Subgroups — #28330795

Order 13568.

2038 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291682	C₂	1	15	yes	maximal, complemented
2	C₂×#16811681	C₂	1	15	yes	maximal, complemented
3	#20289427	C₂	1	1	yes	maximal, characteristic
4	#16811681	C₂²	1	20	yes	complemented
5	#16811681	C₂²	1	60	yes	complemented
6	C₂×#13133291	C₂²	1	45	yes	complemented
7	C₂²×#11719910	C₂²	1	15	yes	complemented
8	#16811390	C₄	1	15	yes	—
9	C₂×#13133224	C₂²	1	15	yes	—
10	C₅₃×#585	C₄	1	1	yes	characteristic
11	#13133291	C₂³	1	60	yes	—
12	C₂×#11719910	C₂³	1	15	yes	—
13	C₂×#11719910	C₂³	1	45	yes	—
14	#13133224	C₂×C₄	1	60	yes	—
15	#13133224	C₂³	1	20	yes	—
16	C₂×#11719917	C₂×C₄	1	45	yes	—
17	C₂²×#88322	C₂³	1	15	yes	—
18	2₋¹⁺⁴×C₅₃	C₈	1	6	yes	—
19	H(5,𝔽₂)×C₅₃	C₈	1	10	yes	—
20	C₂×#41×C₅₃	C₂×C₄	1	15	yes	—
21	#41×C₅₃	C₂×C₈	1	60	yes	—
22	#41×C₅₃	C₂²×C₄	1	20	yes	—
23	C₂×Q₈×C₅₃	C₂×C₈	1	15	yes	—
24	C₂×D₄×C₅₃	C₂×C₈	1	45	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	15	yes	—
26	#11719917	C₂²×C₄	1	60	yes	—
27	C₂×#88322	C₂²×C₄	1	45	yes	—
28	C₂×#88322	C₂⁴	1	15	yes	—
29	#11719910	C₂⁴	1	1	yes	characteristic
30	#11719910	C₂⁴	1	15	yes	—
31	C₂³×C₅₃	C₂²×C₈	1	15	yes	—
32	Q₈×C₅₃	C₂²×C₈	1	20	yes	—
33	D₄×C₅₃	C₂²×C₈	1	60	yes	—
34	C₂×C₄×C₅₃	C₂²×C₈	1	45	yes	—
35	C₂×C₄×C₅₃	C₂³×C₄	1	15	yes	—
36	#88322	C₂³×C₄	1	15	yes	—
37	#88322	C₂⁵	1	1	yes	characteristic
38	#62624	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	—	4	15	no	—
40	C₂²×C₅₃	C₂³×C₈	1	15	yes	—
41	C₄×C₅₃	C₂³×C₈	1	15	yes	—
42	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
43	#3573	—	53	1	no	—
44	#3407	—	53	15	no	—
45	C₂×#504	—	53	15	no	—
46	C₂×C₅₃	—	2	15	no	—
47	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
48	#585	#6157	1	1	yes	characteristic
49	#568	—	53	15	no	—
50	C₂×#131	—	53	15	no	—
51	#504	—	53	20	no	—
52	#504	—	53	60	no	—
53	C₂×M₃₂	—	53	45	no	—
54	C₂²×C₁₆	—	53	15	no	—
55	C₅₃	#62624	1	1	yes	characteristic, complemented, Sylow
56	2₋¹⁺⁴	#88324	1	6	yes	—
57	H(5,𝔽₂)	#88324	1	10	yes	—
58	C₂×#41	C₂×#6157	1	15	yes	—
59	#131	—	53	20	no	—
60	#131	—	53	60	no	—
61	C₂×M₁₆	—	53	45	no	—
62	C₂²×C₈	—	53	15	no	—
63	M₃₂	—	53	60	no	—
64	C₂×C₁₆	—	53	15	no	—
65	C₂×C₁₆	—	53	45	no	—
66	#41	C₂×#88324	1	60	yes	—
67	#41	C₂²×#6157	1	20	yes	—
68	C₂×Q₈	C₂×#88324	1	15	yes	—
69	C₂×D₄	C₂×#88324	1	45	yes	—
70	C₂²×C₄	C₂²×#6157	1	15	yes	—
71	M₁₆	—	53	60	no	—
72	C₂×C₈	—	53	15	no	—
73	C₂×C₈	—	53	45	no	—
74	C₁₆	—	53	1	no	—
75	C₁₆	—	53	15	no	—
76	C₂³	C₂²×#88324	1	15	yes	—
77	Q₈	C₂²×#88324	1	20	yes	—
78	D₄	C₂²×#88324	1	60	yes	—
79	C₂×C₄	C₂²×#88324	1	45	yes	—
80	C₂×C₄	C₂³×#6157	1	15	yes	—
81	C₈	—	53	15	no	—
82	C₈	—	53	1	no	—
83	C₂²	—	4	15	no	—
84	C₂²	C₂³×#88324	1	15	yes	—
85	C₄	C₂³×#88324	1	15	yes	—
86	C₄	C₂⁴×#6157	1	1	yes	characteristic
87	C₂	—	2	15	no	—
88	C₂	C₂⁴×#88324	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×#6157
commutator, socle	C₂×C₅₃	C₂⁴×C₈
Fitting, prime radical	C₅₃×#585	C₄
cosocle	#88322	C₂⁵
hypercenter	#585	#6157
hypocenter	C₅₃	#62624
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #20291682	C₂	semidirect
2. C₂×#16811681	C₂	semidirect
4. #16811681	C₂²	semidirect
5. #16811681	C₂²	semidirect
6. C₂×#13133291	C₂²	semidirect
7. C₂²×#11719910	C₂²	semidirect
55. C₅₃	#62624	semidirect

Maximal subgroups

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

subgroup	index
#20291682	2
C₂×#16811681	2
#20289427	2
#62624	53