Subgroups — C₂₃×#23847 #19171473

Order 5888.

604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2104	C₂	1	8	yes	maximal
2	C₂₃×#1740	C₂	1	7	yes	maximal
3	C₂×C₂₃×#117	C₂²	1	7	yes	—
4	C₂₃×#342	C₂²	1	28	yes	—
5	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
6	C₂₃×#117	—	2	14	no	—
7	C₂₃×#117	—	2	14	no	—
8	C₂×C₄⋊C₄×C₂₃	—	2	7	no	—
9	C₂×C₂²⋊C₄×C₂₃	C₂³	1	14	yes	—
10	C₂×C₄²×C₂₃	—	2	7	no	—
11	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₂₃	—	2	14	no	—
13	C₂²×C₄×C₂₃	#41	1	7	yes	—
14	C₂²×C₄×C₂₃	#41	1	7	yes	—
15	C₄⋊C₄×C₂₃	—	4	14	no	—
16	C₂²⋊C₄×C₂₃	—	2	28	no	—
17	C₄²×C₂₃	—	4	7	no	—
18	C₄²×C₂₃	—	4	7	no	—
19	#23847	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₂₃	—	8	1	no	—
21	C₂³×C₂₃	C₂×#41	1	7	yes	—
22	C₂×C₄×C₂₃	—	4	14	no	—
23	C₂×C₄×C₂₃	—	8	14	no	—
24	C₂×C₄×C₂₃	#126	1	7	yes	—
25	C₂×C₄×C₂₃	#126	1	14	yes	—
26	C₂×C₄×C₂₃	#126	1	7	yes	—
27	#2104	C₂×C₂₃	1	8	yes	—
28	#1740	C₂×C₂₃	1	7	yes	—
29	C₂²×C₂₃	—	4	7	no	—
30	C₂²×C₂₃	C₂×#126	1	7	yes	—
31	C₄×C₂₃	—	8	14	no	—
32	C₄×C₂₃	#401	1	7	yes	—
33	C₄×C₂₃	#401	1	1	yes	characteristic
34	C₂×#117	C₂²×C₂₃	1	7	yes	—
35	#342	C₂²×C₂₃	1	28	yes	—
36	C₂×C₂₃	—	2	7	no	—
37	C₂×C₂₃	C₂×#401	1	1	yes	characteristic
38	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
39	#117	—	2	14	no	—
40	#117	—	2	14	no	—
41	C₂×C₄⋊C₄	—	2	7	no	—
42	C₂×C₂²⋊C₄	C₂³×C₂₃	1	14	yes	—
43	C₂×C₄²	—	2	7	no	—
44	C₂₃	#23847	1	1	yes	characteristic, direct product factor, Sylow
45	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
46	C₂²×C₄	—	2	14	no	—
47	C₂²×C₄	#41×C₂₃	1	7	yes	—
48	C₂²×C₄	#41×C₂₃	1	7	yes	—
49	C₄⋊C₄	—	4	14	no	—
50	C₂²⋊C₄	—	2	28	no	—
51	C₄²	—	4	7	no	—
52	C₄²	—	4	7	no	—
53	C₂³	—	8	1	no	—
54	C₂³	C₂×#41×C₂₃	1	7	yes	—
55	C₂×C₄	—	4	14	no	—
56	C₂×C₄	—	8	14	no	—
57	C₂×C₄	C₂₃×#126	1	7	yes	—
58	C₂×C₄	C₂₃×#126	1	7	yes	—
59	C₂×C₄	C₂₃×#126	1	14	yes	—
60	C₂²	—	4	7	no	—
61	C₂²	C₂×C₂₃×#126	1	7	yes	—
62	C₄	—	8	14	no	—
63	C₄	C₂₃×#401	1	7	yes	—
64	C₄	C₂₃×#401	1	1	yes	characteristic
65	C₂	—	2	7	no	—
66	C₂	C₂×C₂₃×#401	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#401
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂×#401
prime radical	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
19. #23847	C₂₃	direct (N × H)
44. C₂₃	#23847	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×#2104	2
C₂₃×#1740	2
#23847	23

Subgroups — C23×#23847 #19171473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23847 #19171473