Subgroups — C₂₃×#33689 #19181315

Order 5888.

520 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2956	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2239	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2230	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2214	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2191	C₂	1	2	yes	maximal
7	C₂₃×#2180	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#575	C₂²	1	1	yes	characteristic
10	C₂₃×#568	—	2	1	no	—
11	C₂₃×#533	—	2	1	no	—
12	C₂₃×#510	C₂²	1	4	yes	—
13	C₂₃×#498	C₂²	1	1	yes	characteristic, complemented
14	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#487	—	2	1	no	—
16	C₂₃×#470	—	2	1	no	—
17	C₂₃×#470	C₂²	1	2	yes	—
18	C₂₃×#444	—	2	1	no	—
19	C₂₃×#444	C₂²	1	2	yes	—
20	C₂₃×#443	—	2	1	no	—
21	C₂₃×#443	C₂²	1	2	yes	—
22	C₂₃×#441	—	2	1	no	—
23	C₂₃×#441	C₂²	1	2	yes	complemented
24	C₂₃×#430	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#104	—	2	1	no	—
26	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#365	C₂²	1	4	yes	complemented
28	C₂₃×#364	C₂²	1	2	yes	complemented
29	C₂₃×#362	C₂²	1	4	yes	complemented
30	C₂₃×#360	C₂²	1	4	yes	—
31	C₂₃×#349	C₂²	1	2	yes	—
32	C₂₃×#347	C₂²	1	2	yes	complemented
33	C₂×#41×C₂₃	—	2	1	no	—
34	C₂₃×#137	—	2	2	no	—
35	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
36	C₂₃×#135	—	2	1	no	—
37	C₂₃×#135	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₂₃	—	2	1	no	—
39	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
40	C₂₃×#131	—	2	2	no	—
41	C₂₃×#131	—	2	2	no	—
42	C₂₃×#131	—	4	2	no	—
43	C₂×M₁₆×C₂₃	—	2	1	no	—
44	C₂×M₁₆×C₂₃	—	4	2	no	—
45	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×C₄⋊Q₈	—	2	1	no	—
48	C₂₃×C₄⋊Q₈	—	2	1	no	—
49	C₂₃×#125	—	4	1	no	—
50	C₂₃×C₂²⋊Q₈	—	2	2	no	—
51	C₂₃×C₂²⋊Q₈	—	4	1	no	—
52	C₄×Q₈×C₂₃	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	4	1	no	—
54	C₂₃×#117	—	4	1	no	—
55	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
56	C₂₃×Q₃₂	—	2	4	no	—
57	C₂₃×SD₃₂	—	2	4	no	—
58	C₂₃×M₃₂	C₂³	1	4	yes	—
59	C₂₃×#108	—	2	1	no	—
60	C₂₃×#108	C₂³	1	2	yes	—
61	C₂₃×#106	—	2	1	no	—
62	C₂₃×#106	C₂³	1	2	yes	—
63	C₂₃×#104	—	2	2	no	—
64	C₂₃×#104	—	2	2	no	—
65	C₂₃×#103	—	2	2	no	—
66	C₂₃×#103	—	2	2	no	—
67	C₂₃×#97	D₄	1	1	yes	characteristic
68	C₂₃×#97	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	#41×C₂₃	—	2	2	no	—
72	#41×C₂₃	—	2	2	no	—
73	#41×C₂₃	—	4	1	no	—
74	#41×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	—	2	1	no	—
76	C₂×Q₈×C₂₃	—	2	2	no	—
77	C₂×Q₈×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	2	1	no	—
79	C₂×D₄×C₂₃	—	4	1	no	—
80	C₂²×C₄×C₂₃	—	2	1	no	—
81	C₂²×C₄×C₂₃	—	4	1	no	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₂₃	—	2	1	no	—
84	Q₁₆×C₂₃	—	2	4	no	—
85	Q₁₆×C₂₃	—	4	2	no	—
86	Q₁₆×C₂₃	#41	1	2	yes	—
87	SD₁₆×C₂₃	—	2	2	no	—
88	SD₁₆×C₂₃	—	2	2	no	—
89	D₈×C₂₃	—	2	1	no	—
90	D₈×C₂₃	#41	1	2	yes	—
91	M₁₆×C₂₃	—	2	2	no	—
92	M₁₆×C₂₃	—	2	2	no	—
93	M₁₆×C₂₃	—	2	2	no	—
94	M₁₆×C₂₃	—	4	2	no	—
95	M₁₆×C₂₃	—	4	1	no	—
96	M₁₆×C₂₃	—	4	2	no	—
97	C₂×C₈×C₂₃	—	4	2	no	—
98	C₂×C₈×C₂₃	—	4	1	no	—
99	C₂×C₈×C₂₃	—	4	2	no	—
100	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₄⋊C₄×C₂₃	—	4	1	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₄⋊C₄×C₂₃	—	4	1	no	—
109	C₄⋊C₄×C₂₃	—	4	2	no	—
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₄²×C₂₃	—	2	1	no	—
112	C₄²×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₂₃	—	2	4	no	—
116	#33689	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₂₃	—	4	1	no	—
118	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	Q₈×C₂₃	—	2	2	no	—
120	Q₈×C₂₃	—	4	1	no	—
121	Q₈×C₂₃	—	4	2	no	—
122	Q₈×C₂₃	—	4	2	no	—
123	Q₈×C₂₃	—	8	2	no	—
124	D₄×C₂₃	—	2	2	no	—
125	D₄×C₂₃	—	4	1	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	2	1	no	—
129	C₂×C₄×C₂₃	—	2	1	no	—
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	2	1	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	4	1	no	—
135	C₂×C₄×C₂₃	—	4	2	no	—
136	C₂×C₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	4	2	no	—
143	C₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₂₃	C₂×#41	1	2	yes	—
145	#3185	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#2956	C₂×C₂₃	1	1	yes	characteristic
147	#2239	C₂×C₂₃	1	2	yes	complemented
148	#2230	C₂×C₂₃	1	2	yes	complemented
149	#2214	C₂×C₂₃	1	2	yes	complemented
150	#2191	C₂×C₂₃	1	2	yes	—
151	#2180	C₂×C₂₃	1	4	yes	complemented
152	#2165	C₂×C₂₃	1	1	yes	characteristic, complemented
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	—	4	2	no	—
156	C₂²×C₂₃	—	16	1	no	—
157	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₄×C₂₃	—	2	2	no	—
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	4	2	no	—
162	C₄×C₂₃	—	8	2	no	—
163	C₄×C₂₃	—	8	2	no	—
164	C₄×C₂₃	#525	1	1	yes	characteristic
165	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	#575	C₂²×C₂₃	1	1	yes	characteristic
167	#568	—	2	1	no	—
168	#533	—	2	1	no	—
169	#510	C₂²×C₂₃	1	4	yes	—
170	#498	C₂²×C₂₃	1	1	yes	characteristic, complemented
171	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
172	#487	—	2	1	no	—
173	#470	—	2	1	no	—
174	#470	C₂²×C₂₃	1	2	yes	—
175	#444	—	2	1	no	—
176	#444	C₂²×C₂₃	1	2	yes	—
177	#443	—	2	1	no	—
178	#443	C₂²×C₂₃	1	2	yes	—
179	#441	—	2	1	no	—
180	#441	C₂²×C₂₃	1	2	yes	complemented
181	#430	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×#104	—	2	1	no	—
183	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
184	#365	C₂²×C₂₃	1	4	yes	complemented
185	#364	C₂²×C₂₃	1	2	yes	complemented
186	#362	C₂²×C₂₃	1	4	yes	complemented
187	#360	C₂²×C₂₃	1	4	yes	—
188	#349	C₂²×C₂₃	1	2	yes	—
189	#347	C₂²×C₂₃	1	2	yes	complemented
190	C₂×C₂₃	—	2	1	no	—
191	C₂×C₂₃	—	4	1	no	—
192	C₂×C₂₃	—	8	2	no	—
193	C₂×C₂₃	#3109	1	1	yes	characteristic
194	C₂×#41	—	2	1	no	—
195	#137	—	2	2	no	—
196	AGL(1,ℤ/8)	—	2	2	no	—
197	#135	—	2	1	no	—
198	#135	C₂³×C₂₃	1	2	yes	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	4	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	4	2	no	—
206	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
207	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	#125	—	4	1	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	4	1	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	4	1	no	—
215	#117	—	4	1	no	—
216	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
217	Q₃₂	—	2	4	no	—
218	SD₃₂	—	2	4	no	—
219	M₃₂	C₂³×C₂₃	1	4	yes	—
220	#108	—	2	1	no	—
221	#108	C₂³×C₂₃	1	2	yes	—
222	#106	—	2	1	no	—
223	#106	C₂³×C₂₃	1	2	yes	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#97	D₄×C₂₃	1	1	yes	characteristic
229	#97	C₂³×C₂₃	1	1	yes	characteristic
230	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
232	C₂₃	#33689	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
245	Q₁₆	—	2	1	no	—
246	Q₁₆	—	2	4	no	—
247	Q₁₆	—	4	2	no	—
248	Q₁₆	#41×C₂₃	1	2	yes	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	D₈	—	2	1	no	—
252	D₈	#41×C₂₃	1	2	yes	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	2	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
277	C₁₆	—	2	4	no	—
278	C₂³	—	4	1	no	—
279	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	—	4	1	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	4	2	no	—
304	C₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
305	C₈	C₂×#41×C₂₃	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	16	1	no	—
310	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₂₃×#525	1	1	yes	characteristic
318	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	4	1	no	—
321	C₂	—	8	2	no	—
322	C₂	C₂₃×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₄×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3185	C₂	semidirect
3. C₂₃×#2239	C₂	semidirect
4. C₂₃×#2230	C₂	semidirect
5. C₂₃×#2214	C₂	semidirect
7. C₂₃×#2180	C₂	semidirect
8. C₂₃×#2165	C₂	semidirect
13. C₂₃×#498	C₂²	semidirect
14. C₂₃×#494	C₂²	semidirect
23. C₂₃×#441	C₂²	semidirect
27. C₂₃×#365	C₂²	semidirect
28. C₂₃×#364	C₂²	semidirect
29. C₂₃×#362	C₂²	semidirect
32. C₂₃×#347	C₂²	semidirect
116. #33689	C₂₃	direct (N × H)
145. #3185	C₂×C₂₃	semidirect
147. #2239	C₂×C₂₃	semidirect
148. #2230	C₂×C₂₃	semidirect
149. #2214	C₂×C₂₃	semidirect
151. #2180	C₂×C₂₃	semidirect
152. #2165	C₂×C₂₃	semidirect
170. #498	C₂²×C₂₃	semidirect
171. #494	C₂²×C₂₃	semidirect
180. #441	C₂²×C₂₃	semidirect
184. #365	C₂²×C₂₃	semidirect
185. #364	C₂²×C₂₃	semidirect
186. #362	C₂²×C₂₃	semidirect
189. #347	C₂²×C₂₃	semidirect
232. C₂₃	#33689	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×#3185	2
C₂₃×#2956	2
C₂₃×#2239	2
C₂₃×#2230	2
C₂₃×#2214	2
C₂₃×#2191	2
C₂₃×#2180	2
C₂₃×#2165	2
#33689	23

Subgroups — C23×#33689 #19181315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33689 #19181315