Subgroups — #15673540

Order 2944.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092172	C₂	1	6	yes	maximal
2	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₂₃	C₄	1	3	yes	—
4	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
5	C₂×C₄²×C₂₃	C₂²	1	3	yes	—
6	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
7	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
8	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
9	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
10	C₄²×C₂₃	—	2	6	no	—
11	C₄²×C₂₃	—	2	6	no	—
12	C₄²×C₂₃	C₈	1	4	yes	complemented
13	C₂×C₂₃⋊C₈	—	2	12	no	—
14	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₃	—	2	3	no	—
16	C₂×C₄×C₂₃	—	2	3	no	—
17	C₂×C₄×C₂₃	—	2	3	no	—
18	C₂×C₄×C₂₃	—	2	3	no	—
19	C₂×C₄×C₂₃	C₂×C₈	1	6	yes	—
20	C₂×C₄×C₂₃	M₁₆	1	6	yes	—
21	C₂×C₄×C₂₃	#41	1	6	yes	—
22	C₂₃⋊C₈	—	4	4	no	—
23	#1839	—	23	1	no	maximal, Sylow
24	C₂²×C₂₃	#117	1	3	yes	—
25	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
26	C₂²×C₂₃	C₂×M₁₆	1	3	yes	—
27	C₄×C₂₃	—	2	6	no	—
28	C₄×C₂₃	—	2	6	no	—
29	C₄×C₂₃	#126	1	4	yes	—
30	C₄³	D₂₃	1	1	yes	characteristic
31	#336	—	23	6	no	—
32	C₂×C₂₃	#383	1	1	yes	characteristic
33	C₂×C₂₃	#431	1	3	yes	—
34	C₂×C₂₃	#432	1	3	yes	—
35	C₂²×C₈	—	23	4	no	—
36	C₂×C₄²	Dic₂₃	1	3	yes	—
37	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
38	C₂×C₄²	C₂×D₂₃	1	3	yes	—
39	C₂₃	#1839	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₄	C₂×Dic₂₃	1	3	yes	—
41	C₂²×C₄	C₂×Dic₂₃	1	3	yes	—
42	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
43	C₂×C₈	—	46	12	no	—
44	C₄²	—	2	6	no	—
45	C₄²	—	2	6	no	—
46	C₄²	C₂₃⋊C₈	1	4	yes	complemented
47	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	C₂×C₂₃⋊C₈	1	6	yes	—
53	C₂×C₄	#67267	1	6	yes	—
54	C₂×C₄	#67288	1	6	yes	—
55	C₈	—	92	4	no	—
56	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
57	C₂²	C₂×#67267	1	3	yes	—
58	C₂²	#10625097	1	3	yes	—
59	C₄	—	2	6	no	—
60	C₄	—	2	6	no	—
61	C₄	#10625047	1	4	yes	—
62	C₂	#13092285	1	1	yes	characteristic
63	C₂	#13092470	1	3	yes	—
64	C₂	#13092474	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₄³×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄³	D₂₃
hypocenter	C₂₃	#1839
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
12. C₄²×C₂₃	C₈	semidirect
39. C₂₃	#1839	semidirect
46. C₄²	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#13092172	2
C₄³×C₂₃	2
#1839	23