Subgroups — #2810×C₂₂₃ #43251825

Order 28544.

344 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₂₂₃	C₂	1	6	yes	maximal
2	#389×C₂₂₃	C₂	1	3	yes	maximal
3	#387×C₂₂₃	C₂	1	6	yes	maximal
4	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	—
5	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
6	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
7	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
9	#95×C₂₂₃	C₂²	1	6	yes	—
10	#95×C₂₂₃	C₂²	1	6	yes	—
11	#95×C₂₂₃	C₂²	1	6	yes	—
12	#95×C₂₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
14	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
15	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
16	C₄⋊C₄×C₂₂₃	—	2	12	no	—
17	C₄⋊C₄×C₂₂₃	—	2	6	no	—
18	C₄⋊C₄×C₂₂₃	—	4	3	no	—
19	C₄²×C₂₂₃	—	4	1	no	—
20	C₄²×C₂₂₃	—	4	2	no	—
21	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	—	2	3	no	—
23	C₂×C₄×C₂₂₃	—	2	3	no	—
24	C₂×C₄×C₂₂₃	—	2	6	no	—
25	C₂×C₄×C₂₂₃	—	2	6	no	—
26	C₂×C₄×C₂₂₃	—	2	6	no	—
27	C₂×C₄×C₂₂₃	—	2	6	no	—
28	C₂×C₄×C₂₂₃	—	2	6	no	—
29	C₂×C₄×C₂₂₃	—	2	6	no	—
30	C₂×C₄×C₂₂₃	#41	1	6	yes	—
31	C₂²×C₂₂₃	C₂×#41	1	3	yes	—
32	C₂²×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
33	C₂²×C₂₂₃	2₋¹⁺⁴	1	3	yes	—
34	C₄×C₂₂₃	—	2	6	no	—
35	C₄×C₂₂₃	—	4	6	no	—
36	C₄×C₂₂₃	—	4	6	no	—
37	C₂×C₂₂₃	#539	1	3	yes	—
38	C₂×C₂₂₃	#556	1	3	yes	—
39	C₂×C₂₂₃	#564	1	1	yes	characteristic
40	C₂₂₃	#2810	1	1	yes	characteristic, direct product factor, Sylow
41	#2810	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	#400	C₂×C₂₂₃	1	6	yes	—
43	#389	C₂×C₂₂₃	1	3	yes	—
44	#387	C₂×C₂₂₃	1	6	yes	—
45	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	—
46	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
47	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
48	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
49	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
50	#95	C₂²×C₂₂₃	1	6	yes	—
51	#95	C₂²×C₂₂₃	1	6	yes	—
52	#95	C₂²×C₂₂₃	1	6	yes	—
53	#95	C₂²×C₂₂₃	1	2	yes	—
54	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
55	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
56	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
57	C₄⋊C₄	—	2	12	no	—
58	C₄⋊C₄	—	2	6	no	—
59	C₄⋊C₄	—	4	3	no	—
60	C₄²	—	4	1	no	—
61	C₄²	—	4	2	no	—
62	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
63	C₂×C₄	—	2	3	no	—
64	C₂×C₄	—	2	3	no	—
65	C₂×C₄	—	2	6	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	—	2	6	no	—
68	C₂×C₄	—	2	6	no	—
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	#41×C₂₂₃	1	6	yes	—
72	C₂²	C₂×#41×C₂₂₃	1	3	yes	—
73	C₂²	H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic
74	C₂²	2₋¹⁺⁴×C₂₂₃	1	3	yes	—
75	C₄	—	2	6	no	—
76	C₄	—	4	6	no	—
77	C₄	—	4	6	no	—
78	C₂	#539×C₂₂₃	1	3	yes	—
79	C₂	#556×C₂₂₃	1	3	yes	—
80	C₂	#564×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
40. C₂₂₃	#2810	direct (N × H)
41. #2810	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
#400×C₂₂₃	2
#389×C₂₂₃	2
#387×C₂₂₃	2
#2810	223

Subgroups — #2810×C223 #43251825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2810×C₂₂₃ #43251825