Subgroups — #34000748

Order 18688.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625036	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556641	C₄	1	2	yes	—
5	#17989473	C₄	1	1	yes	characteristic
6	C₈×#10596833	C₄	1	1	yes	characteristic
7	C₂×#14556497	C₄	1	1	yes	characteristic
8	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924031	C₈	1	2	yes	—
11	C₄×#10596833	C₈	1	2	yes	—
12	#14556641	C₈	1	4	yes	complemented
13	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
14	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
15	#14556497	C₈	1	2	yes	complemented
16	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₇₃	C₈	1	2	yes	complemented
18	C₂×#10596833	C₁₆	1	4	yes	—
19	#11924031	C₁₆	1	4	yes	—
20	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
21	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
22	#11924003	C₂×C₈	1	2	yes	—
23	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
24	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
27	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
28	#10596833	—	2	4	no	—
29	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
30	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
31	#10596831	C₂×C₁₆	1	2	yes	—
32	C₂²×C₇₃	C₄×C₁₆	1	1	yes	characteristic
33	C₄×C₇₃	C₈²	1	1	yes	characteristic
34	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
35	Dic₇₃	M₆₄	1	4	yes	—
36	#7329	—	73	1	no	maximal, Sylow
37	C₂×C₇₃	C₈×C₁₆	1	1	yes	characteristic
38	C₂×C₇₃	#1397	1	2	yes	—
39	C₄×C₃₂	—	73	1	no	—
40	C₄×C₃₂	—	73	1	no	—
41	C₈×C₁₆	—	73	1	no	—
42	C₇₃	#7329	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃₂	—	73	2	no	—
44	C₂×C₃₂	—	73	1	no	—
45	C₂×C₃₂	#64408	1	1	yes	characteristic
46	C₄×C₁₆	—	73	1	no	—
47	C₄×C₁₆	—	73	1	no	—
48	C₈²	—	73	1	no	—
49	C₂×C₁₆	—	73	2	no	—
50	C₂×C₁₆	—	73	1	no	—
51	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
52	C₄×C₈	—	73	2	no	—
53	C₄×C₈	—	73	1	no	—
54	C₃₂	—	73	4	no	—
55	C₃₂	—	73	2	no	—
56	C₃₂	#10596833	1	2	yes	complemented
57	C₂×C₈	—	73	1	no	—
58	C₂×C₈	—	73	4	no	—
59	C₂×C₈	C₄×#64408	1	1	yes	characteristic
60	C₄²	—	73	1	no	—
61	C₁₆	—	73	4	no	—
62	C₁₆	—	73	2	no	—
63	C₁₆	C₂×#10596833	1	1	yes	characteristic
64	C₁₆	#11924036	1	1	yes	characteristic
65	C₂×C₄	—	73	2	no	—
66	C₂×C₄	C₈×#64408	1	1	yes	characteristic
67	C₈	—	73	2	no	—
68	C₈	—	146	4	no	—
69	C₈	C₈×#64408	1	1	yes	characteristic
70	C₈	C₄×#10596833	1	1	yes	characteristic
71	C₂²	C₁₆×#64408	1	1	yes	characteristic
72	C₄	—	73	4	no	—
73	C₄	C₈×#10596833	1	1	yes	characteristic
74	C₄	C₁₆×#64408	1	1	yes	characteristic
75	C₂	C₁₆×#10596833	1	1	yes	characteristic
76	C₂	#22625037	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×#64408
commutator	C₂×C₇₃	C₈×C₁₆
Fitting	C₂×C₃₂×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₄×C₁₆
cosocle	C₁₆×Dic₇₃	C₂²
hypercenter	C₂×C₃₂	#64408
hypocenter	C₇₃	#7329
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
12. #14556641	C₈	semidirect
15. #14556497	C₈	semidirect
17. C₃₂×C₇₃	C₈	semidirect
42. C₇₃	#7329	semidirect
56. C₃₂	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625036	2
C₁₆×#10596833	2
C₃₂×Dic₇₃	2
#7329	73