Subgroups — C₂×#322×C₂₂₃ #43250463

Order 28544.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×C₂₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#322×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₂₂₃	C₄	1	4	yes	—
5	C₂²×C₈×C₂₂₃	C₄	1	2	yes	—
6	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₂₂₃	C₄	1	4	yes	—
8	C₄×C₈×C₂₂₃	C₄	1	2	yes	—
9	C₄×C₈×C₂₂₃	C₂²	1	4	yes	—
10	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
13	C₂×C₈×C₂₂₃	—	2	8	no	—
14	C₂×C₈×C₂₂₃	C₈	1	8	yes	complemented
15	C₂×C₈×C₂₂₃	C₂×C₄	1	8	yes	—
16	C₂×C₈×C₂₂₃	C₂×C₄	1	4	yes	—
17	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
18	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂³×C₂₂₃	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₄²	1	2	yes	—
23	C₂×C₄×C₂₂₃	C₂×C₈	1	4	yes	—
24	C₂×C₄×C₂₂₃	M₁₆	1	8	yes	—
25	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
27	C₈×C₂₂₃	—	2	8	no	—
28	C₈×C₂₂₃	C₂×C₈	1	8	yes	complemented
29	C₂²×C₂₂₃	C₄×C₈	1	2	yes	—
30	C₂²×C₂₂₃	#97	1	4	yes	—
31	C₂²×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
32	C₄×C₂₂₃	C₄×C₈	1	2	yes	—
33	C₄×C₂₂₃	#97	1	4	yes	—
34	C₄×C₂₂₃	C₂²×C₈	1	2	yes	—
35	C₄×C₂₂₃	C₂×M₁₆	1	4	yes	—
36	C₂×C₂₂₃	#322	1	4	yes	direct product factor
37	C₂×C₂₂₃	C₂×C₄×C₈	1	1	yes	characteristic
38	C₂×C₂₂₃	C₂×#97	1	2	yes	—
39	C₂₂₃	C₂×#322	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×#322	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₄×C₈	C₂×C₂₂₃	1	2	yes	—
42	C₂×C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic
43	#322	C₂×C₂₂₃	1	4	yes	direct product factor
44	C₂²×C₈	C₄×C₂₂₃	1	4	yes	—
45	C₂²×C₈	C₄×C₂₂₃	1	2	yes	—
46	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
47	C₄×C₈	C₄×C₂₂₃	1	4	yes	—
48	C₄×C₈	C₄×C₂₂₃	1	2	yes	—
49	C₄×C₈	C₂²×C₂₂₃	1	4	yes	—
50	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
51	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
52	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
53	C₂×C₈	—	2	8	no	—
54	C₂×C₈	C₈×C₂₂₃	1	8	yes	complemented
55	C₂×C₈	C₂×C₄×C₂₂₃	1	8	yes	—
56	C₂×C₈	C₂×C₄×C₂₂₃	1	4	yes	—
57	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
58	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
59	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
60	C₂³	C₄²×C₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₄²×C₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₄²×C₂₂₃	1	2	yes	—
63	C₂×C₄	C₂×C₈×C₂₂₃	1	4	yes	—
64	C₂×C₄	M₁₆×C₂₂₃	1	8	yes	—
65	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
66	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
67	C₈	—	2	8	no	—
68	C₈	C₂×C₈×C₂₂₃	1	8	yes	complemented
69	C₂²	C₄×C₈×C₂₂₃	1	2	yes	—
70	C₂²	#97×C₂₂₃	1	4	yes	—
71	C₂²	C₂×C₄²×C₂₂₃	1	1	yes	characteristic
72	C₄	C₄×C₈×C₂₂₃	1	2	yes	—
73	C₄	#97×C₂₂₃	1	4	yes	—
74	C₄	C₂²×C₈×C₂₂₃	1	2	yes	—
75	C₄	C₂×M₁₆×C₂₂₃	1	4	yes	—
76	C₂	#322×C₂₂₃	1	4	yes	direct product factor
77	C₂	C₂×C₄×C₈×C₂₂₃	1	1	yes	characteristic
78	C₂	C₂×#97×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₂₂₃	C₂²
commutator	C₂	C₂×C₄×C₈×C₂₂₃
Frattini, cosocle	C₄²	C₂³×C₂₂₃
socle, prime radical	C₂³×C₂₂₃	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #322×C₂₂₃	C₂	direct (N × H)
14. C₂×C₈×C₂₂₃	C₈	semidirect
28. C₈×C₂₂₃	C₂×C₈	semidirect
36. C₂×C₂₂₃	#322	direct (N × H)
39. C₂₂₃	C₂×#322	direct (N × H)
40. C₂×#322	C₂₂₃	direct (N × H)
43. #322	C₂×C₂₂₃	direct (N × H)
54. C₂×C₈	C₈×C₂₂₃	semidirect
68. C₈	C₂×C₈×C₂₂₃	semidirect
76. C₂	#322×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
C₂×C₄×C₈×C₂₂₃	2
#322×C₂₂₃	2
C₂×#322	223

Subgroups — C2×#322×C223 #43250463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#322×C₂₂₃ #43250463