Subgroups — #15675626

Order 2944.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092797	C₂	1	1	yes	maximal, characteristic
2	#13092220	C₂	1	4	yes	maximal
3	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂₃×C₄⋊Q₈	—	2	1	no	—
6	C₂×Dic₉₂	—	2	1	no	—
7	C₂×Dic₉₂	C₂²	1	2	yes	—
8	#10625040	—	2	1	no	—
9	C₂₃×#107	—	2	1	no	—
10	C₂₃×#107	C₂²	1	2	yes	—
11	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
13	C₂×Q₈×C₂₃	—	4	1	no	—
14	C₂×Dic₄₆	—	2	2	no	—
15	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₄⋊C₄×C₂₃	—	4	1	no	—
19	C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₄⋊Dic₂₃	—	4	1	no	—
21	C₄⋊Dic₂₃	—	4	1	no	—
22	Dic₉₂	—	2	4	no	—
23	Dic₉₂	—	4	2	no	—
24	C₂₃⋊C₁₆	—	2	2	no	—
25	Q₈×C₂₃	—	8	2	no	—
26	C₂×C₄×C₂₃	—	4	2	no	—
27	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
28	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₂₃	—	4	2	no	—
31	Dic₄₆	—	4	2	no	—
32	Dic₄₆	—	4	2	no	—
33	C₈×C₂₃	SD₁₆	1	2	yes	—
34	C₈×C₂₃	#41	1	2	yes	—
35	#2241	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
37	C₄×C₂₃	—	8	2	no	—
38	C₄×C₂₃	SD₃₂	1	2	yes	—
39	C₄×C₂₃	Q₃₂	1	2	yes	—
40	C₄×C₂₃	#124	1	1	yes	characteristic
41	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
42	Dic₂₃	—	8	2	no	—
43	#500	D₂₃	1	1	yes	characteristic
44	#494	—	23	1	no	—
45	#358	—	23	4	no	—
46	C₄×C₁₆	—	23	1	no	—
47	C₂×C₂₃	#486	1	1	yes	characteristic
48	C₂×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
49	C₂×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×Q₁₆	—	23	2	no	—
51	C₂×Q₁₆	—	46	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	46	1	no	—
54	C₂×C₁₆	—	23	2	no	—
55	#107	—	2	1	no	—
56	#107	C₂×D₂₃	1	2	yes	—
57	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
58	C₂₃	#2241	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	46	2	no	—
61	Q₁₆	—	46	4	no	—
62	Q₁₆	—	92	2	no	—
63	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	92	1	no	—
68	C₄⋊C₄	—	92	1	no	—
69	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
70	C₁₆	—	46	2	no	—
71	Q₈	—	8	2	no	—
72	Q₈	—	92	2	no	—
73	Q₈	—	92	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	92	2	no	—
76	C₂×C₄	#67272	1	1	yes	characteristic
77	C₂×C₄	#67272	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
79	C₈	#67274	1	2	yes	—
80	C₈	#67292	1	2	yes	—
81	C₂²	C₂×#67272	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	184	2	no	—
84	C₄	#10625009	1	2	yes	—
85	C₄	#10625011	1	2	yes	—
86	C₄	C₂×#67274	1	1	yes	characteristic
87	C₄	#10625118	1	1	yes	characteristic
88	C₂	#13092765	1	1	yes	characteristic
89	C₂	C₂×#10625009	1	1	yes	characteristic
90	C₂	C₂×#10625011	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67272
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#500	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₈
hypercenter	#500	D₂₃
hypocenter	C₂₃	#2241
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
58. C₂₃	#2241	semidirect

Maximal subgroups

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

subgroup	index
#13092797	2
#13092220	2
C₄×C₂₃⋊C₁₆	2
C₂₃×#500	2
#2241	23