Subgroups — #43252966

Order 28544.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383379	C₂	1	2	yes	maximal
2	#491×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297223	—	2	2	no	—
4	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
5	#125×C₂₂₃	C₄	1	1	yes	characteristic
6	#103×C₂₂₃	—	2	2	no	—
7	#97×C₂₂₃	C₄	1	1	yes	characteristic
8	#16813154	—	4	2	no	—
9	C₂×Q₈×C₂₂₃	D₄	1	2	yes	—
10	C₂×C₈×C₂₂₃	—	2	1	no	—
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	C₂₂₃⋊C₈	—	8	2	no	—
16	Q₈×C₂₂₃	—	4	2	no	—
17	C₂×C₄×C₂₂₃	—	2	1	no	—
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	—	4	1	no	—
20	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₈×C₂₂₃	—	4	1	no	—
22	C₂²×C₂₂₃	#99	1	1	yes	characteristic
23	C₄×C₂₂₃	—	2	1	no	—
24	C₄×C₂₂₃	—	4	1	no	—
25	C₄×C₂₂₃	—	4	2	no	—
26	C₄×C₂₂₃	—	8	1	no	—
27	C₂×C₂₂₃	—	2	1	no	—
28	C₂×C₂₂₃	#354	1	1	yes	characteristic
29	C₂₂₃	#1413	1	1	yes	characteristic, complemented, Sylow
30	#1413	—	223	1	no	maximal, Sylow
31	#491	D₂₂₃	1	1	yes	characteristic
32	#356	—	223	2	no	—
33	C₄⋊Q₈	C₂×D₂₂₃	1	1	yes	characteristic
34	#125	Dic₂₂₃	1	1	yes	characteristic
35	#103	—	2	2	no	—
36	#101	—	446	2	no	—
37	#97	Dic₂₂₃	1	1	yes	characteristic
38	C₂×Q₈	C₂²⋊D₂₂₃	1	2	yes	—
39	M₁₆	—	892	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×Dic₂₂₃	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⋊Dic₂₂₃	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	1784	2	no	—
52	C₂²	#20297219	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	#28383375	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383375
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₂₃
Fitting	#491×C₂₂₃	C₂
socle	C₂×C₂₂₃	#354
prime radical	C₂²×C₂₂₃	#99
cosocle	C₄⋊Q₈×C₂₂₃	C₂²
hypercenter	#491	D₂₂₃
hypocenter	C₂₂₃	#1413
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
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
29. C₂₂₃	#1413	semidirect

Maximal subgroups

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

subgroup	index
#28383379	2
#491×C₂₂₃	2
#1413	223