Subgroups — #28383913

Order 14272.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#20297237	C₂	1	2	yes	maximal, complemented
3	#20297228	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	—	2	1	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	—	2	1	no	—
13	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	4	2	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
19	Q₈×C₂₂₃	—	2	2	no	—
20	C₂×C₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	—	2	1	no	—
22	C₂×C₄×C₂₂₃	—	2	1	no	—
23	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	—	2	1	no	—
27	C₂²×C₂₂₃	Q₁₆	1	2	yes	—
28	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₃	—	2	1	no	—
30	C₄×C₂₂₃	—	4	1	no	—
31	C₄×C₂₂₃	—	4	1	no	—
32	C₄×C₂₂₃	#41	1	2	yes	—
33	Dic₂₂₃	—	4	2	no	—
34	C₂×C₂₂₃	—	2	2	no	—
35	C₂×C₂₂₃	#123	1	1	yes	characteristic
36	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₂₂₃	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	223	1	no	maximal, Sylow
40	C₂²⋊Q₈	D₂₂₃	1	1	yes	characteristic
41	C₂×C₄⋊C₄	—	223	1	no	—
42	#107	—	223	2	no	—
43	#103	—	223	2	no	—
44	C₂²⋊C₈	—	223	1	no	—
45	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
46	C₂²×C₄	—	446	1	no	—
47	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×C₈	—	223	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	446	1	no	—
51	C₄⋊C₄	—	223	2	no	—
52	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	892	2	no	—
60	C₂×C₄	—	446	2	no	—
61	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
63	C₈	—	446	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#16813160	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	892	2	no	—
72	C₄	#16813169	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20297285	1	1	yes	characteristic
75	C₂	C₂×#16813160	1	1	yes	characteristic
76	C₂	#20297329	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂²⋊Q₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂³×C₂₂₃	D₄
hypercenter	C₂²⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#484
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20297237	C₂	semidirect
3. #20297228	C₂	semidirect
38. C₂₂₃	#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₂₃	2
#20297237	2
#20297228	2
C₂²⋊(C₂₂₃⋊C₈)	2
C₂²⋊Q₈×C₂₂₃	2
#484	223