Subgroups — C₂₃×#29202 #19176828

Order 5888.

658 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2647	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2327	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2068	C₂	1	2	yes	maximal
4	C₂₃×#2066	C₂	1	2	yes	maximal
5	C₂₃×#1960	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1956	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1865	C₂	1	2	yes	maximal
8	C₂₃×#1863	C₂	1	2	yes	maximal
9	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1836	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
14	C₂×C₂₃×#107	C₂²	1	2	yes	—
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#103	C₂²	1	2	yes	—
19	C₂×C₂₃×#103	C₂²	1	2	yes	—
20	C₂×C₂₃×#103	C₂²	1	2	yes	—
21	C₂×C₂₃×#103	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#400	—	2	2	no	—
25	C₂₃×#398	—	2	2	no	—
26	C₂₃×#391	C₂²	1	2	yes	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#387	—	2	2	no	—
30	C₂₃×#387	—	2	2	no	—
31	C₂₃×#384	C₂²	1	1	yes	characteristic
32	C₂₃×#382	C₂²	1	1	yes	characteristic
33	C₂₃×#382	C₂²	1	1	yes	characteristic
34	C₂₃×#376	C₂²	1	1	yes	characteristic
35	C₂₃×#340	C₂²	1	2	yes	—
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂₃×#336	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×C₄⋊Q₈	—	2	2	no	—
45	C₂₃×C₄⋊Q₈	Q₈	1	4	yes	—
46	C₄×Q₈×C₂₃	—	4	2	no	—
47	C₄×Q₈×C₂₃	—	4	2	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₂₃	—	2	2	no	—
57	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#107	—	4	2	no	—
61	C₂₃×#106	—	4	2	no	—
62	C₂₃×C₄⋊C₈	—	2	2	no	—
63	C₂₃×C₄⋊C₈	—	2	2	no	—
64	C₂₃×#103	—	2	4	no	—
65	C₂₃×#103	—	2	4	no	—
66	C₂₃×#103	—	4	2	no	—
67	C₂₃×#103	—	4	2	no	—
68	C₂₃×#97	—	2	2	no	—
69	C₄×C₈×C₂₃	—	2	2	no	—
70	C₂₃×#95	—	2	2	no	—
71	C₂₃×#95	—	2	2	no	—
72	C₂₃×#95	—	2	2	no	—
73	C₂₃×#95	—	2	2	no	—
74	C₂₃×#95	—	2	2	no	—
75	C₂₃×#95	—	2	2	no	—
76	C₂₃×#95	D₄	1	2	yes	—
77	C₂₃×#95	D₄	1	2	yes	—
78	C₂×Q₈×C₂₃	—	2	2	no	—
79	C₂×Q₈×C₂₃	—	2	2	no	—
80	C₂×Q₈×C₂₃	—	2	2	no	—
81	C₂×Q₈×C₂₃	—	2	2	no	—
82	C₂×Q₈×C₂₃	—	2	2	no	—
83	C₂×Q₈×C₂₃	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
97	C₂×C₈×C₂₃	—	2	1	no	—
98	C₂×C₈×C₂₃	—	2	1	no	—
99	C₂×C₈×C₂₃	—	2	1	no	—
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	2	1	no	—
102	C₂×C₈×C₂₃	—	2	1	no	—
103	C₂×C₈×C₂₃	#41	1	2	yes	—
104	C₂×C₈×C₂₃	#41	1	2	yes	—
105	C₄⋊C₄×C₂₃	—	2	8	no	—
106	C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₄⋊C₄×C₂₃	—	2	2	no	—
108	C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₄⋊C₄×C₂₃	—	2	2	no	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	4	4	no	—
113	C₄⋊C₄×C₂₃	—	8	2	no	—
114	C₄⋊C₄×C₂₃	—	8	2	no	—
115	C₄⋊C₄×C₂₃	—	8	2	no	—
116	C₄²×C₂₃	—	2	1	no	—
117	C₄²×C₂₃	—	2	1	no	—
118	C₄²×C₂₃	—	2	1	no	—
119	C₄²×C₂₃	—	2	1	no	—
120	C₄²×C₂₃	—	4	4	no	—
121	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
122	C₄²×C₂₃	#41	1	2	yes	—
123	#29202	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₂₃	—	2	4	no	—
126	Q₈×C₂₃	—	2	4	no	—
127	Q₈×C₂₃	—	4	2	no	—
128	Q₈×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	4	no	—
132	C₂×C₄×C₂₃	—	2	1	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	4	2	no	—
140	C₂×C₄×C₂₃	—	4	2	no	—
141	C₂×C₄×C₂₃	—	4	2	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	—	4	2	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	—	4	2	no	—
146	C₂×C₄×C₂₃	—	4	2	no	—
147	C₂×C₄×C₂₃	—	4	2	no	—
148	C₂×C₄×C₂₃	—	4	2	no	—
149	C₂×C₄×C₂₃	—	4	2	no	—
150	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
151	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₈×C₂₃	—	2	2	no	—
158	C₈×C₂₃	—	2	2	no	—
159	C₈×C₂₃	—	4	1	no	—
160	C₈×C₂₃	—	4	1	no	—
161	#2647	C₂×C₂₃	1	1	yes	characteristic
162	#2327	C₂×C₂₃	1	1	yes	characteristic
163	#2068	C₂×C₂₃	1	2	yes	—
164	#2066	C₂×C₂₃	1	2	yes	—
165	#1960	C₂×C₂₃	1	1	yes	characteristic
166	#1956	C₂×C₂₃	1	1	yes	characteristic
167	#1865	C₂×C₂₃	1	2	yes	—
168	#1863	C₂×C₂₃	1	2	yes	—
169	#1846	C₂×C₂₃	1	1	yes	characteristic
170	#1836	C₂×C₂₃	1	1	yes	characteristic
171	#1768	C₂×C₂₃	1	1	yes	characteristic
172	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₂²×C₂₃	#532	1	1	yes	characteristic
174	C₂²×C₂₃	#535	1	1	yes	characteristic
175	C₂²×C₂₃	#577	1	2	yes	—
176	C₂²×C₂₃	#578	1	2	yes	—
177	C₄×C₂₃	—	2	4	no	—
178	C₄×C₂₃	—	4	4	no	—
179	C₄×C₂₃	—	4	1	no	—
180	C₄×C₂₃	—	4	1	no	—
181	C₄×C₂₃	—	4	2	no	—
182	C₄×C₂₃	—	8	2	no	—
183	C₄×C₂₃	—	8	2	no	—
184	C₄×C₂₃	—	8	2	no	—
185	C₄×C₂₃	#553	1	1	yes	characteristic
186	C₄×C₂₃	#553	1	1	yes	characteristic
187	C₄×C₂₃	#554	1	1	yes	characteristic
188	C₄×C₂₃	#554	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
190	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
191	C₂×#107	C₂²×C₂₃	1	2	yes	—
192	C₂×#106	C₂²×C₂₃	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
195	C₂×#103	C₂²×C₂₃	1	2	yes	—
196	C₂×#103	C₂²×C₂₃	1	2	yes	—
197	C₂×#103	C₂²×C₂₃	1	2	yes	—
198	C₂×#103	C₂²×C₂₃	1	2	yes	—
199	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
201	#400	—	2	2	no	—
202	#398	—	2	2	no	—
203	#391	C₂²×C₂₃	1	2	yes	—
204	#389	C₂²×C₂₃	1	2	yes	—
205	#389	C₂²×C₂₃	1	2	yes	—
206	#387	—	2	2	no	—
207	#387	—	2	2	no	—
208	#384	C₂²×C₂₃	1	1	yes	characteristic
209	#382	C₂²×C₂₃	1	1	yes	characteristic
210	#382	C₂²×C₂₃	1	1	yes	characteristic
211	#376	C₂²×C₂₃	1	1	yes	characteristic
212	#340	C₂²×C₂₃	1	2	yes	—
213	#340	C₂²×C₂₃	1	2	yes	—
214	#336	C₂²×C₂₃	1	1	yes	characteristic
215	#336	C₂²×C₂₃	1	1	yes	characteristic
216	C₂×C₂₃	#2566	1	1	yes	characteristic
217	C₂×C₂₃	#3082	1	2	yes	—
218	C₂×C₂₃	#3174	1	1	yes	characteristic
219	C₂×C₂₃	#3177	1	1	yes	characteristic
220	C₂×C₂₃	#3225	1	2	yes	—
221	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
222	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	Q₈×C₂₃	1	4	yes	—
228	C₄×Q₈	—	4	2	no	—
229	C₄×Q₈	—	4	2	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
237	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
242	#107	—	4	2	no	—
243	#106	—	4	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	#103	—	2	4	no	—
247	#103	—	2	4	no	—
248	#103	—	4	2	no	—
249	#103	—	4	2	no	—
250	#97	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	#95	D₄×C₂₃	1	2	yes	—
259	#95	D₄×C₂₃	1	2	yes	—
260	C₂₃	#29202	1	1	yes	characteristic, direct product factor, Sylow
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	C₂×Q₈×C₂₃	1	4	yes	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	#41×C₂₃	1	2	yes	—
287	C₂×C₈	#41×C₂₃	1	2	yes	—
288	C₄⋊C₄	—	2	8	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	4	no	—
296	C₄⋊C₄	—	8	2	no	—
297	C₄⋊C₄	—	8	2	no	—
298	C₄⋊C₄	—	8	2	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	4	4	no	—
304	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
305	C₄²	#41×C₂₃	1	2	yes	—
306	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
333	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
337	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
338	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
344	C₂²	C₂₃×#532	1	1	yes	characteristic
345	C₂²	C₂₃×#535	1	1	yes	characteristic
346	C₂²	C₂₃×#577	1	2	yes	—
347	C₂²	C₂₃×#578	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₂₃×#553	1	1	yes	characteristic
357	C₄	C₂₃×#553	1	1	yes	characteristic
358	C₄	C₂₃×#554	1	1	yes	characteristic
359	C₄	C₂₃×#554	1	1	yes	characteristic
360	C₂	C₂₃×#2566	1	1	yes	characteristic
361	C₂	C₂₃×#3082	1	2	yes	—
362	C₂	C₂₃×#3174	1	1	yes	characteristic
363	C₂	C₂₃×#3177	1	1	yes	characteristic
364	C₂	C₂₃×#3225	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
123. #29202	C₂₃	direct (N × H)
260. C₂₃	#29202	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×#2647	2
C₂₃×#2327	2
C₂₃×#2068	2
C₂₃×#2066	2
C₂₃×#1960	2
C₂₃×#1956	2
C₂₃×#1865	2
C₂₃×#1863	2
C₂₃×#1846	2
C₂₃×#1836	2
C₂₃×#1768	2
#29202	23

Subgroups — C23×#29202 #19176828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29202 #19176828