Subgroups — #13093043

Order 1472.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal
3	#10625095	C₂	1	4	yes	maximal, complemented
4	#10625049	C₂	1	8	yes	maximal, complemented
5	C₂⁴×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
7	C₂²×Dic₂₃	C₂²	1	4	yes	—
8	C₂×Dic₄₆	C₂²	1	4	yes	complemented
9	C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
11	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
12	C₄⋊Dic₂₃	C₂²	1	4	yes	complemented
13	Dic₂₃⋊C₄	C₂²	1	8	yes	complemented
14	C₂³×C₂₃	—	4	2	no	—
15	C₂³×C₂₃	Q₈	1	4	yes	complemented
16	C₂³×C₂₃	C₂³	1	1	yes	characteristic
17	C₂³×C₂₃	C₂³	1	2	yes	—
18	C₂×C₄×C₂₃	—	4	2	no	—
19	C₂×C₄×C₂₃	C₂³	1	4	yes	—
20	C₂×Dic₂₃	—	4	4	no	—
21	C₂×Dic₂₃	C₂³	1	8	yes	—
22	Dic₄₆	—	4	4	no	—
23	C₂²×C₂₃	—	2	4	no	—
24	C₂²×C₂₃	—	2	2	no	—
25	C₂²×C₂₃	—	4	4	no	—
26	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
27	C₂²×C₂₃	C₂×Q₈	1	4	yes	—
28	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₃	—	2	4	no	—
30	Dic₂₃	—	2	8	no	—
31	#543	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	—	2	4	no	—
33	C₂×C₂₃	—	2	2	no	—
34	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
36	C₂²⋊Q₈	—	23	4	no	—
37	C₂²⋊Q₈	—	23	8	no	—
38	C₂×C₂²⋊C₄	—	23	2	no	—
39	C₂×C₂²⋊C₄	D₂₃	1	1	yes	characteristic
40	C₂₃	#543	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	C₂×D₂₃	1	1	yes	characteristic
42	C₂×Q₈	—	23	4	no	—
43	C₂²×C₄	—	23	4	no	—
44	C₂²×C₄	C₂×D₂₃	1	2	yes	—
45	C₄⋊C₄	—	23	8	no	—
46	C₄⋊C₄	—	23	4	no	—
47	C₂²⋊C₄	—	23	4	no	—
48	C₂²⋊C₄	—	23	4	no	—
49	C₂²⋊C₄	C₂×D₂₃	1	4	yes	—
50	C₂³	—	4	2	no	—
51	C₂³	Dic₄₆	1	4	yes	complemented
52	C₂³	C₂²×D₂₃	1	1	yes	characteristic
53	C₂³	C₂²×D₂₃	1	2	yes	—
54	Q₈	—	92	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	23	8	no	—
57	C₂×C₄	—	92	4	no	—
58	C₂×C₄	C₂²×D₂₃	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₄₆	1	2	yes	—
63	C₂²	C₂×Dic₄₆	1	4	yes	—
64	C₂²	C₂³×D₂₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	46	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₄₆	1	1	yes	characteristic
70	C₂	#10625155	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₂₃	C₂
prime radical	C₂⁴×C₂₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₂₃
hypocenter	C₂₃	#543
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #10625095	C₂	semidirect
4. #10625049	C₂	semidirect
8. C₂×Dic₄₆	C₂²	semidirect
12. C₄⋊Dic₂₃	C₂²	semidirect
13. Dic₂₃⋊C₄	C₂²	semidirect
15. C₂³×C₂₃	Q₈	semidirect
40. C₂₃	#543	semidirect
51. C₂³	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊C₄×C₂₃	2
C₂×C₂²⋊Dic₂₃	2
#10625095	2
#10625049	2
#543	23