Subgroups — C₂₃×#7538 #19155164

Order 5888.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1423	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₃₂	—	2	1	no	—
5	C₂₃×Q₆₄	—	2	1	no	—
6	C₂₃×Q₆₄	C₄	1	2	yes	complemented
7	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
8	C₂₃×#366	—	2	1	no	—
9	C₂₃×C₆₄	—	2	1	no	—
10	C₂×Q₁₆×C₂₃	—	4	1	no	—
11	C₂₃×Q₃₂	—	2	2	no	—
12	C₂₃×Q₃₂	—	4	1	no	—
13	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
14	C₂₃×#107	—	4	1	no	—
15	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₂₃×C₃₂	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₂₃	—	8	1	no	—
18	Q₁₆×C₂₃	—	4	2	no	—
19	Q₁₆×C₂₃	—	8	1	no	—
20	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₂₃	—	8	1	no	—
22	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₂₃	SD₁₆	1	1	yes	characteristic
24	#7538	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	Q₈×C₂₃	—	8	2	no	—
26	Q₈×C₂₃	—	16	1	no	—
27	C₂×C₄×C₂₃	—	16	1	no	—
28	C₂×C₄×C₂₃	—	16	1	no	—
29	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
30	C₈×C₂₃	#102	1	1	yes	characteristic
31	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
32	C₂×Q₆₄	C₂×C₂₃	1	1	yes	characteristic
33	C₂×C₆₄	C₂×C₂₃	1	1	yes	characteristic
34	#1423	C₂×C₂₃	1	1	yes	characteristic
35	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
36	C₄×C₂₃	—	16	2	no	—
37	C₄×C₂₃	—	32	1	no	—
38	C₄×C₂₃	#357	1	1	yes	characteristic
39	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
40	C₂×Q₃₂	—	2	1	no	—
41	Q₆₄	—	2	1	no	—
42	Q₆₄	C₄×C₂₃	1	2	yes	complemented
43	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
44	#366	—	2	1	no	—
45	C₆₄	—	2	1	no	—
46	C₂×C₂₃	#1415	1	1	yes	characteristic
47	C₂×C₂₃	SD₁₂₈	1	1	yes	characteristic
48	C₂×C₂₃	Q₁₂₈	1	1	yes	characteristic
49	C₂×Q₁₆	—	4	1	no	—
50	Q₃₂	—	2	2	no	—
51	Q₃₂	—	4	1	no	—
52	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
53	#107	—	4	1	no	—
54	C₃₂	C₂×C₄×C₂₃	1	1	yes	characteristic
55	C₃₂	D₄×C₂₃	1	1	yes	characteristic
56	C₂₃	#7538	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
61	C₄⋊C₄	—	8	1	no	—
62	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
63	C₁₆	SD₁₆×C₂₃	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	16	1	no	—
68	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
69	C₈	C₂₃×#102	1	1	yes	characteristic
70	C₈	C₂₃×SD₃₂	1	1	yes	characteristic
71	C₂²	C₂₃×D₃₂	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	32	1	no	—
74	C₄	C₂₃×#357	1	1	yes	characteristic
75	C₄	C₂₃×SD₆₄	1	1	yes	characteristic
76	C₂	C₂₃×#1415	1	1	yes	characteristic
77	C₂	C₂₃×SD₁₂₈	1	1	yes	characteristic
78	C₂	C₂₃×Q₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	D₃₂
commutator	C₃₂	C₂×C₄×C₂₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₂₃
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂₃×Q₆₄	C₄	semidirect
24. #7538	C₂₃	direct (N × H)
42. Q₆₄	C₄×C₂₃	semidirect
56. C₂₃	#7538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×Q₆₄	2
C₂×C₂₃×C₆₄	2
C₂₃×#1423	2
#7538	23

Subgroups — C23×#7538 #19155164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7538 #19155164