Subgroups — C₂₃×#8932 #19156558

Order 5888.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1492	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	—	2	1	no	—
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂₃×#436	—	2	1	no	—
11	C₂₃×#435	—	2	2	no	—
12	C₈×D₄×C₂₃	—	2	1	no	—
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#108	—	2	1	no	—
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂₃×#408	—	2	1	no	—
19	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#341	—	2	1	no	—
22	C₂₃×#334	—	2	1	no	—
23	C₂₃×#334	—	2	1	no	—
24	C₂₃×#332	C₄	1	1	yes	characteristic
25	C₂₃×#332	C₄	1	1	yes	characteristic
26	C₂₃×#331	C₄	1	1	yes	characteristic
27	C₂₃×#331	C₄	1	1	yes	characteristic
28	C₂₃×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₂₃	—	2	1	no	—
30	C₂₃×#131	—	4	2	no	—
31	C₂₃×#131	—	4	2	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	—	2	1	no	—
39	C₂²×C₈×C₂₃	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
41	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
42	C₂₃×#124	—	2	1	no	—
43	C₂₃×#121	—	2	1	no	—
44	C₂₃×#121	Q₈	1	2	yes	—
45	C₄×D₄×C₂₃	—	2	1	no	—
46	C₄×D₄×C₂₃	D₄	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×#108	—	4	2	no	—
49	C₂₃×C₄⋊C₈	—	2	1	no	—
50	C₂₃×C₄⋊C₈	—	2	1	no	—
51	C₂₃×C₄⋊C₈	—	2	1	no	—
52	C₂₃×C₄⋊C₈	—	2	1	no	—
53	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
58	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₂₃×C₂²⋊C₈	—	2	2	no	—
60	C₂₃×C₂²⋊C₈	—	2	2	no	—
61	C₂₃×C₂²⋊C₈	—	2	1	no	—
62	C₂₃×#97	—	2	1	no	—
63	C₄×C₈×C₂₃	—	2	1	no	—
64	C₄×C₈×C₂₃	—	2	1	no	—
65	C₄×C₈×C₂₃	—	2	1	no	—
66	C₄×C₈×C₂₃	—	2	1	no	—
67	#41×C₂₃	—	4	1	no	—
68	#41×C₂₃	—	4	1	no	—
69	#41×C₂₃	—	4	1	no	—
70	#41×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	2	1	no	—
72	C₂×D₄×C₂₃	—	2	2	no	—
73	C₂×D₄×C₂₃	—	2	1	no	—
74	C₂×D₄×C₂₃	—	2	1	no	—
75	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
80	M₁₆×C₂₃	—	2	1	no	—
81	M₁₆×C₂₃	—	2	1	no	—
82	M₁₆×C₂₃	—	2	1	no	—
83	M₁₆×C₂₃	—	4	4	no	—
84	M₁₆×C₂₃	—	4	1	no	—
85	M₁₆×C₂₃	—	4	1	no	—
86	M₁₆×C₂₃	SD₁₆	1	2	yes	—
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	1	no	—
90	C₂×C₈×C₂₃	—	2	1	no	—
91	C₂×C₈×C₂₃	—	2	1	no	—
92	C₂×C₈×C₂₃	—	2	1	no	—
93	C₂×C₈×C₂₃	—	2	1	no	—
94	C₂×C₈×C₂₃	—	2	1	no	—
95	C₂×C₈×C₂₃	—	2	1	no	—
96	C₂×C₈×C₂₃	—	4	4	no	—
97	C₂×C₈×C₂₃	—	4	1	no	—
98	C₂×C₈×C₂₃	—	4	1	no	—
99	C₂×C₈×C₂₃	—	4	1	no	—
100	C₂×C₈×C₂₃	—	4	1	no	—
101	C₂×C₈×C₂₃	—	4	1	no	—
102	C₂×C₈×C₂₃	—	4	1	no	—
103	C₂×C₈×C₂₃	—	4	1	no	—
104	C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₂₃	—	2	2	no	—
107	C₂²⋊C₄×C₂₃	—	2	2	no	—
108	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₂₃	#41	1	1	yes	characteristic
112	#8932	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	—	2	2	no	—
114	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
115	Q₈×C₂₃	—	4	1	no	—
116	Q₈×C₂₃	—	4	1	no	—
117	D₄×C₂₃	—	4	2	no	—
118	D₄×C₂₃	—	4	2	no	—
119	D₄×C₂₃	—	4	2	no	—
120	D₄×C₂₃	—	4	1	no	—
121	D₄×C₂₃	—	4	1	no	—
122	D₄×C₂₃	—	4	1	no	—
123	D₄×C₂₃	—	4	1	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	2	1	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
138	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₈×C₂₃	—	2	2	no	—
141	C₈×C₂₃	—	4	2	no	—
142	C₈×C₂₃	—	4	1	no	—
143	C₈×C₂₃	—	4	1	no	—
144	C₈×C₂₃	—	4	1	no	—
145	C₈×C₂₃	—	4	1	no	—
146	C₈×C₂₃	—	4	1	no	—
147	C₈×C₂₃	—	4	1	no	—
148	C₈×C₂₃	—	4	1	no	—
149	C₈×C₂₃	—	8	1	no	—
150	#2935	C₂×C₂₃	1	1	yes	characteristic
151	#1950	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#1570	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#1540	C₂×C₂₃	1	1	yes	characteristic
154	#1492	C₂×C₂₃	1	1	yes	characteristic
155	#1278	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#1277	C₂×C₂₃	1	1	yes	characteristic, complemented
157	C₂²×C₂₃	—	2	1	no	—
158	C₂²×C₂₃	—	2	1	no	—
159	C₂²×C₂₃	—	4	2	no	—
160	C₂²×C₂₃	—	4	2	no	—
161	C₂²×C₂₃	—	4	2	no	—
162	C₂²×C₂₃	#385	1	1	yes	characteristic
163	C₂²×C₂₃	#476	1	1	yes	characteristic
164	C₂²×C₂₃	#478	1	1	yes	characteristic
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	4	2	no	—
169	C₄×C₂₃	C₄×SD₁₆	1	1	yes	characteristic
170	C₄×C₂₃	#442	1	1	yes	characteristic
171	C₄×C₂₃	#443	1	1	yes	characteristic
172	C₄×C₂₃	#444	1	1	yes	characteristic
173	C₄×C₂₃	#449	1	1	yes	characteristic
174	C₄×C₂₃	#450	1	1	yes	characteristic
175	C₂×#131	—	2	1	no	—
176	#532	C₂²×C₂₃	1	1	yes	characteristic
177	#436	—	2	1	no	—
178	#435	—	2	2	no	—
179	C₈×D₄	—	2	1	no	—
180	#431	C₂²×C₂₃	1	1	yes	characteristic
181	C₂×#108	—	2	1	no	—
182	#423	C₂²×C₂₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
185	#408	—	2	1	no	—
186	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
188	#341	—	2	1	no	—
189	#334	—	2	1	no	—
190	#334	—	2	1	no	—
191	#332	C₄×C₂₃	1	1	yes	characteristic
192	#332	C₄×C₂₃	1	1	yes	characteristic
193	#331	C₄×C₂₃	1	1	yes	characteristic
194	#331	C₄×C₂₃	1	1	yes	characteristic
195	D₄⋊C₈	—	2	2	no	—
196	C₂×C₂₃	—	2	1	no	—
197	C₂×C₂₃	—	2	1	no	—
198	C₂×C₂₃	—	8	2	no	—
199	C₂×C₂₃	#1858	1	1	yes	characteristic
200	C₂×C₂₃	#1864	1	1	yes	characteristic
201	C₂×C₂₃	#1866	1	1	yes	characteristic
202	C₂×#41	—	2	1	no	—
203	#131	—	4	2	no	—
204	#131	—	4	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
208	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
214	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
215	#124	—	2	1	no	—
216	#121	—	2	1	no	—
217	#121	Q₈×C₂₃	1	2	yes	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	D₄×C₂₃	1	2	yes	—
220	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
221	#108	—	4	2	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
227	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
228	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
229	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
230	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
231	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
232	C₂²⋊C₈	—	2	2	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₂₃	#8932	1	1	yes	characteristic, direct product factor, Sylow
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
253	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	SD₁₆×C₂₃	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
285	C₄²	#41×C₂₃	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
307	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
308	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
309	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
310	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
311	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	4	1	no	—
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	8	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	C₂₃×#385	1	1	yes	characteristic
329	C₂²	C₂₃×#476	1	1	yes	characteristic
330	C₂²	C₂₃×#478	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	C₄×SD₁₆×C₂₃	1	1	yes	characteristic
336	C₄	C₂₃×#442	1	1	yes	characteristic
337	C₄	C₂₃×#443	1	1	yes	characteristic
338	C₄	C₂₃×#444	1	1	yes	characteristic
339	C₄	C₂₃×#449	1	1	yes	characteristic
340	C₄	C₂₃×#450	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂₃×#1858	1	1	yes	characteristic
345	C₂	C₂₃×#1864	1	1	yes	characteristic
346	C₂	C₂₃×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#385
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1950	C₂	semidirect
3. C₂₃×#1570	C₂	semidirect
6. C₂₃×#1278	C₂	semidirect
7. C₂₃×#1277	C₂	semidirect
112. #8932	C₂₃	direct (N × H)
151. #1950	C₂×C₂₃	semidirect
152. #1570	C₂×C₂₃	semidirect
155. #1278	C₂×C₂₃	semidirect
156. #1277	C₂×C₂₃	semidirect
240. C₂₃	#8932	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×#2935	2
C₂₃×#1950	2
C₂₃×#1570	2
C₂₃×#1540	2
C₂₃×#1492	2
C₂₃×#1278	2
C₂₃×#1277	2
#8932	23

Subgroups — C23×#8932 #19156558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8932 #19156558