Subgroups — C₂×C₂₃×#346 #15669725

Order 2944.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#346	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₃	C₄	1	8	yes	complemented
8	C₂×C₁₆×C₂₃	C₂²	1	4	yes	—
9	C₄×C₈×C₂₃	C₄	1	2	yes	—
10	C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₃	C₈	1	2	yes	—
12	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₃	C₈	1	4	yes	—
14	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₄²×C₂₃	C₈	1	2	yes	—
19	C₄²×C₂₃	C₂×C₄	1	2	yes	—
20	C₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
21	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₂₃	—	2	4	no	—
23	C₂×C₄×C₂₃	C₄²	1	2	yes	—
24	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₂₃	C₄²	1	2	yes	—
29	C₈×C₂₃	C₂×C₈	1	4	yes	—
30	C₈×C₂₃	C₂²×C₄	1	2	yes	—
31	C₂×#346	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₂₃	C₄×C₈	1	2	yes	—
33	C₂²×C₂₃	M₃₂	1	4	yes	—
34	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
35	C₄×C₂₃	—	2	4	no	—
36	C₄×C₂₃	C₄×C₈	1	2	yes	—
37	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
38	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
39	C₂²×C₁₆	C₂×C₂₃	1	2	yes	—
40	C₂×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
41	#346	C₂×C₂₃	1	4	yes	direct product factor
42	C₂×C₂₃	#346	1	4	yes	direct product factor
43	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
44	C₂×C₂₃	C₂×M₃₂	1	2	yes	—
45	C₂²×C₈	C₄×C₂₃	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
47	C₂×C₄²	C₄×C₂₃	1	1	yes	characteristic
48	C₂×C₁₆	C₄×C₂₃	1	8	yes	complemented
49	C₂×C₁₆	C₂²×C₂₃	1	4	yes	—
50	C₄×C₈	C₄×C₂₃	1	2	yes	—
51	C₄×C₈	C₂²×C₂₃	1	2	yes	—
52	C₂₃	C₂×#346	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₄	C₈×C₂₃	1	2	yes	—
54	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
55	C₂×C₈	C₈×C₂₃	1	4	yes	—
56	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
57	C₂×C₈	C₂×C₄×C₂₃	1	4	yes	—
58	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
59	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
60	C₄²	C₈×C₂₃	1	2	yes	—
61	C₄²	C₂×C₄×C₂₃	1	2	yes	—
62	C₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
63	C₂³	C₂×C₈×C₂₃	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₄²×C₂₃	1	2	yes	—
66	C₂×C₄	C₂×C₈×C₂₃	1	4	yes	—
67	C₂×C₄	C₂×C₈×C₂₃	1	2	yes	—
68	C₂×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
70	C₈	C₄²×C₂₃	1	2	yes	—
71	C₈	C₂×C₈×C₂₃	1	4	yes	—
72	C₈	C₂²×C₄×C₂₃	1	2	yes	—
73	C₂²	C₄×C₈×C₂₃	1	2	yes	—
74	C₂²	C₂₃×M₃₂	1	4	yes	—
75	C₂²	C₂²×C₈×C₂₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	C₄×C₈×C₂₃	1	2	yes	—
78	C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
79	C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
80	C₂	C₂₃×#346	1	4	yes	direct product factor
81	C₂	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
82	C₂	C₂×C₂₃×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂²
commutator	C₂	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#346	C₂	direct (N × H)
7. C₂×C₁₆×C₂₃	C₄	semidirect
20. C₁₆×C₂₃	C₂×C₄	semidirect
31. C₂×#346	C₂₃	direct (N × H)
41. #346	C₂×C₂₃	direct (N × H)
42. C₂×C₂₃	#346	direct (N × H)
48. C₂×C₁₆	C₄×C₂₃	semidirect
52. C₂₃	C₂×#346	direct (N × H)
62. C₁₆	C₂×C₄×C₂₃	semidirect
80. C₂	C₂₃×#346	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×C₁₆×C₂₃	2
C₂×C₄×C₈×C₂₃	2
C₂₃×#346	2
C₂×#346	23

Subgroups — C2×C23×#346 #15669725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#346 #15669725