Group #21423497 — subgroups

Subgroups — #21423497

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	#16837694	C₂	1	2	yes	maximal
2	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
3	#16837155	C₂	1	1	yes	maximal, characteristic
4	#16837151	C₂	1	2	yes	maximal
5	C₅₉×#500	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	#14273660	—	2	2	no	—
8	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	Dic₅₉⋊C₈	—	2	1	no	—
11	#14273650	—	2	2	no	—
12	C₈×Dic₅₉	—	2	1	no	—
13	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
14	#107×C₅₉	—	2	1	no	—
15	#107×C₅₉	—	2	1	no	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	2	no	—
18	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₄⋊C₄×C₅₉	—	2	2	no	—
23	C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
25	Dic₅₉⋊C₄	—	2	1	no	—
26	C₄×Dic₅₉	—	2	1	no	—
27	C₂×C₅₉⋊C₈	—	2	1	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	Q₈×C₅₉	—	8	2	no	—
30	C₂×C₄×C₅₉	—	4	2	no	—
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
34	C₂×Dic₅₉	—	2	1	no	—
35	C₂×Dic₅₉	—	2	1	no	—
36	Dic₁₁₈	—	2	1	no	—
37	Dic₁₁₈	Q₁₆	1	2	yes	—
38	C₈×C₅₉	—	2	1	no	—
39	C₈×C₅₉	Q₁₆	1	2	yes	—
40	C₅₉⋊C₈	—	4	2	no	—
41	C₅₉⋊C₈	—	4	1	no	—
42	C₂²×C₅₉	#121	1	1	yes	characteristic
43	C₄×C₅₉	—	2	1	no	—
44	C₄×C₅₉	—	8	2	no	—
45	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₅₉	#135	1	1	yes	characteristic
48	C₄×C₅₉	#137	1	1	yes	characteristic
49	Dic₅₉	—	2	2	no	—
50	Dic₅₉	—	4	1	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	D₅₉	1	1	yes	characteristic
56	#477	—	59	2	no	—
57	C₈×Q₈	—	59	1	no	—
58	#335	—	59	1	no	—
59	#333	—	59	2	no	—
60	C₅₉	#1676	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
62	C₄×Q₈	—	59	1	no	—
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	—	118	2	no	—
66	C₄⋊C₈	—	59	1	no	—
67	C₄⋊C₈	—	59	2	no	—
68	C₄⋊C₈	—	118	1	no	—
69	#103	—	118	2	no	—
70	C₄×C₈	—	118	1	no	—
71	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	59	1	no	—
74	C₂×C₈	—	118	2	no	—
75	C₂×C₈	—	118	1	no	—
76	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
77	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	59	1	no	—
81	C₄⋊C₄	—	118	1	no	—
82	C₄²	—	118	1	no	—
83	C₄²	C₂²×D₅₉	1	1	yes	characteristic
84	Q₈	—	8	2	no	—
85	Q₈	—	59	2	no	—
86	Q₈	—	118	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	118	1	no	—
89	C₂×C₄	—	118	1	no	—
90	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
91	C₂×C₄	#11745687	1	1	yes	characteristic
92	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	236	2	no	—
95	C₈	—	236	1	no	—
96	C₈	#11745672	1	2	yes	—
97	C₂²	#14273698	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	8	2	no	—
100	C₄	—	118	2	no	—
101	C₄	—	236	1	no	—
102	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
103	C₄	C₂×#11745672	1	1	yes	characteristic
104	C₄	#14273750	1	1	yes	characteristic
105	C₄	#14273758	1	1	yes	characteristic
106	C₂	#16837609	1	1	yes	characteristic
107	C₂	#16837640	1	1	yes	characteristic
108	C₂	#16837669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837694

C₂

yes

maximal

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic

#16837151

C₂

yes

maximal

C₅₉×#500

C₂

yes

maximal, characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273660

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

Dic₅₉⋊C₈

—

#14273650

—

C₈×Dic₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

#107×C₅₉

—

#107×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

Q₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#477

—

C₈×Q₈

—

#335

—

#333

—

C₅₉

#1676

yes

characteristic, complemented, Sylow

C₄⋊Q₈

C₂×D₅₉

yes

—

C₄×Q₈

—

#107

—

#107

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#103

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

118

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745687

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

236

—

C₈

—

236

—

C₈

#11745672

yes

—

C₂²

#14273698

yes

characteristic

C₄

—

C₄

—

100

C₄

—

118

—

101

C₄

—

236

—

102

C₄

Q₁₆×D₅₉

yes

characteristic

103

C₄

C₂×#11745672

yes

characteristic

104

C₄

#14273750

yes

characteristic

105

C₄

#14273758

yes

characteristic

106

C₂

#16837609

yes

characteristic

107

C₂

#16837640

yes

characteristic

108

C₂

#16837669

yes

characteristic

	subgroup	G/H
center	C₂²	#14273698
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	#121
hypercenter	#500	D₅₉
hypocenter	C₅₉	#1676
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14273698

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#500

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

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

H ≅ G/N

type

60. C₅₉

#1676

semidirect

Subgroups — #21423497

Conjugacy classes of subgroups

Complements