Subgroups — C₂×#519×C₂₂₃ #43252441

Order 28544.

1488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#519×C₂₂₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Q₈×C₂₂₃	C₂	1	8	yes	maximal, complemented
4	C₂×#117×C₂₂₃	C₂	1	6	yes	maximal
5	C₂²×Q₈×C₂₂₃	C₄	1	16	yes	complemented
6	C₂²×Q₈×C₂₂₃	C₂²	1	4	yes	—
7	C₂²×Q₈×C₂₂₃	C₂²	1	8	yes	—
8	C₂³×C₄×C₂₂₃	C₂²	1	3	yes	—
9	C₄×Q₈×C₂₂₃	C₂²	1	64	yes	complemented
10	#117×C₂₂₃	C₂²	1	48	yes	—
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	12	yes	—
12	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
13	C₂×C₄²×C₂₂₃	C₂²	1	12	yes	—
14	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
15	C₂×Q₈×C₂₂₃	C₂×C₄	1	32	yes	—
16	C₂×Q₈×C₂₂₃	C₂×C₄	1	64	yes	complemented
17	C₂×Q₈×C₂₂₃	C₂³	1	16	yes	—
18	C₂²×C₄×C₂₂₃	C₂×C₄	1	24	yes	—
19	C₂²×C₄×C₂₂₃	C₂³	1	8	yes	—
20	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
21	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
22	C₄⋊C₄×C₂₂₃	C₂³	1	48	yes	—
23	C₂²⋊C₄×C₂₂₃	C₂³	1	16	yes	—
24	C₄²×C₂₂₃	C₂³	1	48	yes	—
25	C₂³×C₂₂₃	—	2	4	no	—
26	C₂³×C₂₂₃	C₂²×C₄	1	4	yes	—
27	C₂³×C₂₂₃	C₂⁴	1	2	yes	—
28	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	Q₈×C₂₂₃	C₂²×C₄	1	64	yes	—
30	C₂×C₄×C₂₂₃	—	2	16	no	—
31	C₂×C₄×C₂₂₃	C₂²×C₄	1	24	yes	—
32	C₂×C₄×C₂₂₃	C₂²×C₄	1	48	yes	—
33	C₂×C₄×C₂₂₃	C₂⁴	1	16	yes	—
34	C₂×C₄×C₂₂₃	C₂⁴	1	12	yes	—
35	C₂²×C₂₂₃	—	2	8	no	—
36	C₂²×C₂₂₃	—	2	4	no	—
37	C₂²×C₂₂₃	C₂³×C₄	1	4	yes	—
38	C₂²×C₂₂₃	C₂³×C₄	1	2	yes	—
39	C₂²×C₂₂₃	2₋¹⁺⁴	1	4	yes	—
40	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
41	C₄×C₂₂₃	—	2	16	no	—
42	C₄×C₂₂₃	C₂³×C₄	1	24	yes	—
43	C₂×C₂₂₃	—	2	4	no	—
44	C₂×C₂₂₃	#519	1	4	yes	direct product factor
45	C₂×C₂₂₃	C₂⁴×C₄	1	1	yes	characteristic
46	C₂×C₂₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
47	C₂₂₃	C₂×#519	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×#519	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
50	#519	C₂×C₂₂₃	1	16	yes	direct product factor
51	C₂×C₄×Q₈	C₂×C₂₂₃	1	8	yes	complemented
52	C₂×#117	C₂×C₂₂₃	1	6	yes	—
53	C₂²×Q₈	C₄×C₂₂₃	1	16	yes	complemented
54	C₂²×Q₈	C₂²×C₂₂₃	1	4	yes	—
55	C₂²×Q₈	C₂²×C₂₂₃	1	8	yes	—
56	C₂³×C₄	C₂²×C₂₂₃	1	3	yes	—
57	C₄×Q₈	C₂²×C₂₂₃	1	64	yes	complemented
58	#117	C₂²×C₂₂₃	1	48	yes	—
59	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	12	yes	—
60	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	—
61	C₂×C₄²	C₂²×C₂₂₃	1	12	yes	—
62	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₂×Q₈	C₂×C₄×C₂₂₃	1	32	yes	—
64	C₂×Q₈	C₂×C₄×C₂₂₃	1	64	yes	complemented
65	C₂×Q₈	C₂³×C₂₂₃	1	16	yes	—
66	C₂²×C₄	C₂×C₄×C₂₂₃	1	24	yes	—
67	C₂²×C₄	C₂³×C₂₂₃	1	8	yes	—
68	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
69	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
70	C₄⋊C₄	C₂³×C₂₂₃	1	48	yes	—
71	C₂²⋊C₄	C₂³×C₂₂₃	1	16	yes	—
72	C₄²	C₂³×C₂₂₃	1	48	yes	—
73	C₂³	—	2	4	no	—
74	C₂³	C₂²×C₄×C₂₂₃	1	4	yes	—
75	C₂³	C₂⁴×C₂₂₃	1	2	yes	—
76	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
77	Q₈	C₂²×C₄×C₂₂₃	1	64	yes	—
78	C₂×C₄	—	2	16	no	—
79	C₂×C₄	C₂²×C₄×C₂₂₃	1	24	yes	—
80	C₂×C₄	C₂²×C₄×C₂₂₃	1	48	yes	—
81	C₂×C₄	C₂⁴×C₂₂₃	1	16	yes	—
82	C₂×C₄	C₂⁴×C₂₂₃	1	12	yes	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	8	no	—
85	C₂²	C₂³×C₄×C₂₂₃	1	4	yes	—
86	C₂²	C₂³×C₄×C₂₂₃	1	2	yes	—
87	C₂²	2₋¹⁺⁴×C₂₂₃	1	4	yes	—
88	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
89	C₄	—	2	16	no	—
90	C₄	C₂³×C₄×C₂₂₃	1	24	yes	—
91	C₂	—	2	4	no	—
92	C₂	#519×C₂₂₃	1	4	yes	direct product factor
93	C₂	C₂⁴×C₄×C₂₂₃	1	1	yes	characteristic
94	C₂	C₂×2₋¹⁺⁴×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂⁴
commutator	C₂	C₂⁴×C₄×C₂₂₃
Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
prime radical	C₂⁴×C₂₂₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #519×C₂₂₃	C₂	direct (N × H)
3. C₂×C₄×Q₈×C₂₂₃	C₂	semidirect
5. C₂²×Q₈×C₂₂₃	C₄	semidirect
9. C₄×Q₈×C₂₂₃	C₂²	semidirect
16. C₂×Q₈×C₂₂₃	C₂×C₄	semidirect
44. C₂×C₂₂₃	#519	direct (N × H)
47. C₂₂₃	C₂×#519	direct (N × H)
48. C₂×#519	C₂₂₃	direct (N × H)
50. #519	C₂×C₂₂₃	direct (N × H)
51. C₂×C₄×Q₈	C₂×C₂₂₃	semidirect
53. C₂²×Q₈	C₄×C₂₂₃	semidirect
57. C₄×Q₈	C₂²×C₂₂₃	semidirect
64. C₂×Q₈	C₂×C₄×C₂₂₃	semidirect
92. C₂	#519×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
C₂³×Q₈×C₂₂₃	2
#519×C₂₂₃	2
C₂×C₄×Q₈×C₂₂₃	2
C₂×#117×C₂₂₃	2
C₂×#519	223

Subgroups — C2×#519×C223 #43252441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#519×C₂₂₃ #43252441