Subgroups — #2247×C₂₂₃ #43251262

Order 28544.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₂₂₃	C₂	1	4	yes	maximal
2	#494×C₂₂₃	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₂₃	—	2	2	no	—
5	C₂×Q₁₆×C₂₂₃	C₂²	1	4	yes	—
6	C₄⋊Q₈×C₂₂₃	—	2	2	no	—
7	Q₃₂×C₂₂₃	—	2	8	no	—
8	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
9	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×Q₈×C₂₂₃	—	2	4	no	—
11	Q₁₆×C₂₂₃	—	2	8	no	—
12	Q₁₆×C₂₂₃	—	4	4	no	—
13	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
15	C₄⋊C₄×C₂₂₃	—	4	4	no	—
16	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
17	C₁₆×C₂₂₃	D₄	1	4	yes	—
18	Q₈×C₂₂₃	—	4	4	no	—
19	Q₈×C₂₂₃	—	4	4	no	—
20	C₂×C₄×C₂₂₃	—	4	4	no	—
21	C₂×C₄×C₂₂₃	D₈	1	2	yes	—
22	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
23	C₈×C₂₂₃	D₈	1	2	yes	—
24	C₈×C₂₂₃	C₂×D₄	1	2	yes	—
25	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
26	C₄×C₂₂₃	—	8	4	no	—
27	C₄×C₂₂₃	Q₃₂	1	4	yes	complemented
28	C₄×C₂₂₃	#127	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂×D₈	1	1	yes	characteristic
30	C₂×C₂₂₃	#493	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂×Q₃₂	1	2	yes	—
32	C₂₂₃	#2247	1	1	yes	characteristic, direct product factor, Sylow
33	#2247	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×Q₃₂	C₂×C₂₂₃	1	4	yes	—
35	#494	C₂×C₂₂₃	1	2	yes	—
36	C₄×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
37	C₂×Q₁₆	—	2	2	no	—
38	C₂×Q₁₆	C₂²×C₂₂₃	1	4	yes	—
39	C₄⋊Q₈	—	2	2	no	—
40	Q₃₂	—	2	8	no	—
41	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
42	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
43	C₂×Q₈	—	2	4	no	—
44	Q₁₆	—	2	8	no	—
45	Q₁₆	—	4	4	no	—
46	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
47	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
48	C₄⋊C₄	—	4	4	no	—
49	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
50	C₁₆	D₄×C₂₂₃	1	4	yes	—
51	Q₈	—	4	4	no	—
52	Q₈	—	4	4	no	—
53	C₂×C₄	—	4	4	no	—
54	C₂×C₄	D₈×C₂₂₃	1	2	yes	—
55	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
56	C₈	D₈×C₂₂₃	1	2	yes	—
57	C₈	C₂×D₄×C₂₂₃	1	2	yes	—
58	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
59	C₄	—	8	4	no	—
60	C₄	Q₃₂×C₂₂₃	1	4	yes	complemented
61	C₄	#127×C₂₂₃	1	1	yes	characteristic
62	C₄	C₂×D₈×C₂₂₃	1	1	yes	characteristic
63	C₂	#493×C₂₂₃	1	1	yes	characteristic
64	C₂	C₂×Q₃₂×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×C₂₂₃	Q₃₂	semidirect
32. C₂₂₃	#2247	direct (N × H)
33. #2247	C₂₂₃	direct (N × H)
60. C₄	Q₃₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×C₂₂₃	2
#494×C₂₂₃	2
C₄×C₁₆×C₂₂₃	2
#2247	223

Subgroups — #2247×C223 #43251262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2247×C₂₂₃ #43251262