Group #21423722 — subgroups

Subgroups — #21423722

Order 7552.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837800	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837774	C₂	1	2	yes	maximal, complemented
3	#16837748	C₂	1	2	yes	maximal, complemented
4	#16837123	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745670	—	2	4	no	—
7	#14273727	C₂²	1	2	yes	—
8	#14273659	—	2	2	no	—
9	#14273648	—	2	4	no	—
10	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₂×D₈×C₅₉	—	2	2	no	—
12	#127×C₅₉	C₂²	1	1	yes	characteristic
13	#127×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₅₉	—	2	1	no	—
16	C₂×D₄×C₅₉	—	2	1	no	—
17	C₂×D₄×C₅₉	—	2	2	no	—
18	C₂×D₄×C₅₉	—	2	2	no	—
19	C₂×Dic₁₁₈	—	2	2	no	—
20	D₈×C₅₉	—	4	4	no	—
21	C₂×C₈×C₅₉	D₄	1	2	yes	—
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	#11745670	—	4	8	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
27	C₂³×C₅₉	—	4	1	no	—
28	C₂³×C₅₉	—	4	1	no	—
29	D₄×C₅₉	—	4	2	no	—
30	D₄×C₅₉	—	4	2	no	—
31	D₄×C₅₉	—	4	4	no	—
32	D₄×C₅₉	—	4	4	no	—
33	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₉	—	4	2	no	—
36	Dic₁₁₈	—	4	4	no	—
37	C₈×C₅₉	SD₁₆	1	4	yes	complemented
38	C₅₉⋊C₈	—	2	4	no	—
39	C₂²×C₅₉	—	8	1	no	—
40	C₂²×C₅₉	—	8	1	no	—
41	C₂²×C₅₉	—	8	2	no	—
42	C₂²×C₅₉	—	8	2	no	—
43	C₂²×C₅₉	#127	1	1	yes	characteristic
44	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
45	C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₅₉	—	8	2	no	—
47	#1715	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	—	16	1	no	—
49	C₂×C₅₉	—	16	1	no	—
50	C₂×C₅₉	#492	1	1	yes	characteristic
51	C₂×C₅₉	#496	1	2	yes	—
52	#498	—	59	1	no	—
53	#493	D₅₉	1	1	yes	characteristic
54	#492	—	59	2	no	—
55	#486	—	59	2	no	—
56	#322	—	59	1	no	—
57	C₅₉	#1715	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	118	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	59	2	no	—
61	#127	C₂×D₅₉	1	1	yes	characteristic
62	#127	C₂×D₅₉	1	1	yes	characteristic
63	#106	—	118	2	no	—
64	#102	—	118	4	no	—
65	C₄×C₈	—	59	2	no	—
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	118	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	236	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	59	4	no	—
75	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
76	C₄⋊C₄	—	118	2	no	—
77	C₄⋊C₄	—	118	2	no	—
78	C₄²	C₂²×D₅₉	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	236	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	236	2	no	—
87	C₂×C₄	D₄×D₅₉	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
89	C₈	—	118	4	no	—
90	C₈	#11745670	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#14273723	1	1	yes	characteristic
96	C₄	—	472	2	no	—
97	C₄	C₂×#11745670	1	2	yes	—
98	C₄	#14273751	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#16837774	1	1	yes	characteristic
102	C₂	#16837790	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837800

C₂

yes

maximal, characteristic, complemented

#16837774

C₂

yes

maximal, complemented

#16837748

C₂

yes

maximal, complemented

#16837123

C₂

yes

maximal, characteristic, complemented

C₅₉×#493

C₂

yes

maximal, characteristic

C₂×#11745670

—

#14273727

C₂²

yes

—

#14273659

—

#14273648

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×D₈×C₅₉

—

#127×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#127

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

Dic₅₉

—

#1715

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#492

yes

characteristic

C₂×C₅₉

#496

yes

—

#498

—

#493

D₅₉

yes

—

—

—

C₅₉

#1715

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₅₉

yes

characteristic

#127

C₂×D₅₉

yes

characteristic

#106

—

118

—

#102

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

236

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

236

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

236

—

C₂×C₄

D₄×D₅₉

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

118

—

C₈

#11745670

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14273723

yes

characteristic

C₄

—

472

—

C₄

C₂×#11745670

yes

—

C₄

#14273751

yes

—

C₂

—

100

C₂

—

101

C₂

#16837774

yes

characteristic

102

C₂

#16837790

yes

—

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#493	C₂
socle	C₂²×C₅₉	#127
hypercenter	#493	D₅₉
hypocenter	C₅₉	#1715
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#14273723

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#493

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16837800	C₂	semidirect
2. #16837774	C₂	semidirect
3. #16837748	C₂	semidirect
4. #16837123	C₂	semidirect
37. C₈×C₅₉	SD₁₆	semidirect
57. C₅₉	#1715	semidirect
90. C₈	#11745670	semidirect

H ≅ G/N

type

1. #16837800

C₂

semidirect

2. #16837774

C₂

semidirect

3. #16837748

C₂

semidirect

4. #16837123

C₂

semidirect

37. C₈×C₅₉

SD₁₆

semidirect

57. C₅₉

#1715

semidirect

90. C₈

#11745670

semidirect

Subgroups — #21423722

Conjugacy classes of subgroups

Complements