Subgroups — C₂₃×#13259 #19160885

Order 5888.

472 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2145	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2143	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#135	—	2	1	no	—
7	C₂×C₂₃×#135	C₂²	1	2	yes	—
8	C₂₃×#532	—	2	1	no	—
9	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
10	C₂₃×#495	—	2	1	no	—
11	C₂₃×#495	C₄	1	2	yes	complemented
12	C₂₃×#494	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
14	C₂₃×#492	—	2	1	no	—
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂₃×#427	C₂²	1	1	yes	characteristic
17	C₂₃×#405	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#364	—	2	1	no	—
20	C₂₃×#364	—	2	1	no	—
21	C₂₃×#360	—	2	4	no	—
22	C₂₃×#359	—	2	4	no	—
23	C₂₃×#339	—	2	1	no	—
24	C₂×#41×C₂₃	—	2	2	no	—
25	C₂₃×#135	—	2	2	no	—
26	C₂₃×#135	—	2	2	no	—
27	C₂₃×#135	—	2	2	no	—
28	C₂₃×#135	—	2	2	no	—
29	C₂₃×#135	—	4	1	no	—
30	C₂₃×#135	—	4	1	no	—
31	C₂₃×#135	—	4	1	no	—
32	C₂₃×#135	—	4	1	no	—
33	C₂×Q₁₆×C₂₃	—	2	1	no	—
34	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₂₃	—	2	2	no	—
36	C₂×SD₁₆×C₂₃	—	2	2	no	—
37	C₂×D₈×C₂₃	—	2	1	no	—
38	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊Q₈	—	2	1	no	—
43	C₂₃×#127	—	2	1	no	—
44	C₂₃×#124	—	2	2	no	—
45	C₂₃×#121	—	4	2	no	—
46	C₄×D₄×C₂₃	—	4	2	no	—
47	C₂₃×#117	—	2	1	no	—
48	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
49	C₂₃×M₃₂	—	2	2	no	—
50	C₂₃×M₃₂	—	2	2	no	—
51	C₂×C₁₆×C₂₃	—	2	2	no	—
52	C₂₃×#108	—	2	1	no	—
53	C₂₃×#108	D₄	1	2	yes	complemented
54	C₂₃×#107	—	2	1	no	—
55	C₂₃×#106	D₄	1	2	yes	complemented
56	C₂₃×#97	—	4	1	no	—
57	C₄×C₈×C₂₃	—	4	1	no	—
58	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
62	#41×C₂₃	—	4	2	no	—
63	#41×C₂₃	—	4	2	no	—
64	#41×C₂₃	—	4	2	no	—
65	#41×C₂₃	—	4	2	no	—
66	C₂×Q₈×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	—	4	2	no	—
69	C₂×D₄×C₂₃	—	4	2	no	—
70	C₂²×C₄×C₂₃	—	2	1	no	—
71	C₂²×C₄×C₂₃	—	4	2	no	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₂₃	—	2	2	no	—
74	Q₁₆×C₂₃	—	2	2	no	—
75	Q₁₆×C₂₃	—	4	1	no	—
76	Q₁₆×C₂₃	—	4	1	no	—
77	SD₁₆×C₂₃	—	4	2	no	—
78	SD₁₆×C₂₃	—	4	2	no	—
79	SD₁₆×C₂₃	—	4	2	no	—
80	SD₁₆×C₂₃	—	4	2	no	—
81	D₈×C₂₃	—	2	2	no	—
82	D₈×C₂₃	—	2	2	no	—
83	D₈×C₂₃	—	4	1	no	—
84	D₈×C₂₃	—	4	1	no	—
85	M₁₆×C₂₃	—	2	2	no	—
86	M₁₆×C₂₃	—	4	1	no	—
87	C₂×C₈×C₂₃	—	4	1	no	—
88	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
91	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
92	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	4	2	no	—
102	C₂²⋊C₄×C₂₃	—	4	4	no	—
103	C₂²⋊C₄×C₂₃	—	4	1	no	—
104	C₄²×C₂₃	—	2	1	no	—
105	C₄²×C₂₃	—	4	1	no	—
106	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₁₆×C₂₃	—	4	2	no	—
109	#13259	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₂₃	—	4	2	no	—
111	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	Q₈×C₂₃	—	4	2	no	—
113	Q₈×C₂₃	—	4	2	no	—
114	D₄×C₂₃	—	4	2	no	—
115	D₄×C₂₃	—	4	2	no	—
116	D₄×C₂₃	—	8	4	no	—
117	D₄×C₂₃	—	8	2	no	—
118	D₄×C₂₃	—	8	2	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	8	2	no	—
125	C₂×C₄×C₂₃	—	8	2	no	—
126	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₈×C₂₃	—	4	2	no	—
133	C₈×C₂₃	#102	1	1	yes	characteristic
134	C₈×C₂₃	#102	1	1	yes	characteristic
135	C₈×C₂₃	#102	1	1	yes	characteristic
136	C₈×C₂₃	#102	1	1	yes	characteristic
137	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
138	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₂₃	#124	1	1	yes	characteristic
140	C₈×C₂₃	#127	1	1	yes	characteristic
141	#3147	C₂×C₂₃	1	1	yes	characteristic
142	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#2145	C₂×C₂₃	1	2	yes	complemented
144	#2143	C₂×C₂₃	1	2	yes	complemented
145	#1772	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	8	4	no	—
148	C₂²×C₂₃	—	8	2	no	—
149	C₂²×C₂₃	#418	1	1	yes	characteristic
150	C₂²×C₂₃	#459	1	1	yes	characteristic
151	C₂²×C₂₃	#460	1	1	yes	characteristic
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	4	1	no	—
154	C₄×C₂₃	—	8	2	no	—
155	C₄×C₂₃	—	8	2	no	—
156	C₄×C₂₃	#390	1	1	yes	characteristic
157	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₂₃	#459	1	1	yes	characteristic
159	C₄×C₂₃	#460	1	1	yes	characteristic
160	C₂×#135	—	2	1	no	—
161	C₂×#135	C₂²×C₂₃	1	2	yes	—
162	#532	—	2	1	no	—
163	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
164	#495	—	2	1	no	—
165	#495	C₄×C₂₃	1	2	yes	complemented
166	#494	C₄×C₂₃	1	1	yes	characteristic, complemented
167	#493	C₄×C₂₃	1	1	yes	characteristic, complemented
168	#492	—	2	1	no	—
169	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
170	#427	C₂²×C₂₃	1	1	yes	characteristic
171	#405	—	2	1	no	—
172	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
173	#364	—	2	1	no	—
174	#364	—	2	1	no	—
175	#360	—	2	4	no	—
176	#359	—	2	4	no	—
177	#339	—	2	1	no	—
178	C₂×C₂₃	—	2	1	no	—
179	C₂×C₂₃	—	2	1	no	—
180	C₂×C₂₃	—	2	1	no	—
181	C₂×C₂₃	—	16	2	no	—
182	C₂×C₂₃	#1982	1	1	yes	characteristic
183	C₂×#41	—	2	2	no	—
184	#135	—	2	2	no	—
185	#135	—	2	2	no	—
186	#135	—	2	2	no	—
187	#135	—	2	2	no	—
188	#135	—	4	1	no	—
189	#135	—	4	1	no	—
190	#135	—	4	1	no	—
191	#135	—	4	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×SD₁₆	—	2	2	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
201	C₄⋊Q₈	—	2	1	no	—
202	#127	—	2	1	no	—
203	#124	—	2	2	no	—
204	#121	—	4	2	no	—
205	C₄×D₄	—	4	2	no	—
206	#117	—	2	1	no	—
207	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
208	M₃₂	—	2	2	no	—
209	M₃₂	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	#108	—	2	1	no	—
212	#108	D₄×C₂₃	1	2	yes	complemented
213	#107	—	2	1	no	—
214	#106	D₄×C₂₃	1	2	yes	complemented
215	#97	—	4	1	no	—
216	C₄×C₈	—	4	1	no	—
217	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
218	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
219	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
220	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
221	C₂₃	#13259	1	1	yes	characteristic, direct product factor, Sylow
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	4	1	no	—
237	SD₁₆	—	4	2	no	—
238	SD₁₆	—	4	2	no	—
239	SD₁₆	—	4	2	no	—
240	SD₁₆	—	4	2	no	—
241	D₈	—	2	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	4	1	no	—
244	D₈	—	4	1	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
251	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
252	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
253	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
258	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
259	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
267	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
268	C₁₆	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	8	4	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
287	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
288	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
290	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	C₂₃×#102	1	1	yes	characteristic
293	C₈	C₂₃×#102	1	1	yes	characteristic
294	C₈	C₂₃×#102	1	1	yes	characteristic
295	C₈	C₂₃×#102	1	1	yes	characteristic
296	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
297	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
298	C₈	C₂₃×#124	1	1	yes	characteristic
299	C₈	C₂₃×#127	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₂₃×#418	1	1	yes	characteristic
304	C₂²	C₂₃×#459	1	1	yes	characteristic
305	C₂²	C₂₃×#460	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂₃×#390	1	1	yes	characteristic
311	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
312	C₄	C₂₃×#459	1	1	yes	characteristic
313	C₄	C₂₃×#460	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₂₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#459
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1982
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#364	C₂	semidirect
3. C₂₃×#2145	C₂	semidirect
4. C₂₃×#2143	C₂	semidirect
5. C₂₃×#1772	C₂	semidirect
11. C₂₃×#495	C₄	semidirect
12. C₂₃×#494	C₄	semidirect
13. C₂₃×#493	C₄	semidirect
53. C₂₃×#108	D₄	semidirect
55. C₂₃×#106	D₄	semidirect
109. #13259	C₂₃	direct (N × H)
142. C₂×#364	C₂×C₂₃	semidirect
143. #2145	C₂×C₂₃	semidirect
144. #2143	C₂×C₂₃	semidirect
145. #1772	C₂×C₂₃	semidirect
165. #495	C₄×C₂₃	semidirect
166. #494	C₄×C₂₃	semidirect
167. #493	C₄×C₂₃	semidirect
212. #108	D₄×C₂₃	semidirect
214. #106	D₄×C₂₃	semidirect
221. C₂₃	#13259	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×#3147	2
C₂×C₂₃×#364	2
C₂₃×#2145	2
C₂₃×#2143	2
C₂₃×#1772	2
#13259	23

Subgroups — C23×#13259 #19160885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13259 #19160885