Subgroups — #35298183

Order 21248.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907591	C₂	1	6	yes	maximal
2	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
3	#19127571	C₂²	1	4	yes	—
4	#19127464	—	2	12	no	—
5	C₂×C₄×C₈×C₈₃	C₂²	1	3	yes	—
6	C₈²×C₈₃	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₈₃	—	2	12	no	—
8	C₂²×C₈×C₈₃	D₄	1	6	yes	—
9	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₈₃	C₂×C₄	1	6	yes	—
11	C₄×C₈×C₈₃	Q₈	1	6	yes	—
12	C₂²×Dic₈₃	—	4	4	no	—
13	C₄⋊Dic₈₃	—	4	24	no	—
14	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
15	C₂×C₈×C₈₃	—	2	12	no	—
16	C₂×C₈×C₈₃	C₄⋊C₄	1	12	yes	—
17	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₈₃	C₂×Q₈	1	3	yes	—
19	C₂³×C₈₃	#127	1	1	yes	characteristic
20	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
21	C₂×C₄×C₈₃	C₄⋊Q₈	1	3	yes	—
22	C₂×C₄×C₈₃	#135	1	12	yes	—
23	C₂×Dic₈₃	—	8	12	no	—
24	C₈×C₈₃	—	2	12	no	—
25	C₂²×C₈₃	#384	1	1	yes	characteristic
26	C₂²×C₈₃	#495	1	6	yes	—
27	C₄×C₈₃	#427	1	6	yes	—
28	C₄×C₈₃	#499	1	6	yes	—
29	Dic₈₃	—	16	4	no	—
30	#8817	—	83	1	no	maximal, Sylow
31	C₂×C₈₃	#1711	1	4	yes	—
32	C₂×C₈₃	#1846	1	3	yes	—
33	#1834	—	83	6	no	—
34	C₂×C₈²	D₈₃	1	1	yes	characteristic
35	C₈₃	#8817	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄×C₈	C₂×D₈₃	1	3	yes	—
37	#384	—	83	4	no	—
38	#340	—	166	12	no	—
39	C₈²	Dic₈₃	1	4	yes	complemented
40	C₂²×C₈	D₃₃₂	1	6	yes	—
41	C₂×C₄⋊C₄	—	166	12	no	—
42	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
43	C₄×C₈	Dic₁₆₆	1	6	yes	—
44	C₄×C₈	C₂×Dic₈₃	1	6	yes	—
45	C₂²×C₄	—	332	4	no	—
46	C₂²×C₄	C₂×D₃₃₂	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊Dic₈₃	1	12	yes	—
49	C₄⋊C₄	—	332	24	no	—
50	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
51	C₄²	C₂×Dic₁₆₆	1	3	yes	—
52	C₂³	#14581789	1	1	yes	characteristic
53	C₂×C₄	—	664	12	no	—
54	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
55	C₂×C₄	#14581792	1	3	yes	—
56	C₂×C₄	#14581814	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	#19127571	1	1	yes	characteristic
59	C₂²	#19128085	1	6	yes	—
60	C₄	—	1328	4	no	—
61	C₄	#19127739	1	6	yes	—
62	C₄	#19128103	1	6	yes	—
63	C₂	#24907075	1	4	yes	—
64	C₂	#24907627	1	3	yes	—

	subgroup	G/H
center	C₂³	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#8817
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₈₃	C₄	semidirect
35. C₈₃	#8817	semidirect
39. C₈²	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24907591	2
C₂×C₈²×C₈₃	2
#8817	83