Group #21426458 — subgroups

Subgroups — #21426458

Order 7552.

780 conjugacy classes of subgroups; only the 325 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 325 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×C₄×D₂₃₆	C₂	1	3	yes	maximal
2	C₄×C₄⋊Dic₅₉	C₂	1	3	yes	maximal, complemented
3	#16837249	C₂	1	6	yes	maximal, complemented
4	#95×D₅₉	C₂	1	2	yes	maximal, complemented
5	C₅₉×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×D₅₉	C₂²	1	6	yes	—
8	C₄×D₂₃₆	C₄	1	24	yes	complemented
9	C₂×C₄⋊Dic₅₉	C₂²	1	3	yes	—
10	C₂×(C₂×D₅₉)⋊C₄	C₂²	1	6	yes	—
11	C₂×C₄×Dic₅₉	C₂²	1	6	yes	—
12	C₅₉⋊#95	C₂²	1	6	yes	—
13	C₂×C₄²×C₅₉	C₂²	1	3	yes	—
14	#95×C₅₉	C₂²	1	1	yes	characteristic
15	#95×C₅₉	C₂²	1	3	yes	—
16	C₂³×D₅₉	C₂³	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
18	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₉	C₂³	1	6	yes	—
21	C₂×D₂₃₆	C₂×C₄	1	12	yes	—
22	C₂×C₄×D₅₉	—	4	12	no	—
23	C₂×C₄×D₅₉	C₂×C₄	1	24	yes	complemented
24	C₄²×C₅₉	C₂×C₄	1	12	yes	complemented
25	(C₂×D₅₉)⋊C₄	C₂×C₄	1	24	yes	complemented
26	C₄⋊Dic₅₉	C₂×C₄	1	12	yes	complemented
27	C₄×Dic₅₉	—	4	6	no	—
28	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
29	C₂²×D₅₉	—	2	6	no	—
30	C₂²×D₅₉	—	2	2	no	—
31	C₂²×D₅₉	C₂²×C₄	1	12	yes	—
32	C₂×C₄×C₅₉	—	2	3	no	—
33	C₂×C₄×C₅₉	—	2	6	no	—
34	C₂×C₄×C₅₉	—	2	3	no	—
35	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
38	C₂×Dic₅₉	—	2	12	no	—
39	C₂×Dic₅₉	C₂²×C₄	1	12	yes	—
40	D₂₃₆	C₄²	1	16	yes	complemented
41	C₄×D₅₉	—	4	24	no	—
42	C₂²×C₅₉	C₂³×C₄	1	3	yes	—
43	C₂²×C₅₉	H(5,𝔽₂)	1	3	yes	—
44	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
45	C₂×D₅₉	—	2	12	no	—
46	C₂×D₅₉	—	2	12	no	—
47	C₂×D₅₉	C₂×C₄²	1	8	yes	—
48	C₄×C₅₉	—	2	6	no	—
49	C₄×C₅₉	—	2	6	no	—
50	C₄×C₅₉	C₂×C₄²	1	4	yes	—
51	Dic₅₉	—	2	12	no	—
52	#2275	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	C₂²×C₄²	1	1	yes	characteristic
54	C₂×C₅₉	#518	1	3	yes	—
55	C₂×C₅₉	#520	1	3	yes	—
56	D₅₉	—	2	8	no	—
57	C₂×C₄×D₄	—	59	3	no	—
58	C₄×C₄⋊C₄	—	59	3	no	—
59	C₄×C₂²⋊C₄	—	59	6	no	—
60	#376	D₅₉	1	1	yes	characteristic, complemented
61	C₂×#95	—	59	2	no	—
62	C₅₉	#2275	1	1	yes	characteristic, complemented, Sylow
63	C₂²×D₄	—	59	1	no	—
64	C₂³×C₄	—	59	6	no	—
65	C₄×D₄	—	59	24	no	—
66	C₂×C₄⋊C₄	—	59	3	no	—
67	C₂×C₂²⋊C₄	—	59	6	no	—
68	C₂×C₄²	—	59	6	no	—
69	C₂×C₄²	C₂×D₅₉	1	3	yes	—
70	#95	—	59	6	no	—
71	#95	C₂×D₅₉	1	1	yes	characteristic
72	#95	C₂×D₅₉	1	3	yes	—
73	C₂⁴	—	59	2	no	—
74	C₂×D₄	—	59	12	no	—
75	C₂²×C₄	—	59	6	no	—
76	C₂²×C₄	—	59	24	no	—
77	C₂²×C₄	—	236	12	no	—
78	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
79	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
80	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
81	C₄⋊C₄	—	59	12	no	—
82	C₂²⋊C₄	—	59	24	no	—
83	C₄²	—	236	6	no	—
84	C₄²	C₄×D₅₉	1	12	yes	complemented
85	C₂³	—	59	12	no	—
86	C₂³	—	118	6	no	—
87	C₂³	—	118	2	no	—
88	C₂³	C₂³×D₅₉	1	1	yes	characteristic
89	D₄	—	59	16	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	59	12	no	—
94	C₂×C₄	—	118	12	no	—
95	C₂×C₄	—	236	24	no	—
96	C₂×C₄	C₂×C₄×D₅₉	1	6	yes	—
97	C₂×C₄	C₂×C₄×D₅₉	1	6	yes	—
98	C₂×C₄	C₂×C₄×D₅₉	1	6	yes	—
99	C₂²	—	59	8	no	—
100	C₂²	—	118	12	no	—
101	C₂²	—	118	12	no	—
102	C₂²	C₂²×C₄×D₅₉	1	3	yes	—
103	C₂²	#14273773	1	3	yes	—
104	C₂²	#14273774	1	1	yes	characteristic
105	C₄	—	2	6	no	—
106	C₄	—	2	6	no	—
107	C₄	—	118	12	no	—
108	C₄	C₄²×D₅₉	1	4	yes	—
109	C₂	—	118	8	no	—
110	C₂	C₂×C₄²×D₅₉	1	1	yes	characteristic
111	C₂	#16837876	1	3	yes	—
112	C₂	#16837882	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₂₃₆

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, complemented

#16837249

C₂

yes

maximal, complemented

#95×D₅₉

C₂

yes

maximal, complemented

C₅₉×#376

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

characteristic

C₂²×C₄×D₅₉

C₂²

yes

—

C₄×D₂₃₆

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×(C₂×D₅₉)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

—

#95×C₅₉

C₂²

yes

characteristic

#95×C₅₉

C₂²

yes

—

C₂³×D₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

complemented

C₄²×C₅₉

C₂×C₄

yes

complemented

(C₂×D₅₉)⋊C₄

C₂×C₄

yes

complemented

C₄⋊Dic₅₉

C₂×C₄

yes

complemented

C₄×Dic₅₉

—

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²×C₄

yes

—

D₂₃₆

C₄²

yes

complemented

C₄×D₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

—

C₂²×C₅₉

H(5,𝔽₂)

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₂×C₄²

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄²

yes

—

Dic₅₉

—

#2275

—

maximal, Sylow

C₂×C₅₉

C₂²×C₄²

yes

characteristic

C₂×C₅₉

#518

yes

—

C₂×C₅₉

#520

yes

—

D₅₉

—

C₂×C₄×D₄

—

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

#376

D₅₉

yes

characteristic, complemented

C₂×#95

—

C₅₉

#2275

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

—

#95

—

#95

C₂×D₅₉

yes

characteristic

#95

C₂×D₅₉

yes

—

C₂⁴

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

236

—

C₄²

C₄×D₅₉

yes

complemented

C₂³

—

C₂³

—

118

—

C₂³

—

118

—

C₂³

C₂³×D₅₉

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂²

—

100

C₂²

—

118

—

101

C₂²

—

118

—

102

C₂²

C₂²×C₄×D₅₉

yes

—

103

C₂²

#14273773

yes

—

104

C₂²

#14273774

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

118

—

108

C₄

C₄²×D₅₉

yes

—

109

C₂

—

118

—

110

C₂

C₂×C₄²×D₅₉

yes

characteristic

111

C₂

#16837876

yes

—

112

C₂

#16837882

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄²
Fitting	C₅₉×#376	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	#376	D₅₉
hypocenter	C₅₉	#2275
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator

C₂×C₅₉

C₂²×C₄²

Fitting

C₅₉×#376

C₂

socle

C₂³×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
2. C₄×C₄⋊Dic₅₉	C₂	semidirect
3. #16837249	C₂	semidirect
4. #95×D₅₉	C₂	semidirect
5. C₅₉×#376	C₂	semidirect
8. C₄×D₂₃₆	C₄	semidirect
23. C₂×C₄×D₅₉	C₂×C₄	semidirect
24. C₄²×C₅₉	C₂×C₄	semidirect
25. (C₂×D₅₉)⋊C₄	C₂×C₄	semidirect
26. C₄⋊Dic₅₉	C₂×C₄	semidirect
40. D₂₃₆	C₄²	semidirect
60. #376	D₅₉	semidirect
62. C₅₉	#2275	semidirect
84. C₄²	C₄×D₅₉	semidirect

H ≅ G/N

type

2. C₄×C₄⋊Dic₅₉

C₂

semidirect

3. #16837249

C₂

semidirect

4. #95×D₅₉

C₂

semidirect

5. C₅₉×#376

C₂

semidirect

8. C₄×D₂₃₆

C₄

semidirect

23. C₂×C₄×D₅₉

C₂×C₄

semidirect

24. C₄²×C₅₉

C₂×C₄

semidirect

25. (C₂×D₅₉)⋊C₄

C₂×C₄

semidirect

26. C₄⋊Dic₅₉

C₂×C₄

semidirect

40. D₂₃₆

C₄²

semidirect

60. #376

D₅₉

semidirect

62. C₅₉

#2275

semidirect

84. C₄²

C₄×D₅₉

semidirect

Subgroups — #21426458

Conjugacy classes of subgroups

Complements