Subgroups — #34000074

Order 18688.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624960	C₂	1	2	yes	maximal
2	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11924031	—	2	2	no	—
4	C₄×#11924003	C₄	1	1	yes	characteristic
5	C₈×#10596831	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924031	—	4	2	no	—
8	C₂×#11924003	C₈	1	1	yes	characteristic
9	C₂×#11924003	C₈	1	1	yes	characteristic
10	C₄×#10596831	D₄	1	1	yes	characteristic
11	C₄×#10596831	Q₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	—	2	1	no	—
13	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
14	#11924031	—	8	2	no	—
15	C₂×#10596831	—	2	1	no	—
16	C₂×#10596831	D₈	1	1	yes	characteristic
17	C₂×#10596831	Q₁₆	1	1	yes	characteristic
18	#11924003	—	2	1	no	—
19	#11924003	C₁₆	1	2	yes	complemented
20	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
21	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
22	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₃	—	2	2	no	—
24	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
26	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
27	C₈×C₇₃	—	2	1	no	—
28	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
29	C₈×C₇₃	M₃₂	1	1	yes	characteristic
30	#10596831	—	2	2	no	—
31	#10596831	—	2	1	no	—
32	#10596831	D₁₆	1	1	yes	characteristic
33	#10596831	Q₃₂	1	1	yes	characteristic
34	C₂²×C₇₃	#335	1	1	yes	characteristic
35	C₄×C₇₃	—	2	1	no	—
36	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₇₃	#364	1	1	yes	characteristic
38	C₄×C₇₃	#366	1	1	yes	characteristic
39	C₄×C₇₃	#368	1	1	yes	characteristic
40	#7100	—	73	1	no	maximal, Sylow
41	C₂×C₇₃	#1370	1	1	yes	characteristic
42	C₂×C₇₃	#1371	1	1	yes	characteristic
43	C₂×C₇₃	#1373	1	1	yes	characteristic
44	#1370	—	73	2	no	—
45	C₈×C₁₆	—	73	1	no	—
46	C₇₃	#7100	1	1	yes	characteristic, complemented, Sylow
47	C₄⋊C₁₆	—	146	2	no	—
48	C₄×C₁₆	—	73	1	no	—
49	C₄×C₁₆	#64408	1	1	yes	characteristic
50	C₈²	—	73	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	292	2	no	—
53	C₂×C₁₆	—	73	1	no	—
54	C₂×C₁₆	—	73	1	no	—
55	C₄×C₈	—	73	1	no	—
56	C₄×C₈	—	73	1	no	—
57	C₄×C₈	C₂×#64408	1	1	yes	characteristic
58	C₂×C₈	—	73	1	no	—
59	C₂×C₈	—	73	1	no	—
60	C₂×C₈	—	146	1	no	—
61	C₂×C₈	#11924036	1	1	yes	characteristic
62	C₂×C₈	#11924037	1	1	yes	characteristic
63	C₄²	C₄⋊#64408	1	1	yes	characteristic
64	C₁₆	—	2	2	no	—
65	C₁₆	—	584	2	no	—
66	C₁₆	—	73	2	no	—
67	C₁₆	—	146	1	no	—
68	C₂×C₄	#14556658	1	1	yes	characteristic
69	C₂×C₄	#14556660	1	1	yes	characteristic
70	C₂×C₄	#14556661	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	73	1	no	—
73	C₈	—	73	1	no	—
74	C₈	—	146	2	no	—
75	C₈	—	146	1	no	—
76	C₈	#14556664	1	1	yes	characteristic
77	C₈	#14556666	1	1	yes	characteristic
78	C₂²	#17989451	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	#17989508	1	1	yes	characteristic
81	C₄	#17989510	1	1	yes	characteristic
82	C₄	#17989513	1	1	yes	characteristic
83	C₄	#17989516	1	1	yes	characteristic
84	C₂	#22624961	1	1	yes	characteristic
85	C₂	#22624963	1	1	yes	characteristic
86	C₂	#22624967	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556660
commutator	C₈×C₇₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	#335
cosocle	C₈×#10596831	C₂²
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#7100
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
19. #11924003	C₁₆	semidirect
46. C₇₃	#7100	semidirect

Maximal subgroups

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

subgroup	index
#22624960	2
C₁₆×#10596831	2
#7100	73