Subgroups — #34013325

Order 18688.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626272	C₂	1	3	yes	maximal
2	#22626268	C₂	1	3	yes	maximal
3	#22609687	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	3	yes	—
5	C₂²×#11924031	C₂²	1	1	yes	characteristic
6	#17989509	—	2	6	no	—
7	#17989508	—	2	6	no	—
8	#17988545	C₂²	1	3	yes	—
9	#17988344	C₄	1	3	yes	—
10	#338×C₇₃	C₄	1	1	yes	characteristic
11	#14556665	—	2	6	no	—
12	C₂×#11924031	—	2	3	no	—
13	C₂×#11924031	—	2	3	no	—
14	C₂²×#10596831	C₂³	1	1	yes	characteristic
15	#14556550	C₂×C₄	1	3	yes	—
16	#14556501	Q₈	1	6	yes	—
17	C₈×Dic₇₃	D₄	1	6	yes	—
18	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
19	#11924031	—	4	1	no	—
20	#11924031	—	4	3	no	—
21	C₂×#10596831	C₂×D₄	1	3	yes	—
22	C₂×#10596831	C₂×Q₈	1	3	yes	—
23	C₄⋊Dic₇₃	—	2	3	no	—
24	C₂²⋊Dic₇₃	—	2	3	no	—
25	C₄×Dic₇₃	C₄⋊C₄	1	6	yes	—
26	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
27	M₁₆×C₇₃	—	2	6	no	—
28	C₂×C₈×C₇₃	C₄⋊C₄	1	6	yes	—
29	C₂³×C₇₃	#131	1	1	yes	characteristic
30	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
31	C₂×C₄×C₇₃	#131	1	3	yes	—
32	C₂×Dic₇₃	—	2	6	no	—
33	C₈×C₇₃	—	2	6	no	—
34	#10596831	#127	1	1	yes	characteristic
35	#10596831	C₄⋊Q₈	1	3	yes	—
36	C₂²×C₇₃	—	4	1	no	—
37	C₂²×C₇₃	#424	1	3	yes	—
38	C₄×C₇₃	#384	1	1	yes	characteristic
39	C₄×C₇₃	#424	1	3	yes	—
40	Dic₇₃	—	4	6	no	—
41	#12902	—	73	1	no	maximal, Sylow
42	C₂×C₇₃	—	2	3	no	—
43	C₂×C₇₃	#1843	1	1	yes	characteristic
44	#2153	—	73	3	no	—
45	#2151	—	73	3	no	—
46	#1747	—	73	1	no	—
47	C₇₃	#12902	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	73	3	no	—
49	C₂²×C₁₆	—	73	1	no	—
50	#405	—	73	3	no	—
51	#364	—	146	6	no	—
52	C₄⋊C₁₆	—	146	6	no	—
53	#338	#64408	1	1	yes	characteristic
54	H(3,ℤ/4)	—	73	3	no	—
55	C₂×M₁₆	C₂×#64408	1	3	yes	—
56	C₂²×C₈	—	73	1	no	—
57	#117	—	73	3	no	—
58	M₃₂	—	146	6	no	—
59	C₂×C₁₆	—	146	3	no	—
60	C₂×C₁₆	—	146	3	no	—
61	#97	—	73	6	no	—
62	C₄×C₈	—	73	6	no	—
63	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	—	73	3	no	—
66	C₂×C₈	—	73	3	no	—
67	C₂×C₈	C₄⋊#64408	1	6	yes	—
68	C₄⋊C₄	—	146	3	no	—
69	C₂²⋊C₄	—	146	3	no	—
70	C₄²	—	73	6	no	—
71	C₁₆	—	292	1	no	—
72	C₁₆	—	292	3	no	—
73	C₂³	#14556679	1	1	yes	characteristic
74	C₂×C₄	—	146	6	no	—
75	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
76	C₂×C₄	#14556678	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	—	73	1	no	—
79	C₈	—	73	3	no	—
80	C₂²	—	4	1	no	—
81	C₂²	#17989594	1	3	yes	—
82	C₄	—	292	6	no	—
83	C₄	#17989533	1	1	yes	characteristic
84	C₄	#17989593	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#22625865	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17989533
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#338×C₇₃	C₄
socle	C₂×C₇₃	#1843
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	#338	#64408
hypocenter	C₇₃	#12902
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
47. C₇₃	#12902	semidirect

Maximal subgroups

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

subgroup	index
#22626272	2
#22626268	2
#22609687	2
#12902	73