Subgroups — C₂₃×#2842 #15670461

Order 2944.

548 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#399	C₂	1	4	yes	maximal, complemented
2	C₂₃×#396	C₂	1	8	yes	maximal, complemented
3	C₂₃×#379	C₂	1	3	yes	maximal
4	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₂₃	C₂²	1	12	yes	complemented
6	C₂×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
7	C₂×C₂²⋊C₄×C₂₃	C₂²	1	6	yes	—
8	C₂₃×#95	C₂²	1	4	yes	complemented
9	C₂⁴×C₂₃	—	4	6	no	—
10	C₂⁴×C₂₃	Q₈	1	4	yes	—
11	C₂⁴×C₂₃	C₂³	1	2	yes	—
12	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂³	1	12	yes	—
14	C₄⋊C₄×C₂₃	—	4	12	no	—
15	C₂²⋊C₄×C₂₃	—	4	12	no	—
16	C₂²⋊C₄×C₂₃	—	4	6	no	—
17	C₂³×C₂₃	—	2	12	no	—
18	C₂³×C₂₃	—	2	6	no	—
19	C₂³×C₂₃	—	4	12	no	—
20	C₂³×C₂₃	—	4	16	no	—
21	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
22	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
23	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
24	C₂×C₄×C₂₃	—	2	24	no	—
25	C₂×C₄×C₂₃	—	2	12	no	—
26	#2842	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₂₃	—	2	12	no	—
28	C₂²×C₂₃	—	2	6	no	—
29	C₂²×C₂₃	—	4	16	no	—
30	C₂²×C₂₃	—	4	8	no	—
31	C₂²×C₂₃	—	4	4	no	—
32	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
33	C₂²×C₂₃	H(5,𝔽₂)	1	6	yes	—
34	C₄×C₂₃	—	4	12	no	—
35	#399	C₂×C₂₃	1	4	yes	complemented
36	#396	C₂×C₂₃	1	8	yes	complemented
37	#379	C₂×C₂₃	1	3	yes	—
38	C₂×C₂₃	—	4	4	no	—
39	C₂×C₂₃	—	4	2	no	—
40	C₂×C₂₃	#543	1	3	yes	—
41	C₂×C₂₃	#560	1	4	yes	—
42	C₂⁵	C₂²×C₂₃	1	1	yes	characteristic
43	C₂×C₄⋊C₄	C₂²×C₂₃	1	12	yes	complemented
44	C₂×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
45	C₂×C₂²⋊C₄	C₂²×C₂₃	1	6	yes	—
46	#95	C₂²×C₂₃	1	4	yes	complemented
47	C₂₃	#2842	1	1	yes	characteristic, direct product factor, Sylow
48	C₂⁴	—	4	6	no	—
49	C₂⁴	Q₈×C₂₃	1	4	yes	—
50	C₂⁴	C₂³×C₂₃	1	2	yes	—
51	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
52	C₂²×C₄	C₂³×C₂₃	1	12	yes	—
53	C₄⋊C₄	—	4	12	no	—
54	C₂²⋊C₄	—	4	12	no	—
55	C₂²⋊C₄	—	4	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	2	6	no	—
58	C₂³	—	4	12	no	—
59	C₂³	—	4	16	no	—
60	C₂³	C₂×Q₈×C₂₃	1	4	yes	—
61	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
62	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	—	2	12	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	4	16	no	—
68	C₂²	—	4	8	no	—
69	C₂²	—	4	4	no	—
70	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
71	C₂²	C₂₃×H(5,𝔽₂)	1	6	yes	—
72	C₄	—	4	12	no	—
73	C₂	—	4	4	no	—
74	C₂	—	4	2	no	—
75	C₂	C₂₃×#543	1	3	yes	—
76	C₂	C₂₃×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁵×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#399	C₂	semidirect
2. C₂₃×#396	C₂	semidirect
5. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
8. C₂₃×#95	C₂²	semidirect
26. #2842	C₂₃	direct (N × H)
35. #399	C₂×C₂₃	semidirect
36. #396	C₂×C₂₃	semidirect
43. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
46. #95	C₂²×C₂₃	semidirect
47. C₂₃	#2842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#399	2
C₂₃×#396	2
C₂₃×#379	2
#2842	23

Subgroups — C23×#2842 #15670461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2842 #15670461