Subgroups — #1709×C₂₂₃ #43250724

Order 28544.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#494×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#486×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	C₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₂₃	—	2	2	no	—
6	C₂×D₈×C₂₂₃	—	2	2	no	—
7	C₄⋊Q₈×C₂₂₃	C₂²	1	2	yes	—
8	#127×C₂₂₃	C₂²	1	2	yes	—
9	#102×C₂₂₃	—	2	8	no	—
10	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
11	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₂₂₃	—	2	2	no	—
13	C₂×D₄×C₂₂₃	—	2	4	no	—
14	C₂×D₄×C₂₂₃	—	2	2	no	—
15	Q₁₆×C₂₂₃	—	4	4	no	—
16	D₈×C₂₂₃	—	4	4	no	—
17	C₂×C₈×C₂₂₃	D₄	1	4	yes	—
18	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
19	C₄⋊C₄×C₂₂₃	—	2	4	no	—
20	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₂³×C₂₂₃	—	4	2	no	—
22	Q₈×C₂₂₃	—	4	4	no	—
23	D₄×C₂₂₃	—	4	4	no	—
24	D₄×C₂₂₃	—	4	8	no	—
25	C₂×C₄×C₂₂₃	—	4	2	no	—
26	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
28	C₈×C₂₂₃	—	2	4	no	—
29	C₈×C₂₂₃	D₈	1	4	yes	—
30	C₂²×C₂₂₃	—	8	2	no	—
31	C₂²×C₂₂₃	—	8	4	no	—
32	C₂²×C₂₂₃	#127	1	1	yes	characteristic
33	C₄×C₂₂₃	—	8	2	no	—
34	C₄×C₂₂₃	C₂×D₈	1	2	yes	—
35	C₄×C₂₂₃	#135	1	4	yes	—
36	C₂×C₂₂₃	—	16	2	no	—
37	C₂×C₂₂₃	#493	1	1	yes	characteristic
38	C₂×C₂₂₃	#495	1	2	yes	—
39	C₂₂₃	#1709	1	1	yes	characteristic, direct product factor, Sylow
40	#1709	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	#494	C₂×C₂₂₃	1	1	yes	characteristic, complemented
42	#493	C₂×C₂₂₃	1	1	yes	characteristic
43	#486	C₂×C₂₂₃	1	4	yes	complemented
44	C₈²	C₂×C₂₂₃	1	1	yes	characteristic, complemented
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	2	2	no	—
47	C₄⋊Q₈	C₂²×C₂₂₃	1	2	yes	—
48	#127	C₂²×C₂₂₃	1	2	yes	—
49	#102	—	2	8	no	—
50	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
51	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	—	2	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	4	4	no	—
57	C₂×C₈	D₄×C₂₂₃	1	4	yes	—
58	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
61	C₂³	—	4	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	4	4	no	—
64	D₄	—	4	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
67	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
68	C₈	—	2	4	no	—
69	C₈	D₈×C₂₂₃	1	4	yes	—
70	C₂²	—	8	2	no	—
71	C₂²	—	8	4	no	—
72	C₂²	#127×C₂₂₃	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×D₈×C₂₂₃	1	2	yes	—
75	C₄	#135×C₂₂₃	1	4	yes	—
76	C₂	—	16	2	no	—
77	C₂	#493×C₂₂₃	1	1	yes	characteristic
78	C₂	#495×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₂₃
prime radical	#493×C₂₂₃	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #494×C₂₂₃	C₂	semidirect
3. #486×C₂₂₃	C₂	semidirect
4. C₈²×C₂₂₃	C₂	semidirect
39. C₂₂₃	#1709	direct (N × H)
40. #1709	C₂₂₃	direct (N × H)
41. #494	C₂×C₂₂₃	semidirect
43. #486	C₂×C₂₂₃	semidirect
44. C₈²	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#494×C₂₂₃	2
#493×C₂₂₃	2
#486×C₂₂₃	2
C₈²×C₂₂₃	2
#1709	223

Subgroups — #1709×C223 #43250724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1709×C₂₂₃ #43250724