Group #21425601 — subgroups

Subgroups — #21425601

Order 7552.

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

normal subgroups only

This group has 176 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	#16837445	C₂	1	2	yes	maximal
2	C₂×#14273661	C₂	1	1	yes	maximal, characteristic
3	#16837170	C₂	1	2	yes	maximal
4	#16837167	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#424	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	#14273661	—	2	2	no	—
9	#14273660	—	2	2	no	—
10	#14273658	—	2	2	no	—
11	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	#117×C₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₉	—	2	2	no	—
15	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₉	—	2	2	no	—
17	M₁₆×C₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	—	2	2	no	—
19	C₂×C₈×C₅₉	—	2	1	no	—
20	C₂×C₈×C₅₉	D₄	1	2	yes	—
21	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂²⋊C₄×C₅₉	—	2	1	no	—
24	C₄²×C₅₉	—	2	1	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	#11745664	—	2	2	no	—
28	#11745664	Q₈	1	4	yes	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	C₂³×C₅₉	#41	1	1	yes	characteristic
31	C₂×C₄×C₅₉	—	2	1	no	—
32	C₂×C₄×C₅₉	—	2	1	no	—
33	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₅₉	#41	1	2	yes	—
37	C₂×Dic₅₉	—	4	2	no	—
38	C₂×Dic₅₉	—	8	2	no	—
39	C₈×C₅₉	—	2	2	no	—
40	C₈×C₅₉	—	4	1	no	—
41	C₅₉⋊C₈	—	2	4	no	—
42	C₂²×C₅₉	—	2	1	no	—
43	C₂²×C₅₉	—	2	1	no	—
44	C₂²×C₅₉	#121	1	1	yes	characteristic
45	C₂²×C₅₉	#125	1	2	yes	—
46	C₄×C₅₉	—	4	1	no	—
47	C₄×C₅₉	—	4	1	no	—
48	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₅₉	#123	1	1	yes	characteristic
50	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₅₉	—	8	2	no	—
52	#2089	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	2	2	no	—
54	C₂×C₅₉	#398	1	1	yes	characteristic
55	C₂×C₅₉	#472	1	1	yes	characteristic
56	C₂×C₅₉	#473	1	1	yes	characteristic
57	#428	—	59	2	no	—
58	C₂×#107	—	59	1	no	—
59	#424	D₅₉	1	1	yes	characteristic
60	#343	—	59	2	no	—
61	#341	—	59	1	no	—
62	C₅₉	#2089	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	59	2	no	—
64	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
66	#117	C₂×D₅₉	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	59	2	no	—
68	#107	—	118	2	no	—
69	#107	—	118	2	no	—
70	#106	—	118	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	118	2	no	—
73	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	59	4	no	—
76	M₁₆	—	118	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	118	2	no	—
80	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
81	C₄⋊C₄	—	118	2	no	—
82	C₄⋊C₄	—	118	2	no	—
83	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
84	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#11745685	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	472	2	no	—
91	C₂×C₄	—	236	2	no	—
92	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
93	C₂×C₄	#11745687	1	2	yes	—
94	C₂×C₄	Q₈×D₅₉	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	4	1	no	—
98	C₈	—	118	4	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#14273701	1	1	yes	characteristic
102	C₂²	#14273717	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	472	2	no	—
106	C₄	#14273707	1	2	yes	—
107	C₄	#14273708	1	1	yes	characteristic
108	C₄	#14273725	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#16837332	1	1	yes	characteristic
111	C₂	#16837672	1	1	yes	characteristic
112	C₂	#16837677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837445

C₂

yes

maximal

C₂×#14273661

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal

#16837167

C₂

yes

maximal, characteristic

C₅₉×#424

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273661

—

#14273660

—

#14273658

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

#11745664

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#125

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#123

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

#2089

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

#343

—

#341

—

C₅₉

#2089

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

118

—

#107

—

118

—

#106

—

118

—

C₄⋊C₈

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

D₂₃₆

yes

characteristic

C₄⋊C₄

D₂₃₆

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#11745685

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

472

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745687

yes

—

C₂×C₄

Q₈×D₅₉

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

C₈

—

118

—

C₂²

—

100

C₂²

—

101

C₂²

#14273701

yes

characteristic

102

C₂²

#14273717

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

472

—

106

C₄

#14273707

yes

—

107

C₄

#14273708

yes

characteristic

108

C₄

#14273725

yes

characteristic

109

C₂

—

110

C₂

#16837332

yes

characteristic

111

C₂

#16837672

yes

characteristic

112

C₂

#16837677

yes

characteristic

	subgroup	G/H
center	C₂²	#14273701
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#424	C₂
socle	C₂²×C₅₉	#121
hypercenter	#424	D₅₉
hypocenter	C₅₉	#2089
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#14273701

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#424

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
62. C₅₉	#2089	semidirect

H ≅ G/N

type

62. C₅₉

#2089

semidirect

Subgroups — #21425601

Conjugacy classes of subgroups

Complements