Subgroups — #21424233

Order 7552.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837426	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16837166	C₂	1	4	yes	maximal
4	C₅₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	C₂²	1	2	yes	—
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
8	C₄×C₅₉⋊C₈	C₄	1	2	yes	—
9	C₄×C₅₉⋊C₈	C₄	1	2	yes	—
10	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	—	2	2	no	—
13	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
14	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
15	M₁₆×C₅₉	—	2	4	no	—
16	C₂×C₈×C₅₉	—	2	2	no	—
17	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₉	D₄	1	2	yes	—
20	#11745664	—	2	4	no	—
21	C₂×C₅₉⋊C₈	—	2	2	no	—
22	C₂×C₅₉⋊C₈	—	2	2	no	—
23	C₂×C₅₉⋊C₈	—	2	2	no	—
24	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
25	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₅₉	—	2	4	no	—
27	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	#41	1	2	yes	—
32	C₂×C₄×C₅₉	#41	1	2	yes	—
33	C₈×C₅₉	—	4	2	no	—
34	C₅₉⋊C₈	—	2	4	no	—
35	C₅₉⋊C₈	—	4	2	no	—
36	C₂²×C₅₉	—	2	2	no	—
37	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₅₉	#125	1	2	yes	—
39	C₄×C₅₉	—	2	2	no	—
40	C₄×C₅₉	#108	1	4	yes	—
41	C₄×C₅₉	#117	1	2	yes	—
42	C₄×C₅₉	#124	1	1	yes	characteristic
43	C₄×C₅₉	#124	1	1	yes	characteristic
44	#1837	—	59	1	no	maximal, Sylow
45	C₂×C₅₉	—	2	2	no	—
46	C₂×C₅₉	#382	1	1	yes	characteristic
47	C₂×C₅₉	C₂×#108	1	2	yes	—
48	#423	—	59	1	no	—
49	#423	D₅₉	1	1	yes	characteristic
50	C₂×C₄×C₈	—	59	1	no	—
51	#341	—	59	4	no	—
52	C₅₉	#1837	1	1	yes	characteristic, complemented, Sylow
53	C₂×M₁₆	—	59	2	no	—
54	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
55	C₂²×C₈	—	59	2	no	—
56	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	118	2	no	—
59	C₄×C₈	—	59	2	no	—
60	C₄×C₈	—	59	2	no	—
61	C₂²×C₄	Dic₁₁₈	1	2	yes	—
62	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	118	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	59	4	no	—
67	C₂×C₈	—	118	2	no	—
68	C₂×C₈	—	118	2	no	—
69	C₂×C₈	—	118	2	no	—
70	C₄²	C₄×D₅₉	1	1	yes	characteristic
71	C₄²	C₄×D₅₉	1	1	yes	characteristic
72	C₄²	C₂²⋊D₅₉	1	2	yes	—
73	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
76	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
78	C₂×C₄	#11745687	1	2	yes	—
79	C₂×C₄	#11745689	1	2	yes	—
80	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	118	4	no	—
83	C₈	—	236	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
86	C₂²	#14273718	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#14273662	1	4	yes	—
89	C₄	#14273685	1	2	yes	—
90	C₄	#14273714	1	1	yes	characteristic
91	C₄	#14273716	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#16837267	1	1	yes	characteristic
94	C₂	C₂×#14273662	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#423	D₅₉
hypocenter	C₅₉	#1837
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
52. C₅₉	#1837	semidirect