Group #21423689 — subgroups

Subgroups — #21423689

Order 7552.

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

normal subgroups only

This group has 224 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	#16837779	C₂	1	1	yes	maximal, characteristic
2	#16837776	C₂	1	2	yes	maximal, complemented
3	#16837750	C₂	1	2	yes	maximal, complemented
4	C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745671	—	2	4	no	—
7	C₂×D₄₇₂	—	2	2	no	—
8	#14273724	C₂²	1	2	yes	—
9	#14273647	—	2	4	no	—
10	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
13	#107×C₅₉	—	2	2	no	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	2	2	no	—
16	C₂×D₂₃₆	—	2	2	no	—
17	C₂×D₂₃₆	—	2	2	no	—
18	C₂×D₂₃₆	—	2	2	no	—
19	C₂×C₈×C₅₉	D₄	1	2	yes	—
20	C₄⋊C₄×C₅₉	—	2	2	no	—
21	C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	#11745671	—	4	8	no	—
24	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
26	D₄₇₂	—	4	4	no	—
27	C₂²×D₅₉	—	4	2	no	—
28	Q₈×C₅₉	—	4	4	no	—
29	C₂×C₄×C₅₉	—	4	2	no	—
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	D₂₃₆	—	4	4	no	—
34	D₂₃₆	—	4	4	no	—
35	D₂₃₆	—	4	4	no	—
36	C₈×C₅₉	SD₁₆	1	4	yes	complemented
37	C₅₉⋊C₈	—	2	2	no	—
38	C₅₉⋊C₈	D₈	1	4	yes	complemented
39	C₂²×C₅₉	#127	1	1	yes	characteristic
40	C₂×D₅₉	—	8	2	no	—
41	C₂×D₅₉	—	8	2	no	—
42	C₂×D₅₉	—	8	2	no	—
43	C₄×C₅₉	—	8	2	no	—
44	C₄×C₅₉	C₂×D₈	1	2	yes	—
45	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
46	C₄×C₅₉	#135	1	2	yes	—
47	#1706	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	#492	1	1	yes	characteristic
49	C₂×C₅₉	#493	1	1	yes	characteristic
50	C₂×C₅₉	#495	1	1	yes	characteristic
51	D₅₉	—	16	2	no	—
52	#500	D₅₉	1	1	yes	characteristic, complemented
53	#493	—	59	1	no	—
54	#492	—	59	2	no	—
55	#486	—	59	2	no	—
56	C₈²	—	59	1	no	—
57	C₅₉	#1706	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	118	4	no	—
59	C₂×D₈	—	118	2	no	—
60	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
61	#127	—	59	2	no	—
62	#107	—	2	2	no	—
63	#102	—	118	4	no	—
64	C₄×C₈	—	59	1	no	—
65	C₄×C₈	—	59	1	no	—
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	118	2	no	—
69	C₂×D₄	—	118	2	no	—
70	C₂×D₄	—	118	2	no	—
71	SD₁₆	—	236	8	no	—
72	D₈	—	236	4	no	—
73	C₂×C₈	—	59	2	no	—
74	C₂×C₈	—	59	2	no	—
75	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₅₉	1	1	yes	characteristic
79	C₂³	—	236	2	no	—
80	Q₈	—	4	4	no	—
81	D₄	—	236	4	no	—
82	D₄	—	236	4	no	—
83	D₄	—	236	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
86	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
88	C₈	—	59	4	no	—
89	C₈	—	118	2	no	—
90	C₈	#11745671	1	4	yes	complemented
91	C₂²	—	472	2	no	—
92	C₂²	—	472	2	no	—
93	C₂²	—	472	2	no	—
94	C₂²	#14273723	1	1	yes	characteristic
95	C₄	—	8	2	no	—
96	C₄	D₈×D₅₉	1	2	yes	—
97	C₄	C₂×#11745671	1	2	yes	—
98	C₄	#14273750	1	2	yes	—
99	C₂	—	944	2	no	—
100	C₂	#16837776	1	1	yes	characteristic
101	C₂	#16837777	1	1	yes	characteristic
102	C₂	#16837787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837779

C₂

yes

maximal, characteristic

#16837776

C₂

yes

maximal, complemented

#16837750

C₂

yes

maximal, complemented

C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₅₉×#500

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

—

C₂×D₄₇₂

—

#14273724

C₂²

yes

—

#14273647

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

—

#107×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

D₄₇₂

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

D₈

yes

complemented

C₂²×C₅₉

#127

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

#135

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#500

D₅₉

yes

characteristic, complemented

#493

—

#492

—

#486

—

C₈²

—

C₅₉

#1706

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#127

—

#107

—

#102

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

SD₁₆

—

236

—

D₈

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

236

—

Q₈

—

D₄

—

236

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

118

—

C₈

#11745671

yes

complemented

C₂²

—

472

—

C₂²

—

472

—

C₂²

—

472

—

C₂²

#14273723

yes

characteristic

C₄

—

C₄

D₈×D₅₉

yes

—

C₄

C₂×#11745671

yes

—

C₄

#14273750

yes

—

C₂

—

944

—

100

C₂

#16837776

yes

characteristic

101

C₂

#16837777

yes

characteristic

102

C₂

#16837787

yes

characteristic

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	#127
hypercenter	#500	D₅₉
hypocenter	C₅₉	#1706
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14273723

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
2. #16837776	C₂	semidirect
3. #16837750	C₂	semidirect
4. C₈×C₅₉⋊C₈	C₂	semidirect
5. C₅₉×#500	C₂	semidirect
36. C₈×C₅₉	SD₁₆	semidirect
38. C₅₉⋊C₈	D₈	semidirect
52. #500	D₅₉	semidirect
57. C₅₉	#1706	semidirect
90. C₈	#11745671	semidirect

H ≅ G/N

type

2. #16837776

C₂

semidirect

3. #16837750

C₂

semidirect

4. C₈×C₅₉⋊C₈

C₂

semidirect

5. C₅₉×#500

C₂

semidirect

36. C₈×C₅₉

SD₁₆

semidirect

38. C₅₉⋊C₈

D₈

semidirect

52. #500

D₅₉

semidirect

57. C₅₉

#1706

semidirect

90. C₈

#11745671

semidirect

Subgroups — #21423689

Conjugacy classes of subgroups

Complements