Group #21425901 — subgroups

Subgroups — #21425901

Order 7552.

138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16837356	C₂	1	1	yes	maximal, characteristic
2	Dic₅₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₅₉)⋊C₁₆	C₂	1	2	yes	maximal
4	#16837178	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745659	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
8	#14273684	C₄	1	1	yes	characteristic
9	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	#14273638	C₄	1	1	yes	characteristic
11	#14273637	C₄	1	1	yes	characteristic
12	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
18	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	(C₂×D₅₉)⋊C₄	C₈	1	2	yes	—
20	Dic₅₉⋊C₄	C₈	1	2	yes	—
21	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
24	#11745659	—	2	2	no	—
25	C₈×D₅₉	—	2	2	no	—
26	C₁₆×C₅₉	—	2	2	no	—
27	C₅₉⋊C₁₆	—	2	2	no	—
28	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₉	C₂×C₈	1	2	yes	—
32	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
33	C₄×D₅₉	—	2	1	no	—
34	C₄×D₅₉	—	2	1	no	—
35	C₈×C₅₉	M₁₆	1	2	yes	—
36	C₈×C₅₉	#41	1	2	yes	—
37	C₅₉⋊C₈	—	2	1	no	—
38	C₅₉⋊C₈	—	2	1	no	—
39	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₅₉	—	2	1	no	—
41	C₂×D₅₉	—	2	2	no	—
42	C₄×C₅₉	—	2	2	no	—
43	C₄×C₅₉	#117	1	1	yes	characteristic
44	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
45	Dic₅₉	—	2	1	no	—
46	Dic₅₉	—	2	2	no	—
47	#2164	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	#431	1	1	yes	characteristic
49	C₂×C₅₉	#504	1	2	yes	—
50	D₅₉	—	4	1	no	—
51	#405	—	59	1	no	—
52	C₄⋊C₁₆	—	59	2	no	—
53	C₂²⋊C₁₆	—	59	2	no	—
54	#346	—	59	1	no	—
55	C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
56	C₅₉	#2164	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	59	1	no	—
58	C₂²×C₈	—	59	1	no	—
59	#117	—	59	1	no	—
60	C₂×C₁₆	—	59	2	no	—
61	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
62	#97	—	59	1	no	—
63	#97	—	59	1	no	—
64	C₄×C₈	—	59	1	no	—
65	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
66	C₂²×C₄	—	59	1	no	—
67	M₁₆	—	118	2	no	—
68	C₂×C₈	—	59	1	no	—
69	C₂×C₈	—	59	1	no	—
70	C₂×C₈	—	118	2	no	—
71	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
73	C₄⋊C₄	—	59	2	no	—
74	C₂²⋊C₄	—	59	2	no	—
75	C₄²	—	59	1	no	—
76	C₄²	C₄×D₅₉	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	118	2	no	—
79	C₂³	—	59	1	no	—
80	C₂×C₄	—	59	2	no	—
81	C₂×C₄	—	59	1	no	—
82	C₂×C₄	—	118	1	no	—
83	C₂×C₄	—	118	1	no	—
84	C₂×C₄	C₈×D₅₉	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
86	C₈	—	118	1	no	—
87	C₈	—	118	1	no	—
88	C₈	#11745659	1	2	yes	—
89	C₈	#11745685	1	2	yes	—
90	C₂²	—	118	1	no	—
91	C₂²	—	118	2	no	—
92	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	118	1	no	—
95	C₄	—	118	2	no	—
96	C₄	#14273684	1	1	yes	characteristic
97	C₄	C₂×#11745659	1	1	yes	characteristic
98	C₂	—	236	1	no	—
99	C₂	#16837453	1	1	yes	characteristic
100	C₂	#16837818	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837356

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal

#16837178

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745659

C₄

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273684

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

—

#14273638

C₄

yes

characteristic

#14273637

C₄

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

(C₂×D₅₉)⋊C₄

C₈

yes

—

Dic₅₉⋊C₄

C₈

yes

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

—

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

M₁₆

yes

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

D₅₉

yes

characteristic, complemented

C₅₉

#2164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₈×D₅₉

yes

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₈

—

118

—

C₈

—

118

—

C₈

#11745659

yes

—

C₈

#11745685

yes

—

C₂²

—

118

—

C₂²

—

118

—

C₂²

C₂×C₈×D₅₉

yes

characteristic

C₄

—

C₄

—

118

—

C₄

—

118

—

C₄

yes

C₄

yes

C₂

—

236

—

C₂

#16837453

yes

characteristic

100

C₂

#16837818

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₈
Fitting	C₄×C₁₆×C₅₉	C₂
hypercenter	C₄×C₁₆	D₅₉
hypocenter	C₅₉	#2164
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₈

Fitting

C₄×C₁₆×C₅₉

C₂

hypercenter

C₄×C₁₆

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. Dic₅₉⋊C₁₆	C₂	semidirect
4. #16837178	C₂	semidirect
5. C₄×C₁₆×C₅₉	C₂	semidirect
55. C₄×C₁₆	D₅₉	semidirect
56. C₅₉	#2164	semidirect

H ≅ G/N

type

2. Dic₅₉⋊C₁₆

C₂

semidirect

4. #16837178

C₂

semidirect

5. C₄×C₁₆×C₅₉

C₂

semidirect

55. C₄×C₁₆

D₅₉

semidirect

56. C₅₉

#2164

semidirect

Subgroups — #21425901

Conjugacy classes of subgroups

Complements