Group #21425911 — subgroups

Subgroups — #21425911

Order 7552.

122 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16837356	C₂	1	1	yes	maximal, characteristic
2	#16837223	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837183	C₂	1	2	yes	maximal
4	#16837179	C₂	1	2	yes	maximal, complemented
5	C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745659	C₂²	1	1	yes	characteristic
7	C₂×C₈×D₅₉	C₄	1	1	yes	characteristic
8	#14273684	C₄	1	1	yes	characteristic
9	#14273667	C₂²	1	2	yes	—
10	#14273638	C₄	1	1	yes	characteristic
11	#14273637	C₂²	1	1	yes	characteristic
12	C₈×Dic₅₉	C₄	1	1	yes	characteristic
13	M₃₂×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
18	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	(C₂×D₅₉)⋊C₄	—	2	1	no	—
20	Dic₅₉⋊C₄	—	2	1	no	—
21	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
24	#11745659	—	2	1	no	—
25	#11745659	—	2	1	no	—
26	C₈×D₅₉	—	2	2	no	—
27	C₁₆×C₅₉	—	2	2	no	—
28	C₅₉⋊C₁₆	—	2	2	no	—
29	C₂²×D₅₉	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₉	—	2	1	no	—
31	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₉	—	2	1	no	—
33	C₂×Dic₅₉	M₁₆	1	1	yes	characteristic
34	C₄×D₅₉	—	2	1	no	—
35	C₄×D₅₉	—	2	1	no	—
36	C₈×C₅₉	M₁₆	1	2	yes	—
37	C₈×C₅₉	#41	1	1	yes	characteristic
38	C₈×C₅₉	#41	1	1	yes	characteristic
39	C₅₉⋊C₈	—	2	1	no	—
40	C₅₉⋊C₈	—	2	1	no	—
41	C₂²×C₅₉	C₂×M₁₆	1	1	yes	characteristic
42	C₂×D₅₉	—	2	1	no	—
43	C₂×D₅₉	—	4	1	no	—
44	C₄×C₅₉	—	4	1	no	—
45	C₄×C₅₉	#117	1	1	yes	characteristic
46	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
47	Dic₅₉	—	2	1	no	—
48	Dic₅₉	—	4	1	no	—
49	#2166	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	—	2	1	no	—
51	C₂×C₅₉	#432	1	1	yes	characteristic
52	D₅₉	—	4	1	no	—
53	#405	—	59	1	no	—
54	#364	—	59	1	no	—
55	#364	D₅₉	1	1	yes	characteristic, complemented
56	#349	—	59	2	no	—
57	#347	—	59	2	no	—
58	C₅₉	#2166	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	59	1	no	—
60	C₂²×C₈	—	59	1	no	—
61	#117	—	59	1	no	—
62	M₃₂	—	59	2	no	—
63	M₃₂	C₂×D₅₉	1	2	yes	—
64	#97	—	59	1	no	—
65	#97	—	59	1	no	—
66	C₄×C₈	—	59	1	no	—
67	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
68	C₂²×C₄	—	59	1	no	—
69	M₁₆	—	118	1	no	—
70	M₁₆	—	118	1	no	—
71	C₂×C₈	—	59	1	no	—
72	C₂×C₈	—	59	1	no	—
73	C₂×C₈	—	118	2	no	—
74	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
76	C₄⋊C₄	—	118	1	no	—
77	C₂²⋊C₄	—	118	1	no	—
78	C₄²	—	59	1	no	—
79	C₄²	C₄×D₅₉	1	1	yes	characteristic
80	C₁₆	—	2	2	no	—
81	C₁₆	—	118	2	no	—
82	C₂³	—	59	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	59	1	no	—
85	C₂×C₄	—	118	1	no	—
86	C₂×C₄	—	118	1	no	—
87	C₂×C₄	—	118	1	no	—
88	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
89	C₈	—	118	1	no	—
90	C₈	—	118	1	no	—
91	C₈	#11745659	1	2	yes	—
92	C₈	#11745687	1	1	yes	characteristic
93	C₈	#11745689	1	1	yes	characteristic
94	C₂²	—	118	1	no	—
95	C₂²	—	236	1	no	—
96	C₂²	M₁₆×D₅₉	1	1	yes	characteristic
97	C₄	—	4	1	no	—
98	C₄	—	118	1	no	—
99	C₄	—	236	1	no	—
100	C₄	#14273685	1	1	yes	characteristic
101	C₄	C₂×#11745659	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	236	1	no	—
104	C₂	#16837458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837356

C₂

yes

maximal, characteristic

#16837223

C₂

yes

maximal, characteristic, complemented

#16837183

C₂

yes

maximal

#16837179

C₂

yes

maximal, complemented

C₅₉×#364

C₂

yes

maximal, characteristic, complemented

C₂×#11745659

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₄

yes

characteristic

#14273684

C₄

yes

characteristic

#14273667

C₂²

yes

—

#14273638

C₄

yes

characteristic

#14273637

C₂²

yes

characteristic

C₈×Dic₅₉

C₄

yes

characteristic

M₃₂×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

(C₂×D₅₉)⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

#11745659

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

M₁₆

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

M₁₆

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×M₁₆

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2166

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#432

yes

characteristic

D₅₉

—

#405

—

#364

—

#364

D₅₉

yes

characteristic, complemented

#349

—

#347

—

C₅₉

#2166

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

—

C₄²

C₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₈

—

118

—

C₈

—

118

—

C₈

#11745659

yes

—

C₈

yes

C₈

yes

C₂²

—

118

—

C₂²

—

236

—

C₂²

M₁₆×D₅₉

yes

characteristic

C₄

—

C₄

—

118

—

C₄

—

236

—

100

C₄

#14273685

yes

characteristic

101

C₄

C₂×#11745659

yes

characteristic

102

C₂

—

103

C₂

—

236

—

104

C₂

#16837458

yes

characteristic

	subgroup	G/H
center	C₄	#14273685
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#364	C₂
socle	C₂×C₅₉	#432
hypercenter	#364	D₅₉
hypocenter	C₅₉	#2166
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₄

#14273685

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#364

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
2. #16837223	C₂	semidirect
4. #16837179	C₂	semidirect
5. C₅₉×#364	C₂	semidirect
55. #364	D₅₉	semidirect
58. C₅₉	#2166	semidirect

H ≅ G/N

type

2. #16837223

C₂

semidirect

4. #16837179

C₂

semidirect

5. C₅₉×#364

C₂

semidirect

55. #364

D₅₉

semidirect

58. C₅₉

#2166

semidirect

Subgroups — #21425911

Conjugacy classes of subgroups

Complements