Subgroups — #13093252

Order 1472.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625161	C₂	1	6	yes	maximal, complemented
3	#41×D₂₃	C₂	1	6	yes	maximal, complemented
4	#10625155	C₂	1	9	yes	maximal, complemented
5	C₂×#67290	C₂	1	9	yes	maximal, complemented
6	#41×C₂₃	C₂²	1	6	yes	complemented
7	C₂×D₄×C₂₃	C₂²	1	9	yes	complemented
8	C₂×C₂²⋊D₂₃	C₂²	1	18	yes	complemented
9	C₂²×Dic₂₃	C₂²	1	6	yes	complemented
10	#67292	C₂²	1	2	yes	complemented
11	Q₈×D₂₃	C₂²	1	6	yes	complemented
12	#67290	C₂²	1	18	yes	complemented
13	#67290	C₂²	1	36	yes	complemented
14	D₄×D₂₃	C₂²	1	18	yes	complemented
15	#67288	C₂²	1	18	yes	complemented
16	C₂×C₄×D₂₃	C₂²	1	9	yes	complemented
17	C₂×Dic₄₆	C₂²	1	9	yes	complemented
18	C₂³×C₂₃	C₂³	1	6	yes	—
19	C₂²×D₂₃	C₂³	1	9	yes	—
20	Q₈×C₂₃	C₂³	1	2	yes	—
21	D₄×C₂₃	C₂³	1	18	yes	—
22	C₂×C₄×C₂₃	C₂³	1	9	yes	—
23	C₂²⋊D₂₃	C₂³	1	36	yes	—
24	C₂×Dic₂₃	C₂³	1	18	yes	—
25	C₂×Dic₂₃	C₂³	1	9	yes	—
26	D₉₂	C₂³	1	6	yes	—
27	C₄×D₂₃	C₂³	1	6	yes	—
28	C₄×D₂₃	C₂³	1	18	yes	—
29	Dic₄₆	C₂³	1	18	yes	—
30	C₂²×C₂₃	—	4	6	no	—
31	C₂²×C₂₃	C₂⁴	1	9	yes	—
32	C₂×D₂₃	—	4	9	no	—
33	C₂×D₂₃	C₂⁴	1	6	yes	—
34	C₄×C₂₃	C₂⁴	1	6	yes	—
35	Dic₂₃	C₂⁴	1	9	yes	—
36	Dic₂₃	C₂⁴	1	1	yes	characteristic
37	#585	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	—	2	9	no	—
39	C₂×C₂₃	C₂⁵	1	1	yes	characteristic
40	D₂₃	—	2	6	no	—
41	2₋¹⁺⁴	—	23	6	no	—
42	H(5,𝔽₂)	—	23	9	no	—
43	H(5,𝔽₂)	D₂₃	1	1	yes	characteristic, complemented
44	C₂×#41	—	23	6	no	—
45	C₂×#41	—	23	9	no	—
46	C₂₃	#585	1	1	yes	characteristic, complemented, Sylow
47	#41	—	23	2	no	—
48	#41	—	23	18	no	—
49	#41	—	23	36	no	—
50	#41	—	23	18	no	—
51	#41	C₂×D₂₃	1	6	yes	complemented
52	C₂×Q₈	—	23	6	no	—
53	C₂×Q₈	—	23	9	no	—
54	C₂×D₄	—	23	18	no	—
55	C₂×D₄	—	23	18	no	—
56	C₂×D₄	C₂×D₂₃	1	9	yes	complemented
57	C₂²×C₄	—	23	6	no	—
58	C₂²×C₄	—	23	9	no	—
59	C₂³	—	23	9	no	—
60	C₂³	C₂²×D₂₃	1	6	yes	—
61	Q₈	—	23	18	no	—
62	Q₈	C₂²×D₂₃	1	2	yes	—
63	D₄	—	23	36	no	—
64	D₄	—	23	6	no	—
65	D₄	C₂²×D₂₃	1	18	yes	—
66	C₂×C₄	—	23	9	no	—
67	C₂×C₄	—	23	18	no	—
68	C₂×C₄	—	23	6	no	—
69	C₂×C₄	—	23	18	no	—
70	C₂×C₄	C₂²×D₂₃	1	9	yes	—
71	C₂²	—	4	6	no	—
72	C₂²	—	23	6	no	—
73	C₂²	—	92	9	no	—
74	C₂²	C₂³×D₂₃	1	9	yes	—
75	C₄	—	23	1	no	—
76	C₄	—	23	9	no	—
77	C₄	C₂³×D₂₃	1	6	yes	—
78	C₂	—	2	9	no	—
79	C₂	—	46	6	no	—
80	C₂	C₂⁴×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₂₃
commutator, socle, cosocle	C₂×C₂₃	C₂⁵
Fitting	C₂₃×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₂₃
hypocenter	C₂₃	#585
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂₃×H(5,𝔽₂)	C₂	semidirect
2. #10625161	C₂	semidirect
3. #41×D₂₃	C₂	semidirect
4. #10625155	C₂	semidirect
5. C₂×#67290	C₂	semidirect
6. #41×C₂₃	C₂²	semidirect
7. C₂×D₄×C₂₃	C₂²	semidirect
8. C₂×C₂²⋊D₂₃	C₂²	semidirect
9. C₂²×Dic₂₃	C₂²	semidirect
10. #67292	C₂²	semidirect
11. Q₈×D₂₃	C₂²	semidirect
12. #67290	C₂²	semidirect
13. #67290	C₂²	semidirect
14. D₄×D₂₃	C₂²	semidirect
15. #67288	C₂²	semidirect
16. C₂×C₄×D₂₃	C₂²	semidirect
17. C₂×Dic₄₆	C₂²	semidirect
43. H(5,𝔽₂)	D₂₃	semidirect
46. C₂₃	#585	semidirect
51. #41	C₂×D₂₃	semidirect
56. C₂×D₄	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×H(5,𝔽₂)	2
#10625161	2
#41×D₂₃	2
#10625155	2
C₂×#67290	2
#585	23