Subgroups — #28383931

Order 14272.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297304	C₂	1	1	yes	maximal, characteristic
2	#20297304	C₂	1	1	yes	maximal, characteristic
3	#20297229	C₂	1	4	yes	maximal
4	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₄₆	—	2	1	no	—
6	C₂×Dic₄₄₆	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₃	—	2	1	no	—
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	—	2	1	no	—
10	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
11	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
12	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	Dic₄₄₆	—	2	4	no	—
14	Dic₄₄₆	—	4	2	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	2	no	—
21	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₂₃	SD₁₆	1	2	yes	—
23	C₄×C₂₂₃	Q₁₆	1	2	yes	—
24	C₄×C₂₂₃	#41	1	2	yes	—
25	Dic₂₂₃	—	4	2	no	—
26	Dic₂₂₃	—	4	2	no	—
27	C₂×C₂₂₃	#124	1	1	yes	characteristic
28	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
30	C₂₂₃	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	223	1	no	maximal, Sylow
32	C₄⋊Q₈	—	223	1	no	—
33	C₄⋊Q₈	—	223	1	no	—
34	#103	—	223	4	no	—
35	C₄×C₈	D₂₂₃	1	1	yes	characteristic
36	C₂×Q₈	—	446	1	no	—
37	C₂×Q₈	—	223	2	no	—
38	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
39	C₄⋊C₄	—	446	1	no	—
40	C₄⋊C₄	—	446	1	no	—
41	C₄⋊C₄	—	446	1	no	—
42	C₄⋊C₄	—	223	2	no	—
43	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
44	Q₈	—	892	2	no	—
45	Q₈	—	446	4	no	—
46	C₂×C₄	—	446	2	no	—
47	C₂×C₄	—	446	2	no	—
48	C₂×C₄	D₈₉₂	1	1	yes	characteristic
49	C₂×C₄	D₈₉₂	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
53	C₄	—	892	2	no	—
54	C₄	—	892	2	no	—
55	C₄	#16813159	1	2	yes	—
56	C₄	Dic₈₉₂	1	2	yes	—
57	C₄	#16813170	1	2	yes	—
58	C₂	#20297292	1	1	yes	characteristic
59	C₂	C₂×#16813159	1	1	yes	characteristic
60	C₂	C₂×Dic₈₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₉₂
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×C₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
hypercenter	C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#487
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
30. C₂₂₃	#487	semidirect

Maximal subgroups

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

subgroup	index
#20297304	2
#20297229	2
C₄×C₈×C₂₂₃	2
#487	223