Subgroups — #32973795

Order 18688.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608142	C₂	1	2	yes	maximal, complemented
2	C₂×M₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988306	—	2	2	no	—
4	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
5	M₆₄×C₇₃	—	2	2	no	—
6	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₇₃	—	2	1	no	—
10	C₂×C₁₆×C₇₃	—	2	1	no	—
11	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₇₃	—	2	1	no	—
14	C₃₂×C₇₃	D₄	1	2	yes	—
15	C₂²×C₄×C₇₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₇₃	—	2	1	no	—
17	C₂×C₈×C₇₃	—	2	1	no	—
18	C₂×C₈×C₇₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₇₃	—	2	1	no	—
21	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
23	C₂³×C₇₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₇₃	—	2	1	no	—
25	C₂×C₄×C₇₃	—	2	1	no	—
26	C₂×C₄×C₇₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₃	—	2	1	no	—
29	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₇₃	M₃₂	1	1	yes	characteristic
31	C₂²×C₇₃	—	2	1	no	—
32	C₂²×C₇₃	—	4	1	no	—
33	C₂²×C₇₃	C₂×C₃₂	1	1	yes	characteristic
34	C₄×C₇₃	—	2	1	no	—
35	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
36	C₄×C₇₃	M₆₄	1	1	yes	characteristic
37	#7513	—	73	1	no	maximal, Sylow
38	C₂×C₇₃	—	2	1	no	—
39	C₂×C₇₃	—	4	1	no	—
40	C₂×C₇₃	C₂²⋊C₃₂	1	1	yes	characteristic
41	C₂×M₆₄	D₇₃	1	1	yes	characteristic
42	M₁₂₈	—	73	2	no	—
43	C₇₃	#7513	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₁₆	Dic₇₃	1	1	yes	characteristic
45	M₆₄	—	2	2	no	—
46	C₂×C₃₂	Dic₇₃	1	1	yes	characteristic
47	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
48	C₆₄	—	146	2	no	—
49	C₂²×C₈	#10596831	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	#10596831	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
54	C₃₂	—	2	1	no	—
55	C₃₂	C₂²⋊D₇₃	1	2	yes	—
56	C₂²×C₄	#11924003	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	#11924003	1	1	yes	characteristic
60	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	C₂²⋊Dic₇₃	1	1	yes	characteristic
63	C₁₆	#11924012	1	1	yes	characteristic
64	C₂³	#14556497	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#14556497	1	1	yes	characteristic
68	C₂×C₄	C₂×#11924003	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊#10596831	1	1	yes	characteristic
71	C₈	#14556531	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×#14556497	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₂²⋊#11924003	1	1	yes	characteristic
77	C₄	#17988421	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₂²⋊#14556497	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₇₃
commutator	C₂²×C₇₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₃
Fitting	C₂×M₆₄×C₇₃	C₂
socle	C₂×C₇₃	C₂²⋊C₃₂
prime radical	C₂³×C₇₃	C₃₂
cosocle	C₂×C₃₂×C₇₃	C₂²
hypercenter	C₂×M₆₄	D₇₃
hypocenter	C₇₃	#7513
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
1. #22608142	C₂	semidirect
43. C₇₃	#7513	semidirect

Maximal subgroups

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

subgroup	index
#22608142	2
C₂×M₆₄×C₇₃	2
#7513	73