Subgroups — C₂₃×#17654 #19165280

Order 5888.

960 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3518	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×#1531	C₂	1	3	yes	maximal, complemented
5	C₂₃×#1529	C₂	1	3	yes	maximal, complemented
6	C₂₃×#1526	C₂	1	3	yes	maximal, complemented
7	C₂₃×#1524	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×#131	—	2	1	no	—
9	Q₈²×C₂₃	C₄	1	1	yes	characteristic
10	C₂₃×#556	—	2	3	no	—
11	C₂₃×#556	C₄	1	3	yes	—
12	C₂₃×#555	C₄	1	3	yes	—
13	D₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
14	C₂₃×#548	—	2	1	no	—
15	C₂₃×#542	—	2	3	no	—
16	C₂₃×#533	C₂²	1	3	yes	—
17	C₂₃×#529	C₂²	1	3	yes	—
18	C₂₃×#519	—	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	3	yes	—
28	C₂₃×#330	C₂²	1	3	yes	—
29	C₂₃×#330	C₂²	1	6	yes	—
30	C₂₃×#329	C₂²	1	1	yes	characteristic
31	C₂₃×#329	C₂²	1	3	yes	—
32	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₂₃	—	2	1	no	—
34	C₂₃×#131	—	2	8	no	—
35	C₂×M₁₆×C₂₃	—	2	3	no	—
36	C₂×M₁₆×C₂₃	—	2	3	no	—
37	C₂²×C₈×C₂₃	—	2	3	no	—
38	C₂₃×C₄⋊Q₈	—	2	3	no	—
39	C₂₃×C₄⋊Q₈	C₂×C₄	1	3	yes	—
40	C₂₃×C₄⋊Q₈	C₂×C₄	1	3	yes	—
41	C₂₃×#126	—	2	6	no	—
42	C₂₃×#126	—	2	3	no	—
43	C₂₃×#125	C₂³	1	3	yes	—
44	C₂₃×#124	—	2	3	no	—
45	C₂₃×#124	—	2	6	no	—
46	C₂₃×#124	C₂³	1	1	yes	characteristic
47	C₂₃×#123	—	2	3	no	—
48	C₂₃×C₂²⋊Q₈	—	2	3	no	—
49	C₂₃×C₂²⋊Q₈	—	2	3	no	—
50	C₂₃×C₂²⋊Q₈	—	2	3	no	—
51	C₂₃×C₂²⋊Q₈	C₂×C₄	1	3	yes	—
52	C₂₃×#121	—	2	3	no	—
53	C₂₃×#121	C₂×C₄	1	3	yes	—
54	C₄×Q₈×C₂₃	—	2	1	no	—
55	C₄×Q₈×C₂₃	—	2	3	no	—
56	C₄×Q₈×C₂₃	—	2	3	no	—
57	C₄×Q₈×C₂₃	—	2	3	no	—
58	C₄×Q₈×C₂₃	C₂×C₄	1	3	yes	—
59	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
61	C₄×D₄×C₂₃	—	2	3	no	—
62	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
63	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
64	C₄×D₄×C₂₃	D₄	1	3	yes	—
65	C₂₃×#117	—	2	3	no	—
66	C₂₃×#117	—	2	3	no	—
67	C₂₃×#117	C₂×C₄	1	6	yes	—
68	C₂₃×#117	D₄	1	3	yes	—
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	C₂×Q₈×C₂₃	—	2	3	no	—
81	C₂×Q₈×C₂₃	—	2	3	no	—
82	C₂×Q₈×C₂₃	—	4	4	no	—
83	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×D₄×C₂₃	—	2	3	no	—
86	C₂×D₄×C₂₃	C₂²⋊C₄	1	3	yes	—
87	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	—	2	3	no	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
92	M₁₆×C₂₃	—	2	12	no	—
93	M₁₆×C₂₃	—	4	6	no	—
94	C₂×C₈×C₂₃	—	2	12	no	—
95	C₂×C₈×C₂₃	—	2	1	no	—
96	C₂×C₈×C₂₃	—	2	3	no	—
97	C₂×C₈×C₂₃	—	2	1	no	—
98	C₂×C₈×C₂₃	—	2	3	no	—
99	C₄⋊C₄×C₂₃	—	2	3	no	—
100	C₄⋊C₄×C₂₃	—	2	3	no	—
101	C₄⋊C₄×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	C₂²⋊C₄	1	3	yes	—
106	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
107	C₄⋊C₄×C₂₃	C₂²×C₄	1	3	yes	—
108	C₄⋊C₄×C₂₃	C₂²×C₄	1	3	yes	—
109	C₂²⋊C₄×C₂₃	—	2	6	no	—
110	C₂²⋊C₄×C₂₃	—	2	3	no	—
111	C₂²⋊C₄×C₂₃	—	2	3	no	—
112	C₂²⋊C₄×C₂₃	—	2	3	no	—
113	C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
116	C₄²×C₂₃	—	2	6	no	—
117	C₄²×C₂₃	—	2	3	no	—
118	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
119	C₄²×C₂₃	C₂×D₄	1	3	yes	—
120	C₄²×C₂₃	C₂×D₄	1	3	yes	—
121	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
122	#17654	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₂₃	—	2	1	no	—
124	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
125	Q₈×C₂₃	—	4	1	no	—
126	Q₈×C₂₃	—	4	6	no	—
127	Q₈×C₂₃	—	8	3	no	—
128	Q₈×C₂₃	#104	1	4	yes	—
129	D₄×C₂₃	—	2	6	no	—
130	D₄×C₂₃	—	4	1	no	—
131	D₄×C₂₃	—	8	3	no	—
132	C₂×C₄×C₂₃	—	2	6	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	3	no	—
135	C₂×C₄×C₂₃	—	2	3	no	—
136	C₂×C₄×C₂₃	—	4	3	no	—
137	C₂×C₄×C₂₃	—	4	3	no	—
138	C₂×C₄×C₂₃	—	4	6	no	—
139	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
140	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
141	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
142	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₂₃	—	2	2	no	—
146	C₈×C₂₃	—	2	6	no	—
147	C₈×C₂₃	—	2	2	no	—
148	C₈×C₂₃	—	4	3	no	—
149	#3518	C₂×C₂₃	1	1	yes	characteristic
150	#2971	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#2875	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#1531	C₂×C₂₃	1	3	yes	complemented
153	#1529	C₂×C₂₃	1	3	yes	complemented
154	#1526	C₂×C₂₃	1	3	yes	complemented
155	#1524	C₂×C₂₃	1	3	yes	complemented
156	C₂²×C₂₃	—	2	3	no	—
157	C₂²×C₂₃	—	2	3	no	—
158	C₂²×C₂₃	—	2	3	no	—
159	C₂²×C₂₃	—	4	1	no	—
160	C₂²×C₂₃	—	4	1	no	—
161	C₂²×C₂₃	—	4	1	no	—
162	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	2	3	no	—
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	3	no	—
168	C₄×C₂₃	—	4	1	no	—
169	C₄×C₂₃	—	4	3	no	—
170	C₄×C₂₃	—	4	3	no	—
171	C₄×C₂₃	C₂×#104	1	6	yes	—
172	C₂×#131	—	2	1	no	—
173	Q₈²	C₄×C₂₃	1	1	yes	characteristic
174	#556	—	2	3	no	—
175	#556	C₄×C₂₃	1	3	yes	—
176	#555	C₄×C₂₃	1	3	yes	—
177	D₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
178	#548	—	2	1	no	—
179	#542	—	2	3	no	—
180	#533	C₂²×C₂₃	1	3	yes	—
181	#529	C₂²×C₂₃	1	3	yes	—
182	#519	—	2	1	no	—
183	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
184	C₈×Q₈	—	2	1	no	—
185	#436	—	2	3	no	—
186	#432	C₂²×C₂₃	1	3	yes	—
187	#405	—	2	3	no	—
188	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
189	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
190	C₂×#97	C₂²×C₂₃	1	3	yes	—
191	#330	C₂²×C₂₃	1	3	yes	—
192	#330	C₂²×C₂₃	1	3	yes	—
193	#330	C₂²×C₂₃	1	6	yes	—
194	#329	C₂²×C₂₃	1	1	yes	characteristic
195	#329	C₂²×C₂₃	1	3	yes	—
196	C₂×C₂₃	—	4	3	no	—
197	C₂×C₂₃	—	8	1	no	—
198	C₂×C₂₃	#2889	1	1	yes	characteristic
199	C₂×C₂₃	C₂²×#104	1	1	yes	characteristic
200	C₂×C₂₃	#2901	1	1	yes	characteristic
201	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
202	C₂²×Q₈	—	2	1	no	—
203	#131	—	2	8	no	—
204	C₂×M₁₆	—	2	3	no	—
205	C₂×M₁₆	—	2	3	no	—
206	C₂²×C₈	—	2	3	no	—
207	C₄⋊Q₈	—	2	3	no	—
208	C₄⋊Q₈	C₂×C₄×C₂₃	1	3	yes	—
209	C₄⋊Q₈	C₂×C₄×C₂₃	1	3	yes	—
210	#126	—	2	6	no	—
211	#126	—	2	3	no	—
212	#125	C₂³×C₂₃	1	3	yes	—
213	#124	—	2	3	no	—
214	#124	—	2	6	no	—
215	#124	C₂³×C₂₃	1	1	yes	characteristic
216	#123	—	2	3	no	—
217	C₂²⋊Q₈	—	2	3	no	—
218	C₂²⋊Q₈	—	2	3	no	—
219	C₂²⋊Q₈	—	2	3	no	—
220	C₂²⋊Q₈	C₂×C₄×C₂₃	1	3	yes	—
221	#121	—	2	3	no	—
222	#121	C₂×C₄×C₂₃	1	3	yes	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	3	no	—
225	C₄×Q₈	—	2	3	no	—
226	C₄×Q₈	—	2	3	no	—
227	C₄×Q₈	C₂×C₄×C₂₃	1	3	yes	—
228	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
229	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
230	C₄×D₄	—	2	3	no	—
231	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
232	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
233	C₄×D₄	D₄×C₂₃	1	3	yes	—
234	#117	—	2	3	no	—
235	#117	—	2	3	no	—
236	#117	C₂×C₄×C₂₃	1	6	yes	—
237	#117	D₄×C₂₃	1	3	yes	—
238	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
239	C₄⋊C₈	—	2	3	no	—
240	C₂²⋊C₈	—	2	3	no	—
241	#97	—	2	3	no	—
242	#97	C₂³×C₂₃	1	1	yes	characteristic
243	#97	C₂³×C₂₃	1	3	yes	—
244	#97	C₂³×C₂₃	1	1	yes	characteristic
245	#97	C₂³×C₂₃	1	3	yes	—
246	C₄×C₈	—	2	3	no	—
247	C₄×C₈	—	2	3	no	—
248	C₂₃	#17654	1	1	yes	characteristic, direct product factor, Sylow
249	#41	—	2	4	no	—
250	C₂×Q₈	—	2	3	no	—
251	C₂×Q₈	—	2	3	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
254	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
255	C₂×D₄	—	2	3	no	—
256	C₂×D₄	C₂²⋊C₄×C₂₃	1	3	yes	—
257	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
258	C₂²×C₄	—	2	3	no	—
259	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
260	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
261	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
262	M₁₆	—	2	12	no	—
263	M₁₆	—	4	6	no	—
264	C₂×C₈	—	2	12	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	3	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	3	no	—
269	C₄⋊C₄	—	2	3	no	—
270	C₄⋊C₄	—	2	3	no	—
271	C₄⋊C₄	—	2	3	no	—
272	C₄⋊C₄	—	2	3	no	—
273	C₄⋊C₄	—	2	3	no	—
274	C₄⋊C₄	—	2	3	no	—
275	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
276	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
277	C₄⋊C₄	C₂²×C₄×C₂₃	1	3	yes	—
278	C₄⋊C₄	C₂²×C₄×C₂₃	1	3	yes	—
279	C₂²⋊C₄	—	2	6	no	—
280	C₂²⋊C₄	—	2	3	no	—
281	C₂²⋊C₄	—	2	3	no	—
282	C₂²⋊C₄	—	2	3	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
286	C₄²	—	2	6	no	—
287	C₄²	—	2	3	no	—
288	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
289	C₄²	C₂×D₄×C₂₃	1	3	yes	—
290	C₄²	C₂×D₄×C₂₃	1	3	yes	—
291	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	6	no	—
296	Q₈	—	8	3	no	—
297	Q₈	C₂₃×#104	1	4	yes	—
298	D₄	—	2	6	no	—
299	D₄	—	4	1	no	—
300	D₄	—	8	3	no	—
301	C₂×C₄	—	2	6	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	3	no	—
304	C₂×C₄	—	2	3	no	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	3	no	—
307	C₂×C₄	—	4	6	no	—
308	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
309	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
311	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	6	no	—
316	C₈	—	2	2	no	—
317	C₈	—	4	3	no	—
318	C₂²	—	2	3	no	—
319	C₂²	—	2	3	no	—
320	C₂²	—	2	3	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	3	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	3	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	3	no	—
332	C₄	—	4	3	no	—
333	C₄	C₂×C₂₃×#104	1	6	yes	—
334	C₂	—	4	3	no	—
335	C₂	—	8	1	no	—
336	C₂	C₂₃×#2889	1	1	yes	characteristic
337	C₂	C₂²×C₂₃×#104	1	1	yes	characteristic
338	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×#3518	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×#1531	C₂	semidirect
5. C₂₃×#1529	C₂	semidirect
6. C₂₃×#1526	C₂	semidirect
7. C₂₃×#1524	C₂	semidirect
122. #17654	C₂₃	direct (N × H)
150. #2971	C₂×C₂₃	semidirect
151. #2875	C₂×C₂₃	semidirect
152. #1531	C₂×C₂₃	semidirect
153. #1529	C₂×C₂₃	semidirect
154. #1526	C₂×C₂₃	semidirect
155. #1524	C₂×C₂₃	semidirect
248. C₂₃	#17654	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×#3518	2
C₂₃×#2971	2
C₂₃×#2875	2
C₂₃×#1531	2
C₂₃×#1529	2
C₂₃×#1526	2
C₂₃×#1524	2
#17654	23

Subgroups — C23×#17654 #19165280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17654 #19165280