Subgroups — #20291682

Order 6784.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811681	C₂	1	8	yes	maximal, complemented
2	C₂×#13133291	C₂	1	6	yes	maximal, complemented
3	C₂×#13133224	C₂	1	1	yes	maximal, characteristic
4	#13133291	C₂²	1	4	yes	—
5	#13133291	C₂²	1	12	yes	complemented
6	C₂×#11719910	C₂²	1	12	yes	complemented
7	#13133224	C₄	1	4	yes	—
8	#13133224	C₂²	1	4	yes	—
9	C₂×#11719917	C₄	1	3	yes	—
10	C₂²×#88322	C₂²	1	3	yes	—
11	C₂×#41×C₅₃	C₄	1	1	yes	characteristic
12	#41×C₅₃	C₈	1	4	yes	—
13	#41×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₅₃	C₈	1	1	yes	characteristic
15	C₂×D₄×C₅₃	C₈	1	3	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
17	#11719917	C₂×C₄	1	12	yes	—
18	C₂×#88322	C₂×C₄	1	3	yes	—
19	C₂×#88322	C₂×C₄	1	6	yes	—
20	C₂×#88322	C₂³	1	1	yes	characteristic
21	C₂×#88322	C₂³	1	6	yes	—
22	#11719910	C₂³	1	8	yes	—
23	C₂³×C₅₃	C₂×C₈	1	3	yes	—
24	Q₈×C₅₃	C₂×C₈	1	4	yes	—
25	D₄×C₅₃	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
30	#88322	C₂²×C₄	1	1	yes	characteristic
31	#88322	C₂²×C₄	1	6	yes	—
32	#88322	C₂⁴	1	1	yes	characteristic
33	C₂²×C₅₃	—	4	3	no	—
34	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
35	C₂²×C₅₃	C₂²×C₈	1	6	yes	—
36	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₅₃	C₂²×C₈	1	6	yes	—
38	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
39	#3407	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	2	1	no	—
41	C₂×C₅₃	—	2	6	no	—
42	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
43	C₂×#131	—	53	1	no	—
44	#504	—	53	8	no	—
45	C₂×M₃₂	—	53	6	no	—
46	C₅₃	#3407	1	1	yes	characteristic, complemented, Sylow
47	C₂×#41	#6157	1	1	yes	characteristic
48	#131	—	53	4	no	—
49	#131	—	53	4	no	—
50	C₂×M₁₆	—	53	3	no	—
51	C₂²×C₈	—	53	3	no	—
52	M₃₂	—	53	4	no	—
53	M₃₂	—	53	12	no	—
54	C₂×C₁₆	—	53	12	no	—
55	#41	#88324	1	4	yes	—
56	#41	C₂×#6157	1	4	yes	—
57	C₂×Q₈	#88324	1	1	yes	characteristic
58	C₂×D₄	#88324	1	3	yes	—
59	C₂²×C₄	C₂×#6157	1	3	yes	—
60	M₁₆	—	53	12	no	—
61	C₂×C₈	—	53	6	no	—
62	C₂×C₈	—	53	1	no	—
63	C₂×C₈	—	53	3	no	—
64	C₂×C₈	—	53	6	no	—
65	C₁₆	—	53	8	no	—
66	C₂³	C₂×#88324	1	3	yes	—
67	Q₈	C₂×#88324	1	4	yes	—
68	D₄	C₂×#88324	1	12	yes	—
69	C₂×C₄	C₂×#88324	1	3	yes	—
70	C₂×C₄	C₂×#88324	1	6	yes	—
71	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
72	C₂×C₄	C₂²×#6157	1	6	yes	—
73	C₈	—	53	1	no	—
74	C₈	—	53	6	no	—
75	C₈	—	53	1	no	—
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×#88324	1	1	yes	characteristic
78	C₂²	C₂²×#88324	1	6	yes	—
79	C₄	C₂²×#88324	1	1	yes	characteristic
80	C₄	C₂²×#88324	1	6	yes	—
81	C₄	C₂³×#6157	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×#88324	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#6157
commutator, socle	C₂×C₅₃	C₂³×C₈
Fitting	C₂×#41×C₅₃	C₄
hypercenter	C₂×#41	#6157
hypocenter	C₅₃	#3407
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16811681	C₂	semidirect
2. C₂×#13133291	C₂	semidirect
5. #13133291	C₂²	semidirect
6. C₂×#11719910	C₂²	semidirect
46. C₅₃	#3407	semidirect