Subgroups — #28384022

Order 14272.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297327	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297250	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297249	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297247	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297243	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813171	—	2	1	no	—
9	Q₈×D₂₂₃	—	2	1	no	—
10	Dic₈₉₂	C₂²	1	1	yes	characteristic
11	#16813160	—	2	1	no	—
12	#16813157	—	2	1	no	—
13	D₁₇₈₄	C₂²	1	1	yes	characteristic
14	C₈×D₂₂₃	C₂²	1	1	yes	characteristic
15	C₂₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
17	Q₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
19	Dic₄₄₆	—	2	1	no	—
20	Dic₄₄₆	—	4	1	no	—
21	D₈₉₂	—	2	1	no	—
22	D₈₉₂	—	4	1	no	—
23	C₄×D₂₂₃	—	4	1	no	—
24	C₄×D₂₂₃	—	4	1	no	—
25	C₄×D₂₂₃	D₄	1	1	yes	characteristic
26	C₂₂₃⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₂₂₃	—	2	1	no	—
28	Q₈×C₂₂₃	—	2	1	no	—
29	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₂×D₂₂₃	—	4	1	no	—
31	C₂×D₂₂₃	D₈	1	1	yes	characteristic
32	C₄×C₂₂₃	—	4	1	no	—
33	C₄×C₂₂₃	—	4	1	no	—
34	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	Dic₂₂₃	—	4	1	no	—
36	Dic₂₂₃	D₈	1	1	yes	characteristic
37	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
38	D₂₂₃	—	2	1	no	—
39	D₂₂₃	—	8	1	no	—
40	C₂₂₃	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	223	1	no	maximal, Sylow
42	#135	—	223	1	no	—
43	C₂×Q₁₆	—	223	1	no	—
44	Q₃₂	—	223	1	no	—
45	Q₃₂	D₂₂₃	1	1	yes	characteristic, complemented
46	SD₃₂	—	223	1	no	—
47	SD₃₂	—	223	1	no	—
48	M₃₂	—	223	1	no	—
49	#41	—	446	1	no	—
50	C₂×Q₈	—	446	1	no	—
51	Q₁₆	—	446	1	no	—
52	Q₁₆	—	223	1	no	—
53	Q₁₆	C₂×D₂₂₃	1	1	yes	characteristic
54	Q₁₆	C₂×D₂₂₃	1	1	yes	characteristic
55	SD₁₆	—	446	1	no	—
56	D₈	—	223	1	no	—
57	C₂×C₈	—	223	1	no	—
58	C₁₆	—	223	1	no	—
59	C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	892	1	no	—
63	Q₈	—	446	1	no	—
64	D₄	—	892	1	no	—
65	D₄	—	446	1	no	—
66	C₂×C₄	—	892	1	no	—
67	C₂×C₄	—	892	1	no	—
68	C₂×C₄	—	223	1	no	—
69	C₈	—	223	1	no	—
70	C₈	C₂²×D₂₂₃	1	1	yes	characteristic
71	C₂²	—	892	1	no	—
72	C₂²	—	223	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	892	1	no	—
76	C₄	—	223	1	no	—
77	C₄	D₄×D₂₂₃	1	1	yes	characteristic
78	C₂	—	446	1	no	—
79	C₂	—	1784	1	no	—
80	C₂	D₈×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₂₃
commutator, cosocle	C₈×C₂₂₃	C₂³
Frattini	C₈	C₂²×D₂₂₃
Fitting	Q₃₂×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂×D₈
prime radical	#20297327	C₂
hypercenter	Q₃₂	D₂₂₃
hypocenter	C₂₂₃	#510
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₂₂₃	C₂	semidirect
3. #20297250	C₂	semidirect
4. #20297249	C₂	semidirect
5. #20297247	C₂	semidirect
6. #20297243	C₂	semidirect
7. Q₃₂×C₂₂₃	C₂	semidirect
40. C₂₂₃	#510	semidirect
45. Q₃₂	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297327	2
Q₁₆×D₂₂₃	2
#20297250	2
#20297249	2
#20297247	2
#20297243	2
Q₃₂×C₂₂₃	2
#510	223