Subgroups — #33002384

Order 18688.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608009	C₂	1	6	yes	maximal
2	C₄²×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924003	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₇₃	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	—
6	C₄³×C₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924003	—	2	12	no	—
8	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₇₃	C₈	1	3	yes	—
11	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₇₃	—	2	6	no	—
14	C₄×C₈×C₇₃	—	2	6	no	—
15	#11924003	—	4	4	no	—
16	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₇₃	—	2	3	no	—
20	C₂×C₈×C₇₃	—	2	3	no	—
21	C₂×C₈×C₇₃	M₁₆	1	6	yes	—
22	C₂×C₈×C₇₃	#41	1	6	yes	—
23	C₄²×C₇₃	—	2	6	no	—
24	C₄²×C₇₃	—	2	3	no	—
25	C₄²×C₇₃	—	2	3	no	—
26	C₄²×C₇₃	C₁₆	1	4	yes	complemented
27	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇₃	—	2	3	no	—
29	C₂×C₄×C₇₃	—	2	3	no	—
30	C₂×C₄×C₇₃	—	2	3	no	—
31	C₂×C₄×C₇₃	—	2	3	no	—
32	C₂×C₄×C₇₃	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₇₃	M₃₂	1	6	yes	—
34	C₂×C₄×C₇₃	#117	1	3	yes	—
35	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
36	C₈×C₇₃	—	2	6	no	—
37	C₈×C₇₃	#126	1	4	yes	—
38	C₂²×C₇₃	#431	1	3	yes	—
39	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₇₃	C₂×M₃₂	1	3	yes	—
41	C₄×C₇₃	—	2	6	no	—
42	C₄×C₇₃	—	2	6	no	—
43	C₄×C₇₃	#383	1	1	yes	characteristic
44	C₄×C₇₃	#432	1	3	yes	—
45	#12894	—	73	1	no	maximal, Sylow
46	C₂×C₇₃	#1839	1	1	yes	characteristic
47	C₂×C₇₃	#2162	1	3	yes	—
48	C₂×C₇₃	#2163	1	3	yes	—
49	C₄²×C₈	D₇₃	1	1	yes	characteristic
50	#1374	—	73	6	no	—
51	C₇₃	#12894	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	73	4	no	—
53	C₂×C₄×C₈	Dic₇₃	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
55	C₄³	Dic₇₃	1	1	yes	characteristic
56	C₂²×C₈	C₂×Dic₇₃	1	3	yes	—
57	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
58	C₂×C₄²	#10596831	1	3	yes	—
59	C₂×C₄²	#10596831	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₇₃	1	3	yes	—
61	C₂×C₁₆	—	146	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×#10596831	1	3	yes	—
65	C₂²×C₄	C₂×#10596831	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11924012	1	6	yes	—
70	C₂×C₈	#11924028	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	#11924003	1	4	yes	complemented
75	C₁₆	—	292	4	no	—
76	C₂³	C₂²×#10596831	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#11924003	1	6	yes	—
82	C₂×C₄	#14556531	1	6	yes	—
83	C₂×C₄	#14556550	1	3	yes	—
84	C₂×C₄	C₂×#11924012	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#14556585	1	4	yes	—
87	C₂²	#17988640	1	3	yes	—
88	C₂²	C₂²×#11924003	1	1	yes	characteristic
89	C₂²	C₂×#14556531	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#17988455	1	1	yes	characteristic
93	C₄	#17988644	1	3	yes	—
94	C₂	#22610074	1	1	yes	characteristic
95	C₂	#22611730	1	3	yes	—
96	C₂	#22611734	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄²×C₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄²×C₈	D₇₃
hypocenter	C₇₃	#12894
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. C₄²×C₇₃	C₁₆	semidirect
51. C₇₃	#12894	semidirect
74. C₄²	#11924003	semidirect

Maximal subgroups

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

subgroup	index
#22608009	2
C₄²×C₈×C₇₃	2
#12894	73