Subgroups — #15671570

Order 2944.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092216	C₂	1	2	yes	maximal
2	C₂₃×#491	C₂	1	1	yes	maximal, characteristic
3	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
4	C₂₃×#125	C₄	1	1	yes	characteristic
5	C₂₃×#103	—	2	2	no	—
6	C₂₃×#97	C₄	1	1	yes	characteristic
7	#10625018	—	2	2	no	—
8	C₂×Q₈×C₂₃	D₄	1	2	yes	—
9	C₂×C₈×C₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₄⋊C₄×C₂₃	—	4	1	no	—
12	C₄⋊C₄×C₂₃	—	4	1	no	—
13	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
14	#67267	—	4	2	no	—
15	Q₈×C₂₃	—	4	2	no	—
16	C₂×C₄×C₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	—	2	1	no	—
18	C₂×C₄×C₂₃	—	4	1	no	—
19	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₃	—	4	1	no	—
21	C₂₃⋊C₈	—	8	2	no	—
22	#1413	—	23	1	no	maximal, Sylow
23	C₂²×C₂₃	#99	1	1	yes	characteristic
24	C₄×C₂₃	—	2	1	no	—
25	C₄×C₂₃	—	4	1	no	—
26	C₄×C₂₃	—	4	2	no	—
27	C₄×C₂₃	—	8	1	no	—
28	#491	D₂₃	1	1	yes	characteristic
29	#356	—	23	2	no	—
30	C₂×C₂₃	—	2	1	no	—
31	C₂×C₂₃	#354	1	1	yes	characteristic
32	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
33	#125	Dic₂₃	1	1	yes	characteristic
34	#103	—	2	2	no	—
35	#101	—	46	2	no	—
36	#97	Dic₂₃	1	1	yes	characteristic
37	C₂₃	#1413	1	1	yes	characteristic, complemented, Sylow
38	C₂×Q₈	C₂²⋊D₂₃	1	2	yes	—
39	M₁₆	—	92	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
45	Q₈	—	4	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	184	2	no	—
52	C₂²	#10625016	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	4	2	no	—
56	C₄	—	8	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#13092212	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092212
commutator	C₄²×C₂₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₃
Fitting	C₂₃×#491	C₂
socle	C₂×C₂₃	#354
prime radical	C₂²×C₂₃	#99
cosocle	C₂₃×C₄⋊Q₈	C₂²
hypercenter	#491	D₂₃
hypocenter	C₂₃	#1413
LCS	G ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #491

Complements

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

N	H ≅ G/N	type
37. C₂₃	#1413	semidirect

Maximal subgroups

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

subgroup	index
#13092216	2
C₂₃×#491	2
#1413	23