Subgroups — C₂₃×#13636 #19161262

Order 5888.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#368	C₂	1	2	yes	maximal
3	C₂₃×#1425	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×C₂₃×#108	—	2	2	no	—
6	C₂₃×M₆₄	C₄	1	4	yes	—
7	C₂×C₂₃×C₃₂	C₂²	1	2	yes	—
8	C₂₃×#368	—	2	2	no	—
9	C₂₃×#368	C₂²	1	4	yes	—
10	C₂×M₁₆×C₂₃	—	4	2	no	—
11	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₂₃	Q₈	1	2	yes	—
15	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
16	C₂₃×#108	—	2	4	no	—
17	C₂₃×#108	—	4	2	no	—
18	C₂₃×C₃₂	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
20	M₁₆×C₂₃	—	4	4	no	—
21	M₁₆×C₂₃	—	8	2	no	—
22	C₂×C₈×C₂₃	—	8	2	no	—
23	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
25	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
26	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
28	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₁₆×C₂₃	C₂×Q₈	1	1	yes	characteristic
30	#13636	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂³×C₂₃	Q₃₂	1	1	yes	characteristic
32	C₂×C₄×C₂₃	#107	1	2	yes	—
33	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
34	C₂×C₄×C₂₃	Q₃₂	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
36	C₈×C₂₃	—	8	4	no	—
37	C₈×C₂₃	#107	1	2	yes	—
38	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×M₆₄	C₂×C₂₃	1	1	yes	characteristic
41	C₂×#368	C₂×C₂₃	1	2	yes	—
42	#1425	C₂×C₂₃	1	4	yes	complemented
43	C₂²×C₂₃	—	4	1	no	—
44	C₂²×C₂₃	#366	1	2	yes	—
45	C₂²×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
46	C₄×C₂₃	#366	1	2	yes	—
47	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
48	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
49	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
50	C₂×#108	—	2	2	no	—
51	M₆₄	C₄×C₂₃	1	4	yes	—
52	C₂×C₃₂	C₂²×C₂₃	1	2	yes	—
53	#368	—	2	2	no	—
54	#368	C₂²×C₂₃	1	4	yes	—
55	C₂×C₂₃	—	2	2	no	—
56	C₂×C₂₃	—	2	1	no	—
57	C₂×C₂₃	C₂×#366	1	1	yes	characteristic
58	C₂×M₁₆	—	4	2	no	—
59	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
61	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
62	C₂×C₁₆	Q₈×C₂₃	1	2	yes	—
63	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
64	#108	—	2	4	no	—
65	#108	—	4	2	no	—
66	C₃₂	C₂×C₄×C₂₃	1	4	yes	—
67	C₂₃	#13636	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
69	M₁₆	—	4	4	no	—
70	M₁₆	—	8	2	no	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
73	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
74	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
75	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
76	C₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
77	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
78	C₁₆	C₂×Q₈×C₂₃	1	1	yes	characteristic
79	C₂³	C₂₃×Q₃₂	1	1	yes	characteristic
80	C₂×C₄	C₂₃×#107	1	2	yes	—
81	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
82	C₂×C₄	C₂₃×Q₃₂	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
84	C₈	—	8	4	no	—
85	C₈	C₂₃×#107	1	2	yes	—
86	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
87	C₈	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	C₂₃×#366	1	2	yes	—
90	C₂²	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
91	C₄	C₂₃×#366	1	2	yes	—
92	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
93	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	C₂×C₂₃×#366	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂×C₂₃	C₂×#366
prime radical	C₂³×C₂₃	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1425	C₂	semidirect
30. #13636	C₂₃	direct (N × H)
42. #1425	C₂×C₂₃	semidirect
67. C₂₃	#13636	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×M₆₄	2
C₂×C₂₃×#368	2
C₂₃×#1425	2
#13636	23

Subgroups — C23×#13636 #19161262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13636 #19161262