Subgroups — #28383998

Order 14272.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297310	C₂	1	1	yes	maximal, characteristic
2	#20297243	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₂₂₃	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813170	C₄	1	1	yes	characteristic
7	#16813154	C₄	1	1	yes	characteristic
8	#16813153	C₄	1	2	yes	—
9	C₈×D₂₂₃	C₂²	1	2	yes	—
10	C₂₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₁₆×C₂₂₃	C₂²	1	2	yes	—
14	Dic₄₄₆	C₈	1	1	yes	characteristic
15	D₈₉₂	C₈	1	1	yes	characteristic
16	C₂²⋊D₂₂₃	C₈	1	2	yes	—
17	C₄×D₂₂₃	C₂×C₄	1	2	yes	—
18	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₂²×C₂₂₃	C₂×C₈	1	1	yes	characteristic
23	C₂×D₂₂₃	C₂×C₈	1	2	yes	—
24	C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
25	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
26	Dic₂₂₃	C₂×C₈	1	2	yes	—
27	C₂×C₂₂₃	—	2	1	no	—
28	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
29	D₂₂₃	—	2	2	no	—
30	C₂₂₃	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	223	1	no	maximal, Sylow
32	#131	—	223	1	no	—
33	M₃₂	—	223	2	no	—
34	M₃₂	D₂₂₃	1	1	yes	characteristic, complemented
35	C₂×C₁₆	—	223	1	no	—
36	C₂×C₁₆	—	223	2	no	—
37	#41	—	223	1	no	—
38	M₁₆	—	223	1	no	—
39	M₁₆	—	223	2	no	—
40	C₂×C₈	—	223	2	no	—
41	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
42	C₁₆	—	223	1	no	—
43	C₁₆	—	223	1	no	—
44	C₁₆	C₂×D₂₂₃	1	2	yes	—
45	Q₈	—	223	1	no	—
46	D₄	—	223	2	no	—
47	D₄	—	223	1	no	—
48	C₂×C₄	—	223	2	no	—
49	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
50	C₈	—	223	2	no	—
51	C₈	C₄×D₂₂₃	1	1	yes	characteristic
52	C₈	C₂²×D₂₂₃	1	1	yes	characteristic
53	C₂²	—	223	2	no	—
54	C₂²	C₈×D₂₂₃	1	1	yes	characteristic
55	C₄	—	223	2	no	—
56	C₄	C₈×D₂₂₃	1	1	yes	characteristic
57	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	446	2	no	—
60	C₂	C₂×C₈×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₂₂₃
commutator, socle	C₂×C₂₂₃	C₂²×C₈
Fitting	M₃₂×C₂₂₃	C₂
prime radical	#16813170	C₄
cosocle	C₈×C₂₂₃	C₂³
hypercenter	M₃₂	D₂₂₃
hypocenter	C₂₂₃	#504
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20297243	C₂	semidirect
3. C₂×C₂₂₃⋊C₁₆	C₂	semidirect
4. C₁₆×D₂₂₃	C₂	semidirect
5. M₃₂×C₂₂₃	C₂	semidirect
30. C₂₂₃	#504	semidirect
34. M₃₂	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297310	2
#20297243	2
C₂×C₂₂₃⋊C₁₆	2
C₁₆×D₂₂₃	2
M₃₂×C₂₂₃	2
#504	223