Subgroups — #28298865

Order 13568.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133267	C₂	1	2	yes	maximal
2	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133267	C₄	1	4	yes	—
4	C₄×#11719908	C₂²	1	1	yes	characteristic
5	C₈×#88322	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	#13133267	—	2	4	no	—
8	C₂×#11719908	C₂×C₄	1	2	yes	—
9	C₄×#88322	C₈	1	1	yes	characteristic
10	C₄×#88322	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×#88322	C₁₆	1	2	yes	—
17	C₂×#88322	C₁₆	1	2	yes	—
18	C₁₆×C₅₃	—	2	2	no	—
19	#11719908	M₁₆	1	4	yes	—
20	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
23	C₈×C₅₃	#97	1	2	yes	—
24	C₈×C₅₃	M₃₂	1	2	yes	—
25	#88322	—	2	2	no	—
26	#88322	C₃₂	1	4	yes	complemented
27	#7331	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₅₃	#322	1	1	yes	characteristic
30	C₄×C₅₃	#346	1	1	yes	characteristic
31	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
32	C₄×C₅₃	M₆₄	1	2	yes	—
33	C₄×C₃₂	—	53	2	no	—
34	C₈×C₁₆	—	53	1	no	—
35	C₂×C₅₃	#1312	1	1	yes	characteristic
36	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
37	C₂×C₅₃	#1397	1	1	yes	characteristic
38	C₂×C₃₂	—	53	4	no	—
39	C₄×C₁₆	—	53	1	no	—
40	C₄×C₁₆	#6157	1	1	yes	characteristic
41	C₈²	—	53	1	no	—
42	C₅₃	#7331	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	—	53	2	no	—
44	C₂×C₁₆	#88324	1	2	yes	—
45	C₄×C₈	—	53	1	no	—
46	C₄×C₈	—	53	1	no	—
47	C₄×C₈	C₂×#6157	1	1	yes	characteristic
48	C₃₂	—	106	4	no	—
49	C₂×C₈	—	53	2	no	—
50	C₂×C₈	—	53	2	no	—
51	C₂×C₈	C₄×#6157	1	1	yes	characteristic
52	C₂×C₈	C₂×#88324	1	1	yes	characteristic
53	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₁₆	—	53	4	no	—
56	C₂×C₄	C₄×#88324	1	1	yes	characteristic
57	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
58	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
59	C₈	—	53	4	no	—
60	C₈	—	106	2	no	—
61	C₈	#13133271	1	2	yes	—
62	C₈	#13133291	1	2	yes	—
63	C₂²	#16811509	1	1	yes	characteristic
64	C₄	#16811464	1	1	yes	characteristic
65	C₄	#16811511	1	1	yes	characteristic
66	C₄	D₅₃⋊C₃₂	1	2	yes	—
67	C₄	#16811556	1	2	yes	—
68	C₂	#20289630	1	1	yes	characteristic
69	C₂	#20289820	1	1	yes	characteristic
70	C₂	#20289823	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₄×C₃₂
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
cosocle	C₄×#11719908	C₂²
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#7331
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
26. #88322	C₃₂	semidirect
42. C₅₃	#7331	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133267	2
C₁₆×#88322	2
#7331	53