Subgroups — #34000072

Order 18688.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624953	C₂	1	1	yes	maximal, characteristic
2	#22624953	C₂	1	1	yes	maximal, characteristic
3	#22607847	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	1	no	—
5	C₄⋊#11924031	—	2	1	no	—
6	C₄×#11924003	C₄	1	1	yes	characteristic
7	C₈×#10596831	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
9	C₂×#11924031	—	4	1	no	—
10	C₂×#11924031	—	4	1	no	—
11	C₂×#11924003	C₈	1	1	yes	characteristic
12	C₂×#11924003	C₈	1	1	yes	characteristic
13	C₄×#10596831	D₄	1	1	yes	characteristic
14	C₄×#10596831	Q₈	1	1	yes	characteristic
15	C₂×C₁₆×C₇₃	—	2	1	no	—
16	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
17	#11924031	—	8	1	no	—
18	#11924031	—	8	1	no	—
19	C₂×#10596831	—	2	1	no	—
20	C₂×#10596831	SD₁₆	1	1	yes	characteristic
21	C₂×#10596831	SD₁₆	1	1	yes	characteristic
22	#11924003	—	2	1	no	—
23	#11924003	C₁₆	1	2	yes	complemented
24	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
26	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
27	C₁₆×C₇₃	—	2	2	no	—
28	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
30	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
31	C₈×C₇₃	—	2	1	no	—
32	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
33	C₈×C₇₃	M₃₂	1	1	yes	characteristic
34	#10596831	—	2	1	no	—
35	#10596831	—	2	1	no	—
36	#10596831	—	4	1	no	—
37	C₂²×C₇₃	#334	1	1	yes	characteristic
38	C₄×C₇₃	—	2	1	no	—
39	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
40	C₄×C₇₃	#364	1	1	yes	characteristic
41	C₄×C₇₃	#365	1	1	yes	characteristic
42	C₄×C₇₃	#365	1	1	yes	characteristic
43	#7099	—	73	1	no	maximal, Sylow
44	C₂×C₇₃	#1367	1	1	yes	characteristic
45	C₂×C₇₃	#1368	1	1	yes	characteristic
46	C₂×C₇₃	#1369	1	1	yes	characteristic
47	#1367	—	73	1	no	—
48	#1367	—	73	1	no	—
49	#1311	—	73	1	no	—
50	C₇₃	#7099	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊C₁₆	—	146	1	no	—
52	C₄⋊C₁₆	—	146	1	no	—
53	C₄×C₁₆	—	73	1	no	—
54	C₄×C₁₆	#64408	1	1	yes	characteristic
55	C₈²	—	73	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	—	292	1	no	—
58	C₂×C₁₆	—	292	1	no	—
59	C₂×C₁₆	—	73	1	no	—
60	C₂×C₁₆	—	73	1	no	—
61	C₄×C₈	—	73	1	no	—
62	C₄×C₈	—	73	1	no	—
63	C₄×C₈	C₂×#64408	1	1	yes	characteristic
64	C₂×C₈	—	73	1	no	—
65	C₂×C₈	—	73	1	no	—
66	C₂×C₈	—	146	1	no	—
67	C₂×C₈	#11924036	1	1	yes	characteristic
68	C₂×C₈	#11924037	1	1	yes	characteristic
69	C₄²	C₄⋊#64408	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	C₁₆	—	584	1	no	—
72	C₁₆	—	584	1	no	—
73	C₁₆	—	73	2	no	—
74	C₁₆	—	146	1	no	—
75	C₂×C₄	#14556658	1	1	yes	characteristic
76	C₂×C₄	#14556659	1	1	yes	characteristic
77	C₂×C₄	#14556662	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	292	1	no	—
80	C₈	—	146	1	no	—
81	C₈	—	146	1	no	—
82	C₈	#14556664	1	1	yes	characteristic
83	C₈	#14556666	1	1	yes	characteristic
84	C₂²	#17989449	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	#17989508	1	1	yes	characteristic
87	C₄	#17989510	1	1	yes	characteristic
88	C₄	#17989511	1	1	yes	characteristic
89	C₄	#17989512	1	1	yes	characteristic
90	C₂	#22624954	1	1	yes	characteristic
91	C₂	#22624957	1	1	yes	characteristic
92	C₂	#22624959	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556659
commutator	C₈×C₇₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	#334
cosocle	C₈×#10596831	C₂²
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#7099
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
23. #11924003	C₁₆	semidirect
50. C₇₃	#7099	semidirect

Maximal subgroups

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

subgroup	index
#22624953	2
#22607847	2
#7099	73