Group #21426658 — subgroups

Subgroups — #21426658

Order 7552.

924 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 249 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	C₂×#14273724	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273689	C₂	1	6	yes	maximal, complemented
3	#16837303	C₂	1	6	yes	maximal, complemented
4	C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₃₆	C₂²	1	6	yes	—
7	C₂²×C₄×D₅₉	C₂²	1	4	yes	—
8	#14273724	C₄	1	8	yes	complemented
9	#14273689	—	2	24	no	—
10	C₂×(C₂×D₅₉)⋊C₄	C₂²	1	12	yes	—
11	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
12	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₅₉	C₂²	1	6	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂³×D₅₉	C₂³	1	4	yes	—
16	C₂²×C₄×C₅₉	C₂³	1	4	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
18	C₂²×Dic₅₉	C₂³	1	3	yes	—
19	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
20	C₂×D₂₃₆	—	2	12	no	—
21	C₂×D₂₃₆	—	2	12	no	—
22	C₂×D₂₃₆	C₂×C₄	1	24	yes	complemented
23	C₂×C₄×D₅₉	—	2	24	no	—
24	C₄⋊C₄×C₅₉	—	2	12	no	—
25	C₄²×C₅₉	C₂×C₄	1	4	yes	complemented
26	(C₂×D₅₉)⋊C₄	—	2	24	no	—
27	C₄×Dic₅₉	—	2	6	no	—
28	C₄×Dic₅₉	D₄	1	12	yes	complemented
29	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
30	C₂²×D₅₉	—	2	12	no	—
31	C₂²×D₅₉	—	2	12	no	—
32	C₂²×D₅₉	C₂²×C₄	1	8	yes	—
33	C₂×C₄×C₅₉	—	2	12	no	—
34	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₅₉	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₅₉	#41	1	6	yes	—
37	C₂×Dic₅₉	—	2	6	no	—
38	C₂×Dic₅₉	C₂×D₄	1	6	yes	—
39	C₂×Dic₅₉	C₂×D₄	1	6	yes	—
40	D₂₃₆	—	2	48	no	—
41	C₄×D₅₉	—	4	16	no	—
42	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
43	C₂²×C₅₉	C₂²×D₄	1	3	yes	—
44	C₂²×C₅₉	C₂×#41	1	3	yes	—
45	C₂×D₅₉	—	2	24	no	—
46	C₂×D₅₉	—	4	4	no	—
47	C₂×D₅₉	—	4	12	no	—
48	C₄×C₅₉	—	4	4	no	—
49	C₄×C₅₉	C₄×D₄	1	12	yes	complemented
50	Dic₅₉	—	2	6	no	—
51	Dic₅₉	#127	1	4	yes	complemented
52	C₄×#127	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	C₂×C₄×D₄	1	3	yes	—
54	C₂×C₅₉	C₂×#127	1	1	yes	characteristic
55	C₂×C₅₉	#532	1	3	yes	—
56	D₅₉	—	4	8	no	—
57	C₂×#127	—	59	1	no	—
58	C₂×C₄×D₄	—	59	6	no	—
59	#390	—	59	6	no	—
60	#384	D₅₉	1	1	yes	characteristic, complemented
61	C₄³	—	59	1	no	—
62	C₅₉	C₄×#127	1	1	yes	characteristic, complemented, Sylow
63	C₂²×D₄	—	59	6	no	—
64	C₂³×C₄	—	59	4	no	—
65	#127	—	59	8	no	—
66	C₄×D₄	—	118	24	no	—
67	C₂×C₄⋊C₄	C₂×D₅₉	1	6	yes	—
68	C₂×C₂²⋊C₄	—	59	12	no	—
69	C₂×C₄²	—	59	3	no	—
70	C₂×C₄²	—	59	3	no	—
71	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
72	C₂⁴	—	59	4	no	—
73	C₂×D₄	—	59	24	no	—
74	C₂×D₄	—	118	12	no	—
75	C₂×D₄	—	118	12	no	—
76	C₂²×C₄	—	59	1	no	—
77	C₂²×C₄	—	59	3	no	—
78	C₂²×C₄	—	118	24	no	—
79	C₂²×C₄	C₂²×D₅₉	1	4	yes	—
80	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
81	C₄⋊C₄	—	2	12	no	—
82	C₂²⋊C₄	—	118	24	no	—
83	C₄²	—	59	12	no	—
84	C₄²	—	118	6	no	—
85	C₄²	C₄×D₅₉	1	4	yes	complemented
86	C₂³	—	59	8	no	—
87	C₂³	—	118	12	no	—
88	C₂³	—	118	12	no	—
89	C₂³	C₂³×D₅₉	1	1	yes	characteristic
90	D₄	—	118	48	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	59	6	no	—
93	C₂×C₄	—	59	6	no	—
94	C₂×C₄	—	118	6	no	—
95	C₂×C₄	—	236	16	no	—
96	C₂×C₄	C₂×C₄×D₅₉	1	6	yes	—
97	C₂×C₄	D₄×D₅₉	1	6	yes	—
98	C₂×C₄	#11745689	1	6	yes	—
99	C₂²	—	118	24	no	—
100	C₂²	—	236	4	no	—
101	C₂²	—	236	12	no	—
102	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
103	C₂²	C₂×D₄×D₅₉	1	3	yes	—
104	C₂²	C₂×#11745689	1	3	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	59	4	no	—
107	C₄	—	118	6	no	—
108	C₄	#14273689	1	12	yes	complemented
109	C₂	—	236	8	no	—
110	C₂	C₂×#14273689	1	3	yes	—
111	C₂	#127×D₅₉	1	1	yes	characteristic
112	C₂	#16837953	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273724

C₂

yes

maximal, characteristic

C₂×#14273689

C₂

yes

maximal, complemented

#16837303

C₂

yes

maximal, complemented

C₄²×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₅₉×#384

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

—

C₂²×C₄×D₅₉

C₂²

yes

—

#14273724

C₄

yes

complemented

#14273689

—

C₂×(C₂×D₅₉)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂³×D₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

complemented

C₂×C₄×D₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

complemented

(C₂×D₅₉)⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

complemented

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

D₂₃₆

—

C₄×D₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂²×C₅₉

C₂²×D₄

yes

—

C₂²×C₅₉

C₂×#41

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

complemented

Dic₅₉

—

Dic₅₉

yes

—

C₂×C₅₉

C₂×C₄×D₄

yes

—

C₂×C₅₉

C₂×#127

yes

characteristic

C₂×C₅₉

#532

yes

—

D₅₉

—

C₂×#127

—

C₂×C₄×D₄

—

#390

—

#384

D₅₉

yes

characteristic, complemented

C₄³

—

C₅₉

C₄×#127

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

118

—

C₄²

—

C₄²

—

118

—

C₄²

C₄×D₅₉

yes

complemented

C₂³

—

C₂³

—

118

—

C₂³

—

118

—

C₂³

C₂³×D₅₉

yes

characteristic

D₄

—

118

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₄

D₄×D₅₉

yes

—

C₂×C₄

#11745689

yes

—

C₂²

—

118

—

100

C₂²

—

236

—

101

C₂²

—

236

—

102

C₂²

C₂²×C₄×D₅₉

yes

characteristic

103

C₂²

C₂×D₄×D₅₉

yes

—

104

C₂²

C₂×#11745689

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

118

—

108

C₄

#14273689

yes

complemented

109

C₂

—

236

—

110

C₂

C₂×#14273689

yes

—

111

C₂

#127×D₅₉

yes

characteristic

112

C₂

#16837953

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#384	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	#384	D₅₉
hypocenter	C₅₉	C₄×#127
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator

C₂²×C₅₉

C₂³×C₄

Fitting

C₅₉×#384

C₂

socle

C₂³×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
2. C₂×#14273689	C₂	semidirect
3. #16837303	C₂	semidirect
4. C₄²×Dic₅₉	C₂	semidirect
5. C₅₉×#384	C₂	semidirect
8. #14273724	C₄	semidirect
22. C₂×D₂₃₆	C₂×C₄	semidirect
25. C₄²×C₅₉	C₂×C₄	semidirect
28. C₄×Dic₅₉	D₄	semidirect
49. C₄×C₅₉	C₄×D₄	semidirect
51. Dic₅₉	#127	semidirect
60. #384	D₅₉	semidirect
62. C₅₉	C₄×#127	semidirect
85. C₄²	C₄×D₅₉	semidirect
108. C₄	#14273689	semidirect

H ≅ G/N

type

2. C₂×#14273689

C₂

semidirect

3. #16837303

C₂

semidirect

4. C₄²×Dic₅₉

C₂

semidirect

5. C₅₉×#384

C₂

semidirect

8. #14273724

C₄

semidirect

22. C₂×D₂₃₆

C₂×C₄

semidirect

25. C₄²×C₅₉

C₂×C₄

semidirect

28. C₄×Dic₅₉

D₄

semidirect

49. C₄×C₅₉

C₄×D₄

semidirect

51. Dic₅₉

#127

semidirect

60. #384

D₅₉

semidirect

62. C₅₉

C₄×#127

semidirect

85. C₄²

C₄×D₅₉

semidirect

108. C₄

#14273689

semidirect

Subgroups — #21426658

Conjugacy classes of subgroups

Complements