Subgroups — #34032687

Order 18688.

2038 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626898	C₂	1	15	yes	maximal, complemented
2	C₂×#17989645	C₂	1	15	yes	maximal, complemented
3	#22624643	C₂	1	1	yes	maximal, characteristic
4	#17989645	C₂²	1	20	yes	complemented
5	#17989645	C₂²	1	60	yes	complemented
6	C₂×#14556665	C₂²	1	45	yes	complemented
7	C₂²×#11924031	C₂²	1	15	yes	complemented
8	#17989354	C₄	1	15	yes	—
9	C₂×#14556598	C₂²	1	15	yes	—
10	#585×C₇₃	C₄	1	1	yes	characteristic
11	#14556665	C₂³	1	60	yes	—
12	C₂×#11924031	C₂³	1	15	yes	—
13	C₂×#11924031	C₂³	1	45	yes	—
14	#14556598	C₂×C₄	1	60	yes	—
15	#14556598	C₂³	1	20	yes	—
16	C₂×#11924012	C₂×C₄	1	45	yes	—
17	C₂²×#10596831	C₂³	1	15	yes	—
18	2₋¹⁺⁴×C₇₃	C₈	1	6	yes	—
19	H(5,𝔽₂)×C₇₃	C₈	1	10	yes	—
20	C₂×#41×C₇₃	C₂×C₄	1	15	yes	—
21	#11924031	C₂⁴	1	1	yes	characteristic
22	#11924031	C₂⁴	1	15	yes	—
23	#11924012	C₂²×C₄	1	60	yes	—
24	C₂×#10596831	C₂²×C₄	1	45	yes	—
25	C₂×#10596831	C₂⁴	1	15	yes	—
26	#41×C₇₃	C₂×C₈	1	60	yes	—
27	#41×C₇₃	C₂²×C₄	1	20	yes	—
28	C₂×Q₈×C₇₃	C₂×C₈	1	15	yes	—
29	C₂×D₄×C₇₃	C₂×C₈	1	45	yes	—
30	C₂²×C₄×C₇₃	C₂²×C₄	1	15	yes	—
31	C₂³×C₇₃	C₂²×C₈	1	15	yes	—
32	Q₈×C₇₃	C₂²×C₈	1	20	yes	—
33	D₄×C₇₃	C₂²×C₈	1	60	yes	—
34	C₂×C₄×C₇₃	C₂²×C₈	1	45	yes	—
35	C₂×C₄×C₇₃	C₂³×C₄	1	15	yes	—
36	#10596831	C₂³×C₄	1	15	yes	—
37	#10596831	C₂⁵	1	1	yes	characteristic
38	C₂²×C₇₃	—	4	15	no	—
39	C₂²×C₇₃	C₂³×C₈	1	15	yes	—
40	C₄×C₇₃	C₂³×C₈	1	15	yes	—
41	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
42	#62624	—	73	1	no	maximal, Sylow
43	C₂×C₇₃	—	2	15	no	—
44	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
45	#3573	—	73	1	no	—
46	#3407	—	73	15	no	—
47	C₂×#504	—	73	15	no	—
48	C₇₃	#62624	1	1	yes	characteristic, complemented, Sylow
49	#585	#64408	1	1	yes	characteristic
50	#568	—	73	15	no	—
51	C₂×#131	—	73	15	no	—
52	#504	—	73	20	no	—
53	#504	—	73	60	no	—
54	C₂×M₃₂	—	73	45	no	—
55	C₂²×C₁₆	—	73	15	no	—
56	2₋¹⁺⁴	#10596833	1	6	yes	—
57	H(5,𝔽₂)	#10596833	1	10	yes	—
58	C₂×#41	C₂×#64408	1	15	yes	—
59	#131	—	73	20	no	—
60	#131	—	73	60	no	—
61	C₂×M₁₆	—	73	45	no	—
62	C₂²×C₈	—	73	15	no	—
63	M₃₂	—	73	60	no	—
64	C₂×C₁₆	—	73	15	no	—
65	C₂×C₁₆	—	73	45	no	—
66	#41	C₂×#10596833	1	60	yes	—
67	#41	C₂²×#64408	1	20	yes	—
68	C₂×Q₈	C₂×#10596833	1	15	yes	—
69	C₂×D₄	C₂×#10596833	1	45	yes	—
70	C₂²×C₄	C₂²×#64408	1	15	yes	—
71	M₁₆	—	73	60	no	—
72	C₂×C₈	—	73	45	no	—
73	C₂×C₈	—	73	15	no	—
74	C₁₆	—	73	1	no	—
75	C₁₆	—	73	15	no	—
76	C₂³	C₂²×#10596833	1	15	yes	—
77	Q₈	C₂²×#10596833	1	20	yes	—
78	D₄	C₂²×#10596833	1	60	yes	—
79	C₂×C₄	C₂²×#10596833	1	45	yes	—
80	C₂×C₄	C₂³×#64408	1	15	yes	—
81	C₈	—	73	1	no	—
82	C₈	—	73	15	no	—
83	C₂²	—	4	15	no	—
84	C₂²	C₂³×#10596833	1	15	yes	—
85	C₄	C₂³×#10596833	1	15	yes	—
86	C₄	C₂⁴×#64408	1	1	yes	characteristic
87	C₂	—	2	15	no	—
88	C₂	C₂⁴×#10596833	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×#64408
commutator, socle	C₂×C₇₃	C₂⁴×C₈
Fitting, prime radical	#585×C₇₃	C₄
cosocle	#10596831	C₂⁵
hypercenter	#585	#64408
hypocenter	C₇₃	#62624
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #22626898	C₂	semidirect
2. C₂×#17989645	C₂	semidirect
4. #17989645	C₂²	semidirect
5. #17989645	C₂²	semidirect
6. C₂×#14556665	C₂²	semidirect
7. C₂²×#11924031	C₂²	semidirect
48. C₇₃	#62624	semidirect

Maximal subgroups

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

subgroup	index
#22626898	2
C₂×#17989645	2
#22624643	2
#62624	73