Subgroups — #559×C₂₂₃ #28383271

Order 14272.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#124×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#123×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	C₄×Q₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
7	C₂×D₄×C₂₂₃	C₂²	1	4	yes	—
8	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₂₃	C₂²	1	4	yes	—
10	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
11	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
12	C₂²⋊C₄×C₂₂₃	C₂²	1	8	yes	—
13	C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
14	C₄²×C₂₂₃	C₂²	1	4	yes	—
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₂₂₃	C₂³	1	4	yes	—
17	Q₈×C₂₂₃	—	2	4	no	—
18	D₄×C₂₂₃	—	2	8	no	—
19	D₄×C₂₂₃	—	4	2	no	—
20	C₂×C₄×C₂₂₃	—	4	4	no	—
21	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
22	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
25	C₂²×C₂₂₃	—	2	8	no	—
26	C₂²×C₂₂₃	—	2	4	no	—
27	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₂₃	—	2	4	no	—
29	C₄×C₂₂₃	—	2	4	no	—
30	C₄×C₂₂₃	—	2	1	no	—
31	C₄×C₂₂₃	#41	1	4	yes	—
32	C₂×C₂₂₃	—	4	4	no	—
33	C₂×C₂₂₃	C₂×#41	1	2	yes	—
34	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂₂₃	#559	1	1	yes	characteristic, direct product factor, Sylow
36	#559	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	#127	C₂×C₂₂₃	1	1	yes	characteristic
38	#124	C₂×C₂₂₃	1	4	yes	complemented
39	#123	C₂×C₂₂₃	1	4	yes	complemented
40	C₄×Q₈	C₂×C₂₂₃	1	2	yes	complemented
41	C₄×D₄	C₂×C₂₂₃	1	4	yes	complemented
42	C₂×Q₈	C₂²×C₂₂₃	1	2	yes	—
43	C₂×D₄	C₂²×C₂₂₃	1	4	yes	—
44	C₂×D₄	C₂²×C₂₂₃	1	2	yes	—
45	C₂²×C₄	C₂²×C₂₂₃	1	4	yes	—
46	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
47	C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
48	C₂²⋊C₄	C₂²×C₂₂₃	1	8	yes	—
49	C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	—
50	C₄²	C₂²×C₂₂₃	1	4	yes	—
51	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
52	C₂³	C₂³×C₂₂₃	1	4	yes	—
53	Q₈	—	2	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	4	4	no	—
57	C₂×C₄	C₂³×C₂₂₃	1	4	yes	—
58	C₂×C₄	C₂³×C₂₂₃	1	4	yes	—
59	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	2	4	no	—
66	C₄	—	2	1	no	—
67	C₄	#41×C₂₂₃	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#41×C₂₂₃	1	2	yes	—
70	C₂	H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
prime radical	#127×C₂₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #124×C₂₂₃	C₂	semidirect
3. #123×C₂₂₃	C₂	semidirect
4. C₄×Q₈×C₂₂₃	C₂	semidirect
5. C₄×D₄×C₂₂₃	C₂	semidirect
35. C₂₂₃	#559	direct (N × H)
36. #559	C₂₂₃	direct (N × H)
38. #124	C₂×C₂₂₃	semidirect
39. #123	C₂×C₂₂₃	semidirect
40. C₄×Q₈	C₂×C₂₂₃	semidirect
41. C₄×D₄	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#127×C₂₂₃	2
#124×C₂₂₃	2
#123×C₂₂₃	2
C₄×Q₈×C₂₂₃	2
C₄×D₄×C₂₂₃	2
#559	223

Subgroups — #559×C223 #28383271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #559×C₂₂₃ #28383271