Subgroups — #28383674

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₉₂	C₂	1	1	yes	maximal, characteristic
2	#20297269	C₂	1	2	yes	maximal
3	#20297227	C₂	1	2	yes	maximal
4	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#107×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₄₆	C₂²	1	2	yes	—
7	Dic₈₉₂	C₄	1	4	yes	complemented
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	Dic₂₂₃⋊C₄	—	2	2	no	—
10	C₄×Dic₂₂₃	—	2	2	no	—
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	Dic₄₄₆	—	2	2	no	—
15	Dic₄₄₆	C₂×C₄	1	4	yes	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
18	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
19	C₂₂₃⋊C₈	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₈×C₂₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₂₃	—	4	2	no	—
25	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₂₂₃	#41	1	1	yes	characteristic
27	Dic₂₂₃	—	2	4	no	—
28	Dic₂₂₃	—	2	1	no	—
29	Dic₂₂₃	Q₁₆	1	2	yes	complemented
30	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₂₂₃	#135	1	1	yes	characteristic
33	C₂₂₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	223	1	no	maximal, Sylow
35	C₂×Q₁₆	—	223	1	no	—
36	C₄×Q₈	—	223	2	no	—
37	#107	D₂₂₃	1	1	yes	characteristic
38	#103	—	223	2	no	—
39	C₄×C₈	—	223	1	no	—
40	C₂×Q₈	—	223	2	no	—
41	Q₁₆	—	223	4	no	—
42	C₂×C₈	—	223	1	no	—
43	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
44	C₄⋊C₄	—	446	2	no	—
45	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
46	C₄²	—	446	2	no	—
47	C₄²	—	223	1	no	—
48	Q₈	—	446	2	no	—
49	Q₈	—	223	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	446	2	no	—
52	C₂×C₄	—	223	1	no	—
53	C₂×C₄	—	223	1	no	—
54	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
55	C₈	—	446	1	no	—
56	C₈	C₄×D₂₂₃	1	2	yes	—
57	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	446	4	no	—
60	C₄	—	446	1	no	—
61	C₄	—	223	2	no	—
62	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
63	C₄	#16813171	1	1	yes	characteristic
64	C₂	#20297266	1	1	yes	characteristic
65	C₂	Q₁₆×D₂₂₃	1	1	yes	characteristic
66	C₂	#20297323	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#107×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#107	D₂₂₃
hypocenter	C₂₂₃	C₄×Q₁₆
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₈₉₂	C₄	semidirect
29. Dic₂₂₃	Q₁₆	semidirect
33. C₂₂₃	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₉₂	2
#20297269	2
#20297227	2
C₈×Dic₂₂₃	2
#107×C₂₂₃	2
C₄×Q₁₆	223