Subgroups — #34032750

Order 18688.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11924040	C₂	1	2	yes	maximal
2	C₁₆×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₄×#11924040	C₄	1	4	yes	—
4	#17989473	C₄	1	1	yes	characteristic
5	C₈×#10596833	C₂²	1	1	yes	characteristic
6	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924040	C₈	1	8	yes	—
8	C₂×#11924031	C₈	1	2	yes	—
9	C₄×#10596833	C₂×C₄	1	2	yes	—
10	C₂×#11924003	C₈	1	1	yes	characteristic
11	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
13	#11924040	—	2	8	no	—
14	C₂×#10596833	C₂×C₈	1	4	yes	—
15	#11924031	C₁₆	1	4	yes	complemented
16	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
17	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
18	#11924003	C₁₆	1	2	yes	complemented
19	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₇₃	C₁₆	1	2	yes	complemented
21	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
22	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
23	#10596833	M₃₂	1	8	yes	—
24	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
25	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
26	#10596831	C₂×C₁₆	1	2	yes	—
27	C₂²×C₇₃	C₈²	1	1	yes	characteristic
28	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
30	Dic₇₃	#346	1	4	yes	—
31	#7052	—	73	1	no	maximal, Sylow
32	C₂×C₇₃	C₈×C₁₆	1	1	yes	characteristic
33	C₂×C₇₃	#1311	1	2	yes	—
34	C₈×C₁₆	—	73	1	no	—
35	C₈×C₁₆	—	73	2	no	—
36	C₇₃	#7052	1	1	yes	characteristic, complemented, Sylow
37	C₄×C₁₆	—	73	4	no	—
38	C₄×C₁₆	—	73	1	no	—
39	C₄×C₁₆	—	73	1	no	—
40	C₈²	—	73	1	no	—
41	C₂×C₁₆	—	73	2	no	—
42	C₂×C₁₆	—	73	1	no	—
43	C₂×C₁₆	—	73	8	no	—
44	C₂×C₁₆	#10596842	1	1	yes	characteristic
45	C₄×C₈	—	73	2	no	—
46	C₄×C₈	—	73	1	no	—
47	C₂×C₈	—	73	4	no	—
48	C₂×C₈	—	73	1	no	—
49	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
50	C₄²	—	73	1	no	—
51	C₁₆	—	73	4	no	—
52	C₁₆	—	73	2	no	—
53	C₁₆	—	146	8	no	—
54	C₁₆	#11924040	1	2	yes	complemented
55	C₂×C₄	—	73	2	no	—
56	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
57	C₈	—	73	8	no	—
58	C₈	—	73	2	no	—
59	C₈	C₂×#11924040	1	1	yes	characteristic
60	C₈	#14556686	1	1	yes	characteristic
61	C₂²	C₈×#10596842	1	1	yes	characteristic
62	C₄	—	73	4	no	—
63	C₄	C₄×#11924040	1	1	yes	characteristic
64	C₄	#64408⋊C₁₆	1	1	yes	characteristic
65	C₂	C₈×#11924040	1	1	yes	characteristic
66	C₂	#22626946	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#10596842
commutator	C₂×C₇₃	C₈×C₁₆
Fitting	C₂×C₁₆×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	C₈²
cosocle	C₈×#10596833	C₂²
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	#7052
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
15. #11924031	C₁₆	semidirect
18. #11924003	C₁₆	semidirect
20. C₁₆×C₇₃	C₁₆	semidirect
36. C₇₃	#7052	semidirect
54. C₁₆	#11924040	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11924040	2
C₁₆×#10596833	2
#7052	73