Subgroups — #49121007

Order 36224.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₈₃⋊C₈	C₂	1	2	yes	maximal
2	#31826220	C₂	1	4	yes	maximal
3	C₄³×C₂₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₈₃⋊C₈	C₂²	1	4	yes	—
5	C₄×C₂₈₃⋊C₈	C₄	1	8	yes	complemented
6	C₂×C₄²×C₂₈₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₈₃	C₄	1	2	yes	—
8	C₂×C₄²×C₂₈₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₈₃⋊C₈	—	2	8	no	—
11	C₂×C₂₈₃⋊C₈	C₂×C₄	1	8	yes	—
12	C₂²×C₄×C₂₈₃	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₂₈₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₈₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
16	C₄²×C₂₈₃	—	2	4	no	—
17	C₄²×C₂₈₃	—	2	4	no	—
18	C₄²×C₂₈₃	C₈	1	8	yes	complemented
19	C₄²×C₂₈₃	C₂×C₄	1	4	yes	—
20	C₂₈₃⋊C₈	—	2	8	no	—
21	C₂³×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₈₃	—	2	4	no	—
23	C₂×C₄×C₂₈₃	—	2	4	no	—
24	C₂×C₄×C₂₈₃	C₄²	1	2	yes	—
25	C₂×C₄×C₂₈₃	C₄²	1	2	yes	—
26	C₂×C₄×C₂₈₃	C₂×C₈	1	8	yes	—
27	C₂×C₄×C₂₈₃	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₂₈₃	M₁₆	1	4	yes	—
29	C₂×C₄×C₂₈₃	C₂²×C₄	1	2	yes	—
30	C₂×C₄×C₂₈₃	#41	1	4	yes	—
31	C₂²×C₂₈₃	C₂×C₄²	1	1	yes	characteristic
32	C₂²×C₂₈₃	#117	1	2	yes	—
33	C₂²×C₂₈₃	C₂²×C₈	1	2	yes	—
34	C₂²×C₂₈₃	C₂×M₁₆	1	2	yes	—
35	C₄×C₂₈₃	—	2	8	no	—
36	C₄×C₂₈₃	C₄×C₈	1	4	yes	—
37	C₄×C₂₈₃	#97	1	4	yes	—
38	C₄×C₂₈₃	#117	1	4	yes	—
39	C₂×C₂₈₃	#376	1	1	yes	characteristic
40	C₂×C₂₈₃	C₂×C₄×C₈	1	1	yes	characteristic
41	C₂×C₂₈₃	C₂×#97	1	1	yes	characteristic
42	C₂×C₂₈₃	#431	1	4	yes	—
43	C₂₈₃	#1744	1	1	yes	characteristic, complemented, Sylow
44	#1744	—	283	1	no	maximal, Sylow
45	C₂×C₄×C₈	—	283	2	no	—
46	C₄³	D₂₈₃	1	1	yes	characteristic
47	#336	—	283	4	no	—
48	C₂²×C₈	—	283	4	no	—
49	C₂×C₄²	Dic₂₈₃	1	2	yes	—
50	C₂×C₄²	Dic₂₈₃	1	2	yes	—
51	C₂×C₄²	C₂×D₂₈₃	1	2	yes	—
52	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
53	C₄×C₈	—	283	8	no	—
54	C₂²×C₄	C₂×Dic₂₈₃	1	4	yes	—
55	C₂²×C₄	C₂×Dic₂₈₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₂₈₃	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
58	C₂×C₈	—	283	8	no	—
59	C₂×C₈	—	566	8	no	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	4	no	—
62	C₄²	C₂₈₃⋊C₈	1	8	yes	complemented
63	C₄²	C₄×D₂₈₃	1	4	yes	—
64	C₂³	C₂²×Dic₂₈₃	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₄×Dic₂₈₃	1	2	yes	—
68	C₂×C₄	C₄×Dic₂₈₃	1	2	yes	—
69	C₂×C₄	C₂×C₂₈₃⋊C₈	1	8	yes	—
70	C₂×C₄	C₂×C₂₈₃⋊C₈	1	4	yes	—
71	C₂×C₄	#17985376	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₂₈₃	1	2	yes	—
73	C₂×C₄	#17985392	1	4	yes	—
74	C₈	—	566	8	no	—
75	C₂²	C₂×C₄×Dic₂₈₃	1	1	yes	characteristic
76	C₂²	#22583569	1	2	yes	—
77	C₂²	C₂²×C₂₈₃⋊C₈	1	2	yes	—
78	C₂²	C₂×#17985376	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×C₂₈₃⋊C₈	1	4	yes	—
81	C₄	#22583521	1	4	yes	—
82	C₄	#22583567	1	4	yes	—
83	C₂	#31826311	1	1	yes	characteristic
84	C₂	C₂×C₄×C₂₈₃⋊C₈	1	1	yes	characteristic
85	C₂	C₂×#22583521	1	1	yes	characteristic
86	C₂	#31826518	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₂₈₃
commutator	C₂×C₂₈₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	C₄³×C₂₈₃	C₂
socle, prime radical	C₂³×C₂₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₈₃	C₂³
hypercenter	C₄³	D₂₈₃
hypocenter	C₂₈₃	#1744
LCS	G ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. C₄×C₂₈₃⋊C₈	C₄	semidirect
18. C₄²×C₂₈₃	C₈	semidirect
43. C₂₈₃	#1744	semidirect
62. C₄²	C₂₈₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₈₃⋊C₈	2
#31826220	2
C₄³×C₂₈₃	2
#1744	283