Subgroups — C₂²×#117×C₅₉ #21421739

Order 7552.

1880 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₅₉	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₄×C₅₉	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₄×C₅₉	C₂	1	2	yes	maximal
5	C₂²×C₄²×C₅₉	C₂	1	2	yes	maximal, complemented
6	C₂⁵×C₅₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₅₉	C₄	1	16	yes	complemented
8	C₂³×C₄×C₅₉	C₂²	1	4	yes	—
9	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₉	C₂²	1	12	yes	—
11	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
12	#117×C₅₉	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊C₄×C₅₉	C₂²	1	24	yes	complemented
14	C₂×C₂²⋊C₄×C₅₉	C₂²	1	24	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	24	yes	complemented
16	C₂⁴×C₅₉	—	2	8	no	—
17	C₂⁴×C₅₉	C₂×C₄	1	8	yes	—
18	C₂⁴×C₅₉	C₂³	1	6	yes	—
19	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	—	2	16	no	—
21	C₂²×C₄×C₅₉	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₅₉	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	96	yes	complemented
24	C₂²×C₄×C₅₉	C₂³	1	24	yes	—
25	C₂²×C₄×C₅₉	C₂³	1	6	yes	—
26	C₂²×C₄×C₅₉	C₂³	1	6	yes	—
27	C₂²×C₄×C₅₉	C₂³	1	16	yes	—
28	C₄⋊C₄×C₅₉	C₂³	1	32	yes	complemented
29	C₂²⋊C₄×C₅₉	C₂³	1	32	yes	—
30	C₄²×C₅₉	C₂³	1	32	yes	complemented
31	C₂³×C₅₉	—	2	32	no	—
32	C₂³×C₅₉	—	2	24	no	—
33	C₂³×C₅₉	C₂²×C₄	1	24	yes	—
34	C₂³×C₅₉	C₂²×C₄	1	4	yes	—
35	C₂³×C₅₉	#41	1	8	yes	—
36	C₂³×C₅₉	C₂⁴	1	4	yes	—
37	C₂³×C₅₉	C₂⁴	1	3	yes	—
38	C₂×C₄×C₅₉	—	2	48	no	—
39	C₂×C₄×C₅₉	C₂²×C₄	1	24	yes	—
40	C₂×C₄×C₅₉	C₂²×C₄	1	24	yes	—
41	C₂×C₄×C₅₉	C₂²×C₄	1	64	yes	complemented
42	C₂×C₄×C₅₉	C₂⁴	1	16	yes	—
43	C₂×C₄×C₅₉	C₂⁴	1	4	yes	—
44	C₂×C₄×C₅₉	C₂⁴	1	4	yes	—
45	C₂²×C₅₉	—	2	48	no	—
46	C₂²×C₅₉	—	2	8	no	—
47	C₂²×C₅₉	#117	1	16	yes	direct product factor
48	C₂²×C₅₉	C₂³×C₄	1	8	yes	—
49	C₂²×C₅₉	C₂³×C₄	1	6	yes	—
50	C₂²×C₅₉	C₂×#41	1	12	yes	—
51	C₂²×C₅₉	C₂⁵	1	1	yes	characteristic
52	C₄×C₅₉	—	2	16	no	—
53	C₄×C₅₉	C₂³×C₄	1	8	yes	—
54	C₄×C₅₉	C₂³×C₄	1	8	yes	—
55	C₂²×#117	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₅₉	—	2	8	no	—
57	C₂×C₅₉	C₂×#117	1	12	yes	direct product factor
58	C₂×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
59	C₂×C₅₉	C₂²×#41	1	2	yes	—
60	C₂⁴×C₄	C₂×C₅₉	1	1	yes	characteristic
61	C₂×#117	C₂×C₅₉	1	24	yes	direct product factor
62	C₂²×C₄⋊C₄	C₂×C₅₉	1	2	yes	complemented
63	C₂²×C₂²⋊C₄	C₂×C₅₉	1	2	yes	—
64	C₂²×C₄²	C₂×C₅₉	1	2	yes	complemented
65	C₅₉	C₂²×#117	1	1	yes	characteristic, direct product factor, Sylow
66	C₂⁵	C₂²×C₅₉	1	1	yes	characteristic
67	C₂³×C₄	C₄×C₅₉	1	16	yes	complemented
68	C₂³×C₄	C₂²×C₅₉	1	4	yes	—
69	C₂³×C₄	C₂²×C₅₉	1	1	yes	characteristic
70	C₂³×C₄	C₂²×C₅₉	1	1	yes	characteristic
71	C₂³×C₄	C₂²×C₅₉	1	12	yes	—
72	#117	C₂²×C₅₉	1	64	yes	direct product factor
73	C₂×C₄⋊C₄	C₂²×C₅₉	1	24	yes	complemented
74	C₂×C₂²⋊C₄	C₂²×C₅₉	1	24	yes	—
75	C₂×C₄²	C₂²×C₅₉	1	24	yes	complemented
76	C₂⁴	—	2	8	no	—
77	C₂⁴	C₂×C₄×C₅₉	1	8	yes	—
78	C₂⁴	C₂³×C₅₉	1	6	yes	—
79	C₂⁴	C₂³×C₅₉	1	1	yes	characteristic
80	C₂²×C₄	—	2	16	no	—
81	C₂²×C₄	C₂×C₄×C₅₉	1	8	yes	—
82	C₂²×C₄	C₂×C₄×C₅₉	1	8	yes	—
83	C₂²×C₄	C₂×C₄×C₅₉	1	96	yes	complemented
84	C₂²×C₄	C₂³×C₅₉	1	24	yes	—
85	C₂²×C₄	C₂³×C₅₉	1	6	yes	—
86	C₂²×C₄	C₂³×C₅₉	1	6	yes	—
87	C₂²×C₄	C₂³×C₅₉	1	16	yes	—
88	C₄⋊C₄	C₂³×C₅₉	1	32	yes	complemented
89	C₂²⋊C₄	C₂³×C₅₉	1	32	yes	—
90	C₄²	C₂³×C₅₉	1	32	yes	complemented
91	C₂³	—	2	32	no	—
92	C₂³	—	2	24	no	—
93	C₂³	C₂²×C₄×C₅₉	1	24	yes	—
94	C₂³	C₂²×C₄×C₅₉	1	4	yes	—
95	C₂³	#41×C₅₉	1	8	yes	—
96	C₂³	C₂⁴×C₅₉	1	4	yes	—
97	C₂³	C₂⁴×C₅₉	1	3	yes	—
98	C₂×C₄	—	2	48	no	—
99	C₂×C₄	C₂²×C₄×C₅₉	1	24	yes	—
100	C₂×C₄	C₂²×C₄×C₅₉	1	24	yes	—
101	C₂×C₄	C₂²×C₄×C₅₉	1	64	yes	complemented
102	C₂×C₄	C₂⁴×C₅₉	1	16	yes	—
103	C₂×C₄	C₂⁴×C₅₉	1	4	yes	—
104	C₂×C₄	C₂⁴×C₅₉	1	4	yes	—
105	C₂²	—	2	48	no	—
106	C₂²	—	2	8	no	—
107	C₂²	#117×C₅₉	1	16	yes	direct product factor
108	C₂²	C₂³×C₄×C₅₉	1	8	yes	—
109	C₂²	C₂³×C₄×C₅₉	1	6	yes	—
110	C₂²	C₂×#41×C₅₉	1	12	yes	—
111	C₂²	C₂⁵×C₅₉	1	1	yes	characteristic
112	C₄	—	2	16	no	—
113	C₄	C₂³×C₄×C₅₉	1	8	yes	—
114	C₄	C₂³×C₄×C₅₉	1	8	yes	—
115	C₂	—	2	8	no	—
116	C₂	C₂×#117×C₅₉	1	12	yes	direct product factor
117	C₂	C₂⁴×C₄×C₅₉	1	1	yes	characteristic
118	C₂	C₂²×#41×C₅₉	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₉	C₂²
commutator	C₂	C₂⁴×C₄×C₅₉
Frattini, cosocle	C₂²	C₂⁵×C₅₉
socle	C₂⁴×C₅₉	C₂³
prime radical	C₂⁵×C₅₉	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#117×C₅₉	C₂	direct (N × H)
3. C₂²×C₄⋊C₄×C₅₉	C₂	semidirect
5. C₂²×C₄²×C₅₉	C₂	semidirect
7. C₂³×C₄×C₅₉	C₄	semidirect
12. #117×C₅₉	C₂²	direct (N × H)
13. C₂×C₄⋊C₄×C₅₉	C₂²	semidirect
15. C₂×C₄²×C₅₉	C₂²	semidirect
23. C₂²×C₄×C₅₉	C₂×C₄	semidirect
28. C₄⋊C₄×C₅₉	C₂³	semidirect
30. C₄²×C₅₉	C₂³	semidirect
41. C₂×C₄×C₅₉	C₂²×C₄	semidirect
47. C₂²×C₅₉	#117	direct (N × H)
55. C₂²×#117	C₅₉	direct (N × H)
57. C₂×C₅₉	C₂×#117	direct (N × H)
61. C₂×#117	C₂×C₅₉	direct (N × H)
62. C₂²×C₄⋊C₄	C₂×C₅₉	semidirect
64. C₂²×C₄²	C₂×C₅₉	semidirect
65. C₅₉	C₂²×#117	direct (N × H)
67. C₂³×C₄	C₄×C₅₉	semidirect
72. #117	C₂²×C₅₉	direct (N × H)
73. C₂×C₄⋊C₄	C₂²×C₅₉	semidirect
75. C₂×C₄²	C₂²×C₅₉	semidirect
83. C₂²×C₄	C₂×C₄×C₅₉	semidirect
88. C₄⋊C₄	C₂³×C₅₉	semidirect
90. C₄²	C₂³×C₅₉	semidirect
101. C₂×C₄	C₂²×C₄×C₅₉	semidirect
107. C₂²	#117×C₅₉	direct (N × H)
116. C₂	C₂×#117×C₅₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄×C₅₉	2
C₂×#117×C₅₉	2
C₂²×C₄⋊C₄×C₅₉	2
C₂²×C₂²⋊C₄×C₅₉	2
C₂²×C₄²×C₅₉	2
C₂²×#117	59

Subgroups — C22×#117×C59 #21421739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#117×C₅₉ #21421739