Subgroups — C₂₃×#17655 #19165281

Order 5888.

1016 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3517	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1530	C₂	1	3	yes	maximal, complemented
5	C₂₃×#1529	C₂	1	3	yes	maximal, complemented
6	C₂₃×#1525	C₂	1	3	yes	maximal, complemented
7	C₂₃×#1524	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×#131	—	2	1	no	—
9	C₂₃×#557	C₄	1	1	yes	characteristic
10	C₂₃×#553	—	2	3	no	—
11	C₂₃×#550	—	2	1	no	—
12	C₂₃×#550	C₄	1	3	yes	—
13	C₂₃×#550	C₄	1	1	yes	characteristic
14	C₂₃×#547	C₄	1	3	yes	—
15	C₂₃×#540	—	2	3	no	—
16	C₂₃×#532	C₂²	1	3	yes	—
17	C₂₃×#529	C₂²	1	3	yes	—
18	C₂₃×#520	—	2	1	no	—
19	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₂₃	—	2	1	no	—
21	C₂₃×#436	—	2	3	no	—
22	C₂₃×#432	C₂²	1	3	yes	—
23	C₂₃×#405	—	2	3	no	—
24	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
25	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
26	C₂×C₂₃×#97	C₂²	1	3	yes	—
27	C₂₃×#330	C₂²	1	1	yes	characteristic
28	C₂₃×#330	C₂²	1	3	yes	—
29	C₂₃×#329	C₂²	1	3	yes	—
30	C₂₃×#329	C₂²	1	3	yes	—
31	C₂₃×#329	C₂²	1	6	yes	—
32	C₂×#41×C₂₃	—	2	1	no	—
33	C₂×#41×C₂₃	C₂×C₄	1	6	yes	—
34	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
35	C₂₃×#131	—	2	8	no	—
36	C₂×M₁₆×C₂₃	—	2	3	no	—
37	C₂×M₁₆×C₂₃	—	2	3	no	—
38	C₂²×C₈×C₂₃	—	2	3	no	—
39	C₂₃×C₄⋊Q₈	C₂×C₄	1	3	yes	—
40	C₂₃×#127	—	2	3	no	—
41	C₂₃×#127	C₂×C₄	1	3	yes	—
42	C₂₃×#126	—	2	3	no	—
43	C₂₃×#125	—	2	3	no	—
44	C₂₃×#125	C₂³	1	1	yes	characteristic
45	C₂₃×#124	C₂³	1	3	yes	—
46	C₂₃×#123	—	2	3	no	—
47	C₂₃×#123	—	2	6	no	—
48	C₂₃×C₂²⋊Q₈	C₂×C₄	1	3	yes	—
49	C₂₃×#121	—	2	3	no	—
50	C₂₃×#121	—	2	3	no	—
51	C₂₃×#121	—	2	3	no	—
52	C₂₃×#121	—	2	3	no	—
53	C₂₃×#121	—	2	6	no	—
54	C₂₃×#121	C₂×C₄	1	3	yes	—
55	C₄×Q₈×C₂₃	—	2	1	no	—
56	C₄×Q₈×C₂₃	C₂×C₄	1	3	yes	—
57	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
59	C₄×D₄×C₂₃	—	2	3	no	—
60	C₄×D₄×C₂₃	—	2	3	no	—
61	C₄×D₄×C₂₃	—	2	3	no	—
62	C₄×D₄×C₂₃	—	2	3	no	—
63	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
64	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
65	C₄×D₄×C₂₃	D₄	1	3	yes	—
66	C₂₃×#117	—	2	3	no	—
67	C₂₃×#117	D₄	1	3	yes	—
68	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
69	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
70	C₂₃×C₄⋊C₈	—	2	3	no	—
71	C₂₃×C₂²⋊C₈	—	2	3	no	—
72	C₂₃×#97	—	2	3	no	—
73	C₂₃×#97	C₂³	1	1	yes	characteristic
74	C₂₃×#97	C₂³	1	3	yes	—
75	C₂₃×#97	C₂³	1	1	yes	characteristic
76	C₂₃×#97	C₂³	1	3	yes	—
77	C₄×C₈×C₂₃	—	2	3	no	—
78	C₄×C₈×C₂₃	—	2	3	no	—
79	#41×C₂₃	—	2	4	no	—
80	#41×C₂₃	—	4	4	no	—
81	#41×C₂₃	—	8	6	no	—
82	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×Q₈×C₂₃	C₂²×C₄	1	3	yes	—
84	C₂×D₄×C₂₃	—	2	3	no	—
85	C₂×D₄×C₂₃	—	2	6	no	—
86	C₂×D₄×C₂₃	—	2	3	no	—
87	C₂×D₄×C₂₃	—	2	3	no	—
88	C₂×D₄×C₂₃	C₂²⋊C₄	1	3	yes	—
89	C₂×D₄×C₂₃	C₂²×C₄	1	3	yes	—
90	C₂²×C₄×C₂₃	—	2	3	no	—
91	C₂²×C₄×C₂₃	—	2	6	no	—
92	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
93	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
95	M₁₆×C₂₃	—	2	12	no	—
96	M₁₆×C₂₃	—	4	6	no	—
97	C₂×C₈×C₂₃	—	2	12	no	—
98	C₂×C₈×C₂₃	—	2	1	no	—
99	C₂×C₈×C₂₃	—	2	3	no	—
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	2	3	no	—
102	C₄⋊C₄×C₂₃	—	2	3	no	—
103	C₄⋊C₄×C₂₃	—	2	3	no	—
104	C₄⋊C₄×C₂₃	—	2	3	no	—
105	C₄⋊C₄×C₂₃	—	2	3	no	—
106	C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
109	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
110	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²⋊C₄×C₂₃	—	2	3	no	—
113	C₂²⋊C₄×C₂₃	—	2	3	no	—
114	C₂²⋊C₄×C₂₃	—	2	3	no	—
115	C₂²⋊C₄×C₂₃	—	2	3	no	—
116	C₂²⋊C₄×C₂₃	—	2	3	no	—
117	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
118	C₄²×C₂₃	—	2	3	no	—
119	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
120	C₄²×C₂₃	C₂×D₄	1	3	yes	—
121	C₄²×C₂₃	C₂×D₄	1	3	yes	—
122	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
123	#17655	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₂₃	—	2	3	no	—
125	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
126	Q₈×C₂₃	—	4	3	no	—
127	Q₈×C₂₃	#104	1	4	yes	—
128	D₄×C₂₃	—	2	6	no	—
129	D₄×C₂₃	—	4	3	no	—
130	D₄×C₂₃	—	4	6	no	—
131	D₄×C₂₃	—	8	3	no	—
132	D₄×C₂₃	—	8	3	no	—
133	D₄×C₂₃	—	8	6	no	—
134	C₂×C₄×C₂₃	—	2	6	no	—
135	C₂×C₄×C₂₃	—	2	1	no	—
136	C₂×C₄×C₂₃	—	2	3	no	—
137	C₂×C₄×C₂₃	—	2	1	no	—
138	C₂×C₄×C₂₃	—	4	3	no	—
139	C₂×C₄×C₂₃	—	4	3	no	—
140	C₂×C₄×C₂₃	—	4	6	no	—
141	C₂×C₄×C₂₃	—	8	6	no	—
142	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
143	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
144	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
145	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₂₃	—	2	2	no	—
149	C₈×C₂₃	—	2	6	no	—
150	C₈×C₂₃	—	2	2	no	—
151	C₈×C₂₃	—	4	3	no	—
152	#3517	C₂×C₂₃	1	1	yes	characteristic
153	#2971	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#2875	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#1530	C₂×C₂₃	1	3	yes	complemented
156	#1529	C₂×C₂₃	1	3	yes	complemented
157	#1525	C₂×C₂₃	1	3	yes	complemented
158	#1524	C₂×C₂₃	1	3	yes	complemented
159	C₂²×C₂₃	—	2	3	no	—
160	C₂²×C₂₃	—	2	3	no	—
161	C₂²×C₂₃	—	2	3	no	—
162	C₂²×C₂₃	—	4	3	no	—
163	C₂²×C₂₃	—	4	3	no	—
164	C₂²×C₂₃	—	4	3	no	—
165	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	3	no	—
168	C₄×C₂₃	—	2	1	no	—
169	C₄×C₂₃	—	2	1	no	—
170	C₄×C₂₃	—	2	3	no	—
171	C₄×C₂₃	—	4	3	no	—
172	C₄×C₂₃	—	4	1	no	—
173	C₄×C₂₃	—	4	1	no	—
174	C₄×C₂₃	C₂×#104	1	6	yes	—
175	C₂×#131	—	2	1	no	—
176	#557	C₄×C₂₃	1	1	yes	characteristic
177	#553	—	2	3	no	—
178	#550	—	2	1	no	—
179	#550	C₄×C₂₃	1	3	yes	—
180	#550	C₄×C₂₃	1	1	yes	characteristic
181	#547	C₄×C₂₃	1	3	yes	—
182	#540	—	2	3	no	—
183	#532	C₂²×C₂₃	1	3	yes	—
184	#529	C₂²×C₂₃	1	3	yes	—
185	#520	—	2	1	no	—
186	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
187	C₈×Q₈	—	2	1	no	—
188	#436	—	2	3	no	—
189	#432	C₂²×C₂₃	1	3	yes	—
190	#405	—	2	3	no	—
191	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
192	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
193	C₂×#97	C₂²×C₂₃	1	3	yes	—
194	#330	C₂²×C₂₃	1	1	yes	characteristic
195	#330	C₂²×C₂₃	1	3	yes	—
196	#329	C₂²×C₂₃	1	3	yes	—
197	#329	C₂²×C₂₃	1	3	yes	—
198	#329	C₂²×C₂₃	1	6	yes	—
199	C₂×C₂₃	—	4	3	no	—
200	C₂×C₂₃	—	8	3	no	—
201	C₂×C₂₃	#2888	1	1	yes	characteristic
202	C₂×C₂₃	C₂²×#104	1	1	yes	characteristic
203	C₂×C₂₃	#2901	1	1	yes	characteristic
204	C₂×#41	—	2	1	no	—
205	C₂×#41	C₂×C₄×C₂₃	1	6	yes	—
206	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
207	#131	—	2	8	no	—
208	C₂×M₁₆	—	2	3	no	—
209	C₂×M₁₆	—	2	3	no	—
210	C₂²×C₈	—	2	3	no	—
211	C₄⋊Q₈	C₂×C₄×C₂₃	1	3	yes	—
212	#127	—	2	3	no	—
213	#127	C₂×C₄×C₂₃	1	3	yes	—
214	#126	—	2	3	no	—
215	#125	—	2	3	no	—
216	#125	C₂³×C₂₃	1	1	yes	characteristic
217	#124	C₂³×C₂₃	1	3	yes	—
218	#123	—	2	3	no	—
219	#123	—	2	6	no	—
220	C₂²⋊Q₈	C₂×C₄×C₂₃	1	3	yes	—
221	#121	—	2	3	no	—
222	#121	—	2	3	no	—
223	#121	—	2	3	no	—
224	#121	—	2	3	no	—
225	#121	—	2	6	no	—
226	#121	C₂×C₄×C₂₃	1	3	yes	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	C₂×C₄×C₂₃	1	3	yes	—
229	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
230	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
231	C₄×D₄	—	2	3	no	—
232	C₄×D₄	—	2	3	no	—
233	C₄×D₄	—	2	3	no	—
234	C₄×D₄	—	2	3	no	—
235	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
236	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
237	C₄×D₄	D₄×C₂₃	1	3	yes	—
238	#117	—	2	3	no	—
239	#117	D₄×C₂₃	1	3	yes	—
240	C₂×C₄⋊C₄	—	2	3	no	—
241	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
242	C₄⋊C₈	—	2	3	no	—
243	C₂²⋊C₈	—	2	3	no	—
244	#97	—	2	3	no	—
245	#97	C₂³×C₂₃	1	1	yes	characteristic
246	#97	C₂³×C₂₃	1	3	yes	—
247	#97	C₂³×C₂₃	1	1	yes	characteristic
248	#97	C₂³×C₂₃	1	3	yes	—
249	C₄×C₈	—	2	3	no	—
250	C₄×C₈	—	2	3	no	—
251	C₂₃	#17655	1	1	yes	characteristic, direct product factor, Sylow
252	#41	—	2	4	no	—
253	#41	—	4	4	no	—
254	#41	—	8	6	no	—
255	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
256	C₂×Q₈	C₂²×C₄×C₂₃	1	3	yes	—
257	C₂×D₄	—	2	3	no	—
258	C₂×D₄	—	2	6	no	—
259	C₂×D₄	—	2	3	no	—
260	C₂×D₄	—	2	3	no	—
261	C₂×D₄	C₂²⋊C₄×C₂₃	1	3	yes	—
262	C₂×D₄	C₂²×C₄×C₂₃	1	3	yes	—
263	C₂²×C₄	—	2	3	no	—
264	C₂²×C₄	—	2	6	no	—
265	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
266	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
267	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
268	M₁₆	—	2	12	no	—
269	M₁₆	—	4	6	no	—
270	C₂×C₈	—	2	12	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	3	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	3	no	—
275	C₄⋊C₄	—	2	3	no	—
276	C₄⋊C₄	—	2	3	no	—
277	C₄⋊C₄	—	2	3	no	—
278	C₄⋊C₄	—	2	3	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
282	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
283	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	3	no	—
286	C₂²⋊C₄	—	2	3	no	—
287	C₂²⋊C₄	—	2	3	no	—
288	C₂²⋊C₄	—	2	3	no	—
289	C₂²⋊C₄	—	2	3	no	—
290	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
291	C₄²	—	2	3	no	—
292	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
293	C₄²	C₂×D₄×C₂₃	1	3	yes	—
294	C₄²	C₂×D₄×C₂₃	1	3	yes	—
295	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
296	C₂³	—	2	3	no	—
297	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
298	Q₈	—	4	3	no	—
299	Q₈	C₂₃×#104	1	4	yes	—
300	D₄	—	2	6	no	—
301	D₄	—	4	3	no	—
302	D₄	—	4	6	no	—
303	D₄	—	8	3	no	—
304	D₄	—	8	6	no	—
305	D₄	—	8	3	no	—
306	C₂×C₄	—	2	6	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	3	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	3	no	—
311	C₂×C₄	—	4	3	no	—
312	C₂×C₄	—	4	6	no	—
313	C₂×C₄	—	8	6	no	—
314	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
315	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
316	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
317	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	6	no	—
322	C₈	—	2	2	no	—
323	C₈	—	4	3	no	—
324	C₂²	—	2	3	no	—
325	C₂²	—	2	3	no	—
326	C₂²	—	2	3	no	—
327	C₂²	—	4	3	no	—
328	C₂²	—	4	3	no	—
329	C₂²	—	4	3	no	—
330	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	3	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	3	no	—
336	C₄	—	4	3	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	C₂×C₂₃×#104	1	6	yes	—
340	C₂	—	4	3	no	—
341	C₂	—	8	3	no	—
342	C₂	C₂₃×#2888	1	1	yes	characteristic
343	C₂	C₂²×C₂₃×#104	1	1	yes	characteristic
344	C₂	C₂₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3517	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2971	C₂	semidirect
3. C₂₃×#2875	C₂	semidirect
4. C₂₃×#1530	C₂	semidirect
5. C₂₃×#1529	C₂	semidirect
6. C₂₃×#1525	C₂	semidirect
7. C₂₃×#1524	C₂	semidirect
123. #17655	C₂₃	direct (N × H)
153. #2971	C₂×C₂₃	semidirect
154. #2875	C₂×C₂₃	semidirect
155. #1530	C₂×C₂₃	semidirect
156. #1529	C₂×C₂₃	semidirect
157. #1525	C₂×C₂₃	semidirect
158. #1524	C₂×C₂₃	semidirect
251. C₂₃	#17655	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×#3517	2
C₂₃×#2971	2
C₂₃×#2875	2
C₂₃×#1530	2
C₂₃×#1529	2
C₂₃×#1525	2
C₂₃×#1524	2
#17655	23

Subgroups — C23×#17655 #19165281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17655 #19165281