Subgroups — #34000070

Order 18688.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624953	C₂	1	2	yes	maximal
2	#22607847	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	—	2	1	no	—
9	C₄×#10596831	D₄	1	1	yes	characteristic
10	C₄×#10596831	Q₈	1	1	yes	characteristic
11	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
13	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
14	#11924031	—	8	2	no	—
15	C₂×#10596831	—	2	1	no	—
16	C₂×#10596831	SD₁₆	1	2	yes	—
17	#11924003	—	2	2	no	—
18	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
20	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
21	C₁₆×C₇₃	—	2	1	no	—
22	C₁₆×C₇₃	C₁₆	1	2	yes	complemented
23	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
25	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
26	C₈×C₇₃	—	2	1	no	—
27	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₃	M₃₂	1	1	yes	characteristic
29	#10596831	—	2	2	no	—
30	#10596831	—	4	1	no	—
31	C₂²×C₇₃	#334	1	1	yes	characteristic
32	C₄×C₇₃	—	2	1	no	—
33	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₇₃	#364	1	1	yes	characteristic
35	C₄×C₇₃	#365	1	2	yes	—
36	#7099	—	73	1	no	maximal, Sylow
37	C₂×C₇₃	#1367	1	1	yes	characteristic
38	C₂×C₇₃	#1368	1	1	yes	characteristic
39	C₂×C₇₃	#1369	1	1	yes	characteristic
40	#1367	—	73	2	no	—
41	#1311	—	73	1	no	—
42	C₇₃	#7099	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊C₁₆	—	146	2	no	—
44	C₄×C₁₆	—	73	1	no	—
45	C₄×C₁₆	#64408	1	1	yes	characteristic
46	C₈²	—	73	1	no	—
47	C₂×C₁₆	—	292	2	no	—
48	C₂×C₁₆	—	146	1	no	—
49	C₂×C₁₆	#10596833	1	1	yes	characteristic
50	C₂×C₁₆	#10596833	1	1	yes	characteristic
51	C₄×C₈	—	73	1	no	—
52	C₄×C₈	—	73	1	no	—
53	C₄×C₈	C₂×#64408	1	1	yes	characteristic
54	C₂×C₈	—	73	2	no	—
55	C₂×C₈	—	146	1	no	—
56	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
57	C₂×C₈	#11924038	1	1	yes	characteristic
58	C₄²	C₄⋊#64408	1	1	yes	characteristic
59	C₁₆	—	2	1	no	—
60	C₁₆	—	584	2	no	—
61	C₁₆	—	146	2	no	—
62	C₁₆	#11924031	1	2	yes	complemented
63	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
64	C₂×C₄	#14556659	1	1	yes	characteristic
65	C₂×C₄	#14556661	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	292	1	no	—
68	C₈	—	146	2	no	—
69	C₈	C₂×#11924031	1	1	yes	characteristic
70	C₈	#14556665	1	1	yes	characteristic
71	C₂²	#17989448	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₄⋊#11924031	1	1	yes	characteristic
74	C₄	#17989509	1	1	yes	characteristic
75	C₄	#17989511	1	2	yes	—
76	C₂	#22624953	1	1	yes	characteristic
77	C₂	#22624955	1	1	yes	characteristic
78	C₂	#22624958	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
22. C₁₆×C₇₃	C₁₆	semidirect
42. C₇₃	#7099	semidirect
62. C₁₆	#11924031	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