Subgroups — #28384309

Order 14272.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297304	C₂	1	3	yes	maximal
2	#20297295	C₂	1	6	yes	maximal
3	Q₈×Dic₂₂₃	C₂	1	2	yes	maximal
4	C₄×Dic₄₄₆	C₂	1	3	yes	maximal
5	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₄₆	C₂²	1	3	yes	—
7	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₂₃	C₂²	1	6	yes	—
9	C₄⋊Dic₂₂₃	C₂²	1	6	yes	—
10	Dic₂₂₃⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₂₂₃	C₂²	1	6	yes	—
12	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
14	C₄²×C₂₂₃	C₂²	1	3	yes	—
15	Dic₄₄₆	—	2	6	no	—
16	C₂×Dic₂₂₃	C₂³	1	2	yes	—
17	C₂×Dic₂₂₃	C₂³	1	6	yes	—
18	Q₈×C₂₂₃	Q₈	1	4	yes	complemented
19	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
20	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
21	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₂₃	—	2	3	no	—
24	C₄×C₂₂₃	C₂×Q₈	1	6	yes	—
25	C₄×C₂₂₃	#41	1	2	yes	—
26	Dic₂₂₃	—	2	6	no	—
27	Dic₂₂₃	—	2	2	no	—
28	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
30	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
31	C₂₂₃	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	223	1	no	maximal, Sylow
33	C₄⋊Q₈	—	223	3	no	—
34	#125	—	223	6	no	—
35	C₄×Q₈	—	223	2	no	—
36	C₄×Q₈	—	223	3	no	—
37	C₄×Q₈	D₂₂₃	1	1	yes	characteristic
38	C₂×Q₈	—	223	3	no	—
39	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
40	C₄⋊C₄	—	223	6	no	—
41	C₄⋊C₄	—	223	1	no	—
42	C₄⋊C₄	—	223	6	no	—
43	C₄⋊C₄	—	223	6	no	—
44	C₄⋊C₄	C₂×D₂₂₃	1	3	yes	—
45	C₄²	—	223	6	no	—
46	C₄²	C₂×D₂₂₃	1	3	yes	—
47	Q₈	—	446	6	no	—
48	Q₈	Dic₄₄₆	1	4	yes	complemented
49	C₂×C₄	—	223	2	no	—
50	C₂×C₄	—	223	6	no	—
51	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
52	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
53	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
54	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	446	6	no	—
57	C₄	—	446	2	no	—
58	C₄	C₂×Dic₄₄₆	1	6	yes	—
59	C₄	#16813171	1	2	yes	—
60	C₂	C₂²×Dic₄₄₆	1	1	yes	characteristic
61	C₂	C₂×#16813171	1	1	yes	characteristic
62	C₂	#20297351	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₄×Q₈×C₂₂₃	C₂
hypercenter	C₄×Q₈	D₂₂₃
hypocenter	C₂₂₃	#557
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₂₂₃	Q₈	semidirect
31. C₂₂₃	#557	semidirect
48. Q₈	Dic₄₄₆	semidirect

Maximal subgroups

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

subgroup	index
#20297304	2
#20297295	2
Q₈×Dic₂₂₃	2
C₄×Dic₄₄₆	2
C₄×Q₈×C₂₂₃	2
#557	223