Subgroups — C₂₃×#2884 #15670503

Order 2944.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#519	C₂	1	2	yes	maximal, complemented
3	C₂₃×#410	C₂	1	12	yes	maximal, complemented
4	C₂₃×2₋¹⁺⁴	—	2	8	no	—
5	C₂×#41×C₂₃	C₄	1	4	yes	complemented
6	C₂×#41×C₂₃	C₂²	1	6	yes	—
7	C₂²×Q₈×C₂₃	C₄	1	4	yes	complemented
8	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₂₃	—	2	8	no	—
10	C₂₃×#117	C₂²	1	12	yes	—
11	C₂₃×#99	C₂²	1	16	yes	—
12	#41×C₂₃	—	2	16	no	—
13	#41×C₂₃	—	2	24	no	—
14	C₂×Q₈×C₂₃	—	2	16	no	—
15	C₂×Q₈×C₂₃	C₂×C₄	1	6	yes	—
16	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
17	C₂×Q₈×C₂₃	D₄	1	8	yes	—
18	C₂×D₄×C₂₃	C₂×C₄	1	6	yes	—
19	C₂×D₄×C₂₃	C₂³	1	4	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	12	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
22	C₄⋊C₄×C₂₃	—	2	12	no	—
23	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
24	C₄²×C₂₃	—	2	12	no	—
25	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
26	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
27	Q₈×C₂₃	—	2	12	no	—
28	Q₈×C₂₃	C₂²⋊C₄	1	16	yes	—
29	D₄×C₂₃	—	2	8	no	—
30	D₄×C₂₃	—	2	12	no	—
31	C₂×C₄×C₂₃	—	2	24	no	—
32	C₂×C₄×C₂₃	—	2	8	no	—
33	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
34	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	—
36	#2884	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₂₃	—	2	4	no	—
38	C₂²×C₂₃	—	4	1	no	—
39	C₂²×C₂₃	—	4	4	no	—
40	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
41	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
42	C₄×C₂₃	—	2	4	no	—
43	C₄×C₂₃	—	4	8	no	—
44	C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
45	C₂×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic
46	#519	C₂×C₂₃	1	2	yes	complemented
47	#410	C₂×C₂₃	1	12	yes	complemented
48	C₂×C₂₃	—	2	2	no	—
49	C₂×C₂₃	—	2	1	no	—
50	C₂×C₂₃	—	4	4	no	—
51	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
52	2₋¹⁺⁴	—	2	8	no	—
53	C₂×#41	C₄×C₂₃	1	4	yes	complemented
54	C₂×#41	C₂²×C₂₃	1	6	yes	—
55	C₂²×Q₈	C₄×C₂₃	1	4	yes	complemented
56	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
57	C₄×Q₈	—	2	8	no	—
58	#117	C₂²×C₂₃	1	12	yes	—
59	#99	C₂²×C₂₃	1	16	yes	—
60	C₂₃	#2884	1	1	yes	characteristic, direct product factor, Sylow
61	#41	—	2	16	no	—
62	#41	—	2	24	no	—
63	C₂×Q₈	—	2	16	no	—
64	C₂×Q₈	C₂×C₄×C₂₃	1	6	yes	—
65	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	—
66	C₂×Q₈	D₄×C₂₃	1	8	yes	—
67	C₂×D₄	C₂×C₄×C₂₃	1	6	yes	—
68	C₂×D₄	C₂³×C₂₃	1	4	yes	—
69	C₂²×C₄	C₂×C₄×C₂₃	1	12	yes	—
70	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
71	C₄⋊C₄	—	2	12	no	—
72	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
73	C₄²	—	2	12	no	—
74	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
75	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
76	Q₈	—	2	12	no	—
77	Q₈	C₂²⋊C₄×C₂₃	1	16	yes	—
78	D₄	—	2	8	no	—
79	D₄	—	2	12	no	—
80	C₂×C₄	—	2	24	no	—
81	C₂×C₄	—	2	8	no	—
82	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
83	C₂×C₄	C₂²×C₄×C₂₃	1	6	yes	—
84	C₂×C₄	C₂×D₄×C₂₃	1	12	yes	—
85	C₂²	—	2	4	no	—
86	C₂²	—	4	1	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
89	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	4	8	no	—
92	C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
93	C₂	—	2	2	no	—
94	C₂	—	2	1	no	—
95	C₂	—	4	4	no	—
96	C₂	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂×C₂₃×2₋¹⁺⁴	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#519	C₂	semidirect
3. C₂₃×#410	C₂	semidirect
5. C₂×#41×C₂₃	C₄	semidirect
7. C₂²×Q₈×C₂₃	C₄	semidirect
36. #2884	C₂₃	direct (N × H)
46. #519	C₂×C₂₃	semidirect
47. #410	C₂×C₂₃	semidirect
53. C₂×#41	C₄×C₂₃	semidirect
55. C₂²×Q₈	C₄×C₂₃	semidirect
60. C₂₃	#2884	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×C₂₃×2₋¹⁺⁴	2
C₂₃×#519	2
C₂₃×#410	2
#2884	23

Subgroups — C23×#2884 #15670503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2884 #15670503