Subgroups — C₂₃×#13287 #19160913

Order 5888.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1386	C₂	1	4	yes	maximal
5	C₂₃×#533	—	2	1	no	—
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂₃×#500	C₄	1	2	yes	complemented
8	C₂₃×#499	—	2	1	no	—
9	C₂₃×#499	C₄	1	2	yes	complemented
10	C₂₃×#498	—	2	1	no	—
11	C₂₃×#427	—	2	1	no	—
12	C₂₃×#427	C₂²	1	2	yes	—
13	C₂×C₂₃×#106	—	2	1	no	—
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#384	—	2	1	no	—
17	C₂₃×#346	—	2	1	no	—
18	C₄×C₁₆×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×C₄⋊Q₈	—	2	2	no	—
22	C₂₃×#125	—	2	2	no	—
23	C₂₃×C₂²⋊Q₈	—	4	2	no	—
24	C₄×Q₈×C₂₃	—	4	2	no	—
25	C₂₃×#117	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
28	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
29	C₂₃×M₃₂	—	2	2	no	—
30	C₂₃×M₃₂	Q₈	1	4	yes	complemented
31	C₂×C₁₆×C₂₃	—	2	2	no	—
32	C₂₃×#107	—	2	2	no	—
33	C₂₃×#107	C₂×C₄	1	4	yes	—
34	C₂₃×#106	—	2	2	no	—
35	C₂₃×#106	—	2	2	no	—
36	C₂₃×#106	—	2	4	no	—
37	C₂₃×#106	—	4	2	no	—
38	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₂₃	—	4	2	no	—
43	C₂²×C₄×C₂₃	—	4	2	no	—
44	C₂²×C₄×C₂₃	D₈	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₂₃	#41	1	2	yes	—
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	4	4	no	—
56	C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₄⋊C₄×C₂₃	—	4	2	no	—
58	C₄⋊C₄×C₂₃	—	4	2	no	—
59	C₄⋊C₄×C₂₃	—	8	4	no	—
60	C₂²⋊C₄×C₂₃	—	4	2	no	—
61	C₄²×C₂₃	—	4	2	no	—
62	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₂₃	SD₁₆	1	2	yes	—
65	C₁₆×C₂₃	—	2	4	no	—
66	#13287	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₂₃	—	8	4	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	8	4	no	—
72	C₂×C₄×C₂₃	—	8	2	no	—
73	C₂×C₄×C₂₃	#102	1	2	yes	—
74	C₂×C₄×C₂₃	#102	1	2	yes	—
75	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₈×C₂₃	#103	1	2	yes	—
82	C₈×C₂₃	#103	1	2	yes	—
83	C₈×C₂₃	C₄×Q₈	1	2	yes	—
84	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
85	#3160	C₂×C₂₃	1	1	yes	characteristic
86	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
87	#1844	C₂×C₂₃	1	1	yes	characteristic
88	#1386	C₂×C₂₃	1	4	yes	—
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
91	C₂²×C₂₃	#486	1	1	yes	characteristic
92	C₂²×C₂₃	#500	1	1	yes	characteristic
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	8	4	no	—
95	C₄×C₂₃	—	16	2	no	—
96	C₄×C₂₃	#360	1	4	yes	—
97	C₄×C₂₃	#391	1	1	yes	characteristic
98	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
99	C₄×C₂₃	#487	1	1	yes	characteristic
100	C₄×C₂₃	#498	1	1	yes	characteristic
101	#533	—	2	1	no	—
102	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
103	#500	C₄×C₂₃	1	2	yes	complemented
104	#499	—	2	1	no	—
105	#499	C₄×C₂₃	1	2	yes	complemented
106	#498	—	2	1	no	—
107	#427	—	2	1	no	—
108	#427	C₂²×C₂₃	1	2	yes	—
109	C₂×#106	—	2	1	no	—
110	C₂×#106	C₂²×C₂₃	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
112	#384	—	2	1	no	—
113	#346	—	2	1	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₂×C₂₃	—	2	1	no	—
116	C₂×C₂₃	—	2	1	no	—
117	C₂×C₂₃	#1990	1	1	yes	characteristic
118	C₂×C₂₃	C₂×#360	1	2	yes	—
119	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	2	no	—
122	#125	—	2	2	no	—
123	C₂²⋊Q₈	—	4	2	no	—
124	C₄×Q₈	—	4	2	no	—
125	#117	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	Q₈×C₂₃	1	4	yes	complemented
131	C₂×C₁₆	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	C₂×C₄×C₂₃	1	4	yes	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	2	4	no	—
137	#106	—	4	2	no	—
138	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
139	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
141	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
142	C₂₃	#13287	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	D₈×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
150	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
151	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
152	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
153	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
154	C₂×C₈	#41×C₂₃	1	2	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	8	4	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
164	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
165	C₄²	SD₁₆×C₂₃	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	C₂₃×#102	1	2	yes	—
174	C₂×C₄	C₂₃×#102	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
177	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
181	C₈	C₂₃×#103	1	2	yes	—
182	C₈	C₂₃×#103	1	2	yes	—
183	C₈	C₄×Q₈×C₂₃	1	2	yes	—
184	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
187	C₂²	C₂₃×#486	1	1	yes	characteristic
188	C₂²	C₂₃×#500	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	8	4	no	—
191	C₄	—	16	2	no	—
192	C₄	C₂₃×#360	1	4	yes	—
193	C₄	C₂₃×#391	1	1	yes	characteristic
194	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
195	C₄	C₂₃×#487	1	1	yes	characteristic
196	C₄	C₂₃×#498	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	C₂₃×#1990	1	1	yes	characteristic
200	C₂	C₂×C₂₃×#360	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
7. C₂₃×#500	C₄	semidirect
9. C₂₃×#499	C₄	semidirect
30. C₂₃×M₃₂	Q₈	semidirect
66. #13287	C₂₃	direct (N × H)
103. #500	C₄×C₂₃	semidirect
105. #499	C₄×C₂₃	semidirect
130. M₃₂	Q₈×C₂₃	semidirect
142. C₂₃	#13287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3160	2
C₄×C₂₃×M₃₂	2
C₂₃×#1844	2
C₂₃×#1386	2
#13287	23

Subgroups — C23×#13287 #19160913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13287 #19160913