Subgroups — #35319113

Order 21248.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909692	C₂	1	4	yes	maximal
2	#24907081	C₂	1	1	yes	maximal, characteristic
3	#24905380	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1710	C₂	1	1	yes	maximal, characteristic
5	#19128106	—	2	2	no	—
6	#19128081	C₂²	1	2	yes	—
7	#19127504	—	2	8	no	—
8	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₈₃	C₂²	1	2	yes	—
10	#487×C₈₃	—	2	2	no	—
11	C₈²×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₃₂	—	2	4	no	—
13	#14581792	—	2	2	no	—
14	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
15	#14581726	—	4	4	no	—
16	C₄⋊Q₈×C₈₃	—	2	2	no	—
17	#107×C₈₃	—	2	4	no	—
18	#103×C₈₃	—	4	4	no	—
19	C₄×C₈×C₈₃	D₄	1	2	yes	—
20	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₆₆	—	4	2	no	—
22	Dic₃₃₂	—	4	8	no	—
23	C₄⋊Dic₈₃	—	4	4	no	—
24	C₈₃⋊C₁₆	—	2	4	no	—
25	C₂×Q₈×C₈₃	—	4	2	no	—
26	C₂×C₈×C₈₃	D₈	1	4	yes	—
27	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
28	C₄⋊C₄×C₈₃	—	4	2	no	—
29	C₄⋊C₄×C₈₃	—	4	2	no	—
30	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₈₃	—	8	4	no	—
32	C₂×C₄×C₈₃	—	8	2	no	—
33	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
35	C₂×Dic₈₃	—	8	2	no	—
36	Dic₁₆₆	—	8	4	no	—
37	C₈×C₈₃	—	2	4	no	—
38	C₈×C₈₃	#135	1	4	yes	—
39	C₂²×C₈₃	#493	1	1	yes	characteristic
40	C₄×C₈₃	—	16	2	no	—
41	C₄×C₈₃	#495	1	2	yes	—
42	C₄×C₈₃	#510	1	4	yes	—
43	Dic₈₃	—	16	2	no	—
44	#12514	—	83	1	no	maximal, Sylow
45	C₂×C₈₃	#1709	1	1	yes	characteristic
46	C₂×C₈₃	#2251	1	2	yes	—
47	#2241	—	83	4	no	—
48	#1713	—	83	1	no	—
49	#1710	D₈₃	1	1	yes	characteristic
50	#1311	—	83	1	no	—
51	C₈₃	#12514	1	1	yes	characteristic, complemented, Sylow
52	#500	—	166	2	no	—
53	#500	C₂×D₈₃	1	2	yes	—
54	#494	—	83	2	no	—
55	#487	—	2	2	no	—
56	#358	—	166	8	no	—
57	C₄×C₁₆	—	83	2	no	—
58	C₈²	C₂×D₈₃	1	1	yes	characteristic
59	C₂×Q₁₆	—	166	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	166	2	no	—
62	C₂×C₁₆	—	83	4	no	—
63	#107	—	2	4	no	—
64	#107	—	332	4	no	—
65	#103	—	4	4	no	—
66	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
67	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	332	2	no	—
70	Q₁₆	—	332	8	no	—
71	C₂×C₈	#13050575	1	4	yes	—
72	C₂×C₈	D₄×D₈₃	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	332	4	no	—
76	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
77	C₁₆	—	166	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	664	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	664	2	no	—
82	C₂×C₄	#14581788	1	1	yes	characteristic
83	C₂×C₄	C₂×#13050575	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#14581815	1	4	yes	—
86	C₂²	#19128077	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	1328	2	no	—
89	C₄	#19128086	1	2	yes	—
90	C₄	#19128141	1	4	yes	—
91	C₂	#24907070	1	1	yes	characteristic
92	C₂	#24909735	1	2	yes	—

	subgroup	G/H
center	C₂²	#19128077
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1710	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#1710	D₈₃
hypocenter	C₈₃	#12514
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

Complements

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

N	H ≅ G/N	type
51. C₈₃	#12514	semidirect

Maximal subgroups

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

subgroup	index
#24909692	2
#24907081	2
#24905380	2
C₈₃×#1710	2
#12514	83