Subgroups — #34005493

Order 18688.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625867	C₂	1	6	yes	maximal
2	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556660	—	2	12	no	—
4	#17989533	C₂²	1	4	yes	—
5	C₄×C₈×D₇₃	C₂²	1	3	yes	—
6	C₈×#10596831	C₄	1	3	yes	complemented
7	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄⋊#64408	—	2	12	no	—
9	#14556660	—	4	24	no	—
10	C₂×C₈×D₇₃	D₄	1	6	yes	—
11	C₄²×D₇₃	C₂³	1	1	yes	characteristic
12	C₄×#10596831	C₂×C₄	1	3	yes	—
13	C₈×Dic₇₃	Q₈	1	6	yes	—
14	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
15	C₂²×#64408	—	4	4	no	—
16	C₄⋊#64408	—	4	24	no	—
17	C₂×C₄×D₇₃	C₂×D₄	1	3	yes	—
18	C₂×#10596831	C₄⋊C₄	1	6	yes	—
19	C₈×D₇₃	D₈	1	12	yes	—
20	C₈×D₇₃	Q₁₆	1	12	yes	—
21	C₄×Dic₇₃	C₂×Q₈	1	3	yes	—
22	C₂×C₈×C₇₃	C₄⋊C₄	1	6	yes	—
23	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
24	C₂²×D₇₃	#127	1	1	yes	characteristic
25	C₂×#64408	—	8	12	no	—
26	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
27	C₂×Dic₇₃	C₄⋊Q₈	1	3	yes	—
28	C₄×D₇₃	C₂×D₈	1	6	yes	—
29	C₄×D₇₃	C₂×Q₁₆	1	6	yes	—
30	C₈×C₇₃	#107	1	12	yes	complemented
31	#10596831	#107	1	12	yes	complemented
32	C₂²×C₇₃	#384	1	1	yes	characteristic
33	C₂×D₇₃	#493	1	3	yes	—
34	C₂×D₇₃	#494	1	3	yes	—
35	#64408	—	16	4	no	—
36	C₄×C₇₃	C₂×#107	1	6	yes	—
37	Dic₇₃	#500	1	6	yes	—
38	#8818	—	73	1	no	maximal, Sylow
39	C₂×C₇₃	#1845	1	3	yes	—
40	D₇₃	#1712	1	1	yes	characteristic
41	D₇₃	#1713	1	3	yes	—
42	#1845	—	73	6	no	—
43	C₂×C₈²	—	73	1	no	—
44	C₇₃	#8818	1	1	yes	characteristic, complemented, Sylow
45	C₂×#107	—	146	12	no	—
46	C₂×C₄×C₈	—	73	3	no	—
47	#384	—	73	4	no	—
48	C₈²	—	73	3	no	—
49	C₈²	#64408	1	1	yes	characteristic, complemented
50	C₂²×C₈	—	73	6	no	—
51	C₂×C₄⋊C₄	—	146	12	no	—
52	C₂×C₄²	—	73	1	no	—
53	#107	—	292	24	no	—
54	C₄×C₈	—	73	6	no	—
55	C₄×C₈	—	73	3	no	—
56	C₄×C₈	C₂×#64408	1	3	yes	—
57	C₂²×C₄	—	292	4	no	—
58	C₂²×C₄	—	73	3	no	—
59	C₂×C₈	—	73	6	no	—
60	C₂×C₈	—	73	12	no	—
61	C₂×C₈	—	73	12	no	—
62	C₂×C₈	C₄⋊#64408	1	6	yes	—
63	C₄⋊C₄	—	292	24	no	—
64	C₄²	—	73	3	no	—
65	C₄²	C₂²×#64408	1	1	yes	characteristic
66	C₂³	—	73	1	no	—
67	C₂×C₄	—	584	12	no	—
68	C₂×C₄	—	73	3	no	—
69	C₂×C₄	—	73	6	no	—
70	C₂×C₄	—	73	6	no	—
71	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
72	C₈	—	73	12	no	—
73	C₈	#14556660	1	12	yes	complemented
74	C₂²	—	73	3	no	—
75	C₂²	—	73	3	no	—
76	C₂²	#17989533	1	1	yes	characteristic
77	C₄	—	73	6	no	—
78	C₄	—	1168	4	no	—
79	C₄	C₂×#14556660	1	6	yes	—
80	C₂	—	73	1	no	—
81	C₂	—	73	3	no	—
82	C₂	#22625867	1	3	yes	—

	subgroup	G/H
center	C₂²	#17989533
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#384
prime radical	C₂²×D₇₃	#127
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8818
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
6. C₈×#10596831	C₄	semidirect
7. C₈²×C₇₃	C₄	semidirect
30. C₈×C₇₃	#107	semidirect
31. #10596831	#107	semidirect
44. C₇₃	#8818	semidirect
49. C₈²	#64408	semidirect
73. C₈	#14556660	semidirect

Maximal subgroups

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

subgroup	index
#22625867	2
C₈²×D₇₃	2
#8818	73