Subgroups — C₂₃×#3162 #15670781

Order 2944.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#533	C₂	1	2	yes	maximal
2	C₂₃×#501	C₂	1	8	yes	maximal, complemented
3	C₂₃×#427	C₂	1	4	yes	maximal
4	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
7	C₂₃×#125	C₂²	1	4	yes	—
8	C₂₃×C₂²⋊Q₈	—	2	4	no	—
9	C₄×Q₈×C₂₃	—	2	4	no	—
10	C₂₃×#117	C₂²	1	4	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#107	C₂²	1	8	yes	—
13	C₂₃×#106	C₂²	1	8	yes	—
14	C₂₃×#97	C₂²	1	4	yes	—
15	C₂×Q₈×C₂₃	—	2	4	no	—
16	C₂²×C₄×C₂₃	D₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₃	Q₈	1	8	yes	complemented
19	C₂×C₈×C₂₃	C₂³	1	4	yes	—
20	C₄⋊C₄×C₂₃	—	2	8	no	—
21	C₄⋊C₄×C₂₃	—	2	4	no	—
22	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
23	C₂²⋊C₄×C₂₃	—	2	4	no	—
24	C₄²×C₂₃	—	2	4	no	—
25	C₄²×C₂₃	D₄	1	2	yes	—
26	C₄²×C₂₃	C₂³	1	2	yes	—
27	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
28	Q₈×C₂₃	—	4	8	no	—
29	C₂×C₄×C₂₃	—	2	8	no	—
30	C₂×C₄×C₂₃	—	4	2	no	—
31	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
32	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
35	C₈×C₂₃	C₂×Q₈	1	8	yes	—
36	#3162	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₂₃	—	2	2	no	—
38	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
39	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
40	C₄×C₂₃	—	2	4	no	—
41	C₄×C₂₃	—	4	8	no	—
42	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
43	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
44	#533	C₂×C₂₃	1	2	yes	—
45	#501	C₂×C₂₃	1	8	yes	complemented
46	#427	C₂×C₂₃	1	4	yes	—
47	C₂×#97	C₂×C₂₃	1	1	yes	characteristic
48	C₂×C₂₃	—	2	2	no	—
49	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₂₃	#575	1	2	yes	—
51	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
52	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
53	#125	C₂²×C₂₃	1	4	yes	—
54	C₂²⋊Q₈	—	2	4	no	—
55	C₄×Q₈	—	2	4	no	—
56	#117	C₂²×C₂₃	1	4	yes	—
57	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
58	#107	C₂²×C₂₃	1	8	yes	—
59	#106	C₂²×C₂₃	1	8	yes	—
60	#97	C₂²×C₂₃	1	4	yes	—
61	C₂₃	#3162	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×Q₈	—	2	4	no	—
63	C₂²×C₄	D₄×C₂₃	1	2	yes	—
64	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
65	C₂×C₈	Q₈×C₂₃	1	8	yes	complemented
66	C₂×C₈	C₂³×C₂₃	1	4	yes	—
67	C₄⋊C₄	—	2	8	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
70	C₂²⋊C₄	—	2	4	no	—
71	C₄²	—	2	4	no	—
72	C₄²	D₄×C₂₃	1	2	yes	—
73	C₄²	C₂³×C₂₃	1	2	yes	—
74	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
75	Q₈	—	4	8	no	—
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
79	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
80	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
81	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
82	C₈	C₂×Q₈×C₂₃	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	C₂₃×C₄⋊Q₈	1	2	yes	—
85	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	4	8	no	—
88	C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
89	C₄	C₂²×Q₈×C₂₃	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
92	C₂	C₂₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#501	C₂	semidirect
18. C₂×C₈×C₂₃	Q₈	semidirect
36. #3162	C₂₃	direct (N × H)
45. #501	C₂×C₂₃	semidirect
61. C₂₃	#3162	direct (N × H)
65. C₂×C₈	Q₈×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#533	2
C₂₃×#501	2
C₂₃×#427	2
C₂×C₂₃×#97	2
#3162	23

Subgroups — C23×#3162 #15670781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3162 #15670781