅₉	—	8	3	no	—
47	#2032	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	—	2	3	no	—
49	C₂×C₅₉	—	2	1	no	—
50	C₂×C₅₉	—	2	3	no	—
51	C₂×C₅₉	#393	1	1	yes	characteristic
52	D₅₉	—	8	1	no	—
53	#543	D₅₉	1	1	yes	characteristic, complemented
54	C₂×#99	—	59	3	no	—
55	#342	—	59	3	no	—
56	C₅₉	#2032	1	1	yes	characteristic, complemented, Sylow
57	C₂²×D₄	—	59	1	no	—
58	C₂²⋊Q₈	—	2	6	no	—
59	C₂×C₂²⋊C₄	—	59	3	no	—
60	C₂×C₂²⋊C₄	C₂×D₅₉	1	3	yes	—
61	#99	—	118	6	no	—
62	C₂⁴	—	59	1	no	—
63	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
64	C₂×Q₈	—	2	2	no	—
65	C₂×D₄	—	118	3	no	—
66	C₂×D₄	—	118	3	no	—
67	C₂²×C₄	—	2	3	no	—
68	C₂²×C₄	—	59	1	no	—
69	C₂²×C₄	—	118	3	no	—
70	C₄⋊C₄	—	2	6	no	—
71	C₂²⋊C₄	—	2	3	no	—
72	C₂²⋊C₄	—	118	3	no	—
73	C₂²⋊C₄	—	118	3	no	—
74	C₂²⋊C₄	C₂²⋊D₅₉	1	6	yes	—
75	C₂³	—	8	1	no	—
76	C₂³	—	118	3	no	—
77	C₂³	—	472	1	no	—
78	C₂³	Q₈×D₅₉	1	1	yes	characteristic
79	C₂³	#11745689	1	3	yes	—
80	C₂³	C₂×C₂²⋊D₅₉	1	3	yes	—
81	Q₈	—	8	2	no	—
82	D₄	—	236	1	no	—
83	D₄	—	236	3	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	—	118	3	no	—
87	C₂×C₄	—	472	3	no	—
88	C₂×C₄	—	236	3	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	4	3	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	472	3	no	—
93	C₂²	—	236	1	no	—
94	C₂²	#14273696	1	1	yes	characteristic
95	C₂²	#14273703	1	3	yes	—
96	C₂²	#14273716	1	3	yes	—
97	C₄	—	4	6	no	—
98	C₄	—	472	3	no	—
99	C₄	—	236	1	no	—
100	C₂	—	2	1	no	—
101	C₂	—	2	3	no	—
102	C₂	—	2	3	no	—
103	C₂	—	472	1	no	—
104	C₂	#16837313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273641

C₂

yes

maximal

#16837168

C₂

yes

maximal, complemented

C₅₉×#543

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₅₉

C₂²

yes

characteristic

C₂×C₂²⋊Dic₅₉

C₂²

yes

—

#14273641

—

C₂²⋊Q₈×C₅₉

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

—

C₂⁴×C₅₉

C₂³

yes

characteristic

C₂³×D₅₉

Q₈

yes

characteristic, complemented

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

Q₈

yes

characteristic, complemented

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂³×C₅₉

#41

yes

—

C₂²×D₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

#124

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

#2032

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#393

yes

characteristic

D₅₉

—

#543

D₅₉

yes

characteristic, complemented

C₂×#99

—

#342

—

C₅₉

#2032

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₉

yes

—

#99

—

118

—

C₂⁴

—

C₂⁴

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₂²⋊C₄

C₂²⋊D₅₉

yes

—

C₂³

—

C₂³

—

118

—

C₂³

—

472

—

C₂³

Q₈×D₅₉

yes

characteristic

C₂³

#11745689

yes

—

C₂³

C₂×C₂²⋊D₅₉

yes

—

Q₈

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

472

—

C₂×C₄

—

236

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

472

—

C₂²

—

236

—

C₂²

#14273696

yes

characteristic

C₂²

#14273703

yes

—

C₂²

#14273716

yes

—

C₄

—

C₄

—

472

—

C₄

—

236

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

—

472

—

104

C₂

#16837313

yes

characteristic

	subgroup	G/H
center	C₂	#16837313
commutator	C₂⁴×C₅₉	C₂³
Frattini	C₂⁴	C₂²×D₅₉
Fitting	C₅₉×#543	C₂
socle	C₂×C₅₉	#393
hypercenter	#543	D₅₉
hypocenter	C₅₉	#2032
LCS	G ▷ C₂⁴×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂

#16837313

commutator

C₂⁴×C₅₉

C₂³

Frattini

C₂⁴

C₂²×D₅₉

Fitting

C₅₉×#543

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂⁴×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #16837168	C₂	semidirect
3. C₅₉×#543	C₂	semidirect
10. C₂³×D₅₉	Q₈	semidirect
16. C₂²×Dic₅₉	Q₈	semidirect
53. #543	D₅₉	semidirect
56. C₅₉	#2032	semidirect

H ≅ G/N

type

2. #16837168

C₂

semidirect

3. C₅₉×#543

C₂

semidirect

10. C₂³×D₅₉

Q₈

semidirect

16. C₂²×Dic₅₉

Q₈

semidirect

53. #543

D₅₉

semidirect

56. C₅₉

#2032

semidirect

Subgroups — #21425272

Conjugacy classes of subgroups

Complements