Subgroups — C₂³×#41×C₂₂₃ #43252605

Order 28544.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₂₃	C₂	1	56	yes	maximal, direct product factor
2	C₂³×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×D₄×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₄×C₂₂₃	C₂	1	3	yes	maximal, complemented
5	C₂⁵×C₂₂₃	C₂²	1	3	yes	—
6	C₂×#41×C₂₂₃	C₂²	1	448	yes	direct product factor
7	C₂²×Q₈×C₂₂₃	C₂²	1	28	yes	complemented
8	C₂²×D₄×C₂₂₃	C₂²	1	84	yes	complemented
9	C₂³×C₄×C₂₂₃	C₂²	1	3	yes	—
10	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₂₃	C₂²	1	84	yes	complemented
12	C₂⁴×C₂₂₃	—	2	24	no	—
13	C₂⁴×C₂₂₃	C₂³	1	42	yes	—
14	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
15	#41×C₂₂₃	C₂³	1	512	yes	direct product factor
16	C₂×Q₈×C₂₂₃	C₂³	1	112	yes	complemented
17	C₂×D₄×C₂₂₃	C₂³	1	336	yes	complemented
18	C₂²×C₄×C₂₂₃	C₂³	1	42	yes	—
19	C₂²×C₄×C₂₂₃	C₂³	1	14	yes	—
20	C₂²×C₄×C₂₂₃	C₂³	1	336	yes	complemented
21	C₂³×C₂₂₃	—	2	168	no	—
22	C₂³×C₂₂₃	#41	1	8	yes	direct product factor
23	C₂³×C₂₂₃	C₂⁴	1	84	yes	—
24	C₂³×C₂₂₃	C₂⁴	1	7	yes	—
25	Q₈×C₂₂₃	C₂⁴	1	64	yes	complemented
26	D₄×C₂₂₃	C₂⁴	1	192	yes	complemented
27	C₂×C₄×C₂₂₃	C₂⁴	1	28	yes	—
28	C₂×C₄×C₂₂₃	C₂⁴	1	84	yes	—
29	C₂×C₄×C₂₂₃	C₂⁴	1	192	yes	complemented
30	C₂²×C₂₂₃	—	2	168	no	—
31	C₂²×C₂₂₃	C₂×#41	1	28	yes	direct product factor
32	C₂²×C₂₂₃	C₂⁵	1	24	yes	—
33	C₂²×C₂₂₃	C₂⁵	1	7	yes	—
34	C₄×C₂₂₃	C₂⁵	1	8	yes	—
35	C₄×C₂₂₃	C₂⁵	1	24	yes	—
36	C₂×C₂₂₃	—	2	24	no	—
37	C₂×C₂₂₃	C₂²×#41	1	14	yes	direct product factor
38	C₂×C₂₂₃	C₂⁶	1	1	yes	characteristic
39	C₂₂₃	C₂³×#41	1	1	yes	characteristic, direct product factor, Sylow
40	C₂³×#41	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×#41	C₂×C₂₂₃	1	56	yes	direct product factor
42	C₂³×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
43	C₂³×D₄	C₂×C₂₂₃	1	3	yes	complemented
44	C₂⁴×C₄	C₂×C₂₂₃	1	3	yes	complemented
45	C₂⁵	C₂²×C₂₂₃	1	3	yes	—
46	C₂×#41	C₂²×C₂₂₃	1	448	yes	direct product factor
47	C₂²×Q₈	C₂²×C₂₂₃	1	28	yes	complemented
48	C₂²×D₄	C₂²×C₂₂₃	1	84	yes	complemented
49	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
50	C₂³×C₄	C₂²×C₂₂₃	1	3	yes	—
51	C₂³×C₄	C₂²×C₂₂₃	1	84	yes	complemented
52	C₂⁴	—	2	24	no	—
53	C₂⁴	C₂³×C₂₂₃	1	42	yes	—
54	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
55	#41	C₂³×C₂₂₃	1	512	yes	direct product factor
56	C₂×Q₈	C₂³×C₂₂₃	1	112	yes	complemented
57	C₂×D₄	C₂³×C₂₂₃	1	336	yes	complemented
58	C₂²×C₄	C₂³×C₂₂₃	1	14	yes	—
59	C₂²×C₄	C₂³×C₂₂₃	1	42	yes	—
60	C₂²×C₄	C₂³×C₂₂₃	1	336	yes	complemented
61	C₂³	—	2	168	no	—
62	C₂³	#41×C₂₂₃	1	8	yes	direct product factor
63	C₂³	C₂⁴×C₂₂₃	1	84	yes	—
64	C₂³	C₂⁴×C₂₂₃	1	7	yes	—
65	Q₈	C₂⁴×C₂₂₃	1	64	yes	complemented
66	D₄	C₂⁴×C₂₂₃	1	192	yes	complemented
67	C₂×C₄	C₂⁴×C₂₂₃	1	28	yes	—
68	C₂×C₄	C₂⁴×C₂₂₃	1	84	yes	—
69	C₂×C₄	C₂⁴×C₂₂₃	1	192	yes	complemented
70	C₂²	—	2	168	no	—
71	C₂²	C₂×#41×C₂₂₃	1	28	yes	direct product factor
72	C₂²	C₂⁵×C₂₂₃	1	24	yes	—
73	C₂²	C₂⁵×C₂₂₃	1	7	yes	—
74	C₄	C₂⁵×C₂₂₃	1	8	yes	—
75	C₄	C₂⁵×C₂₂₃	1	24	yes	—
76	C₂	—	2	24	no	—
77	C₂	C₂²×#41×C₂₂₃	1	14	yes	direct product factor
78	C₂	C₂⁶×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₂₃	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₂₂₃
socle	C₂⁴×C₂₂₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₂₃	C₂	direct (N × H)
2. C₂³×Q₈×C₂₂₃	C₂	semidirect
3. C₂³×D₄×C₂₂₃	C₂	semidirect
4. C₂⁴×C₄×C₂₂₃	C₂	semidirect
6. C₂×#41×C₂₂₃	C₂²	direct (N × H)
7. C₂²×Q₈×C₂₂₃	C₂²	semidirect
8. C₂²×D₄×C₂₂₃	C₂²	semidirect
11. C₂³×C₄×C₂₂₃	C₂²	semidirect
15. #41×C₂₂₃	C₂³	direct (N × H)
16. C₂×Q₈×C₂₂₃	C₂³	semidirect
17. C₂×D₄×C₂₂₃	C₂³	semidirect
20. C₂²×C₄×C₂₂₃	C₂³	semidirect
22. C₂³×C₂₂₃	#41	direct (N × H)
25. Q₈×C₂₂₃	C₂⁴	semidirect
26. D₄×C₂₂₃	C₂⁴	semidirect
29. C₂×C₄×C₂₂₃	C₂⁴	semidirect
31. C₂²×C₂₂₃	C₂×#41	direct (N × H)
37. C₂×C₂₂₃	C₂²×#41	direct (N × H)
39. C₂₂₃	C₂³×#41	direct (N × H)
40. C₂³×#41	C₂₂₃	direct (N × H)
41. C₂²×#41	C₂×C₂₂₃	direct (N × H)
42. C₂³×Q₈	C₂×C₂₂₃	semidirect
43. C₂³×D₄	C₂×C₂₂₃	semidirect
44. C₂⁴×C₄	C₂×C₂₂₃	semidirect
46. C₂×#41	C₂²×C₂₂₃	direct (N × H)
47. C₂²×Q₈	C₂²×C₂₂₃	semidirect
48. C₂²×D₄	C₂²×C₂₂₃	semidirect
51. C₂³×C₄	C₂²×C₂₂₃	semidirect
55. #41	C₂³×C₂₂₃	direct (N × H)
56. C₂×Q₈	C₂³×C₂₂₃	semidirect
57. C₂×D₄	C₂³×C₂₂₃	semidirect
60. C₂²×C₄	C₂³×C₂₂₃	semidirect
62. C₂³	#41×C₂₂₃	direct (N × H)
65. Q₈	C₂⁴×C₂₂₃	semidirect
66. D₄	C₂⁴×C₂₂₃	semidirect
69. C₂×C₄	C₂⁴×C₂₂₃	semidirect
71. C₂²	C₂×#41×C₂₂₃	direct (N × H)
77. C₂	C₂²×#41×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₂₃	2
C₂³×Q₈×C₂₂₃	2
C₂³×D₄×C₂₂₃	2
C₂⁴×C₄×C₂₂₃	2
C₂³×#41	223

Subgroups — C23×#41×C223 #43252605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#41×C₂₂₃ #43252605