Subgroups — C₂²×#10625047 #15686653

Order 2944.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625047	C₂	1	24	yes	maximal, direct product factor
2	C₂²×Dic₂₃⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂²×(C₂×D₂₃)⋊C₄	C₂	1	3	yes	maximal
4	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₄×C₂₃	C₂²	1	3	yes	—
7	C₂³×Dic₂₃	C₂²	1	3	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	12	yes	complemented
9	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	36	yes	—
10	C₂×Dic₂₃⋊C₄	C₂²	1	36	yes	complemented
11	#10625047	C₂²	1	64	yes	direct product factor
12	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
13	C₂³×D₂₃	—	2	12	no	—
14	C₂³×D₂₃	C₂³	1	6	yes	—
15	C₂²×C₄×C₂₃	—	2	12	no	—
16	C₂²×C₄×C₂₃	C₂³	1	18	yes	—
17	C₂²×Dic₂₃	—	2	12	no	—
18	C₂²×Dic₂₃	C₂³	1	18	yes	—
19	C₄²×C₂₃	C₂³	1	16	yes	complemented
20	(C₂×D₂₃)⋊C₄	C₂³	1	48	yes	—
21	Dic₂₃⋊C₄	C₂³	1	48	yes	complemented
22	C₂³×C₂₃	#41	1	12	yes	—
23	C₂³×C₂₃	C₂⁴	1	3	yes	—
24	C₂²×D₂₃	—	2	36	no	—
25	C₂²×D₂₃	—	4	16	no	—
26	C₂²×D₂₃	C₂⁴	1	4	yes	—
27	C₂×C₄×C₂₃	—	2	36	no	—
28	C₂×C₄×C₂₃	C₂⁴	1	12	yes	—
29	C₂×Dic₂₃	—	2	36	no	—
30	C₂×Dic₂₃	C₂⁴	1	12	yes	—
31	C₂²×#126	—	23	1	no	maximal, Sylow
32	C₂²×C₂₃	#126	1	16	yes	complemented
33	C₂²×C₂₃	C₂×#41	1	18	yes	—
34	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
35	C₂×D₂₃	—	2	12	no	—
36	C₂×D₂₃	—	4	24	no	—
37	C₄×C₂₃	—	2	12	no	—
38	Dic₂₃	—	2	12	no	—
39	C₂×#126	—	23	24	no	—
40	C₂²×C₄⋊C₄	—	23	3	no	—
41	C₂²×C₂²⋊C₄	—	23	3	no	—
42	C₂²×C₄²	D₂₃	1	1	yes	characteristic, complemented
43	C₂×C₂₃	C₂×#126	1	12	yes	complemented
44	C₂×C₂₃	C₂²×#41	1	3	yes	—
45	D₂₃	—	4	4	no	—
46	C₂⁵	—	23	1	no	—
47	C₂³×C₄	—	23	3	no	—
48	C₂³×C₄	C₂×D₂₃	1	3	yes	—
49	#126	—	23	64	no	—
50	C₂×C₄⋊C₄	—	23	36	no	—
51	C₂×C₂²⋊C₄	—	23	36	no	—
52	C₂×C₄²	C₂×D₂₃	1	12	yes	complemented
53	C₂₃	C₂²×#126	1	1	yes	characteristic, complemented, Sylow
54	C₂⁴	—	23	6	no	—
55	C₂⁴	—	46	12	no	—
56	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
57	C₂²×C₄	—	2	12	no	—
58	C₂²×C₄	—	23	18	no	—
59	C₂²×C₄	—	46	12	no	—
60	C₂²×C₄	C₂²×D₂₃	1	18	yes	—
61	C₄⋊C₄	—	23	48	no	—
62	C₂²⋊C₄	—	23	48	no	—
63	C₄²	C₂²×D₂₃	1	16	yes	complemented
64	C₂³	—	23	4	no	—
65	C₂³	—	46	36	no	—
66	C₂³	—	92	16	no	—
67	C₂³	#67288	1	12	yes	—
68	C₂³	C₂³×D₂₃	1	3	yes	—
69	C₂×C₄	—	2	36	no	—
70	C₂×C₄	—	23	12	no	—
71	C₂×C₄	—	46	36	no	—
72	C₂×C₄	C₂³×D₂₃	1	12	yes	—
73	C₂²	—	46	12	no	—
74	C₂²	—	92	24	no	—
75	C₂²	#10625047	1	16	yes	direct product factor
76	C₂²	C₂×#67288	1	18	yes	—
77	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
78	C₄	—	2	12	no	—
79	C₄	—	46	12	no	—
80	C₂	—	92	4	no	—
81	C₂	C₂×#10625047	1	12	yes	direct product factor
82	C₂	C₂²×#67288	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂²×C₄²×C₂₃	C₂
socle	C₂⁴×C₂₃	C₂³
prime radical	C₂⁴×D₂₃	C₂²
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	C₂²×#126
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#10625047	C₂	direct (N × H)
2. C₂²×Dic₂₃⋊C₄	C₂	semidirect
4. C₂²×C₄²×C₂₃	C₂	semidirect
8. C₂×C₄²×C₂₃	C₂²	semidirect
10. C₂×Dic₂₃⋊C₄	C₂²	semidirect
11. #10625047	C₂²	direct (N × H)
19. C₄²×C₂₃	C₂³	semidirect
21. Dic₂₃⋊C₄	C₂³	semidirect
32. C₂²×C₂₃	#126	semidirect
42. C₂²×C₄²	D₂₃	semidirect
43. C₂×C₂₃	C₂×#126	semidirect
52. C₂×C₄²	C₂×D₂₃	semidirect
53. C₂₃	C₂²×#126	semidirect
63. C₄²	C₂²×D₂₃	semidirect
75. C₂²	#10625047	direct (N × H)
81. C₂	C₂×#10625047	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×#10625047	2
C₂²×Dic₂₃⋊C₄	2
C₂²×(C₂×D₂₃)⋊C₄	2
C₂²×C₄²×C₂₃	2
C₂²×#126	23

Subgroups — C22×#10625047 #15686653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10625047 #15686653