Subgroups — #20291608

Order 6784.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#88324	C₂	1	8	yes	maximal, complemented
2	#16811639	C₂	1	6	yes	maximal
3	C₂×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88324	C₂²	1	12	yes	—
5	C₂²⋊#88324	C₂²	1	4	yes	—
6	C₄×#88324	C₂²	1	12	yes	—
7	Q₈×Dic₅₃	C₄	1	4	yes	—
8	Q₈×Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₅₃	C₄	1	3	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
11	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×Q₈×C₅₃	C₈	1	8	yes	complemented
13	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
15	C₂²×Dic₅₃	C₂×C₄	1	3	yes	—
16	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
17	C₂×#88324	C₂³	1	8	yes	—
18	C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
19	C₄×Dic₅₃	C₂×C₄	1	6	yes	—
20	C₄×Dic₅₃	C₂³	1	6	yes	—
21	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	Q₈×C₅₃	C₂×C₈	1	16	yes	—
23	C₂×C₄×C₅₃	C₂×C₈	1	12	yes	—
24	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
25	C₂×Dic₅₃	—	2	2	no	—
26	C₂×Dic₅₃	—	4	3	no	—
27	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
28	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
30	#88324	—	2	8	no	—
31	C₂²×C₅₃	—	2	1	no	—
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
34	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₅₃	C₂²×C₈	1	12	yes	—
36	Dic₅₃	—	2	1	no	—
37	Dic₅₃	—	2	6	no	—
38	Dic₅₃	2₋¹⁺⁴	1	2	yes	—
39	#2982	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	2	2	no	—
41	C₂×C₅₃	#519	1	1	yes	characteristic
42	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
43	C₂×C₅₃	#568	1	1	yes	characteristic
44	C₂×C₄×Q₈	—	53	1	no	—
45	C₈×Q₈	—	53	8	no	—
46	#431	—	53	6	no	—
47	C₅₃	#2982	1	1	yes	characteristic, complemented, Sylow
48	C₂²×Q₈	#6157	1	1	yes	characteristic
49	C₄×Q₈	—	53	4	no	—
50	C₄×Q₈	—	53	4	no	—
51	C₂×C₄⋊C₄	—	53	3	no	—
52	C₂×C₄²	—	53	3	no	—
53	C₄⋊C₈	—	53	12	no	—
54	C₂²⋊C₈	—	53	4	no	—
55	C₄×C₈	—	53	12	no	—
56	C₂×Q₈	#88324	1	8	yes	complemented
57	C₂×Q₈	C₂×#6157	1	4	yes	—
58	C₂²×C₄	—	53	1	no	—
59	C₂²×C₄	—	53	3	no	—
60	C₂²×C₄	C₂×#6157	1	3	yes	—
61	C₂×C₈	—	53	8	no	—
62	C₄⋊C₄	—	53	12	no	—
63	C₄²	—	53	6	no	—
64	C₄²	—	53	6	no	—
65	C₂³	C₂²×#6157	1	1	yes	characteristic
66	Q₈	C₂×#88324	1	16	yes	—
67	C₂×C₄	—	53	1	no	—
68	C₂×C₄	—	53	1	no	—
69	C₂×C₄	—	53	6	no	—
70	C₂×C₄	—	106	2	no	—
71	C₂×C₄	—	212	3	no	—
72	C₂×C₄	C₂×#88324	1	12	yes	—
73	C₂×C₄	C₂²×#6157	1	6	yes	—
74	C₈	—	106	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×#88324	1	2	yes	—
78	C₂²	C₂³×#6157	1	1	yes	characteristic
79	C₄	—	53	2	no	—
80	C₄	—	106	1	no	—
81	C₄	—	106	6	no	—
82	C₄	C₂²×#88324	1	12	yes	—
83	C₂	—	2	2	no	—
84	C₂	#16811691	1	1	yes	characteristic
85	C₂	C₂³×#88324	1	1	yes	characteristic
86	C₂	#16811703	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator	C₂×C₅₃	C₂³×C₈
Fitting	C₂²×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂²×Q₈	#6157
hypocenter	C₅₃	#2982
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#88324	C₂	semidirect
12. C₂×Q₈×C₅₃	C₈	semidirect
47. C₅₃	#2982	semidirect
56. C₂×Q₈	#88324	semidirect