Subgroups — #43252846

Order 28544.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383982	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#334×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297296	—	2	1	no	—
5	C₂×C₂₂₃⋊C₁₆	—	2	1	no	—
6	#20297238	C₄	1	1	yes	characteristic
7	#20297238	C₄	1	1	yes	characteristic
8	#20297236	—	2	1	no	—
9	C₄⋊C₈×C₂₂₃	—	2	1	no	—
10	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	—	2	1	no	—
12	C₄⋊Dic₂₂₃	—	2	1	no	—
13	Dic₂₂₃⋊C₄	—	4	1	no	—
14	Dic₂₂₃⋊C₄	—	4	1	no	—
15	C₂₂₃⋊C₁₆	—	4	1	no	—
16	C₂×C₈×C₂₂₃	—	4	1	no	—
17	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
19	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	—	4	1	no	—
21	C₂×Dic₂₂₃	—	4	1	no	—
22	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
25	C₈×C₂₂₃	—	2	1	no	—
26	C₈×C₂₂₃	—	8	1	no	—
27	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
28	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
29	C₂²×C₂₂₃	#102	1	1	yes	characteristic
30	C₄×C₂₂₃	—	2	1	no	—
31	C₄×C₂₂₃	—	2	1	no	—
32	C₄×C₂₂₃	#101	1	1	yes	characteristic
33	C₄×C₂₂₃	#103	1	1	yes	characteristic
34	Dic₂₂₃	—	8	1	no	—
35	Dic₂₂₃	—	8	1	no	—
36	C₂×C₂₂₃	#332	1	1	yes	characteristic
37	C₂×C₂₂₃	#359	1	1	yes	characteristic
38	C₂×C₂₂₃	#360	1	1	yes	characteristic
39	C₂₂₃	#1363	1	1	yes	characteristic, complemented, Sylow
40	#1363	—	223	1	no	maximal, Sylow
41	#499	—	223	1	no	—
42	C₄⋊C₁₆	—	223	1	no	—
43	#334	D₂₂₃	1	1	yes	characteristic
44	#125	—	446	1	no	—
45	C₂×C₁₆	—	446	1	no	—
46	#107	—	223	1	no	—
47	#107	—	223	1	no	—
48	#106	—	446	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
53	C₂×C₈	D₈₉₂	1	1	yes	characteristic
54	C₄⋊C₄	—	892	1	no	—
55	C₄⋊C₄	—	892	1	no	—
56	C₄⋊C₄	—	446	1	no	—
57	C₄⋊C₄	—	446	1	no	—
58	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
59	C₁₆	—	892	1	no	—
60	C₂×C₄	—	892	1	no	—
61	C₂×C₄	—	892	1	no	—
62	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
63	C₂×C₄	#16813155	1	1	yes	characteristic
64	C₂×C₄	#16813157	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	8	1	no	—
67	C₈	#16813159	1	1	yes	characteristic
68	C₈	Dic₈₉₂	1	1	yes	characteristic
69	C₂²	#20297224	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	1784	1	no	—
73	C₄	—	1784	1	no	—
74	C₄	#20297222	1	1	yes	characteristic
75	C₄	#20297229	1	1	yes	characteristic
76	C₂	#28383327	1	1	yes	characteristic
77	C₂	#28383386	1	1	yes	characteristic
78	C₂	#28383389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20297224
commutator	C₂×C₈×C₂₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#334×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	#102
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#334	D₂₂₃
hypocenter	C₂₂₃	#1363
LCS	G ▷ C₂×C₈×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
39. C₂₂₃	#1363	semidirect

Maximal subgroups

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

subgroup	index
#28383982	2
C₄⋊(C₂₂₃⋊C₁₆)	2
#334×C₂₂₃	2
#1363	223