Subgroups — C₂₃×#3556 #15671175

Order 2944.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#564	C₂	1	4	yes	maximal, complemented
2	C₂₃×#557	C₂	1	12	yes	maximal, complemented
3	C₂₃×#541	C₂	1	12	yes	maximal
4	C₂₃×#533	C₂	1	3	yes	maximal
5	C₂₃×C₄⋊Q₈	C₂²	1	6	yes	—
6	C₂₃×C₄⋊Q₈	C₂²	1	12	yes	—
7	C₂₃×#126	C₂²	1	16	yes	—
8	C₂₃×#125	C₂²	1	12	yes	—
9	C₂₃×#125	C₂²	1	48	yes	—
10	C₂₃×#125	C₂²	1	6	yes	—
11	C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	—
12	C₄×Q₈×C₂₃	C₂²	1	12	yes	—
13	C₄×Q₈×C₂₃	C₂²	1	24	yes	—
14	C₂₃×#117	C₂²	1	6	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₃	C₂³	1	12	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
18	C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
19	C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
20	C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
21	C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
22	C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
23	C₄²×C₂₃	C₂³	1	3	yes	—
24	C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	C₄²×C₂₃	C₂³	1	12	yes	—
26	C₄²×C₂₃	C₂³	1	16	yes	—
27	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
28	Q₈×C₂₃	—	2	24	no	—
29	C₂×C₄×C₂₃	—	2	6	no	—
30	C₂×C₄×C₂₃	C₂⁴	1	24	yes	—
31	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
32	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
33	#3556	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₂₃	—	2	3	no	—
35	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
36	C₄×C₂₃	—	2	3	no	—
37	C₄×C₂₃	—	2	24	no	—
38	C₄×C₂₃	2₋¹⁺⁴	1	6	yes	—
39	#564	C₂×C₂₃	1	4	yes	complemented
40	#557	C₂×C₂₃	1	12	yes	complemented
41	#541	C₂×C₂₃	1	12	yes	—
42	#533	C₂×C₂₃	1	3	yes	—
43	C₂×C₂₃	—	4	1	no	—
44	C₂×C₂₃	C₂×2₋¹⁺⁴	1	3	yes	—
45	C₄⋊Q₈	C₂²×C₂₃	1	6	yes	—
46	C₄⋊Q₈	C₂²×C₂₃	1	12	yes	—
47	#126	C₂²×C₂₃	1	16	yes	—
48	#125	C₂²×C₂₃	1	6	yes	—
49	#125	C₂²×C₂₃	1	12	yes	—
50	#125	C₂²×C₂₃	1	48	yes	—
51	C₂²⋊Q₈	C₂²×C₂₃	1	12	yes	—
52	C₄×Q₈	C₂²×C₂₃	1	12	yes	—
53	C₄×Q₈	C₂²×C₂₃	1	24	yes	—
54	#117	C₂²×C₂₃	1	6	yes	—
55	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
56	C₂₃	#3556	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×Q₈	C₂³×C₂₃	1	12	yes	—
58	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
59	C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
60	C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
61	C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
62	C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
63	C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
64	C₄²	C₂³×C₂₃	1	3	yes	—
65	C₄²	C₂³×C₂₃	1	1	yes	characteristic
66	C₄²	C₂³×C₂₃	1	12	yes	—
67	C₄²	C₂³×C₂₃	1	16	yes	—
68	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
69	Q₈	—	2	24	no	—
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	C₂⁴×C₂₃	1	24	yes	—
72	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
73	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
74	C₂²	—	2	3	no	—
75	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
76	C₄	—	2	3	no	—
77	C₄	—	2	24	no	—
78	C₄	C₂₃×2₋¹⁺⁴	1	6	yes	—
79	C₂	—	4	1	no	—
80	C₂	C₂×C₂₃×2₋¹⁺⁴	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#564	C₂	semidirect
2. C₂₃×#557	C₂	semidirect
33. #3556	C₂₃	direct (N × H)
39. #564	C₂×C₂₃	semidirect
40. #557	C₂×C₂₃	semidirect
56. C₂₃	#3556	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×#564	2
C₂₃×#557	2
C₂₃×#541	2
C₂₃×#533	2
#3556	23

Subgroups — C23×#3556 #15671175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3556 #15671175