Subgroups — C₂₃×#1708 #15669327

Order 2944.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#492	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#487	C₂	1	2	yes	maximal, complemented
4	C₂₃×#486	C₂	1	2	yes	maximal
5	C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₂₃	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂₃×#106	—	2	2	no	—
11	C₂₃×#103	—	2	4	no	—
12	C₂₃×#102	—	2	4	no	—
13	C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	1	no	—
17	C₂×D₄×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	1	no	—
19	SD₁₆×C₂₃	—	4	4	no	—
20	C₂×C₈×C₂₃	D₄	1	4	yes	—
21	C₂×C₈×C₂₃	D₄	1	2	yes	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂³×C₂₃	—	4	1	no	—
27	Q₈×C₂₃	—	4	2	no	—
28	Q₈×C₂₃	—	4	4	no	—
29	D₄×C₂₃	—	4	2	no	—
30	D₄×C₂₃	—	4	4	no	—
31	C₂×C₄×C₂₃	—	4	1	no	—
32	C₂×C₄×C₂₃	—	4	2	no	—
33	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
35	C₈×C₂₃	—	2	4	no	—
36	C₈×C₂₃	SD₁₆	1	4	yes	—
37	#1708	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₂₃	—	8	1	no	—
39	C₂²×C₂₃	—	8	2	no	—
40	C₂²×C₂₃	#127	1	1	yes	characteristic
41	C₄×C₂₃	—	8	1	no	—
42	C₄×C₂₃	—	8	2	no	—
43	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
44	C₄×C₂₃	#135	1	4	yes	—
45	#498	C₂×C₂₃	1	1	yes	characteristic, complemented
46	#492	C₂×C₂₃	1	1	yes	characteristic
47	#487	C₂×C₂₃	1	2	yes	complemented
48	#486	C₂×C₂₃	1	2	yes	—
49	C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
50	C₂×C₂₃	—	16	1	no	—
51	C₂×C₂₃	#492	1	1	yes	characteristic
52	C₂×C₂₃	#495	1	2	yes	—
53	C₂×SD₁₆	—	2	2	no	—
54	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
55	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
56	#127	C₂²×C₂₃	1	1	yes	characteristic
57	#106	—	2	2	no	—
58	#103	—	2	4	no	—
59	#102	—	2	4	no	—
60	C₄×C₈	C₂²×C₂₃	1	2	yes	—
61	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
62	C₂₃	#1708	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	2	1	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	2	1	no	—
67	SD₁₆	—	4	4	no	—
68	C₂×C₈	D₄×C₂₃	1	4	yes	—
69	C₂×C₈	D₄×C₂₃	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄²	C₂³×C₂₃	1	1	yes	characteristic
74	C₂³	—	4	1	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	4	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	4	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	SD₁₆×C₂₃	1	4	yes	—
85	C₂²	—	8	1	no	—
86	C₂²	—	8	2	no	—
87	C₂²	C₂₃×#127	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	8	2	no	—
90	C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
91	C₄	C₂₃×#135	1	4	yes	—
92	C₂	—	16	1	no	—
93	C₂	C₂₃×#492	1	1	yes	characteristic
94	C₂	C₂₃×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₃
prime radical	C₂₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#498	C₂	semidirect
3. C₂₃×#487	C₂	semidirect
5. C₈²×C₂₃	C₂	semidirect
37. #1708	C₂₃	direct (N × H)
45. #498	C₂×C₂₃	semidirect
47. #487	C₂×C₂₃	semidirect
49. C₈²	C₂×C₂₃	semidirect
62. C₂₃	#1708	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×#498	2
C₂₃×#492	2
C₂₃×#487	2
C₂₃×#486	2
C₈²×C₂₃	2
#1708	23

Subgroups — C23×#1708 #15669327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1708 #15669327