Subgroups — #32973865

Order 18688.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988306	C₂	1	1	yes	maximal, characteristic
2	#22608131	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988413	—	2	1	no	—
5	#17988306	—	2	1	no	—
6	C₂×Q₃₂×C₇₃	—	2	1	no	—
7	Q₆₄×C₇₃	—	2	1	no	—
8	Q₆₄×C₇₃	C₄	1	2	yes	complemented
9	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
10	#14556525	—	4	1	no	—
11	C₂×Q₁₆×C₇₃	—	4	1	no	—
12	Q₃₂×C₇₃	—	2	2	no	—
13	Q₃₂×C₇₃	—	4	1	no	—
14	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
15	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₃	D₄	1	1	yes	characteristic
17	C₄⋊Dic₇₃	—	8	1	no	—
18	C₂×Q₈×C₇₃	—	8	1	no	—
19	Q₁₆×C₇₃	—	4	2	no	—
20	Q₁₆×C₇₃	—	8	1	no	—
21	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
22	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
24	Q₈×C₇₃	—	8	2	no	—
25	Q₈×C₇₃	—	16	1	no	—
26	C₂×C₄×C₇₃	—	16	1	no	—
27	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
28	C₂×Dic₇₃	—	16	1	no	—
29	C₈×C₇₃	#102	1	1	yes	characteristic
30	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
31	C₂²×C₇₃	D₃₂	1	1	yes	characteristic
32	C₄×C₇₃	—	16	2	no	—
33	C₄×C₇₃	#357	1	1	yes	characteristic
34	C₄×C₇₃	SD₆₄	1	1	yes	characteristic
35	Dic₇₃	—	32	1	no	—
36	#7538	—	73	1	no	maximal, Sylow
37	C₂×C₇₃	#1415	1	1	yes	characteristic
38	C₂×C₇₃	SD₁₂₈	1	1	yes	characteristic
39	C₂×C₇₃	Q₁₂₈	1	1	yes	characteristic
40	C₂×Q₆₄	D₇₃	1	1	yes	characteristic
41	C₂×C₆₄	—	73	1	no	—
42	#1423	—	73	1	no	—
43	C₇₃	#7538	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	2	1	no	—
45	Q₆₄	—	2	1	no	—
46	Q₆₄	Dic₇₃	1	2	yes	complemented
47	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
48	#366	—	146	1	no	—
49	C₆₄	—	146	1	no	—
50	C₂×Q₁₆	—	4	1	no	—
51	Q₃₂	—	2	2	no	—
52	Q₃₂	—	4	1	no	—
53	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
54	#107	—	292	1	no	—
55	C₃₂	C₂×Dic₇₃	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₇₃	1	1	yes	characteristic
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	#11924013	1	1	yes	characteristic
61	C₄⋊C₄	—	584	1	no	—
62	C₁₆	C₂²⋊Dic₇₃	1	1	yes	characteristic
63	C₁₆	#11924016	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	1168	1	no	—
68	C₂×C₄	#14556532	1	1	yes	characteristic
69	C₈	#14556512	1	1	yes	characteristic
70	C₈	#14556535	1	1	yes	characteristic
71	C₂²	#17988422	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	2336	1	no	—
74	C₄	#17988388	1	1	yes	characteristic
75	C₄	#17988425	1	1	yes	characteristic
76	C₂	#22608109	1	1	yes	characteristic
77	C₂	#22608145	1	1	yes	characteristic
78	C₂	#22608148	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988422
commutator	C₃₂×C₇₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₃
Fitting	C₂×Q₆₄×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	D₃₂
cosocle	C₂×C₃₂×C₇₃	C₂²
hypercenter	C₂×Q₆₄	D₇₃
hypocenter	C₇₃	#7538
LCS	G ▷ C₃₂×C₇₃ ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
8. Q₆₄×C₇₃	C₄	semidirect
43. C₇₃	#7538	semidirect
46. Q₆₄	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988306	2
#22608131	2
C₂×Q₆₄×C₇₃	2
#7538	73