Subgroups — C₂₃×#11999 #19159625

Order 5888.

234 conjugacy classes of subgroups.

The subgroup lattice has 134 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	2	yes	maximal
3	C₂₃×#1368	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂₃×#431	—	2	2	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#346	C₄	1	2	yes	—
9	C₂₃×#346	C₂²	1	2	yes	—
10	C₂₃×#335	—	2	2	no	—
11	C₂₃×#334	C₂²	1	4	yes	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₂₃	—	2	2	no	—
16	C₂×C₁₆×C₂₃	C₈	1	4	yes	complemented
17	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂₃×C₄⋊C₈	—	2	4	no	—
20	C₂₃×C₂²⋊C₈	—	4	2	no	—
21	C₄×C₈×C₂₃	—	4	2	no	—
22	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	4	4	no	—
30	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
34	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₂₃	SD₁₆	1	2	yes	—
38	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₂₃	—	2	2	no	—
40	C₁₆×C₂₃	C₂×C₈	1	4	yes	—
41	#11999	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂³×C₂₃	#106	1	1	yes	characteristic
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
45	C₂×C₄×C₂₃	#106	1	2	yes	—
46	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#108	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
50	C₈×C₂₃	—	2	1	no	—
51	C₈×C₂₃	—	2	1	no	—
52	C₈×C₂₃	—	8	4	no	—
53	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
54	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
57	#1564	C₂×C₂₃	1	2	yes	—
58	#1368	C₂×C₂₃	1	4	yes	complemented
59	C₂²×C₂₃	—	2	2	no	—
60	C₂²×C₂₃	#334	1	2	yes	—
61	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	#334	1	2	yes	—
65	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
66	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
67	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
68	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
69	#431	—	2	2	no	—
70	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
71	#346	C₄×C₂₃	1	2	yes	—
72	#346	C₂²×C₂₃	1	2	yes	—
73	#335	—	2	2	no	—
74	#334	C₂²×C₂₃	1	4	yes	—
75	C₂×C₂₃	—	2	2	no	—
76	C₂×C₂₃	C₂×#334	1	1	yes	characteristic
77	C₂×C₂₃	#2155	1	2	yes	—
78	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
79	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
80	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₈×C₂₃	1	4	yes	complemented
83	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
84	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
85	C₄⋊C₈	—	2	4	no	—
86	C₂²⋊C₈	—	4	2	no	—
87	C₄×C₈	—	4	2	no	—
88	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
89	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
90	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
91	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
92	C₂₃	#11999	1	1	yes	characteristic, direct product factor, Sylow
93	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
94	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
98	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
100	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
101	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
102	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
103	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
104	C₄²	SD₁₆×C₂₃	1	2	yes	—
105	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂×C₈×C₂₃	1	4	yes	—
108	C₂³	C₂₃×#106	1	1	yes	characteristic
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
111	C₂×C₄	C₂₃×#106	1	2	yes	—
112	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
113	C₂×C₄	C₂₃×#108	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
115	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	8	4	no	—
119	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
120	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	C₂₃×#334	1	2	yes	—
124	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	C₂₃×#334	1	2	yes	—
128	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
129	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×C₂₃×#334	1	1	yes	characteristic
132	C₂	C₂₃×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#106
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#106
prime radical	C₂³×C₂₃	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1368	C₂	semidirect
16. C₂×C₁₆×C₂₃	C₈	semidirect
41. #11999	C₂₃	direct (N × H)
58. #1368	C₂×C₂₃	semidirect
82. C₂×C₁₆	C₈×C₂₃	semidirect
92. C₂₃	#11999	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×#346	2
C₂₃×#1564	2
C₂₃×#1368	2
#11999	23

Subgroups — C23×#11999 #19159625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11999 #19159625