Subgroups — #43252735

Order 28544.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₂₂₃⋊C₁₆)	C₂	1	2	yes	maximal
2	#424×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₂₃⋊C₁₆	—	2	2	no	—
4	C₂×M₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
6	#117×C₂₂₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₂₂₃	—	2	2	no	—
8	C₂₂₃⋊C₁₆	—	4	2	no	—
9	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
10	M₁₆×C₂₂₃	—	4	1	no	—
11	C₂×C₈×C₂₂₃	—	2	1	no	—
12	C₂×C₈×C₂₂₃	—	4	1	no	—
13	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
14	C₄⋊C₄×C₂₂₃	C₈	1	2	yes	—
15	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
16	C₄²×C₂₂₃	—	2	1	no	—
17	C₂³×C₂₂₃	M₁₆	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	2	1	no	—
19	C₂×C₄×C₂₂₃	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
23	C₈×C₂₂₃	—	2	2	no	—
24	C₈×C₂₂₃	—	4	1	no	—
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	—	2	2	no	—
27	C₂²×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	—	4	2	no	—
30	C₄×C₂₂₃	#99	1	1	yes	characteristic
31	C₄×C₂₂₃	#101	1	1	yes	characteristic
32	C₂×C₂₂₃	—	4	1	no	—
33	C₂×C₂₂₃	#324	1	1	yes	characteristic
34	C₂×C₂₂₃	#350	1	2	yes	—
35	C₂₂₃	#1323	1	1	yes	characteristic, complemented, Sylow
36	#1323	—	223	1	no	maximal, Sylow
37	#424	D₂₂₃	1	1	yes	characteristic
38	C₂²⋊C₁₆	—	223	2	no	—
39	C₂×M₁₆	Dic₂₂₃	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₂₂₃	1	1	yes	characteristic
41	#117	Dic₂₂₃	1	1	yes	characteristic
42	C₂×C₁₆	—	446	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂²⋊D₂₂₃	1	2	yes	—
49	C₄⋊C₄	C₂₂₃⋊C₈	1	2	yes	—
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	892	2	no	—
53	C₂³	#16813154	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
58	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	2	no	—
63	C₂²	C₂²⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#20297219	1	1	yes	characteristic
67	C₄	#20297223	1	1	yes	characteristic
68	C₂	—	4	1	no	—
69	C₂	#28383306	1	1	yes	characteristic
70	C₂	#28383364	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₂₃
commutator	C₂×C₄×C₂₂₃	C₂×C₈
Frattini	C₂²×C₈	C₂×D₂₂₃
Fitting	#424×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₈
prime radical	C₂³×C₂₂₃	M₁₆
cosocle	C₂²×C₈×C₂₂₃	C₂²
hypercenter	#424	D₂₂₃
hypocenter	C₂₂₃	#1323
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

Complements

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

N	H ≅ G/N	type
35. C₂₂₃	#1323	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₂₂₃⋊C₁₆)	2
#424×C₂₂₃	2
#1323	223