Group #21425939 — subgroups

Subgroups — #21425939

Order 7552.

144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837818	C₂	1	2	yes	maximal, complemented
2	#16837520	C₂	1	1	yes	maximal, characteristic
3	#16837223	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837185	C₂	1	2	yes	maximal, complemented
5	C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273749	C₄	1	1	yes	characteristic
7	#14273733	C₂²	1	2	yes	—
8	#14273667	C₂²	1	2	yes	—
9	#14273666	—	2	2	no	—
10	C₁₆×D₅₉	—	2	2	no	—
11	#14273663	C₄	1	1	yes	characteristic
12	#14273655	C₄	1	2	yes	—
13	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	#11745685	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
19	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	#11745664	C₂×C₄	1	2	yes	—
21	Dic₂₃₆	C₈	1	1	yes	characteristic
22	D₄₇₂	C₈	1	1	yes	characteristic
23	#11745660	C₈	1	2	yes	—
24	#11745659	—	2	2	no	—
25	C₈×D₅₉	—	2	2	no	—
26	C₁₆×C₅₉	—	2	1	no	—
27	C₁₆×C₅₉	D₄	1	2	yes	—
28	C₅₉⋊C₁₆	—	2	2	no	—
29	C₂×C₄×C₅₉	—	2	1	no	—
30	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²⋊D₅₉	—	2	2	no	—
32	D₂₃₆	C₂×C₈	1	2	yes	—
33	C₄×D₅₉	—	2	2	no	—
34	Dic₁₁₈	C₂×C₈	1	2	yes	—
35	C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
36	C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
37	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₉	#41	1	1	yes	characteristic
39	C₅₉⋊C₈	—	2	2	no	—
40	C₂²×C₅₉	#131	1	1	yes	characteristic
41	C₂×D₅₉	—	2	2	no	—
42	C₄×C₅₉	—	4	1	no	—
43	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
44	C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
45	Dic₅₉	—	2	2	no	—
46	#2171	—	59	1	no	maximal, Sylow
47	C₂×C₅₉	—	2	1	no	—
48	C₂×C₅₉	C₈×D₄	1	1	yes	characteristic
49	D₅₉	—	4	2	no	—
50	#504	—	59	2	no	—
51	#443	—	59	1	no	—
52	#364	—	59	1	no	—
53	#350	—	59	2	no	—
54	#346	D₅₉	1	1	yes	characteristic, complemented
55	C₅₉	#2171	1	1	yes	characteristic, complemented, Sylow
56	#135	—	59	1	no	—
57	#131	—	59	2	no	—
58	M₃₂	—	59	2	no	—
59	M₃₂	—	118	2	no	—
60	C₂×C₁₆	—	118	2	no	—
61	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
62	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
63	#108	—	59	1	no	—
64	#104	—	59	2	no	—
65	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
66	#41	—	59	2	no	—
67	Q₁₆	—	59	1	no	—
68	SD₁₆	—	59	2	no	—
69	D₈	—	59	1	no	—
70	M₁₆	—	59	2	no	—
71	M₁₆	—	118	2	no	—
72	C₂×C₈	—	118	2	no	—
73	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
75	C₄²	C₄×D₅₉	1	1	yes	characteristic
76	C₁₆	—	2	1	no	—
77	C₁₆	—	118	2	no	—
78	C₁₆	D₂₃₆	1	2	yes	—
79	Q₈	—	59	2	no	—
80	D₄	—	59	2	no	—
81	D₄	—	118	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	118	2	no	—
84	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
85	C₈	—	118	2	no	—
86	C₈	C₈×D₅₉	1	1	yes	characteristic
87	C₈	C₈×D₅₉	1	1	yes	characteristic
88	C₈	C₂×D₂₃₆	1	1	yes	characteristic
89	C₈	#11745685	1	1	yes	characteristic
90	C₂²	—	118	2	no	—
91	C₂²	#14273733	1	1	yes	characteristic
92	C₄	—	4	1	no	—
93	C₄	—	118	2	no	—
94	C₄	C₄×D₂₃₆	1	1	yes	characteristic
95	C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	236	2	no	—
98	C₂	C₈×D₂₃₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837818

C₂

yes

maximal, complemented

#16837520

C₂

yes

maximal, characteristic

#16837223

C₂

yes

maximal, characteristic, complemented

#16837185

C₂

yes

maximal, complemented

C₅₉×#346

C₂

yes

maximal, characteristic, complemented

#14273749

C₄

yes

characteristic

#14273733

C₂²

yes

—

#14273667

C₂²

yes

—

#14273666

—

C₁₆×D₅₉

—

#14273663

C₄

yes

characteristic

#14273655

C₄

yes

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

#11745685

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

#11745664

C₂×C₄

yes

—

Dic₂₃₆

C₈

yes

characteristic

D₄₇₂

C₈

yes

characteristic

#11745660

C₈

yes

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

D₄

yes

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²⋊D₅₉

—

D₂₃₆

C₂×C₈

yes

—

C₄×D₅₉

—

Dic₁₁₈

C₂×C₈

yes

—

C₈×C₅₉

C₂×C₈

yes

characteristic

C₈×C₅₉

C₂×C₈

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

#131

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₂²×C₈

yes

characteristic

Dic₅₉

—

#2171

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₈×D₄

yes

characteristic

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2171

yes

characteristic, complemented, Sylow

#135

—

#131

—

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#108

—

#104

—

C₄×C₈

C₂×D₅₉

yes

characteristic

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₁₆

D₂₃₆

yes

—

Q₈

—

D₄

—

D₄

—

118

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₈

—

118

—

C₈

C₈×D₅₉

yes

characteristic

C₈

C₈×D₅₉

yes

characteristic

C₈

C₂×D₂₃₆

yes

characteristic

C₈

#11745685

yes

characteristic

C₂²

—

118

—

C₂²

#14273733

yes

characteristic

C₄

—

C₄

—

118

—

C₄

C₄×D₂₃₆

yes

characteristic

C₄

C₂×C₈×D₅₉

yes

characteristic

C₂

—

C₂

—

236

—

C₂

C₈×D₂₃₆

yes

characteristic

	subgroup	G/H
center	C₈	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#346	C₂
socle	C₂×C₅₉	C₈×D₄
hypercenter	#346	D₅₉
hypocenter	C₅₉	#2171
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

subgroup

G/H

center

C₈

C₂×D₂₃₆

commutator

C₄×C₅₉

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#346

C₂

socle

C₂×C₅₉

C₈×D₄

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

N	H ≅ G/N	type
1. #16837818	C₂	semidirect
3. #16837223	C₂	semidirect
4. #16837185	C₂	semidirect
5. C₅₉×#346	C₂	semidirect
54. #346	D₅₉	semidirect
55. C₅₉	#2171	semidirect

H ≅ G/N

type

1. #16837818

C₂

semidirect

3. #16837223

C₂

semidirect

4. #16837185

C₂

semidirect

5. C₅₉×#346

C₂

semidirect

54. #346

D₅₉

semidirect

55. C₅₉

#2171

semidirect

Subgroups — #21425939

Conjugacy classes of subgroups

Complements