Subgroups — #32973869

Order 18688.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612240	C₂	1	1	yes	maximal, characteristic
2	#22608135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989019	—	2	1	no	—
5	Dic₁₁₆₈	C₄	1	1	yes	characteristic
6	#17988426	—	2	1	no	—
7	D₂₃₃₆	C₄	1	1	yes	characteristic
8	#17988417	—	2	1	no	—
9	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
10	C₆₄×C₇₃	—	2	1	no	—
11	#14556613	—	4	1	no	—
12	Dic₅₈₄	—	2	1	no	—
13	#14556536	—	4	1	no	—
14	D₁₁₆₈	—	2	1	no	—
15	#14556527	—	4	1	no	—
16	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
17	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₃	D₄	1	1	yes	characteristic
19	#11924028	—	8	1	no	—
20	Dic₂₉₂	—	4	1	no	—
21	#11924017	—	8	1	no	—
22	D₅₈₄	—	4	1	no	—
23	#11924012	—	8	1	no	—
24	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
25	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₃	—	16	1	no	—
29	D₂₉₂	—	8	1	no	—
30	C₄×D₇₃	—	16	1	no	—
31	Dic₁₄₆	—	8	1	no	—
32	C₈×C₇₃	#102	1	1	yes	characteristic
33	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
34	#10596831	—	16	1	no	—
35	C₂²×C₇₃	SD₆₄	1	1	yes	characteristic
36	C₂×D₇₃	—	16	1	no	—
37	C₄×C₇₃	#357	1	1	yes	characteristic
38	C₄×C₇₃	D₃₂	1	1	yes	characteristic
39	Dic₇₃	—	16	1	no	—
40	#7539	—	73	1	no	maximal, Sylow
41	C₂×C₇₃	—	2	1	no	—
42	C₂×C₇₃	#1415	1	1	yes	characteristic
43	D₇₃	—	32	1	no	—
44	#2262	—	73	1	no	—
45	C₂×C₆₄	D₇₃	1	1	yes	characteristic, complemented
46	#1425	—	73	1	no	—
47	C₇₃	#7539	1	1	yes	characteristic, complemented, Sylow
48	#508	—	146	1	no	—
49	Q₆₄	—	73	1	no	—
50	SD₆₄	—	146	1	no	—
51	D₃₂	—	73	1	no	—
52	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
53	#368	—	146	1	no	—
54	C₆₄	—	2	1	no	—
55	#135	—	292	1	no	—
56	Q₃₂	—	146	1	no	—
57	SD₃₂	—	292	1	no	—
58	D₁₆	—	146	1	no	—
59	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
60	#108	—	292	1	no	—
61	C₃₂	C₄×D₇₃	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₇₃	1	1	yes	characteristic
63	#41	—	584	1	no	—
64	Q₁₆	—	292	1	no	—
65	SD₁₆	—	584	1	no	—
66	D₈	—	292	1	no	—
67	M₁₆	—	584	1	no	—
68	C₂×C₈	D₅₈₄	1	1	yes	characteristic
69	C₁₆	#11924005	1	1	yes	characteristic
70	C₁₆	#11924017	1	1	yes	characteristic
71	Q₈	—	584	1	no	—
72	D₄	—	584	1	no	—
73	D₄	—	1168	1	no	—
74	C₂×C₄	—	1168	1	no	—
75	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
76	C₈	—	1168	1	no	—
77	C₈	#14556513	1	1	yes	characteristic
78	C₈	#14556536	1	1	yes	characteristic
79	C₂²	—	1168	1	no	—
80	C₂²	#17988426	1	1	yes	characteristic
81	C₄	—	1168	1	no	—
82	C₄	#17988389	1	1	yes	characteristic
83	C₄	D₂₃₃₆	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2336	1	no	—
86	C₂	#22608110	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₃₃₆
commutator	C₃₂×C₇₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₃
Fitting	C₂×C₆₄×C₇₃	C₂
socle	C₂×C₇₃	#1415
prime radical	#22612240	C₂
cosocle	C₂×C₃₂×C₇₃	C₂²
hypercenter	C₂×C₆₄	D₇₃
hypocenter	C₇₃	#7539
LCS	G ▷ C₃₂×C₇₃ ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #22608135	C₂	semidirect
3. C₂×C₆₄×C₇₃	C₂	semidirect
45. C₂×C₆₄	D₇₃	semidirect
47. C₇₃	#7539	semidirect

Maximal subgroups

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

subgroup	index
#22612240	2
#22608135	2
C₂×C₆₄×C₇₃	2
#7539	73