Subgroups — #20289928

Order 6784.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811639	C₂	1	2	yes	maximal
2	#88324⋊C₈	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88324	C₄	1	4	yes	—
5	C₂²⋊#88324	C₄	1	4	yes	—
6	C₄×#88324	C₂²	1	4	yes	—
7	C₂²×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₈×Dic₅₃	C₄	1	2	yes	—
10	C₈×Dic₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
14	C₂×#88324	C₂×C₄	1	8	yes	—
15	C₂×C₈×C₅₃	—	2	2	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88322	—	2	2	no	—
21	C₂×#88322	C₂×C₄	1	2	yes	—
22	C₂³×C₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
25	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
26	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	2	4	no	—
28	C₂×Dic₅₃	C₄²	1	2	yes	—
29	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
30	#88324	—	2	8	no	—
31	C₈×C₅₃	—	2	2	no	—
32	#88322	—	2	2	no	—
33	C₂²×C₅₃	—	2	2	no	—
34	C₂²×C₅₃	#97	1	2	yes	—
35	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
36	C₄×C₅₃	#97	1	2	yes	—
37	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
38	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
39	Dic₅₃	—	2	2	no	—
40	Dic₅₃	#131	1	4	yes	—
41	#1452	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	2	no	—
43	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
44	C₂×C₅₃	#405	1	2	yes	—
45	#431	—	53	2	no	—
46	C₂×C₄×C₈	—	53	1	no	—
47	#322	—	53	4	no	—
48	C₅₃	#1452	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	—	53	1	no	—
50	C₂²×C₈	#6157	1	1	yes	characteristic
51	C₂×C₄²	—	53	1	no	—
52	C₄⋊C₈	—	53	4	no	—
53	C₂²⋊C₈	—	53	4	no	—
54	C₄×C₈	—	53	4	no	—
55	C₄×C₈	—	53	2	no	—
56	C₄×C₈	—	53	2	no	—
57	C₂²×C₄	—	53	2	no	—
58	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	53	8	no	—
61	C₂×C₈	—	53	2	no	—
62	C₂×C₈	—	106	2	no	—
63	C₂×C₈	C₂×#6157	1	2	yes	—
64	C₄²	—	53	2	no	—
65	C₄²	—	53	1	no	—
66	C₄²	—	53	1	no	—
67	C₂³	C₄×#6157	1	1	yes	characteristic
68	C₂×C₄	—	53	2	no	—
69	C₂×C₄	—	53	2	no	—
70	C₂×C₄	—	106	4	no	—
71	C₂×C₄	#11719936	1	2	yes	—
72	C₂×C₄	C₄×#6157	1	1	yes	characteristic
73	C₂×C₄	#11719940	1	2	yes	—
74	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	106	8	no	—
77	C₈	—	106	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	#13133272	1	2	yes	—
80	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
81	C₄	—	53	4	no	—
82	C₄	—	106	2	no	—
83	C₄	#13133272	1	2	yes	—
84	C₄	C₂×#11719940	1	1	yes	characteristic
85	C₄	C₂×#11719936	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	C₂×#13133272	1	1	yes	characteristic
88	C₂	#16811586	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₂²×C₈×C₅₃	C₄
hypercenter	C₂²×C₈	#6157
hypocenter	C₅₃	#1452
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #88324⋊C₈	C₂	semidirect
48. C₅₃	#1452	semidirect