Subgroups — C₂²×#97×C₂₂₃ #43251885

Order 28544.

680 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₂₂₃	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₂₂₃	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₂₂₃	C₄	1	2	yes	—
5	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₂₂₃	C₄	1	16	yes	complemented
7	C₂²×C₈×C₂₂₃	C₂²	1	12	yes	—
8	C₂×C₄²×C₂₂₃	C₄	1	6	yes	—
9	C₂×C₄²×C₂₂₃	C₂²	1	6	yes	—
10	#97×C₂₂₃	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	—	2	8	no	—
13	C₂²×C₄×C₂₂₃	C₂×C₄	1	12	yes	—
14	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₂₂₃	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
17	C₂×C₈×C₂₂₃	C₂×C₄	1	48	yes	complemented
18	C₂×C₈×C₂₂₃	C₂³	1	8	yes	—
19	C₄²×C₂₂₃	C₂×C₄	1	12	yes	—
20	C₄²×C₂₂₃	C₂³	1	4	yes	—
21	C₂³×C₂₂₃	C₄²	1	4	yes	—
22	C₂³×C₂₂₃	M₁₆	1	8	yes	—
23	C₂³×C₂₂₃	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₂₂₃	—	2	24	no	—
25	C₂×C₄×C₂₂₃	C₄²	1	12	yes	—
26	C₂×C₄×C₂₂₃	C₂²×C₄	1	8	yes	—
27	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
28	C₂×C₄×C₂₂₃	C₂²×C₄	1	12	yes	—
29	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
30	C₈×C₂₂₃	C₂²×C₄	1	16	yes	complemented
31	C₂²×C₂₂₃	#97	1	16	yes	direct product factor
32	C₂²×C₂₂₃	C₂×C₄²	1	6	yes	—
33	C₂²×C₂₂₃	C₂×M₁₆	1	12	yes	—
34	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	8	no	—
36	C₄×C₂₂₃	C₂×C₄²	1	6	yes	—
37	C₄×C₂₂₃	C₂³×C₄	1	2	yes	—
38	C₂×C₂₂₃	C₂×#97	1	12	yes	direct product factor
39	C₂×C₂₂₃	C₂²×C₄²	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂²×M₁₆	1	2	yes	—
41	C₂₂₃	C₂²×#97	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×#97	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₈	C₂×C₂₂₃	1	2	yes	—
44	C₂²×C₄²	C₂×C₂₂₃	1	1	yes	characteristic
45	C₂×#97	C₂×C₂₂₃	1	12	yes	direct product factor
46	C₂³×C₄	C₄×C₂₂₃	1	2	yes	—
47	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
48	C₂²×C₈	C₄×C₂₂₃	1	16	yes	complemented
49	C₂²×C₈	C₂²×C₂₂₃	1	12	yes	—
50	C₂×C₄²	C₄×C₂₂₃	1	6	yes	—
51	C₂×C₄²	C₂²×C₂₂₃	1	6	yes	—
52	#97	C₂²×C₂₂₃	1	16	yes	direct product factor
53	C₂⁴	C₂×C₄×C₂₂₃	1	1	yes	characteristic
54	C₂²×C₄	—	2	8	no	—
55	C₂²×C₄	C₂×C₄×C₂₂₃	1	12	yes	—
56	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
57	C₂²×C₄	C₂×C₄×C₂₂₃	1	8	yes	—
58	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
59	C₂×C₈	C₂×C₄×C₂₂₃	1	48	yes	complemented
60	C₂×C₈	C₂³×C₂₂₃	1	8	yes	—
61	C₄²	C₂×C₄×C₂₂₃	1	12	yes	—
62	C₄²	C₂³×C₂₂₃	1	4	yes	—
63	C₂³	C₄²×C₂₂₃	1	4	yes	—
64	C₂³	M₁₆×C₂₂₃	1	8	yes	—
65	C₂³	C₂²×C₄×C₂₂₃	1	3	yes	—
66	C₂×C₄	—	2	24	no	—
67	C₂×C₄	C₄²×C₂₂₃	1	12	yes	—
68	C₂×C₄	C₂²×C₄×C₂₂₃	1	8	yes	—
69	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
70	C₂×C₄	C₂²×C₄×C₂₂₃	1	12	yes	—
71	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
72	C₈	C₂²×C₄×C₂₂₃	1	16	yes	complemented
73	C₂²	#97×C₂₂₃	1	16	yes	direct product factor
74	C₂²	C₂×C₄²×C₂₂₃	1	6	yes	—
75	C₂²	C₂×M₁₆×C₂₂₃	1	12	yes	—
76	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
77	C₄	—	2	8	no	—
78	C₄	C₂×C₄²×C₂₂₃	1	6	yes	—
79	C₄	C₂³×C₄×C₂₂₃	1	2	yes	—
80	C₂	C₂×#97×C₂₂₃	1	12	yes	direct product factor
81	C₂	C₂²×C₄²×C₂₂₃	1	1	yes	characteristic
82	C₂	C₂²×M₁₆×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₂₃	C₂²
commutator	C₂	C₂²×C₄²×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle, prime radical	C₂⁴×C₂₂₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#97×C₂₂₃	C₂	direct (N × H)
6. C₂²×C₈×C₂₂₃	C₄	semidirect
10. #97×C₂₂₃	C₂²	direct (N × H)
17. C₂×C₈×C₂₂₃	C₂×C₄	semidirect
30. C₈×C₂₂₃	C₂²×C₄	semidirect
31. C₂²×C₂₂₃	#97	direct (N × H)
38. C₂×C₂₂₃	C₂×#97	direct (N × H)
41. C₂₂₃	C₂²×#97	direct (N × H)
42. C₂²×#97	C₂₂₃	direct (N × H)
45. C₂×#97	C₂×C₂₂₃	direct (N × H)
48. C₂²×C₈	C₄×C₂₂₃	semidirect
52. #97	C₂²×C₂₂₃	direct (N × H)
59. C₂×C₈	C₂×C₄×C₂₂₃	semidirect
72. C₈	C₂²×C₄×C₂₂₃	semidirect
73. C₂²	#97×C₂₂₃	direct (N × H)
80. C₂	C₂×#97×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×C₈×C₂₂₃	2
C₂²×C₄²×C₂₂₃	2
C₂×#97×C₂₂₃	2
C₂²×#97	223

Subgroups — C22×#97×C223 #43251885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#97×C₂₂₃ #43251885