Subgroups — #33000698

Order 18688.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612183	C₂	1	4	yes	maximal
2	#22609549	C₂	1	1	yes	maximal, characteristic
3	C₈×#11924003	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556537	—	2	8	no	—
6	#17988992	—	2	2	no	—
7	#17988967	C₂²	1	2	yes	—
8	C₄×#11924003	C₂²	1	2	yes	—
9	#500×C₇₃	—	2	2	no	—
10	#494×C₇₃	C₂²	1	2	yes	—
11	C₈²×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₉₂	—	2	4	no	—
13	#14556591	—	2	2	no	—
14	#14556537	—	4	16	no	—
15	C₂×#11924003	D₄	1	4	yes	—
16	#14556525	—	4	4	no	—
17	C₂×Q₁₆×C₇₃	—	2	4	no	—
18	C₄⋊Q₈×C₇₃	—	2	2	no	—
19	#107×C₇₃	—	4	4	no	—
20	C₄×C₈×C₇₃	D₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₄₆	—	4	2	no	—
23	Dic₂₉₂	—	4	8	no	—
24	C₄⋊Dic₇₃	—	4	4	no	—
25	#11924003	D₈	1	8	yes	—
26	C₂×Q₈×C₇₃	—	4	2	no	—
27	Q₁₆×C₇₃	—	4	8	no	—
28	C₂×C₈×C₇₃	D₈	1	4	yes	—
29	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
30	C₄⋊C₄×C₇₃	—	4	4	no	—
31	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₇₃	—	8	4	no	—
33	C₂×C₄×C₇₃	—	8	2	no	—
34	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
36	C₂×Dic₇₃	—	8	2	no	—
37	Dic₁₄₆	—	8	4	no	—
38	C₈×C₇₃	Q₃₂	1	8	yes	complemented
39	C₈×C₇₃	C₂×D₈	1	4	yes	—
40	C₂²×C₇₃	#493	1	1	yes	characteristic
41	C₄×C₇₃	—	16	2	no	—
42	C₄×C₇₃	#493	1	2	yes	—
43	C₄×C₇₃	C₂×Q₃₂	1	4	yes	—
44	Dic₇₃	—	16	2	no	—
45	#12516	—	73	1	no	maximal, Sylow
46	C₂×C₇₃	#1712	1	1	yes	characteristic
47	C₂×C₇₃	#2247	1	2	yes	—
48	#2247	—	73	4	no	—
49	#1713	—	73	1	no	—
50	#1713	D₇₃	1	1	yes	characteristic
51	C₈×C₁₆	—	73	1	no	—
52	C₇₃	#12516	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₃₂	—	146	8	no	—
54	#500	—	2	2	no	—
55	#500	—	146	2	no	—
56	#494	—	73	2	no	—
57	#494	C₂×D₇₃	1	2	yes	—
58	C₄×C₁₆	—	73	2	no	—
59	C₈²	C₂×D₇₃	1	1	yes	characteristic
60	C₂×Q₁₆	—	2	4	no	—
61	C₂×Q₁₆	—	146	4	no	—
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	146	2	no	—
64	Q₃₂	—	292	16	no	—
65	C₂×C₁₆	—	73	4	no	—
66	#107	—	4	4	no	—
67	#107	—	292	4	no	—
68	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
69	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
70	C₂×Q₈	—	4	2	no	—
71	C₂×Q₈	—	292	2	no	—
72	Q₁₆	—	4	8	no	—
73	Q₁₆	—	292	8	no	—
74	C₂×C₈	#11924013	1	4	yes	—
75	C₂×C₈	D₄×D₇₃	1	2	yes	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	292	4	no	—
78	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
79	C₁₆	—	73	8	no	—
80	Q₈	—	8	4	no	—
81	Q₈	—	584	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	584	2	no	—
84	C₂×C₄	#14556587	1	1	yes	characteristic
85	C₂×C₄	C₂×#11924013	1	2	yes	—
86	C₈	#14556537	1	8	yes	complemented
87	C₈	D₈×D₇₃	1	4	yes	—
88	C₂²	#17988963	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	1168	2	no	—
91	C₄	#17988962	1	2	yes	—
92	C₄	C₂×#14556537	1	4	yes	—
93	C₂	#22609545	1	1	yes	characteristic
94	C₂	#22612183	1	2	yes	—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1713	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1713	D₇₃
hypocenter	C₇₃	#12516
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
38. C₈×C₇₃	Q₃₂	semidirect
52. C₇₃	#12516	semidirect
86. C₈	#14556537	semidirect

Maximal subgroups

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

subgroup	index
#22612183	2
#22609549	2
C₈×#11924003	2
C₇₃×#1713	2
#12516	73