Subgroups — C₂²×#95×C₂₂₃ #43251281

Order 28544.

1880 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₂₂₃	C₂	1	3	yes	maximal
2	C₂×#95×C₂₂₃	C₂	1	12	yes	maximal, direct product factor
3	C₂⁵×C₂₂₃	C₂²	1	1	yes	characteristic
4	C₂³×C₄×C₂₂₃	—	2	24	no	—
5	C₂³×C₄×C₂₂₃	C₄	1	24	yes	complemented
6	C₂³×C₄×C₂₂₃	C₂²	1	18	yes	—
7	#95×C₂₂₃	C₂²	1	16	yes	direct product factor
8	C₂⁴×C₂₂₃	C₂×C₄	1	12	yes	—
9	C₂⁴×C₂₂₃	D₄	1	12	yes	—
10	C₂⁴×C₂₂₃	Q₈	1	4	yes	—
11	C₂⁴×C₂₂₃	C₂³	1	3	yes	—
12	C₂²×C₄×C₂₂₃	—	2	96	no	—
13	C₂²×C₄×C₂₂₃	—	2	72	no	—
14	C₂²×C₄×C₂₂₃	C₂×C₄	1	72	yes	complemented
15	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
16	C₂³×C₂₂₃	C₄²	1	16	yes	—
17	C₂³×C₂₂₃	C₂²⋊C₄	1	48	yes	—
18	C₂³×C₂₂₃	C₄⋊C₄	1	48	yes	—
19	C₂³×C₂₂₃	C₂²×C₄	1	18	yes	—
20	C₂³×C₂₂₃	C₂×D₄	1	18	yes	—
21	C₂³×C₂₂₃	C₂×Q₈	1	6	yes	—
22	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	—	2	144	no	—
24	C₂×C₄×C₂₂₃	—	2	24	no	—
25	C₂×C₄×C₂₂₃	C₂²×C₄	1	24	yes	complemented
26	C₂²×C₂₂₃	#95	1	64	yes	direct product factor
27	C₂²×C₂₂₃	C₂×C₄²	1	12	yes	—
28	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	36	yes	—
29	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	36	yes	—
30	C₂²×C₂₂₃	C₂³×C₄	1	3	yes	—
31	C₂²×C₂₂₃	C₂²×D₄	1	3	yes	—
32	C₂²×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
33	C₄×C₂₂₃	—	2	24	no	—
34	C₂×C₂₂₃	C₂×#95	1	24	yes	direct product factor
35	C₂×C₂₂₃	C₂²×C₄²	1	1	yes	characteristic
36	C₂×C₂₂₃	C₂²×C₂²⋊C₄	1	3	yes	—
37	C₂×C₂₂₃	C₂²×C₄⋊C₄	1	3	yes	—
38	C₂₂₃	C₂²×#95	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×#95	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₄	C₂×C₂₂₃	1	3	yes	—
41	C₂×#95	C₂×C₂₂₃	1	12	yes	direct product factor
42	C₂⁵	C₂²×C₂₂₃	1	1	yes	characteristic
43	C₂³×C₄	—	2	24	no	—
44	C₂³×C₄	C₄×C₂₂₃	1	24	yes	complemented
45	C₂³×C₄	C₂²×C₂₂₃	1	18	yes	—
46	#95	C₂²×C₂₂₃	1	16	yes	direct product factor
47	C₂⁴	C₂×C₄×C₂₂₃	1	12	yes	—
48	C₂⁴	D₄×C₂₂₃	1	12	yes	—
49	C₂⁴	Q₈×C₂₂₃	1	4	yes	—
50	C₂⁴	C₂³×C₂₂₃	1	3	yes	—
51	C₂²×C₄	—	2	96	no	—
52	C₂²×C₄	—	2	72	no	—
53	C₂²×C₄	C₂×C₄×C₂₂₃	1	72	yes	complemented
54	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
55	C₂³	C₄²×C₂₂₃	1	16	yes	—
56	C₂³	C₂²⋊C₄×C₂₂₃	1	48	yes	—
57	C₂³	C₄⋊C₄×C₂₂₃	1	48	yes	—
58	C₂³	C₂²×C₄×C₂₂₃	1	18	yes	—
59	C₂³	C₂×D₄×C₂₂₃	1	18	yes	—
60	C₂³	C₂×Q₈×C₂₂₃	1	6	yes	—
61	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
62	C₂×C₄	—	2	144	no	—
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	C₂²×C₄×C₂₂₃	1	24	yes	complemented
65	C₂²	#95×C₂₂₃	1	64	yes	direct product factor
66	C₂²	C₂×C₄²×C₂₂₃	1	12	yes	—
67	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	36	yes	—
68	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	36	yes	—
69	C₂²	C₂³×C₄×C₂₂₃	1	3	yes	—
70	C₂²	C₂²×D₄×C₂₂₃	1	3	yes	—
71	C₂²	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
72	C₄	—	2	24	no	—
73	C₂	C₂×#95×C₂₂₃	1	24	yes	direct product factor
74	C₂	C₂²×C₄²×C₂₂₃	1	1	yes	characteristic
75	C₂	C₂²×C₂²⋊C₄×C₂₂₃	1	3	yes	—
76	C₂	C₂²×C₄⋊C₄×C₂₂₃	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₂₃	C₂²
commutator	C₂	C₂²×C₄²×C₂₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#95×C₂₂₃	C₂	direct (N × H)
5. C₂³×C₄×C₂₂₃	C₄	semidirect
7. #95×C₂₂₃	C₂²	direct (N × H)
14. C₂²×C₄×C₂₂₃	C₂×C₄	semidirect
25. C₂×C₄×C₂₂₃	C₂²×C₄	semidirect
26. C₂²×C₂₂₃	#95	direct (N × H)
34. C₂×C₂₂₃	C₂×#95	direct (N × H)
38. C₂₂₃	C₂²×#95	direct (N × H)
39. C₂²×#95	C₂₂₃	direct (N × H)
41. C₂×#95	C₂×C₂₂₃	direct (N × H)
44. C₂³×C₄	C₄×C₂₂₃	semidirect
46. #95	C₂²×C₂₂₃	direct (N × H)
53. C₂²×C₄	C₂×C₄×C₂₂₃	semidirect
64. C₂×C₄	C₂²×C₄×C₂₂₃	semidirect
65. C₂²	#95×C₂₂₃	direct (N × H)
73. C₂	C₂×#95×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
C₂⁴×C₄×C₂₂₃	2
C₂×#95×C₂₂₃	2
C₂²×#95	223

Subgroups — C22×#95×C223 #43251281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#95×C₂₂₃ #43251281