Group #21429815 — subgroups

Subgroups — #21429815

Order 7552.

782 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14273710	C₂	1	3	yes	maximal
2	#16837321	C₂	1	4	yes	maximal, complemented
3	#16837313	C₂	1	4	yes	maximal, complemented
4	#16837265	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
7	#14273710	—	2	12	no	—
8	C₂×Dic₅₉⋊C₄	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₅₉	C₂²	1	3	yes	—
10	C₂×(C₂×D₅₉)⋊C₄	C₂²	1	6	yes	—
11	C₅₉⋊#95	C₂²	1	12	yes	—
12	C₂²×Q₈×C₅₉	—	2	12	no	—
13	C₂²×Q₈×C₅₉	C₂²	1	4	yes	—
14	C₂³×C₄×C₅₉	C₂²	1	3	yes	—
15	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
16	C₂³×D₅₉	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	32	no	—
18	C₂×Q₈×C₅₉	—	2	24	no	—
19	C₂²×C₄×C₅₉	—	2	12	no	—
20	C₂²×C₄×C₅₉	D₄	1	12	yes	—
21	C₂²×C₄×C₅₉	C₂³	1	6	yes	—
22	C₂×C₂²⋊D₅₉	—	2	6	no	—
23	C₂²×Dic₅₉	C₂³	1	6	yes	—
24	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
25	C₂²⋊Dic₅₉	—	2	6	no	—
26	(C₂×D₅₉)⋊C₄	—	2	12	no	—
27	Dic₅₉⋊C₄	—	2	12	no	—
28	C₂³×C₅₉	—	2	4	no	—
29	C₂³×C₅₉	C₂×D₄	1	6	yes	—
30	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
31	C₂²×D₅₉	—	2	3	no	—
32	C₂²×D₅₉	—	2	4	no	—
33	Q₈×C₅₉	—	2	32	no	—
34	C₂×C₄×C₅₉	—	2	24	no	—
35	C₂×C₄×C₅₉	—	2	12	no	—
36	C₂×C₄×C₅₉	C₂×D₄	1	12	yes	—
37	C₂²⋊D₅₉	—	4	4	no	—
38	C₂×Dic₅₉	—	2	3	no	—
39	C₂×Dic₅₉	—	2	6	no	—
40	C₂×Dic₅₉	—	2	12	no	—
41	C₂²×C₅₉	—	2	12	no	—
42	C₂²×C₅₉	#120	1	4	yes	—
43	C₂²×C₅₉	C₂²×D₄	1	3	yes	—
44	C₂²×C₅₉	2₋¹⁺⁴	1	4	yes	—
45	C₂×D₅₉	—	4	1	no	—
46	C₂×D₅₉	—	4	6	no	—
47	C₄×C₅₉	—	2	12	no	—
48	Dic₅₉	—	4	1	no	—
49	Dic₅₉	—	4	6	no	—
50	#2614	—	59	1	no	maximal, Sylow
51	C₂×C₅₉	—	2	4	no	—
52	C₂×C₅₉	C₂×#120	1	1	yes	characteristic
53	C₂×C₅₉	#536	1	6	yes	—
54	D₅₉	—	8	1	no	—
55	C₂³×Q₈	D₅₉	1	1	yes	characteristic, complemented
56	C₂×#123	—	59	3	no	—
57	#395	—	59	4	no	—
58	#393	—	59	4	no	—
59	#381	—	59	3	no	—
60	C₅₉	#2614	1	1	yes	characteristic, complemented, Sylow
61	C₂²×Q₈	—	2	12	no	—
62	C₂²×Q₈	C₂×D₅₉	1	4	yes	—
63	C₂²×D₄	—	59	1	no	—
64	C₂³×C₄	C₂×D₅₉	1	3	yes	—
65	#123	—	118	12	no	—
66	C₂×C₄⋊C₄	—	59	6	no	—
67	C₂×C₂²⋊C₄	—	59	3	no	—
68	C₂×C₂²⋊C₄	—	59	6	no	—
69	#95	—	59	12	no	—
70	C₂⁴	—	59	1	no	—
71	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
72	C₂×Q₈	—	2	32	no	—
73	C₂×Q₈	—	2	24	no	—
74	C₂×D₄	—	118	6	no	—
75	C₂²×C₄	—	2	12	no	—
76	C₂²×C₄	—	59	6	no	—
77	C₂²×C₄	—	59	1	no	—
78	C₂²×C₄	C₂²⋊D₅₉	1	12	yes	—
79	C₂²×C₄	C₂²×D₅₉	1	6	yes	—
80	C₄⋊C₄	—	118	12	no	—
81	C₂²⋊C₄	—	118	6	no	—
82	C₂²⋊C₄	—	118	12	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	118	3	no	—
85	C₂³	—	118	4	no	—
86	C₂³	C₂×C₂²⋊D₅₉	1	6	yes	—
87	C₂³	C₂³×D₅₉	1	1	yes	characteristic
88	Q₈	—	2	32	no	—
89	D₄	—	236	4	no	—
90	C₂×C₄	—	2	24	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	118	6	no	—
93	C₂×C₄	—	118	3	no	—
94	C₂×C₄	—	118	12	no	—
95	C₂×C₄	C₂×C₂²⋊D₅₉	1	12	yes	—
96	C₂²	—	2	12	no	—
97	C₂²	—	236	1	no	—
98	C₂²	—	236	6	no	—
99	C₂²	#14273696	1	4	yes	—
100	C₂²	C₂²×C₂²⋊D₅₉	1	3	yes	—
101	C₂²	#14273775	1	4	yes	—
102	C₄	—	2	12	no	—
103	C₄	—	236	1	no	—
104	C₄	—	236	6	no	—
105	C₂	—	2	4	no	—
106	C₂	—	472	1	no	—
107	C₂	C₂×#14273696	1	1	yes	characteristic
108	C₂	#16837976	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273710

C₂

yes

maximal

#16837321

C₂

yes

maximal, complemented

#16837313

C₂

yes

maximal, complemented

#16837265

C₂

yes

maximal, complemented

C₂³×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₅₉

C₂²

yes

characteristic

#14273710

—

C₂×Dic₅₉⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₅₉

C₂²

yes

—

C₂×(C₂×D₅₉)⋊C₄

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₅₉

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₂³×C₄×C₅₉

C₂²

yes

—

C₂⁴×C₅₉

C₂³

yes

characteristic

C₂³×D₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

(C₂×D₅₉)⋊C₄

—

Dic₅₉⋊C₄

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

—

C₂²×C₅₉

C₂²×D₄

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

#2614

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#120

yes

characteristic

C₂×C₅₉

#536

yes

—

D₅₉

—

C₂³×Q₈

D₅₉

yes

characteristic, complemented

—

—

—

—

C₅₉

#2614

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₉

yes

—

C₂²×D₄

—

C₂³×C₄

C₂×D₅₉

yes

—

#123

—

118

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₂³

—

C₂³

—

118

—

C₂³

—

118

—

C₂³

C₂×C₂²⋊D₅₉

yes

—

C₂³

C₂³×D₅₉

yes

characteristic

Q₈

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

C₂²

—

C₂²

—

236

—

C₂²

—

236

—

C₂²

#14273696

yes

—

100

C₂²

C₂²×C₂²⋊D₅₉

yes

—

101

C₂²

#14273775

yes

—

102

C₄

—

103

C₄

—

236

—

104

C₄

—

236

—

105

C₂

—

106

C₂

—

472

—

107

C₂

C₂×#14273696

yes

characteristic

108

C₂

#16837976

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₂³×Q₈×C₅₉	C₂
hypercenter	C₂³×Q₈	D₅₉
hypocenter	C₅₉	#2614
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator, socle

C₂³×C₅₉

C₂⁴

Fitting

C₂³×Q₈×C₅₉

C₂

hypercenter

C₂³×Q₈

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #16837321	C₂	semidirect
3. #16837313	C₂	semidirect
4. #16837265	C₂	semidirect
5. C₂³×Q₈×C₅₉	C₂	semidirect
55. C₂³×Q₈	D₅₉	semidirect
60. C₅₉	#2614	semidirect

H ≅ G/N

type

2. #16837321

C₂

semidirect

3. #16837313

C₂

semidirect

4. #16837265

C₂

semidirect

5. C₂³×Q₈×C₅₉

C₂

semidirect

55. C₂³×Q₈

D₅₉

semidirect

60. C₅₉

#2614

semidirect

Subgroups — #21429815

Conjugacy classes of subgroups

Complements