Subgroups — #34032994

Order 18688.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64408⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#10596842	C₂	1	1	yes	maximal, characteristic
3	C₃₂×#64408	C₂	1	1	yes	maximal, characteristic
4	#17989694	C₄	1	2	yes	—
5	#64408⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×#10596842	C₄	1	1	yes	characteristic
7	#17989518	C₄	1	1	yes	characteristic
8	C₁₆×#64408	C₂²	1	1	yes	characteristic
9	C₃₂×D₇₃	C₄	1	1	yes	characteristic
10	#14556686	C₈	1	2	yes	—
11	C₄×#10596842	C₈	1	2	yes	—
12	#14556681	C₈	1	4	yes	complemented
13	#14556664	C₂×C₄	1	1	yes	characteristic
14	C₈×#64408	C₂×C₄	1	1	yes	characteristic
15	#14556641	C₈	1	2	yes	complemented
16	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
17	#14556497	C₈	1	1	yes	characteristic, complemented
18	C₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
19	C₂×#10596842	C₁₆	1	4	yes	—
20	#11924040	C₁₆	1	4	yes	—
21	#11924036	C₂×C₈	1	1	yes	characteristic
22	C₄×#64408	C₂×C₈	1	1	yes	characteristic
23	#11924031	C₂×C₈	1	2	yes	—
24	C₈×D₇₃	C₄²	1	1	yes	characteristic
25	#11924003	C₂×C₈	1	1	yes	characteristic
26	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
27	C₂×#64408	C₂×C₁₆	1	2	yes	—
28	#10596842	—	2	4	no	—
29	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
30	#10596833	C₂×C₁₆	1	2	yes	—
31	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
32	#10596831	C₄×C₈	1	1	yes	characteristic
33	C₂×D₇₃	C₄×C₁₆	1	1	yes	characteristic
34	#64408	M₆₄	1	4	yes	—
35	C₄×C₇₃	C₈²	1	1	yes	characteristic
36	Dic₇₃	C₄×C₁₆	1	1	yes	characteristic
37	#7329	—	73	1	no	maximal, Sylow
38	C₂×C₇₃	C₈×C₁₆	1	1	yes	characteristic
39	D₇₃	#1397	1	2	yes	—
40	C₄×C₃₂	—	73	1	no	—
41	C₄×C₃₂	—	73	1	no	—
42	C₈×C₁₆	—	73	1	no	—
43	C₇₃	#7329	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃₂	—	73	2	no	—
45	C₂×C₃₂	—	73	1	no	—
46	C₂×C₃₂	—	73	1	no	—
47	C₄×C₁₆	—	73	1	no	—
48	C₄×C₁₆	—	73	1	no	—
49	C₈²	—	73	1	no	—
50	C₂×C₁₆	—	73	2	no	—
51	C₂×C₁₆	—	73	1	no	—
52	C₂×C₁₆	—	73	1	no	—
53	C₄×C₈	—	73	2	no	—
54	C₄×C₈	—	73	1	no	—
55	C₃₂	—	73	4	no	—
56	C₃₂	—	73	2	no	—
57	C₃₂	—	73	1	no	—
58	C₃₂	#10596842	1	1	yes	characteristic, complemented
59	C₂×C₈	—	73	1	no	—
60	C₂×C₈	—	73	1	no	—
61	C₂×C₈	—	73	4	no	—
62	C₄²	—	73	1	no	—
63	C₁₆	—	73	2	no	—
64	C₁₆	—	73	4	no	—
65	C₁₆	—	73	1	no	—
66	C₁₆	C₂×#10596842	1	1	yes	characteristic
67	C₂×C₄	—	73	2	no	—
68	C₂×C₄	—	73	1	no	—
69	C₈	—	73	2	no	—
70	C₈	—	73	1	no	—
71	C₈	—	146	4	no	—
72	C₈	C₄×#10596842	1	1	yes	characteristic
73	C₂²	—	73	1	no	—
74	C₄	—	73	4	no	—
75	C₄	—	73	1	no	—
76	C₄	C₈×#10596842	1	1	yes	characteristic
77	C₂	—	73	2	no	—
78	C₂	C₁₆×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂×#10596842
commutator, socle	C₂×C₇₃	C₈×C₁₆
Fitting	C₃₂×C₇₃	C₈
prime radical	C₂×D₇₃	C₄×C₁₆
cosocle	C₁₆×#64408	C₂²
hypercenter	C₃₂	#10596842
hypocenter	C₇₃	#7329
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
12. #14556681	C₈	semidirect
15. #14556641	C₈	semidirect
17. #14556497	C₈	semidirect
18. C₃₂×C₇₃	C₈	semidirect
43. C₇₃	#7329	semidirect
58. C₃₂	#10596842	semidirect

Maximal subgroups

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

subgroup	index
#64408⋊C₃₂	2
C₁₆×#10596842	2
C₃₂×#64408	2
#7329	73