Subgroups — C₂²×#10625040 #15686673

Order 2944.

1624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₄₆	C₂	1	2	yes	maximal
2	C₂×#10625040	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₂₃	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
7	C₂³×Dic₂₃	C₂²	1	4	yes	—
8	C₂²×Dic₄₆	—	2	16	no	—
9	C₂²×Dic₄₆	C₂²	1	24	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	12	yes	—
11	C₂×C₄⋊Dic₂₃	C₂²	1	48	yes	—
12	#10625040	C₂²	1	64	yes	direct product factor
13	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₂₃	D₄	1	8	yes	—
15	C₂²×C₄×C₂₃	Q₈	1	16	yes	complemented
16	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	12	yes	—
18	C₂²×Dic₂₃	—	2	16	no	—
19	C₂²×Dic₂₃	C₂³	1	24	yes	—
20	C₂×Dic₄₆	—	2	96	no	—
21	C₂×Dic₄₆	C₂³	1	32	yes	—
22	C₄²×C₂₃	C₂³	1	16	yes	—
23	C₄⋊Dic₂₃	C₂³	1	64	yes	—
24	C₂³×C₂₃	C₂×D₄	1	4	yes	—
25	C₂³×C₂₃	C₂×Q₈	1	8	yes	—
26	C₂³×C₂₃	C₂⁴	1	3	yes	—
27	C₂×C₄×C₂₃	C₂×D₄	1	24	yes	—
28	C₂×C₄×C₂₃	C₂×Q₈	1	48	yes	complemented
29	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
30	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
31	C₂×Dic₂₃	—	2	48	no	—
32	C₂×Dic₂₃	C₂⁴	1	16	yes	—
33	Dic₄₆	—	2	64	no	—
34	C₂²×C₄⋊Q₈	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	C₄⋊Q₈	1	16	yes	complemented
36	C₂²×C₂₃	C₂²×D₄	1	6	yes	—
37	C₂²×C₂₃	C₂²×Q₈	1	12	yes	—
38	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
39	C₄×C₂₃	C₂²×D₄	1	8	yes	—
40	C₄×C₂₃	C₂²×Q₈	1	16	yes	complemented
41	Dic₂₃	—	2	16	no	—
42	C₂³×Q₈	—	23	2	no	—
43	C₂×C₄⋊Q₈	—	23	24	no	—
44	C₂²×C₄⋊C₄	—	23	4	no	—
45	C₂²×C₄²	D₂₃	1	1	yes	characteristic
46	C₂×C₂₃	C₂×C₄⋊Q₈	1	12	yes	complemented
47	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
48	C₂×C₂₃	C₂³×Q₈	1	2	yes	—
49	C₂²×Q₈	—	23	24	no	—
50	C₂²×Q₈	—	46	16	no	—
51	C₂³×C₄	—	23	4	no	—
52	C₂³×C₄	C₂×D₂₃	1	2	yes	—
53	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
54	C₄⋊Q₈	—	23	64	no	—
55	C₂×C₄⋊C₄	—	23	48	no	—
56	C₂×C₄²	C₂×D₂₃	1	12	yes	—
57	C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
58	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
59	C₂×Q₈	—	23	32	no	—
60	C₂×Q₈	—	46	96	no	—
61	C₂²×C₄	—	23	24	no	—
62	C₂²×C₄	—	46	16	no	—
63	C₂²×C₄	Dic₄₆	1	16	yes	complemented
64	C₂²×C₄	D₉₂	1	8	yes	—
65	C₂²×C₄	C₂²×D₂₃	1	12	yes	—
66	C₂²×C₄	C₂²×D₂₃	1	6	yes	—
67	C₄⋊C₄	—	23	64	no	—
68	C₄²	C₂²×D₂₃	1	16	yes	—
69	C₂³	C₂×Dic₄₆	1	8	yes	—
70	C₂³	C₂×D₉₂	1	4	yes	—
71	C₂³	C₂³×D₂₃	1	3	yes	—
72	Q₈	—	46	64	no	—
73	C₂×C₄	—	23	16	no	—
74	C₂×C₄	—	46	48	no	—
75	C₂×C₄	C₂×Dic₄₆	1	48	yes	complemented
76	C₂×C₄	C₂×D₉₂	1	24	yes	—
77	C₂×C₄	C₂³×D₂₃	1	8	yes	—
78	C₂×C₄	C₂³×D₂₃	1	4	yes	—
79	C₂²	#10625040	1	16	yes	direct product factor
80	C₂²	C₂²×Dic₄₆	1	12	yes	—
81	C₂²	C₂²×D₉₂	1	6	yes	—
82	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
83	C₄	—	46	16	no	—
84	C₄	C₂²×Dic₄₆	1	16	yes	complemented
85	C₄	C₂²×D₉₂	1	8	yes	—
86	C₂	C₂×#10625040	1	12	yes	direct product factor
87	C₂	C₂³×D₉₂	1	1	yes	characteristic
88	C₂	C₂³×Dic₄₆	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂²×C₄²×C₂₃	C₂
socle, prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	C₂²×C₄⋊Q₈
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
2. C₂×#10625040	C₂	direct (N × H)
12. #10625040	C₂²	direct (N × H)
15. C₂²×C₄×C₂₃	Q₈	semidirect
28. C₂×C₄×C₂₃	C₂×Q₈	semidirect
35. C₂²×C₂₃	C₄⋊Q₈	semidirect
40. C₄×C₂₃	C₂²×Q₈	semidirect
46. C₂×C₂₃	C₂×C₄⋊Q₈	semidirect
57. C₂₃	C₂²×C₄⋊Q₈	semidirect
63. C₂²×C₄	Dic₄₆	semidirect
75. C₂×C₄	C₂×Dic₄₆	semidirect
79. C₂²	#10625040	direct (N × H)
84. C₄	C₂²×Dic₄₆	semidirect
86. C₂	C₂×#10625040	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×Dic₄₆	2
C₂×#10625040	2
C₂²×C₄⋊Dic₂₃	2
C₂²×C₄²×C₂₃	2
C₂²×C₄⋊Q₈	23

Subgroups — C22×#10625040 #15686673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10625040 #15686673