Group #21423004 — subgroups

Subgroups — #21423004

Order 7552.

148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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	#16837529	C₂	1	2	yes	maximal
2	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#16837132	C₂	1	2	yes	maximal
4	#16837123	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₄	1	1	yes	characteristic
7	C₄×Dic₁₁₈	C₂²	1	2	yes	—
8	#14273661	C₄	1	1	yes	characteristic
9	Dic₅₉⋊C₈	—	2	2	no	—
10	#14273652	C₄	1	2	yes	—
11	#14273637	—	2	2	no	—
12	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₉	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
22	Dic₅₉⋊C₄	—	2	2	no	—
23	C₄×Dic₅₉	—	2	2	no	—
24	C₂×C₅₉⋊C₈	—	2	1	no	—
25	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
26	Dic₂₃₆	—	2	2	no	—
27	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
30	C₂×Dic₅₉	—	2	2	no	—
31	C₂×Dic₅₉	—	2	2	no	—
32	Dic₁₁₈	—	2	2	no	—
33	Dic₁₁₈	—	2	2	no	—
34	C₈×C₅₉	—	2	1	no	—
35	C₈×C₅₉	—	4	2	no	—
36	C₈×C₅₉	M₁₆	1	2	yes	—
37	C₅₉⋊C₈	—	2	2	no	—
38	C₅₉⋊C₈	—	4	1	no	—
39	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
40	C₄×C₅₉	—	2	1	no	—
41	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₅₉	#131	1	1	yes	characteristic
43	C₄×C₅₉	#137	1	2	yes	—
44	Dic₅₉	—	4	2	no	—
45	Dic₅₉	—	4	2	no	—
46	#1593	—	59	1	no	maximal, Sylow
47	C₂×C₅₉	#436	1	1	yes	characteristic
48	C₂×C₅₉	#441	1	1	yes	characteristic
49	C₂×C₅₉	#444	1	1	yes	characteristic
50	#446	—	59	2	no	—
51	C₄×Q₁₆	—	59	1	no	—
52	#334	D₅₉	1	1	yes	characteristic
53	Q₈⋊C₈	—	59	2	no	—
54	#322	—	59	1	no	—
55	C₅₉	#1593	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	—	59	1	no	—
57	C₄×Q₈	—	59	2	no	—
58	#107	—	59	1	no	—
59	C₄⋊C₈	—	118	2	no	—
60	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
61	#103	—	59	2	no	—
62	#97	—	118	2	no	—
63	C₄×C₈	—	59	1	no	—
64	C₄×C₈	—	59	1	no	—
65	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
66	C₂×Q₈	—	59	2	no	—
67	Q₁₆	—	118	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	59	2	no	—
70	C₂×C₈	—	118	1	no	—
71	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
72	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
73	C₄⋊C₄	—	59	2	no	—
74	C₄⋊C₄	—	118	2	no	—
75	C₄²	—	118	2	no	—
76	C₄²	C₂²×D₅₉	1	1	yes	characteristic
77	Q₈	—	118	2	no	—
78	Q₈	—	118	2	no	—
79	C₂×C₄	—	118	2	no	—
80	C₂×C₄	—	118	2	no	—
81	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
82	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
83	C₂×C₄	#11745689	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	4	2	no	—
86	C₈	—	118	2	no	—
87	C₈	—	236	1	no	—
88	C₈	#11745659	1	2	yes	—
89	C₂²	#14273689	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	236	2	no	—
92	C₄	—	236	2	no	—
93	C₄	C₂×#11745659	1	1	yes	characteristic
94	C₄	#14273735	1	1	yes	characteristic
95	C₄	#14273756	1	2	yes	—
96	C₂	#16837479	1	1	yes	characteristic
97	C₂	#16837508	1	1	yes	characteristic
98	C₂	#16837523	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837529

C₂

yes

maximal

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837132

C₂

yes

maximal

#16837123

C₂

yes

maximal, characteristic

C₅₉×#334

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₄

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

—

#14273661

C₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

C₄

yes

—

#14273637

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

Dic₂₃₆

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₄×Q₁₆

—

#334

D₅₉

yes

characteristic

Q₈⋊C₈

—

#322

—

C₅₉

#1593

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

#97

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

118

—

Q₈

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745689

yes

characteristic

C₈

—

C₈

—

C₈

—

118

—

C₈

—

236

—

C₈

#11745659

yes

—

C₂²

#14273689

yes

characteristic

C₄

—

C₄

—

236

—

C₄

—

236

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#334	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1593
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#334

C₂

socle

C₂²×C₅₉

C₄×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
55. C₅₉	#1593	semidirect

H ≅ G/N

type

55. C₅₉

#1593

semidirect

Subgroups — #21423004

Conjugacy classes of subgroups

Complements