Group #21426327 — subgroups

Subgroups — #21426327

Order 7552.

138 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837804	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837206	C₂	1	2	yes	maximal, complemented
3	#16837203	C₂	1	2	yes	maximal
4	C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#495	C₂	1	1	yes	maximal, characteristic
6	#14273719	—	2	1	no	—
7	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	#14273661	C₂²	1	2	yes	—
9	#14273659	—	2	1	no	—
10	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₅₉	—	2	1	no	—
12	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
13	#124×C₅₉	—	2	1	no	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	2	1	no	—
16	C₂×D₄×C₅₉	—	2	1	no	—
17	Q₁₆×C₅₉	—	2	2	no	—
18	SD₁₆×C₅₉	—	4	2	no	—
19	D₈×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²⋊C₄×C₅₉	—	4	2	no	—
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₄⋊Dic₅₉	—	2	2	no	—
25	Dic₅₉⋊C₄	—	4	2	no	—
26	C₅₉⋊C₁₆	—	2	2	no	—
27	C₂³×C₅₉	—	4	1	no	—
28	Q₈×C₅₉	—	4	1	no	—
29	Q₈×C₅₉	—	4	1	no	—
30	D₄×C₅₉	—	4	1	no	—
31	D₄×C₅₉	—	4	1	no	—
32	C₂×C₄×C₅₉	—	4	1	no	—
33	C₂×C₄×C₅₉	D₈	1	2	yes	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₉	—	4	2	no	—
36	C₈×C₅₉	SD₁₆	1	2	yes	—
37	C₈×C₅₉	#41	1	2	yes	—
38	C₂²×C₅₉	—	8	1	no	—
39	C₂²×C₅₉	—	8	2	no	—
40	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
41	C₄×C₅₉	—	2	2	no	—
42	C₄×C₅₉	—	8	1	no	—
43	C₄×C₅₉	#124	1	1	yes	characteristic
44	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
45	Dic₅₉	—	8	2	no	—
46	#2242	—	59	1	no	maximal, Sylow
47	C₂×C₅₉	—	16	1	no	—
48	C₂×C₅₉	#486	1	1	yes	characteristic
49	C₂×C₅₉	#508	1	2	yes	—
50	#499	—	59	1	no	—
51	#495	D₅₉	1	1	yes	characteristic
52	#358	—	59	2	no	—
53	#357	—	59	2	no	—
54	C₄×C₁₆	—	59	1	no	—
55	C₅₉	#2242	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
57	C₂×SD₁₆	—	2	1	no	—
58	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
59	#125	—	118	1	no	—
60	#124	—	2	1	no	—
61	C₂×C₁₆	—	59	2	no	—
62	#107	—	59	2	no	—
63	#106	—	118	1	no	—
64	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
65	C₂×Q₈	—	2	1	no	—
66	C₂×D₄	—	2	1	no	—
67	Q₁₆	—	2	2	no	—
68	SD₁₆	—	4	2	no	—
69	D₈	—	2	2	no	—
70	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
72	C₄⋊C₄	—	118	2	no	—
73	C₄⋊C₄	—	236	2	no	—
74	C₂²⋊C₄	—	4	2	no	—
75	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
76	C₁₆	—	118	2	no	—
77	C₂³	—	4	1	no	—
78	Q₈	—	4	1	no	—
79	Q₈	—	4	1	no	—
80	D₄	—	4	1	no	—
81	D₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	236	2	no	—
84	C₂×C₄	#11745669	1	2	yes	—
85	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
86	C₈	#11745670	1	2	yes	—
87	C₈	#11745687	1	2	yes	—
88	C₂²	—	8	1	no	—
89	C₂²	—	8	2	no	—
90	C₂²	C₂×#11745669	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	8	1	no	—
93	C₄	—	472	2	no	—
94	C₄	#14273714	1	1	yes	characteristic
95	C₄	C₂×#11745670	1	1	yes	characteristic
96	C₂	—	16	1	no	—
97	C₂	#16837748	1	1	yes	characteristic
98	C₂	#16837832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837804

C₂

yes

maximal, characteristic, complemented

#16837206

C₂

yes

maximal, complemented

#16837203

C₂

yes

maximal

C₄×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₅₉×#495

C₂

yes

maximal, characteristic

#14273719

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

#14273661

C₂²

yes

—

#14273659

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×SD₁₆×C₅₉

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

#124×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

Q₁₆×C₅₉

—

SD₁₆×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

SD₁₆

yes

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

#2242

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#486

yes

characteristic

C₂×C₅₉

#508

yes

—

#499

—

#495

D₅₉

yes

characteristic

#358

—

#357

—

C₄×C₁₆

—

C₅₉

#2242

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#125

—

118

—

#124

—

C₂×C₁₆

—

#107

—

#106

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₉

yes

characteristic

C₁₆

—

118

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

#11745669

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

#11745670

yes

—

C₈

#11745687

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#11745669

yes

characteristic

C₄

—

C₄

—

C₄

—

472

—

C₄

yes

C₄

yes

C₂

—

C₂

#16837748

yes

characteristic

C₂

#16837832

yes

—

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#495	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#495	D₅₉
hypocenter	C₅₉	#2242
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#495

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. #16837804	C₂	semidirect
2. #16837206	C₂	semidirect
4. C₄×C₅₉⋊C₁₆	C₂	semidirect
55. C₅₉	#2242	semidirect

H ≅ G/N

type

1. #16837804

C₂

semidirect

2. #16837206

C₂

semidirect

4. C₄×C₅₉⋊C₁₆

C₂

semidirect

55. C₅₉

#2242

semidirect

Subgroups — #21426327

Conjugacy classes of subgroups

Complements