Subgroups — #34001028

Order 18688.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14556641	C₂	1	2	yes	maximal
2	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556641	—	2	2	no	—
4	C₄×#11924003	C₂²	1	1	yes	characteristic
5	C₈×#10596831	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
7	#14556641	—	4	2	no	—
8	C₂×#11924003	D₄	1	1	yes	characteristic
9	C₂×#11924003	Q₈	1	1	yes	characteristic
10	C₄×#10596831	C₈	1	1	yes	characteristic
11	C₄×#10596831	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	—	2	1	no	—
13	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×#10596831	—	2	1	no	—
15	C₂×#10596831	C₁₆	1	1	yes	characteristic
16	C₂×#10596831	C₁₆	1	1	yes	characteristic
17	#11924003	—	2	1	no	—
18	#11924003	SD₁₆	1	2	yes	—
19	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
21	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₇₃	—	2	2	no	—
23	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
25	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
26	C₈×C₇₃	—	2	1	no	—
27	C₈×C₇₃	#106	1	1	yes	characteristic
28	C₈×C₇₃	#108	1	1	yes	characteristic
29	#10596831	—	2	2	no	—
30	#10596831	—	2	1	no	—
31	#10596831	C₃₂	1	2	yes	complemented
32	C₂²×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
33	C₄×C₇₃	—	2	1	no	—
34	C₄×C₇₃	#334	1	1	yes	characteristic
35	C₄×C₇₃	#364	1	1	yes	characteristic
36	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₇₃	M₆₄	1	1	yes	characteristic
38	#7455	—	73	1	no	maximal, Sylow
39	C₂×C₇₃	#1367	1	1	yes	characteristic
40	C₂×C₇₃	C₄⋊C₃₂	1	1	yes	characteristic
41	C₂×C₇₃	#1422	1	1	yes	characteristic
42	C₄⋊C₃₂	—	73	2	no	—
43	C₈×C₁₆	—	73	1	no	—
44	C₇₃	#7455	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃₂	—	146	2	no	—
46	C₄×C₁₆	—	73	1	no	—
47	C₄×C₁₆	#64408	1	1	yes	characteristic
48	C₈²	—	73	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	73	1	no	—
51	C₂×C₁₆	—	73	1	no	—
52	C₄×C₈	—	73	1	no	—
53	C₄×C₈	—	73	1	no	—
54	C₄×C₈	C₂×#64408	1	1	yes	characteristic
55	C₃₂	—	292	2	no	—
56	C₂×C₈	—	73	1	no	—
57	C₂×C₈	—	73	1	no	—
58	C₂×C₈	—	146	1	no	—
59	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
60	C₂×C₈	#11924037	1	1	yes	characteristic
61	C₄²	#11924036	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	C₁₆	—	73	2	no	—
64	C₁₆	—	146	1	no	—
65	C₂×C₄	#14556658	1	1	yes	characteristic
66	C₂×C₄	#14556664	1	1	yes	characteristic
67	C₂×C₄	#14556666	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	73	2	no	—
70	C₈	—	146	2	no	—
71	C₈	—	146	1	no	—
72	C₈	#14556659	1	1	yes	characteristic
73	C₈	#14556662	1	1	yes	characteristic
74	C₂²	#17989508	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	#17989449	1	1	yes	characteristic
77	C₄	#17989510	1	1	yes	characteristic
78	C₄	#17989518	1	1	yes	characteristic
79	C₄	#17989520	1	1	yes	characteristic
80	C₂	#22624954	1	1	yes	characteristic
81	C₂	#22625089	1	1	yes	characteristic
82	C₂	#22625091	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₃₂
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₄⋊C₁₆
cosocle	C₄×#11924003	C₂²
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#7455
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
31. #10596831	C₃₂	semidirect
44. C₇₃	#7455	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14556641	2
C₁₆×#10596831	2
#7455	73