Group #21425671 — subgroups

Subgroups — #21425671

Order 7552.

158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837451	C₂	1	1	yes	maximal, characteristic
2	#16837174	C₂	1	2	yes	maximal
3	#16837162	C₂	1	2	yes	maximal
4	#16837159	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	#14273686	C₂²	1	2	yes	—
8	#14273663	—	2	2	no	—
9	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	—	2	2	no	—
13	C₂²×C₄×C₅₉	—	2	1	no	—
14	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
15	M₁₆×C₅₉	—	2	2	no	—
16	M₁₆×C₅₉	—	2	2	no	—
17	C₂×C₈×C₅₉	—	2	1	no	—
18	C₂×C₈×C₅₉	D₄	1	2	yes	—
19	C₄²×C₅₉	—	2	1	no	—
20	C₄²×C₅₉	Q₈	1	1	yes	characteristic
21	C₄²×C₅₉	Q₈	1	1	yes	characteristic
22	C₂²⋊Dic₅₉	—	2	2	no	—
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₄×Dic₅₉	—	2	2	no	—
25	#11745664	—	2	2	no	—
26	#11745664	—	2	2	no	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂³×C₅₉	#41	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	2	1	no	—
30	C₂×C₄×C₅₉	—	2	1	no	—
31	C₂×C₄×C₅₉	—	4	2	no	—
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₅₉	#41	1	2	yes	—
35	C₂×C₄×C₅₉	#41	1	2	yes	—
36	C₂×Dic₅₉	—	4	2	no	—
37	C₈×C₅₉	—	2	2	no	—
38	C₈×C₅₉	—	4	1	no	—
39	C₅₉⋊C₈	—	4	2	no	—
40	C₂²×C₅₉	—	4	1	no	—
41	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₅₉	#126	1	2	yes	—
43	C₄×C₅₉	—	2	2	no	—
44	C₄×C₅₉	—	4	1	no	—
45	C₄×C₅₉	#123	1	1	yes	characteristic
46	C₄×C₅₉	#124	1	1	yes	characteristic
47	C₄×C₅₉	#125	1	2	yes	—
48	Dic₅₉	—	8	2	no	—
49	#2102	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	—	2	2	no	—
51	C₂×C₅₉	—	2	1	no	—
52	C₂×C₅₉	#400	1	1	yes	characteristic
53	#430	—	59	1	no	—
54	#423	D₅₉	1	1	yes	characteristic
55	#344	—	59	2	no	—
56	#339	—	59	2	no	—
57	H(3,ℤ/4)	—	59	1	no	—
58	C₅₉	#2102	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	59	2	no	—
60	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
61	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
62	#117	—	59	2	no	—
63	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
64	#108	—	118	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	118	2	no	—
71	M₁₆	—	118	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	118	2	no	—
74	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
75	C₄⋊C₄	—	118	2	no	—
76	C₂²⋊C₄	—	118	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	—	118	2	no	—
79	C₄²	Dic₁₁₈	1	1	yes	characteristic
80	C₄²	Dic₁₁₈	1	1	yes	characteristic
81	C₂³	#11745685	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	236	2	no	—
86	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
87	C₂×C₄	#11745687	1	2	yes	—
88	C₂×C₄	#11745689	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	4	1	no	—
92	C₈	—	236	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#14273706	1	1	yes	characteristic
95	C₂²	#14273722	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	4	1	no	—
98	C₄	—	472	2	no	—
99	C₄	#14273708	1	1	yes	characteristic
100	C₄	#14273716	1	1	yes	characteristic
101	C₄	#14273718	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#16837341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837451

C₂

yes

maximal, characteristic

#16837174

C₂

yes

maximal

#16837162

C₂

yes

maximal

#16837159

C₂

yes

maximal, characteristic

C₅₉×#423

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

#14273686

C₂²

yes

—

#14273663

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

#126

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#2102

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

C₅₉

#2102

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#108

—

118

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

—

C₄²

—

118

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

Dic₁₁₈

yes

characteristic

C₂³

#11745685

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

#11745687

yes

—

C₂×C₄

#11745689

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

C₈

—

236

—

C₂²

—

C₂²

#14273706

yes

characteristic

C₂²

#14273722

yes

—

C₄

—

C₄

—

C₄

—

472

—

C₄

#14273708

yes

characteristic

100

C₄

#14273716

yes

characteristic

101

C₄

#14273718

yes

—

102

C₂

—

103

C₂

—

104

C₂

#16837341

yes

characteristic

	subgroup	G/H
center	C₄	#14273716
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂×C₅₉	#400
hypercenter	#423	D₅₉
hypocenter	C₅₉	#2102
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#14273716

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#423

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
58. C₅₉	#2102	semidirect

H ≅ G/N

type

58. C₅₉

#2102

semidirect

Subgroups — #21425671

Conjugacy classes of subgroups

Complements