Subgroups — #34001275

Order 18688.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625101	C₂	1	1	yes	maximal, characteristic
2	#22625095	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989515	—	2	1	no	—
5	#17989426	—	2	1	no	—
6	Q₃₂×D₇₃	—	2	1	no	—
7	Dic₁₁₆₈	C₄	1	1	yes	characteristic
8	#17988427	—	2	1	no	—
9	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
10	Q₆₄×C₇₃	C₄	1	1	yes	characteristic
11	#14556662	—	4	1	no	—
12	Q₁₆×D₇₃	—	4	1	no	—
13	Dic₅₈₄	—	2	1	no	—
14	#14556537	—	4	1	no	—
15	C₁₆×D₇₃	D₄	1	1	yes	characteristic
16	#14556497	D₄	1	1	yes	characteristic
17	Q₃₂×C₇₃	—	2	1	no	—
18	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
19	#11924038	—	8	1	no	—
20	Q₈×D₇₃	—	8	1	no	—
21	Dic₂₉₂	—	4	1	no	—
22	#11924018	—	8	1	no	—
23	C₈×D₇₃	D₈	1	1	yes	characteristic
24	#11924003	SD₁₆	1	1	yes	characteristic
25	Q₁₆×C₇₃	—	4	1	no	—
26	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₇₃	—	8	1	no	—
28	C₄×D₇₃	—	16	1	no	—
29	C₄×D₇₃	D₁₆	1	1	yes	characteristic
30	Dic₁₄₆	—	8	1	no	—
31	Dic₁₄₆	—	16	1	no	—
32	#10596833	—	16	1	no	—
33	C₈×C₇₃	#102	1	1	yes	characteristic
34	#10596831	SD₃₂	1	1	yes	characteristic
35	C₂×D₇₃	SD₆₄	1	1	yes	characteristic
36	C₄×C₇₃	—	16	1	no	—
37	C₄×C₇₃	#357	1	1	yes	characteristic
38	Dic₇₃	—	16	1	no	—
39	Dic₇₃	D₃₂	1	1	yes	characteristic
40	#7542	—	73	1	no	maximal, Sylow
41	C₂×C₇₃	#1415	1	1	yes	characteristic
42	D₇₃	—	2	1	no	—
43	C₂×Q₆₄	—	73	1	no	—
44	M₁₂₈	—	73	1	no	—
45	#1425	—	73	1	no	—
46	C₇₃	#7542	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₃₂	—	146	1	no	—
48	Q₆₄	—	73	1	no	—
49	Q₆₄	—	146	1	no	—
50	Q₆₄	#64408	1	1	yes	characteristic
51	C₂×C₃₂	—	73	1	no	—
52	#368	—	146	1	no	—
53	C₆₄	—	146	1	no	—
54	C₂×Q₁₆	—	292	1	no	—
55	Q₃₂	—	2	1	no	—
56	Q₃₂	—	292	1	no	—
57	Q₃₂	—	146	1	no	—
58	C₂×C₁₆	—	73	1	no	—
59	#108	—	292	1	no	—
60	C₃₂	—	73	1	no	—
61	C₃₂	C₂×#64408	1	1	yes	characteristic
62	C₂×Q₈	—	584	1	no	—
63	Q₁₆	—	4	1	no	—
64	Q₁₆	—	292	1	no	—
65	Q₁₆	—	584	1	no	—
66	M₁₆	—	584	1	no	—
67	C₂×C₈	—	73	1	no	—
68	C₁₆	—	73	1	no	—
69	C₁₆	C₂²⋊#64408	1	1	yes	characteristic
70	Q₈	—	8	1	no	—
71	Q₈	—	584	1	no	—
72	Q₈	—	1168	1	no	—
73	C₂×C₄	—	73	1	no	—
74	C₂×C₄	—	1168	1	no	—
75	C₈	—	73	1	no	—
76	C₈	—	1168	1	no	—
77	C₈	#14556653	1	1	yes	characteristic
78	C₂²	—	73	1	no	—
79	C₄	—	16	1	no	—
80	C₄	—	73	1	no	—
81	C₄	—	1168	1	no	—
82	C₄	#17989499	1	1	yes	characteristic
83	C₂	—	146	1	no	—
84	C₂	#22625080	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22625080
commutator	C₃₂×C₇₃	C₂×C₄
Frattini	C₃₂	C₂×#64408
Fitting	Q₆₄×C₇₃	C₄
socle	C₂×C₇₃	#1415
prime radical	C₂×D₇₃	SD₆₄
cosocle	C₃₂×D₇₃	C₂²
hypercenter	Q₆₄	#64408
hypocenter	C₇₃	#7542
LCS	G ▷ C₃₂×C₇₃ ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
46. C₇₃	#7542	semidirect

Maximal subgroups

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

subgroup	index
#22625101	2
#22625095	2
Q₆₄×D₇₃	2
#7542	73