Subgroups — #43252828

Order 28544.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383358	C₂	1	2	yes	maximal
2	#491×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297244	—	2	2	no	—
4	C₄⋊Q₈×C₂₂₃	C₄	1	1	yes	characteristic
5	#125×C₂₂₃	C₄	1	1	yes	characteristic
6	#103×C₂₂₃	—	2	2	no	—
7	#97×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂₂₃⋊C₁₆	—	2	4	no	—
9	C₂×Q₈×C₂₂₃	—	2	1	no	—
10	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
11	C₄⋊C₄×C₂₂₃	—	2	1	no	—
12	C₄⋊C₄×C₂₂₃	—	4	1	no	—
13	C₄⋊C₄×C₂₂₃	—	4	1	no	—
14	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	Q₈×C₂₂₃	—	4	2	no	—
16	C₂×C₄×C₂₂₃	—	2	1	no	—
17	C₂×C₄×C₂₂₃	—	4	1	no	—
18	C₂×C₄×C₂₂₃	—	4	1	no	—
19	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	2	no	—
21	C₂²×C₂₂₃	#100	1	1	yes	characteristic
22	C₄×C₂₂₃	—	4	1	no	—
23	C₄×C₂₂₃	—	8	1	no	—
24	C₄×C₂₂₃	—	8	1	no	—
25	C₄×C₂₂₃	#104	1	2	yes	—
26	C₂×C₂₂₃	—	2	1	no	—
27	C₂×C₂₂₃	#329	1	1	yes	characteristic
28	C₂₂₃	#1356	1	1	yes	characteristic, complemented, Sylow
29	#1356	—	223	1	no	maximal, Sylow
30	#491	D₂₂₃	1	1	yes	characteristic
31	#347	—	223	2	no	—
32	C₄⋊Q₈	Dic₂₂₃	1	1	yes	characteristic
33	#125	Dic₂₂₃	1	1	yes	characteristic
34	M₃₂	—	446	2	no	—
35	#103	—	2	2	no	—
36	#97	C₂×D₂₂₃	1	1	yes	characteristic
37	C₂×Q₈	—	2	1	no	—
38	C₂×C₈	C₂²⋊D₂₂₃	1	2	yes	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	4	1	no	—
42	C₄²	C₂×Dic₂₂₃	1	1	yes	characteristic
43	C₁₆	—	446	4	no	—
44	Q₈	—	4	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₂²	#20297221	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	8	1	no	—
54	C₄	#20297231	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#28383320	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383320
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	#97	C₂×D₂₂₃
Fitting	#491×C₂₂₃	C₂
socle	C₂×C₂₂₃	#329
prime radical	C₂²×C₂₂₃	#100
cosocle	#97×C₂₂₃	C₂²
hypercenter	#491	D₂₂₃
hypocenter	C₂₂₃	#1356
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #491

Complements

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

N	H ≅ G/N	type
28. C₂₂₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
#28383358	2
#491×C₂₂₃	2
#1356	223