Subgroups — #33000696

Order 18688.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612181	C₂	1	4	yes	maximal, complemented
2	#22609546	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁₆₈	—	2	8	no	—
5	#17988964	—	2	4	no	—
6	#17988964	C₂²	1	4	yes	—
7	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
8	C₈²×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×D₅₈₄	—	2	8	no	—
10	C₂×D₅₈₄	—	4	8	no	—
11	#14556588	—	2	4	no	—
12	D₁₁₆₈	—	4	16	no	—
13	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
14	C₄×C₈×C₇₃	D₄	1	2	yes	—
15	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
16	C₂×D₂₉₂	—	4	4	no	—
17	C₂×D₂₉₂	—	4	8	no	—
18	D₅₈₄	—	4	16	no	—
19	D₅₈₄	—	8	16	no	—
20	C₂×C₈×C₇₃	D₈	1	4	yes	—
21	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
22	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
23	C₁₆×C₇₃	D₈	1	8	yes	complemented
24	C₂²×D₇₃	—	8	4	no	—
25	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
26	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
27	D₂₉₂	—	8	8	no	—
28	D₂₉₂	—	8	16	no	—
29	C₈×C₇₃	D₁₆	1	8	yes	complemented
30	C₈×C₇₃	C₂×D₈	1	4	yes	—
31	C₂²×C₇₃	#493	1	1	yes	characteristic
32	C₂×D₇₃	—	16	4	no	—
33	C₂×D₇₃	—	16	8	no	—
34	C₄×C₇₃	#493	1	2	yes	—
35	C₄×C₇₃	C₂×D₁₆	1	4	yes	—
36	#12515	—	73	1	no	maximal, Sylow
37	C₂×C₇₃	#1712	1	1	yes	characteristic
38	C₂×C₇₃	#2246	1	2	yes	—
39	D₇₃	—	32	4	no	—
40	#2246	—	73	4	no	—
41	#1712	—	73	2	no	—
42	C₈×C₁₆	D₇₃	1	1	yes	characteristic, complemented
43	C₇₃	#12515	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₁₆	—	146	8	no	—
45	#493	—	73	4	no	—
46	#493	—	146	4	no	—
47	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
48	C₈²	C₂×D₇₃	1	1	yes	characteristic
49	C₂×D₈	—	292	8	no	—
50	C₂×D₈	—	146	8	no	—
51	#127	—	146	4	no	—
52	D₁₆	—	292	16	no	—
53	C₂×C₁₆	D₂₉₂	1	4	yes	—
54	C₄×C₈	D₂₉₂	1	2	yes	—
55	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
56	C₂×D₄	—	292	4	no	—
57	C₂×D₄	—	292	8	no	—
58	D₈	—	292	16	no	—
59	D₈	—	584	16	no	—
60	C₂×C₈	D₅₈₄	1	4	yes	—
61	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
62	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
63	C₁₆	D₅₈₄	1	8	yes	complemented
64	C₂³	—	584	4	no	—
65	D₄	—	584	8	no	—
66	D₄	—	584	16	no	—
67	C₂×C₄	#14556588	1	1	yes	characteristic
68	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
69	C₈	D₁₁₆₈	1	8	yes	complemented
70	C₈	C₂×D₅₈₄	1	4	yes	—
71	C₂²	—	1168	4	no	—
72	C₂²	—	1168	8	no	—
73	C₂²	#17988964	1	1	yes	characteristic
74	C₄	#17988964	1	2	yes	—
75	C₄	C₂×D₁₁₆₈	1	4	yes	—
76	C₂	—	2336	4	no	—
77	C₂	#22609546	1	1	yes	characteristic
78	C₂	#22612181	1	2	yes	—

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₈×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#493
hypercenter	C₈×C₁₆	D₇₃
hypocenter	C₇₃	#12515
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #22612181	C₂	semidirect
2. #22609546	C₂	semidirect
3. C₈×C₁₆×C₇₃	C₂	semidirect
23. C₁₆×C₇₃	D₈	semidirect
29. C₈×C₇₃	D₁₆	semidirect
42. C₈×C₁₆	D₇₃	semidirect
43. C₇₃	#12515	semidirect
63. C₁₆	D₅₈₄	semidirect
69. C₈	D₁₁₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22612181	2
#22609546	2
C₈×C₁₆×C₇₃	2
#12515	73