Subgroups — C₂×C₂₃×#1712 #19167227

Order 5888.

1342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#493	C₂	1	6	yes	maximal, complemented
2	C₂₃×#1712	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈×C₂₃	—	2	12	no	—
5	C₂×C₂₃×#127	C₂²	1	4	yes	—
6	C₂₃×#493	—	2	12	no	—
7	C₂₃×#493	C₂²	1	24	yes	complemented
8	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
9	C₈²×C₂₃	C₂²	1	4	yes	complemented
10	C₂²×D₄×C₂₃	—	2	12	no	—
11	C₂×D₈×C₂₃	—	2	48	no	—
12	C₂×D₈×C₂₃	—	4	48	no	—
13	C₂²×C₈×C₂₃	D₄	1	6	yes	—
14	C₂₃×#127	—	2	12	no	—
15	C₂₃×#127	C₂³	1	8	yes	—
16	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₂₃	D₄	1	6	yes	—
18	C₄×C₈×C₂₃	C₂³	1	6	yes	—
19	C₂⁴×C₂₃	—	4	4	no	—
20	C₂×D₄×C₂₃	—	2	24	no	—
21	C₂×D₄×C₂₃	—	4	12	no	—
22	C₂×D₄×C₂₃	—	4	48	no	—
23	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
24	D₈×C₂₃	—	4	96	no	—
25	C₂×C₈×C₂₃	D₈	1	24	yes	complemented
26	C₂×C₈×C₂₃	C₂×D₄	1	12	yes	—
27	C₄²×C₂₃	C₂×D₄	1	3	yes	—
28	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
29	C₂×#1712	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₂₃	—	4	8	no	—
31	C₂³×C₂₃	—	8	24	no	—
32	C₂³×C₂₃	#127	1	1	yes	characteristic
33	D₄×C₂₃	—	4	48	no	—
34	D₄×C₂₃	—	8	24	no	—
35	C₂×C₄×C₂₃	#127	1	3	yes	—
36	C₂×C₄×C₂₃	C₂×D₈	1	12	yes	—
37	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
38	C₈×C₂₃	C₂×D₈	1	24	yes	complemented
39	C₂×#493	C₂×C₂₃	1	6	yes	complemented
40	#1712	C₂×C₂₃	1	8	yes	direct product factor
41	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
42	C₂²×C₂₃	—	8	24	no	—
43	C₂²×C₂₃	—	16	16	no	—
44	C₂²×C₂₃	#493	1	6	yes	—
45	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
46	C₄×C₂₃	#493	1	6	yes	—
47	C₄×C₂₃	C₂²×D₈	1	6	yes	—
48	C₂²×D₈	—	2	12	no	—
49	C₂×#127	C₂²×C₂₃	1	4	yes	—
50	#493	—	2	12	no	—
51	#493	C₂²×C₂₃	1	24	yes	complemented
52	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
53	C₈²	C₂²×C₂₃	1	4	yes	complemented
54	C₂×C₂₃	—	16	8	no	—
55	C₂×C₂₃	#1712	1	4	yes	direct product factor
56	C₂×C₂₃	C₂×#493	1	3	yes	—
57	C₂²×D₄	—	2	12	no	—
58	C₂×D₈	—	2	48	no	—
59	C₂×D₈	—	4	48	no	—
60	C₂²×C₈	D₄×C₂₃	1	6	yes	—
61	#127	—	2	12	no	—
62	#127	C₂³×C₂₃	1	8	yes	—
63	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
64	C₄×C₈	D₄×C₂₃	1	6	yes	—
65	C₄×C₈	C₂³×C₂₃	1	6	yes	—
66	C₂₃	C₂×#1712	1	1	yes	characteristic, direct product factor, Sylow
67	C₂⁴	—	4	4	no	—
68	C₂×D₄	—	2	24	no	—
69	C₂×D₄	—	4	12	no	—
70	C₂×D₄	—	4	48	no	—
71	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
72	D₈	—	4	96	no	—
73	C₂×C₈	D₈×C₂₃	1	24	yes	complemented
74	C₂×C₈	C₂×D₄×C₂₃	1	12	yes	—
75	C₄²	C₂×D₄×C₂₃	1	3	yes	—
76	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
77	C₂³	—	4	8	no	—
78	C₂³	—	8	24	no	—
79	C₂³	C₂₃×#127	1	1	yes	characteristic
80	D₄	—	4	48	no	—
81	D₄	—	8	24	no	—
82	C₂×C₄	C₂₃×#127	1	3	yes	—
83	C₂×C₄	C₂×D₈×C₂₃	1	12	yes	—
84	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
85	C₈	C₂×D₈×C₂₃	1	24	yes	complemented
86	C₂²	—	8	24	no	—
87	C₂²	—	16	16	no	—
88	C₂²	C₂₃×#493	1	6	yes	—
89	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
90	C₄	C₂₃×#493	1	6	yes	—
91	C₄	C₂²×D₈×C₂₃	1	6	yes	—
92	C₂	—	16	8	no	—
93	C₂	C₂₃×#1712	1	4	yes	direct product factor
94	C₂	C₂×C₂₃×#493	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#493	C₂	semidirect
2. C₂₃×#1712	C₂	direct (N × H)
3. C₂×C₈²×C₂₃	C₂	semidirect
7. C₂₃×#493	C₂²	semidirect
9. C₈²×C₂₃	C₂²	semidirect
25. C₂×C₈×C₂₃	D₈	semidirect
29. C₂×#1712	C₂₃	direct (N × H)
38. C₈×C₂₃	C₂×D₈	semidirect
39. C₂×#493	C₂×C₂₃	semidirect
40. #1712	C₂×C₂₃	direct (N × H)
41. C₂×C₈²	C₂×C₂₃	semidirect
51. #493	C₂²×C₂₃	semidirect
53. C₈²	C₂²×C₂₃	semidirect
55. C₂×C₂₃	#1712	direct (N × H)
66. C₂₃	C₂×#1712	direct (N × H)
73. C₂×C₈	D₈×C₂₃	semidirect
85. C₈	C₂×D₈×C₂₃	semidirect
93. C₂	C₂₃×#1712	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×C₂₃×#493	2
C₂₃×#1712	2
C₂×C₈²×C₂₃	2
C₂×#1712	23

Subgroups — C2×C23×#1712 #19167227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1712 #19167227