Subgroups — C₂²×#10625000 #15681972

Order 2944.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₃	C₂	1	2	yes	maximal
2	C₂×#10625000	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂³×Dic₂₃	—	2	2	no	—
6	C₂²×C₈×C₂₃	C₄	1	8	yes	complemented
7	C₂²×C₈×C₂₃	C₂²	1	6	yes	—
8	C₂×C₄⋊Dic₂₃	—	2	6	no	—
9	C₂×C₄⋊Dic₂₃	C₂²	1	12	yes	—
10	#10625000	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₂₃	D₄	1	3	yes	—
14	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
15	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
16	C₂²×Dic₂₃	—	2	12	no	—
17	C₂²×Dic₂₃	—	4	8	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	24	yes	complemented
19	C₂×C₈×C₂₃	C₂³	1	4	yes	—
20	C₄⋊Dic₂₃	—	2	12	no	—
21	C₄⋊Dic₂₃	C₂³	1	8	yes	—
22	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
23	C₂³×C₂₃	D₈	1	4	yes	—
24	C₂³×C₂₃	Q₁₆	1	4	yes	—
25	C₂³×C₂₃	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₂₃	C₄⋊C₄	1	12	yes	—
27	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
28	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
29	C₂×C₄×C₂₃	C₂×Q₈	1	6	yes	—
30	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
31	C₂×Dic₂₃	—	2	8	no	—
32	C₂×Dic₂₃	—	4	24	no	—
33	C₈×C₂₃	C₂²×C₄	1	8	yes	complemented
34	C₂²×#107	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	#107	1	16	yes	complemented
36	C₂²×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
37	C₂²×C₂₃	C₂×D₈	1	6	yes	—
38	C₂²×C₂₃	C₂×Q₁₆	1	6	yes	—
39	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
40	C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
41	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
43	Dic₂₃	—	4	8	no	—
44	C₂³×C₈	D₂₃	1	1	yes	characteristic
45	C₂²×C₄⋊C₄	—	23	2	no	—
46	C₂×#107	—	23	12	no	—
47	C₂×C₂₃	C₂×#107	1	12	yes	complemented
48	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
50	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
51	C₂³×C₄	—	46	2	no	—
52	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
53	C₂²×C₈	Dic₂₃	1	8	yes	complemented
54	C₂²×C₈	C₂×D₂₃	1	6	yes	—
55	C₂×C₄⋊C₄	—	23	12	no	—
56	C₂×C₄⋊C₄	—	46	6	no	—
57	#107	—	23	16	no	—
58	C₂₃	C₂²×#107	1	1	yes	characteristic, complemented, Sylow
59	C₂⁴	D₉₂	1	1	yes	characteristic
60	C₂²×C₄	—	46	12	no	—
61	C₂²×C₄	—	92	8	no	—
62	C₂²×C₄	Dic₄₆	1	4	yes	—
63	C₂²×C₄	D₉₂	1	3	yes	—
64	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
65	C₂²×C₄	C₂²×D₂₃	1	3	yes	—
66	C₂×C₈	C₂×Dic₂₃	1	24	yes	complemented
67	C₂×C₈	C₂²×D₂₃	1	4	yes	—
68	C₄⋊C₄	—	23	8	no	—
69	C₄⋊C₄	—	46	12	no	—
70	C₂³	D₁₈₄	1	4	yes	—
71	C₂³	Dic₉₂	1	4	yes	—
72	C₂³	C₄⋊Dic₂₃	1	4	yes	—
73	C₂³	C₂×D₉₂	1	3	yes	—
74	C₂×C₄	—	46	8	no	—
75	C₂×C₄	—	92	24	no	—
76	C₂×C₄	C₄⋊Dic₂₃	1	12	yes	—
77	C₂×C₄	C₂×Dic₄₆	1	6	yes	—
78	C₂×C₄	C₂×D₉₂	1	3	yes	—
79	C₂×C₄	C₂²×Dic₂₃	1	6	yes	—
80	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
81	C₈	C₂²×Dic₂₃	1	8	yes	complemented
82	C₂²	#10625000	1	16	yes	direct product factor
83	C₂²	C₂×D₁₈₄	1	6	yes	—
84	C₂²	C₂×Dic₉₂	1	6	yes	—
85	C₂²	C₂×C₄⋊Dic₂₃	1	6	yes	—
86	C₂²	C₂²×D₉₂	1	1	yes	characteristic
87	C₄	—	92	8	no	—
88	C₄	C₂×C₄⋊Dic₂₃	1	6	yes	—
89	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
90	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
91	C₂	C₂×#10625000	1	12	yes	direct product factor
92	C₂	C₂²×C₄⋊Dic₂₃	1	1	yes	characteristic
93	C₂	C₂²×D₁₈₄	1	1	yes	characteristic
94	C₂	C₂²×Dic₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₉₂
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂³×C₈×C₂₃	C₂
socle, prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂³×C₈	D₂₃
hypocenter	C₂₃	C₂²×#107
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#10625000	C₂	direct (N × H)
6. C₂²×C₈×C₂₃	C₄	semidirect
10. #10625000	C₂²	direct (N × H)
18. C₂×C₈×C₂₃	C₂×C₄	semidirect
33. C₈×C₂₃	C₂²×C₄	semidirect
35. C₂²×C₂₃	#107	semidirect
47. C₂×C₂₃	C₂×#107	semidirect
53. C₂²×C₈	Dic₂₃	semidirect
58. C₂₃	C₂²×#107	semidirect
66. C₂×C₈	C₂×Dic₂₃	semidirect
81. C₈	C₂²×Dic₂₃	semidirect
82. C₂²	#10625000	direct (N × H)
91. C₂	C₂×#10625000	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⋊Dic₂₃	2
C₂×#10625000	2
C₂³×C₈×C₂₃	2
C₂²×#107	23

Subgroups — C22×#10625000 #15681972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10625000 #15681972