Subgroups — #28298087

Order 13568.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719910	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719910	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
4	C₄×#11719910	C₄	1	2	yes	—
5	C₄×#11719910	C₄	1	2	yes	—
6	C₄×#11719908	C₄	1	1	yes	characteristic
7	C₈×#88322	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂×#11719910	C₈	1	4	yes	—
10	C₂×#11719910	C₈	1	4	yes	—
11	C₂×#11719908	C₈	1	2	yes	—
12	C₄×#88322	C₂×C₄	1	1	yes	characteristic
13	C₄×#88322	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₅₃	C₄²	1	1	yes	characteristic
19	C₂×#88322	C₂×C₈	1	2	yes	—
20	C₂×#88322	C₂×C₈	1	2	yes	—
21	#11719910	—	2	4	no	—
22	#11719910	C₁₆	1	8	yes	complemented
23	C₁₆×C₅₃	—	2	2	no	—
24	#11719908	—	2	2	no	—
25	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
28	C₈×C₅₃	M₃₂	1	2	yes	—
29	C₈×C₅₃	M₃₂	1	2	yes	—
30	#88322	C₂×C₁₆	1	4	yes	—
31	#88322	M₃₂	1	4	yes	—
32	#7052	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	C₈²	1	1	yes	characteristic
34	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
35	C₄×C₅₃	#346	1	2	yes	—
36	C₄×C₅₃	#346	1	1	yes	characteristic
37	C₄×C₅₃	#346	1	1	yes	characteristic
38	C₈×C₁₆	—	53	1	no	—
39	C₈×C₁₆	—	53	1	no	—
40	C₈×C₁₆	—	53	1	no	—
41	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
42	C₂×C₅₃	#1311	1	1	yes	characteristic
43	C₂×C₅₃	#1311	1	1	yes	characteristic
44	C₄×C₁₆	—	53	2	no	—
45	C₄×C₁₆	—	53	2	no	—
46	C₄×C₁₆	—	53	1	no	—
47	C₄×C₁₆	#6157	1	1	yes	characteristic
48	C₈²	—	53	1	no	—
49	C₅₃	#7052	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	53	4	no	—
51	C₂×C₁₆	—	53	4	no	—
52	C₂×C₁₆	—	53	2	no	—
53	C₂×C₁₆	#88324	1	2	yes	—
54	C₄×C₈	—	53	1	no	—
55	C₄×C₈	—	53	1	no	—
56	C₄×C₈	C₂×#6157	1	1	yes	characteristic
57	C₂×C₈	—	53	2	no	—
58	C₂×C₈	—	53	2	no	—
59	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
60	C₂×C₈	C₂×#88324	1	1	yes	characteristic
61	C₄²	C₄×#6157	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	C₁₆	—	53	8	no	—
64	C₁₆	—	106	4	no	—
65	C₁₆	—	106	2	no	—
66	C₂×C₄	C₈×#6157	1	1	yes	characteristic
67	C₂×C₄	C₈×#6157	1	1	yes	characteristic
68	C₂×C₄	C₄×#88324	1	1	yes	characteristic
69	C₈	—	53	4	no	—
70	C₈	—	53	4	no	—
71	C₈	#13133291	1	2	yes	—
72	C₈	#13133292	1	2	yes	—
73	C₂²	C₈×#88324	1	1	yes	characteristic
74	C₄	C₁₆×#6157	1	2	yes	—
75	C₄	#16811510	1	2	yes	—
76	C₄	#16811511	1	1	yes	characteristic
77	C₄	#16811512	1	1	yes	characteristic
78	C₂	C₁₆×#88324	1	1	yes	characteristic
79	C₂	#20289626	1	1	yes	characteristic
80	C₂	#20289627	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₈×C₁₆
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₈²
cosocle	C₈×#88322	C₂²
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#7052
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
22. #11719910	C₁₆	semidirect
49. C₅₃	#7052	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈×#11719910	2
C₁₆×#88322	2
#7052	53