Subgroups — #2245×C₂₂₃ #43251260

Order 28544.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#494×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#358×C₂₂₃	C₂	1	4	yes	maximal
4	#346×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₂₂₃	—	2	1	no	—
6	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₂₃	—	2	1	no	—
8	#125×C₂₂₃	—	2	1	no	—
9	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
10	#107×C₂₂₃	C₂²	1	2	yes	—
11	#106×C₂₂₃	—	2	1	no	—
12	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₂₂₃	—	2	2	no	—
14	Q₁₆×C₂₂₃	—	2	4	no	—
15	Q₁₆×C₂₂₃	—	4	2	no	—
16	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₂₂₃	—	2	2	no	—
19	C₄⋊C₄×C₂₂₃	—	4	2	no	—
20	C₄⋊C₄×C₂₂₃	—	4	2	no	—
21	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
22	C₁₆×C₂₂₃	—	2	2	no	—
23	Q₈×C₂₂₃	—	4	2	no	—
24	Q₈×C₂₂₃	—	4	2	no	—
25	C₂×C₄×C₂₂₃	—	4	2	no	—
26	C₂×C₄×C₂₂₃	—	4	2	no	—
27	C₂×C₄×C₂₂₃	D₈	1	2	yes	—
28	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₂₃	SD₁₆	1	2	yes	—
30	C₈×C₂₂₃	#41	1	2	yes	—
31	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	2	no	—
33	C₄×C₂₂₃	—	8	2	no	—
34	C₄×C₂₂₃	—	8	2	no	—
35	C₄×C₂₂₃	#124	1	1	yes	characteristic
36	C₄×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
37	C₂×C₂₂₃	#486	1	1	yes	characteristic
38	C₂×C₂₂₃	#510	1	2	yes	—
39	C₂₂₃	#2245	1	1	yes	characteristic, direct product factor, Sylow
40	#2245	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	#499	C₂×C₂₂₃	1	1	yes	characteristic
42	#494	C₂×C₂₂₃	1	1	yes	characteristic
43	#358	C₂×C₂₂₃	1	4	yes	—
44	#346	C₂×C₂₂₃	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	1	no	—
46	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
47	C₄⋊Q₈	—	2	1	no	—
48	#125	—	2	1	no	—
49	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
50	#107	C₂²×C₂₂₃	1	2	yes	—
51	#106	—	2	1	no	—
52	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
53	C₂×Q₈	—	2	2	no	—
54	Q₁₆	—	2	4	no	—
55	Q₁₆	—	4	2	no	—
56	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
57	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	4	2	no	—
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	D₈×C₂₂₃	1	2	yes	—
68	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
69	C₈	SD₁₆×C₂₂₃	1	2	yes	—
70	C₈	#41×C₂₂₃	1	2	yes	—
71	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	8	2	no	—
75	C₄	#124×C₂₂₃	1	1	yes	characteristic
76	C₄	C₂×SD₁₆×C₂₂₃	1	1	yes	characteristic
77	C₂	#486×C₂₂₃	1	1	yes	characteristic
78	C₂	#510×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
39. C₂₂₃	#2245	direct (N × H)
40. #2245	C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
#499×C₂₂₃	2
#494×C₂₂₃	2
#358×C₂₂₃	2
#346×C₂₂₃	2
#2245	223

Subgroups — #2245×C223 #43251260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2245×C₂₂₃ #43251260