Subgroups — #43252936

Order 28544.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383996	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297310	C₄	1	1	yes	characteristic
5	#20297244	C₄	1	1	yes	characteristic
6	#20297243	—	2	1	no	—
7	C₁₆×D₂₂₃	—	2	1	no	—
8	C₂₂₃⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₂₃	—	2	1	no	—
11	#16813170	C₈	1	1	yes	characteristic
12	#16813154	C₈	1	1	yes	characteristic
13	#16813153	—	2	1	no	—
14	C₈×D₂₂₃	—	2	1	no	—
15	C₂₂₃⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
18	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
19	Dic₄₄₆	C₁₆	1	1	yes	characteristic
20	D₈₉₂	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₂₂₃	—	2	1	no	—
22	C₄×D₂₂₃	—	2	1	no	—
23	C₂₂₃⋊C₈	—	2	1	no	—
24	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
25	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₂₂₃	M₃₂	1	1	yes	characteristic
28	C₂×D₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
31	Dic₂₂₃	—	2	1	no	—
32	C₂×C₂₂₃	—	2	1	no	—
33	C₂×C₂₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₂₂₃	—	4	1	no	—
35	C₂₂₃	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	223	1	no	maximal, Sylow
37	#504	—	223	1	no	—
38	M₆₄	D₂₂₃	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	223	1	no	—
40	#131	—	223	1	no	—
41	M₃₂	—	446	1	no	—
42	M₃₂	—	223	1	no	—
43	C₂×C₁₆	—	446	1	no	—
44	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	446	1	no	—
47	#41	—	223	1	no	—
48	M₁₆	—	446	1	no	—
49	M₁₆	—	223	1	no	—
50	C₂×C₈	—	446	1	no	—
51	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
52	C₁₆	—	446	1	no	—
53	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
54	C₁₆	D₈₉₂	1	1	yes	characteristic
55	Q₈	—	223	1	no	—
56	D₄	—	446	1	no	—
57	D₄	—	223	1	no	—
58	C₂×C₄	—	446	1	no	—
59	C₂×C₄	C₈×D₂₂₃	1	1	yes	characteristic
60	C₈	—	446	1	no	—
61	C₈	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
62	C₈	#16813153	1	1	yes	characteristic
63	C₂²	—	446	1	no	—
64	C₂²	#20297243	1	1	yes	characteristic
65	C₄	—	446	1	no	—
66	C₄	(C₂×D₂₂₃)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₂₂₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	892	1	no	—
70	C₂	(C₂×D₂₂₃)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	M₆₄×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂²⋊C₁₆
prime radical	#16813170	C₈
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	M₆₄	D₂₂₃
hypocenter	C₂₂₃	#1401
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₂₃⋊C₃₂	C₂	semidirect
3. M₆₄×C₂₂₃	C₂	semidirect
35. C₂₂₃	#1401	semidirect
38. M₆₄	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#28383996	2
C₂×C₂₂₃⋊C₃₂	2
M₆₄×C₂₂₃	2
#1401	223