Subgroups — #20291130

Order 6784.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#11719910	C₂	1	2	yes	maximal
3	#16811511	C₂	1	2	yes	maximal
4	C₄×#11719910	C₂	1	1	yes	maximal, characteristic
5	Q₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719910	C₂²	1	1	yes	characteristic
7	C₂×#11719910	C₂²	1	2	yes	—
8	C₂×#11719910	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₄	1	2	yes	—
10	C₄⋊#88322	C₄	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₄×#88322	C₂²	1	2	yes	—
13	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	C₈	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	C₈	1	2	yes	—
16	C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
17	C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×#88322	C₂×C₄	1	2	yes	—
21	C₂×#88322	C₂³	1	1	yes	characteristic
22	#11719910	—	2	1	no	—
23	#11719910	—	2	2	no	—
24	#11719910	Q₈	1	2	yes	complemented
25	Q₈×C₅₃	—	2	2	no	—
26	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	#88322	—	2	1	no	—
32	#88322	—	2	2	no	—
33	#88322	C₂×Q₈	1	1	yes	characteristic
34	#88322	#41	1	1	yes	characteristic
35	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₅₃	—	2	2	no	—
37	C₄×C₅₃	—	2	2	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	M₃₂	1	2	yes	—
40	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
41	C₄×C₅₃	#131	1	1	yes	characteristic
42	#2183	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
44	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
45	C₂×C₅₃	#504	1	1	yes	characteristic
46	C₈×Q₈	—	53	1	no	—
47	C₄⋊C₁₆	—	53	1	no	—
48	C₄⋊C₁₆	—	53	2	no	—
49	#346	—	53	2	no	—
50	C₄×C₁₆	—	53	1	no	—
51	C₅₃	#2183	1	1	yes	characteristic, complemented, Sylow
52	C₄×Q₈	#6157	1	1	yes	characteristic
53	C₂×C₁₆	—	53	1	no	—
54	C₂×C₁₆	—	53	1	no	—
55	C₂×C₁₆	—	53	2	no	—
56	C₄⋊C₈	—	53	2	no	—
57	C₄⋊C₈	—	53	1	no	—
58	C₄×C₈	—	53	1	no	—
59	C₄×C₈	—	53	2	no	—
60	C₂×Q₈	#88324	1	1	yes	characteristic
61	C₂×C₈	—	53	1	no	—
62	C₂×C₈	—	53	1	no	—
63	C₂×C₈	—	53	2	no	—
64	C₄⋊C₄	#88324	1	2	yes	—
65	C₄⋊C₄	#88324	1	1	yes	characteristic
66	C₄²	C₂×#6157	1	2	yes	—
67	C₄²	C₂×#6157	1	1	yes	characteristic
68	C₁₆	—	53	2	no	—
69	C₁₆	—	106	1	no	—
70	C₁₆	—	106	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×#88324	1	2	yes	—
73	C₂×C₄	C₂×#88324	1	1	yes	characteristic
74	C₂×C₄	C₂×#88324	1	2	yes	—
75	C₂×C₄	C₂×#88324	1	1	yes	characteristic
76	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
77	C₈	—	53	1	no	—
78	C₈	—	53	1	no	—
79	C₈	—	106	1	no	—
80	C₈	—	106	2	no	—
81	C₂²	C₂²×#88324	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#13133291	1	2	yes	—
86	C₄	Q₈×#6157	1	1	yes	characteristic
87	C₄	#13133305	1	1	yes	characteristic
88	C₂	Q₈×#88324	1	1	yes	characteristic
89	C₂	C₂×#13133291	1	1	yes	characteristic
90	C₂	#16811681	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₄×Q₈×C₅₃	C₄
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	#2183
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
24. #11719910	Q₈	semidirect
51. C₅₃	#2183	semidirect