Subgroups — C₂×C₂₃×#384 #15669903

Order 2944.

888 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₂₃	C₂	1	6	yes	maximal
2	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#384	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₄×C₂₃	C₂²	1	4	yes	—
5	C₂³×C₄×C₂₃	C₂²	1	3	yes	—
6	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	24	yes	—
8	C₂×C₄²×C₂₃	C₄	1	8	yes	complemented
9	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
10	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	—	2	24	no	—
12	C₂²×C₄×C₂₃	C₂×C₄	1	12	yes	—
13	C₂²×C₄×C₂₃	D₄	1	12	yes	—
14	C₂²×C₄×C₂₃	Q₈	1	12	yes	—
15	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
16	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
17	C₄⋊C₄×C₂₃	—	2	48	no	—
18	C₄²×C₂₃	C₂×C₄	1	16	yes	complemented
19	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
20	C₂³×C₂₃	C₂×D₄	1	6	yes	—
21	C₂³×C₂₃	C₂×Q₈	1	6	yes	—
22	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₂₃	—	2	24	no	—
24	C₂×C₄×C₂₃	—	4	16	no	—
25	C₂×C₄×C₂₃	C₄⋊C₄	1	48	yes	complemented
26	C₂×C₄×C₂₃	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	—
28	C₂×C₄×C₂₃	C₂×Q₈	1	12	yes	—
29	C₂×#384	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
31	C₂²×C₂₃	#127	1	4	yes	—
32	C₂²×C₂₃	C₄⋊Q₈	1	12	yes	—
33	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
34	C₂²×C₂₃	C₂²×D₄	1	3	yes	—
35	C₂²×C₂₃	C₂²×Q₈	1	3	yes	—
36	C₄×C₂₃	—	4	8	no	—
37	C₄×C₂₃	C₂×C₄⋊C₄	1	24	yes	complemented
38	C₂²×C₄⋊C₄	C₂×C₂₃	1	6	yes	—
39	C₂²×C₄²	C₂×C₂₃	1	1	yes	characteristic
40	#384	C₂×C₂₃	1	8	yes	direct product factor
41	C₂×C₂₃	#384	1	8	yes	direct product factor
42	C₂×C₂₃	C₂²×C₄⋊C₄	1	3	yes	—
43	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
44	C₂×C₂₃	C₂×C₄⋊Q₈	1	3	yes	—
45	C₂³×C₄	C₂²×C₂₃	1	4	yes	—
46	C₂³×C₄	C₂²×C₂₃	1	3	yes	—
47	C₂×C₄⋊C₄	—	2	24	no	—
48	C₂×C₄⋊C₄	C₂²×C₂₃	1	24	yes	—
49	C₂×C₄²	C₄×C₂₃	1	8	yes	complemented
50	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
51	C₂₃	C₂×#384	1	1	yes	characteristic, direct product factor, Sylow
52	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
53	C₂²×C₄	—	2	24	no	—
54	C₂²×C₄	C₂×C₄×C₂₃	1	12	yes	—
55	C₂²×C₄	D₄×C₂₃	1	12	yes	—
56	C₂²×C₄	Q₈×C₂₃	1	12	yes	—
57	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
58	C₂²×C₄	C₂³×C₂₃	1	6	yes	—
59	C₄⋊C₄	—	2	48	no	—
60	C₄²	C₂×C₄×C₂₃	1	16	yes	complemented
61	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
62	C₂³	C₂×D₄×C₂₃	1	6	yes	—
63	C₂³	C₂×Q₈×C₂₃	1	6	yes	—
64	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
65	C₂×C₄	—	2	24	no	—
66	C₂×C₄	—	4	16	no	—
67	C₂×C₄	C₄⋊C₄×C₂₃	1	48	yes	complemented
68	C₂×C₄	C₂²×C₄×C₂₃	1	12	yes	—
69	C₂×C₄	C₂×D₄×C₂₃	1	12	yes	—
70	C₂×C₄	C₂×Q₈×C₂₃	1	12	yes	—
71	C₂²	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
72	C₂²	C₂₃×#127	1	4	yes	—
73	C₂²	C₂₃×C₄⋊Q₈	1	12	yes	—
74	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
75	C₂²	C₂²×D₄×C₂₃	1	3	yes	—
76	C₂²	C₂²×Q₈×C₂₃	1	3	yes	—
77	C₄	—	4	8	no	—
78	C₄	C₂×C₄⋊C₄×C₂₃	1	24	yes	complemented
79	C₂	C₂₃×#384	1	8	yes	direct product factor
80	C₂	C₂²×C₄⋊C₄×C₂₃	1	3	yes	—
81	C₂	C₂×C₂₃×#127	1	1	yes	characteristic
82	C₂	C₂×C₂₃×C₄⋊Q₈	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#384	C₂	direct (N × H)
8. C₂×C₄²×C₂₃	C₄	semidirect
18. C₄²×C₂₃	C₂×C₄	semidirect
25. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
29. C₂×#384	C₂₃	direct (N × H)
37. C₄×C₂₃	C₂×C₄⋊C₄	semidirect
40. #384	C₂×C₂₃	direct (N × H)
41. C₂×C₂₃	#384	direct (N × H)
49. C₂×C₄²	C₄×C₂₃	semidirect
51. C₂₃	C₂×#384	direct (N × H)
60. C₄²	C₂×C₄×C₂₃	semidirect
67. C₂×C₄	C₄⋊C₄×C₂₃	semidirect
78. C₄	C₂×C₄⋊C₄×C₂₃	semidirect
79. C₂	C₂₃×#384	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₄×C₂₃	2
C₂²×C₄²×C₂₃	2
C₂₃×#384	2
C₂×#384	23

Subgroups — C2×C23×#384 #15669903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#384 #15669903