Subgroups — C₂₃×#2017 #15669636

Order 2944.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#575	C₂	1	3	yes	maximal, complemented
2	C₂₃×#408	C₂	1	3	yes	maximal, complemented
3	C₂₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#41×C₂₃	C₂²	1	3	yes	—
5	C₂₃×#137	—	2	6	no	—
6	C₂₃×AGL(1,ℤ/8)	—	2	6	no	—
7	C₂₃×#135	—	2	6	no	—
8	C₂×M₁₆×C₂₃	C₂²	1	3	yes	—
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×C₂²⋊C₈	—	2	6	no	—
11	#41×C₂₃	—	2	6	no	—
12	#41×C₂₃	—	2	6	no	—
13	C₂×Q₈×C₂₃	D₄	1	3	yes	—
14	C₂×D₄×C₂₃	—	2	3	no	—
15	C₂×D₄×C₂₃	D₄	1	3	yes	—
16	C₂²×C₄×C₂₃	—	2	3	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	Q₁₆×C₂₃	—	2	6	no	—
19	SD₁₆×C₂₃	—	2	12	no	—
20	D₈×C₂₃	—	2	6	no	—
21	M₁₆×C₂₃	—	2	6	no	—
22	C₂×C₈×C₂₃	—	2	3	no	—
23	C₂×C₈×C₂₃	D₄	1	6	yes	—
24	C₂³×C₂₃	—	2	3	no	—
25	C₂³×C₂₃	#41	1	1	yes	characteristic
26	Q₈×C₂₃	—	2	6	no	—
27	D₄×C₂₃	—	2	6	no	—
28	D₄×C₂₃	—	4	6	no	—
29	C₂×C₄×C₂₃	—	2	3	no	—
30	C₂×C₄×C₂₃	—	4	3	no	—
31	C₂×C₄×C₂₃	—	4	3	no	—
32	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
34	C₈×C₂₃	—	2	6	no	—
35	C₈×C₂₃	—	4	1	no	—
36	#2017	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₂₃	—	4	1	no	—
38	C₂²×C₂₃	—	4	6	no	—
39	C₂²×C₂₃	—	4	3	no	—
40	C₂²×C₂₃	#121	1	3	yes	—
41	C₄×C₂₃	—	4	3	no	—
42	C₄×C₂₃	#120	1	3	yes	—
43	C₄×C₂₃	#127	1	1	yes	characteristic
44	#575	C₂×C₂₃	1	3	yes	complemented
45	#408	C₂×C₂₃	1	3	yes	complemented
46	#338	C₂×C₂₃	1	1	yes	characteristic, complemented
47	C₂×C₂₃	—	2	3	no	—
48	C₂×C₂₃	—	8	3	no	—
49	C₂×C₂₃	#392	1	1	yes	characteristic
50	C₂×#41	C₂²×C₂₃	1	3	yes	—
51	#137	—	2	6	no	—
52	AGL(1,ℤ/8)	—	2	6	no	—
53	#135	—	2	6	no	—
54	C₂×M₁₆	C₂²×C₂₃	1	3	yes	—
55	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
56	C₂²⋊C₈	—	2	6	no	—
57	C₂₃	#2017	1	1	yes	characteristic, direct product factor, Sylow
58	#41	—	2	6	no	—
59	#41	—	2	6	no	—
60	C₂×Q₈	D₄×C₂₃	1	3	yes	—
61	C₂×D₄	—	2	3	no	—
62	C₂×D₄	D₄×C₂₃	1	3	yes	—
63	C₂²×C₄	—	2	3	no	—
64	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
65	Q₁₆	—	2	6	no	—
66	SD₁₆	—	2	12	no	—
67	D₈	—	2	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	2	3	no	—
70	C₂×C₈	D₄×C₂₃	1	6	yes	—
71	C₂³	—	2	3	no	—
72	C₂³	#41×C₂₃	1	1	yes	characteristic
73	Q₈	—	2	6	no	—
74	D₄	—	2	6	no	—
75	D₄	—	4	6	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	4	3	no	—
78	C₂×C₄	—	4	3	no	—
79	C₂×C₄	C₂×D₄×C₂₃	1	3	yes	—
80	C₂×C₄	C₂×D₄×C₂₃	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	6	no	—
85	C₂²	—	4	3	no	—
86	C₂²	C₂₃×#121	1	3	yes	—
87	C₄	—	4	3	no	—
88	C₄	C₂₃×#120	1	3	yes	—
89	C₄	C₂₃×#127	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	8	3	no	—
92	C₂	C₂₃×#392	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂×C₂₃	#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#575	C₂	semidirect
2. C₂₃×#408	C₂	semidirect
3. C₂₃×#338	C₂	semidirect
36. #2017	C₂₃	direct (N × H)
44. #575	C₂×C₂₃	semidirect
45. #408	C₂×C₂₃	semidirect
46. #338	C₂×C₂₃	semidirect
57. C₂₃	#2017	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×#575	2
C₂₃×#408	2
C₂₃×#338	2
#2017	23

Subgroups — C23×#2017 #15669636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2017 #15669636