Group #21423501 — subgroups

Subgroups — #21423501

Order 7552.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837807	C₂	1	1	yes	maximal, characteristic
2	#16837698	C₂	1	2	yes	maximal
3	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
4	#16837150	C₂	1	2	yes	maximal
5	C₅₉×#335	C₂	1	1	yes	maximal, characteristic
6	#14273727	C₂²	1	2	yes	—
7	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
8	#14273661	—	2	1	no	—
9	#14273661	—	2	1	no	—
10	#14273661	—	2	2	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₈	—	2	1	no	—
13	#14273652	—	2	2	no	—
14	C₈×Dic₅₉	—	2	1	no	—
15	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₁₈	—	2	2	no	—
18	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
26	Dic₅₉⋊C₄	—	2	1	no	—
27	C₄×Dic₅₉	—	2	1	no	—
28	C₂×C₅₉⋊C₈	—	2	1	no	—
29	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
32	C₂×Dic₅₉	—	2	1	no	—
33	C₂×Dic₅₉	—	2	1	no	—
34	C₂×Dic₅₉	—	4	2	no	—
35	Dic₁₁₈	—	2	1	no	—
36	Dic₁₁₈	—	8	2	no	—
37	Dic₁₁₈	Q₁₆	1	2	yes	—
38	C₈×C₅₉	—	2	1	no	—
39	C₈×C₅₉	—	4	2	no	—
40	C₈×C₅₉	Q₁₆	1	2	yes	—
41	C₅₉⋊C₈	—	4	1	no	—
42	C₂²×C₅₉	#121	1	1	yes	characteristic
43	C₄×C₅₉	—	2	1	no	—
44	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
45	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₉	#135	1	1	yes	characteristic
47	C₄×C₅₉	#137	1	1	yes	characteristic
48	Dic₅₉	—	2	2	no	—
49	Dic₅₉	—	4	1	no	—
50	Dic₅₉	—	8	2	no	—
51	#1676	—	59	1	no	maximal, Sylow
52	C₂×C₅₉	#462	1	1	yes	characteristic
53	C₂×C₅₉	#467	1	1	yes	characteristic
54	C₂×C₅₉	#472	1	1	yes	characteristic
55	#500	—	59	1	no	—
56	#477	—	59	2	no	—
57	C₈×Q₈	—	59	1	no	—
58	#335	D₅₉	1	1	yes	characteristic
59	#333	—	59	2	no	—
60	C₅₉	#1676	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	—	59	2	no	—
62	C₄×Q₈	—	59	1	no	—
63	#107	—	118	2	no	—
64	#107	—	118	1	no	—
65	#107	—	118	1	no	—
66	C₄⋊C₈	—	59	1	no	—
67	C₄⋊C₈	—	118	1	no	—
68	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
69	#103	—	118	2	no	—
70	C₄×C₈	—	118	1	no	—
71	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
72	C₂×Q₈	—	59	1	no	—
73	C₂×Q₈	—	118	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	118	1	no	—
76	C₂×C₈	D₂₃₆	1	1	yes	characteristic
77	C₂×C₈	D₂₃₆	1	1	yes	characteristic
78	C₄⋊C₄	—	59	1	no	—
79	C₄⋊C₄	—	118	1	no	—
80	C₄⋊C₄	—	118	2	no	—
81	C₄⋊C₄	—	118	2	no	—
82	C₄²	—	118	1	no	—
83	C₄²	C₂²×D₅₉	1	1	yes	characteristic
84	Q₈	—	59	2	no	—
85	Q₈	—	118	1	no	—
86	Q₈	—	472	2	no	—
87	C₂×C₄	—	118	1	no	—
88	C₂×C₄	—	118	1	no	—
89	C₂×C₄	—	236	2	no	—
90	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
91	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
92	C₂×C₄	#11745689	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	2	no	—
95	C₈	—	236	1	no	—
96	C₈	Dic₂₃₆	1	2	yes	—
97	C₂²	#14273702	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	118	2	no	—
100	C₄	—	472	2	no	—
101	C₄	—	236	1	no	—
102	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
103	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
104	C₄	#14273746	1	1	yes	characteristic
105	C₄	#14273760	1	1	yes	characteristic
106	C₂	#16837613	1	1	yes	characteristic
107	C₂	#16837644	1	1	yes	characteristic
108	C₂	#16837673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic

#16837698

C₂

yes

maximal

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837150

C₂

yes

maximal

C₅₉×#335

C₂

yes

maximal, characteristic

#14273727

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

—

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

Q₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

C₈×Q₈

—

#335

D₅₉

yes

characteristic

#333

—

C₅₉

#1676

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

118

—

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

Q₈

—

118

—

Q₈

—

472

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745689

yes

characteristic

C₈

—

C₈

—

C₈

—

236

—

C₈

Dic₂₃₆

yes

—

C₂²

#14273702

yes

characteristic

C₄

—

C₄

—

118

—

100

C₄

—

472

—

101

C₄

—

236

—

102

C₄

Q₁₆×D₅₉

yes

characteristic

103

C₄

C₂×Dic₂₃₆

yes

characteristic

104

C₄

#14273746

yes

characteristic

105

C₄

#14273760

yes

characteristic

106

C₂

#16837613

yes

characteristic

107

C₂

#16837644

yes

characteristic

108

C₂

#16837673

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#335	C₂
socle	C₂²×C₅₉	#121
hypercenter	#335	D₅₉
hypocenter	C₅₉	#1676
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#335

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
60. C₅₉	#1676	semidirect

H ≅ G/N

type

60. C₅₉

#1676

semidirect

Subgroups — #21423501

Conjugacy classes of subgroups

Complements