Subgroups — #28383499

Order 14272.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₂₃⋊C₄	C₂	1	3	yes	maximal
2	C₂₂₃⋊#95	C₂	1	3	yes	maximal
3	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
5	C₂²×Dic₂₂₃	C₂²	1	3	yes	—
6	Dic₂₂₃⋊C₄	—	2	6	no	—
7	C₂×Q₈×C₂₂₃	—	2	6	no	—
8	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
9	C₂×Dic₂₂₃	—	2	3	no	—
10	C₂×Dic₂₂₃	—	2	3	no	—
11	C₂×Dic₂₂₃	—	2	3	no	—
12	C₂×Dic₂₂₃	Q₈	1	6	yes	complemented
13	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
14	Q₈×C₂₂₃	—	2	8	no	—
15	C₂×C₄×C₂₂₃	—	2	6	no	—
16	C₂×C₄×C₂₂₃	D₄	1	6	yes	—
17	C₂²×C₂₂₃	C₂×D₄	1	3	yes	—
18	C₂²×C₂₂₃	C₂×Q₈	1	3	yes	—
19	C₂²×C₂₂₃	#41	1	1	yes	characteristic
20	C₄×C₂₂₃	—	2	6	no	—
21	Dic₂₂₃	—	2	6	no	—
22	Dic₂₂₃	—	4	1	no	—
23	C₂×C₂₂₃	#120	1	1	yes	characteristic
24	C₂×C₂₂₃	C₂²⋊Q₈	1	3	yes	—
25	C₂×C₂₂₃	C₄⋊Q₈	1	3	yes	—
26	C₂₂₃	#395	1	1	yes	characteristic, complemented, Sylow
27	#395	—	223	1	no	maximal, Sylow
28	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic
29	C₂×C₄⋊C₄	—	223	3	no	—
30	#95	—	223	3	no	—
31	C₂×Q₈	—	2	6	no	—
32	C₂²×C₄	—	223	1	no	—
33	C₂²×C₄	—	223	3	no	—
34	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
35	C₄⋊C₄	—	446	6	no	—
36	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
37	Q₈	—	2	8	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	—	446	3	no	—
40	C₂×C₄	—	446	3	no	—
41	C₂×C₄	—	446	3	no	—
42	C₂×C₄	—	223	6	no	—
43	C₂×C₄	C₂²⋊D₂₂₃	1	6	yes	—
44	C₂²	C₂×C₂²⋊D₂₂₃	1	3	yes	—
45	C₂²	Q₈×D₂₂₃	1	3	yes	—
46	C₂²	#16813171	1	1	yes	characteristic
47	C₄	—	2	6	no	—
48	C₄	—	892	1	no	—
49	C₄	—	446	6	no	—
50	C₂	#20297273	1	1	yes	characteristic
51	C₂	#20297280	1	3	yes	—
52	C₂	#20297302	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator, socle, prime radical, cosocle	C₂³×C₂₂₃	C₂³
Fitting	C₂²×Q₈×C₂₂₃	C₂
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	#395
LCS	G ▷ C₂³×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
12. C₂×Dic₂₂₃	Q₈	semidirect
26. C₂₂₃	#395	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₂₃⋊C₄	2
C₂₂₃⋊#95	2
C₂²×Q₈×C₂₂₃	2
#395	223