Subgroups — #43252831

Order 28544.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383960	C₂	1	1	yes	maximal, characteristic
2	#28383401	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₉₂	—	2	1	no	—
5	C₂×Dic₈₉₂	—	2	1	no	—
6	#20297304	C₄	1	2	yes	—
7	#20297244	—	2	1	no	—
8	M₃₂×C₂₂₃	—	2	1	no	—
9	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₄₆	—	2	2	no	—
11	Dic₈₉₂	—	4	1	no	—
12	Dic₈₉₂	—	4	1	no	—
13	Dic₈₉₂	—	4	1	no	—
14	Dic₈₉₂	—	4	1	no	—
15	C₄⋊Dic₂₂₃	—	4	2	no	—
16	C₂₂₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
19	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₂₂₃	—	4	1	no	—
21	Dic₄₄₆	—	4	2	no	—
22	Dic₄₄₆	—	4	2	no	—
23	C₂×Dic₂₂₃	—	4	2	no	—
24	C₂×C₄×C₂₂₃	—	2	1	no	—
25	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	D₈	1	1	yes	characteristic
27	C₈×C₂₂₃	D₈	1	1	yes	characteristic
28	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
29	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
30	C₂²×C₂₂₃	#100	1	1	yes	characteristic
31	C₄×C₂₂₃	—	2	2	no	—
32	C₄×C₂₂₃	#102	1	1	yes	characteristic
33	C₄×C₂₂₃	#102	1	1	yes	characteristic
34	Dic₂₂₃	—	8	2	no	—
35	C₂×C₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	#331	1	1	yes	characteristic
37	C₂₂₃	#1358	1	1	yes	characteristic, complemented, Sylow
38	#1358	—	223	1	no	maximal, Sylow
39	#494	—	223	1	no	—
40	#364	—	223	1	no	—
41	#364	D₂₂₃	1	1	yes	characteristic
42	C₂×Q₁₆	—	446	1	no	—
43	C₂×Q₁₆	—	446	1	no	—
44	C₄⋊Q₈	—	223	2	no	—
45	M₃₂	—	2	1	no	—
46	M₃₂	—	446	1	no	—
47	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×Q₈	—	446	2	no	—
49	Q₁₆	—	892	1	no	—
50	Q₁₆	—	892	1	no	—
51	Q₁₆	—	892	1	no	—
52	Q₁₆	—	892	1	no	—
53	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
54	C₂×C₈	D₈₉₂	1	1	yes	characteristic
55	C₄⋊C₄	—	892	2	no	—
56	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
57	C₁₆	—	4	1	no	—
58	C₁₆	—	892	1	no	—
59	Q₈	—	892	2	no	—
60	Q₈	—	892	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	892	2	no	—
63	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
64	C₈	#16813155	1	1	yes	characteristic
65	C₈	D₁₇₈₄	1	1	yes	characteristic
66	C₈	#16813157	1	1	yes	characteristic
67	C₈	#16813159	1	1	yes	characteristic
68	C₂²	#20297220	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	1784	2	no	—
71	C₄	#20297224	1	1	yes	characteristic
72	C₄	#20297226	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#28383323	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383323
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#364×C₂₂₃	C₂
socle	C₂×C₂₂₃	#331
prime radical	C₂²×C₂₂₃	#100
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#364	D₂₂₃
hypocenter	C₂₂₃	#1358
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
37. C₂₂₃	#1358	semidirect

Maximal subgroups

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

subgroup	index
#28383960	2
#28383401	2
#364×C₂₂₃	2
#1358	223