Subgroups — C₂₃×#10153 #19157779

Order 5888.

620 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1965	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₂₃	—	2	2	no	—
9	C₂₃×#533	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
12	C₂₃×#501	—	2	2	no	—
13	C₂₃×#500	—	2	1	no	—
14	C₂₃×#498	—	2	1	no	—
15	C₂₃×#492	—	2	2	no	—
16	C₂₃×#492	—	2	1	no	—
17	C₂₃×#492	—	2	2	no	—
18	C₂₃×#486	—	2	1	no	—
19	C₂₃×#460	—	2	2	no	—
20	C₂₃×#450	—	2	2	no	—
21	C₂₃×#431	C₂²	1	1	yes	characteristic
22	C₂₃×#428	—	2	2	no	—
23	C₂₃×#423	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#100	—	2	2	no	—
25	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#332	—	2	2	no	—
28	C₂₃×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₂₃	—	2	1	no	—
30	C₂²×D₄×C₂₃	—	2	1	no	—
31	C₂²×D₄×C₂₃	—	2	1	no	—
32	C₂²×D₄×C₂₃	D₄	1	2	yes	—
33	C₂×SD₁₆×C₂₃	—	2	8	no	—
34	C₂×SD₁₆×C₂₃	—	2	4	no	—
35	C₂×SD₁₆×C₂₃	—	4	8	no	—
36	C₂×M₁₆×C₂₃	—	2	1	no	—
37	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
38	C₂²×C₈×C₂₃	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	—	2	1	no	—
41	C₂₃×C₄⋊Q₈	—	2	1	no	—
42	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
45	C₂₃×#127	—	2	1	no	—
46	C₂₃×#127	—	2	1	no	—
47	C₂₃×#127	—	2	1	no	—
48	C₂₃×#127	—	2	1	no	—
49	C₂₃×#125	—	2	1	no	—
50	C₂₃×C₂²⋊Q₈	—	2	1	no	—
51	C₂₃×C₂²⋊Q₈	—	2	1	no	—
52	C₄×Q₈×C₂₃	—	2	1	no	—
53	C₄×Q₈×C₂₃	—	2	1	no	—
54	C₂₃×#117	—	2	1	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×#107	—	2	4	no	—
59	C₂₃×#106	—	2	4	no	—
60	C₂₃×C₄⋊C₈	—	2	1	no	—
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₂₃×#102	—	2	4	no	—
64	C₂₃×#100	—	4	4	no	—
65	C₂₃×C₂²⋊C₈	—	2	1	no	—
66	C₂₃×#97	—	2	1	no	—
67	C₄×C₈×C₂₃	—	2	1	no	—
68	C₄×C₈×C₂₃	—	2	1	no	—
69	C₄×C₈×C₂₃	—	2	1	no	—
70	C₄×C₈×C₂₃	D₄	1	2	yes	—
71	C₂⁴×C₂₃	—	2	2	no	—
72	C₂×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	2	2	no	—
74	C₂×Q₈×C₂₃	—	2	1	no	—
75	C₂×Q₈×C₂₃	—	4	4	no	—
76	C₂×Q₈×C₂₃	—	4	1	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	2	2	no	—
79	C₂×D₄×C₂₃	—	4	4	no	—
80	C₂×D₄×C₂₃	—	4	1	no	—
81	C₂×D₄×C₂₃	—	4	1	no	—
82	C₂×D₄×C₂₃	—	4	1	no	—
83	C₂×D₄×C₂₃	—	4	1	no	—
84	C₂×D₄×C₂₃	—	4	2	no	—
85	C₂×D₄×C₂₃	—	4	2	no	—
86	C₂×D₄×C₂₃	—	8	2	no	—
87	C₂²×C₄×C₂₃	—	4	1	no	—
88	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
91	SD₁₆×C₂₃	—	4	16	no	—
92	SD₁₆×C₂₃	—	8	4	no	—
93	M₁₆×C₂₃	—	2	2	no	—
94	M₁₆×C₂₃	—	4	2	no	—
95	M₁₆×C₂₃	SD₁₆	1	4	yes	complemented
96	C₂×C₈×C₂₃	—	2	4	no	—
97	C₂×C₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	2	2	no	—
99	C₂×C₈×C₂₃	—	2	2	no	—
100	C₂×C₈×C₂₃	—	4	1	no	—
101	C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₄⋊C₄×C₂₃	—	4	1	no	—
108	C₄⋊C₄×C₂₃	—	4	1	no	—
109	C₄⋊C₄×C₂₃	—	4	1	no	—
110	C₄⋊C₄×C₂₃	—	4	1	no	—
111	C₂²⋊C₄×C₂₃	—	4	1	no	—
112	C₄²×C₂₃	—	4	1	no	—
113	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	#10153	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₂₃	—	4	2	no	—
119	C₂³×C₂₃	—	4	4	no	—
120	C₂³×C₂₃	—	8	2	no	—
121	C₂³×C₂₃	—	8	2	no	—
122	C₂³×C₂₃	#121	1	1	yes	characteristic
123	Q₈×C₂₃	—	4	4	no	—
124	Q₈×C₂₃	—	4	2	no	—
125	Q₈×C₂₃	—	4	2	no	—
126	Q₈×C₂₃	—	8	2	no	—
127	D₄×C₂₃	—	4	4	no	—
128	D₄×C₂₃	—	4	4	no	—
129	D₄×C₂₃	—	8	2	no	—
130	D₄×C₂₃	—	8	2	no	—
131	D₄×C₂₃	—	8	2	no	—
132	D₄×C₂₃	—	8	1	no	—
133	D₄×C₂₃	—	8	1	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	4	1	no	—
137	C₂×C₄×C₂₃	—	4	1	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	8	2	no	—
140	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
142	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₂₃	#137	1	2	yes	—
148	C₈×C₂₃	—	2	4	no	—
149	C₈×C₂₃	—	2	4	no	—
150	C₈×C₂₃	—	4	2	no	—
151	C₈×C₂₃	—	8	1	no	—
152	#3208	C₂×C₂₃	1	1	yes	characteristic
153	#3165	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂×#492	C₂×C₂₃	1	1	yes	characteristic
155	#1965	C₂×C₂₃	1	1	yes	characteristic
156	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
158	#1278	C₂×C₂₃	1	1	yes	characteristic, complemented
159	C₂²×C₂₃	—	2	1	no	—
160	C₂²×C₂₃	—	2	1	no	—
161	C₂²×C₂₃	—	8	2	no	—
162	C₂²×C₂₃	—	8	2	no	—
163	C₂²×C₂₃	—	8	2	no	—
164	C₂²×C₂₃	—	16	4	no	—
165	C₂²×C₂₃	#392	1	1	yes	characteristic
166	C₂²×C₂₃	#461	1	2	yes	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	2	1	no	—
169	C₄×C₂₃	—	8	2	no	—
170	C₄×C₂₃	—	8	1	no	—
171	C₄×C₂₃	—	8	1	no	—
172	C₄×C₂₃	#448	1	2	yes	—
173	C₄×C₂₃	#453	1	2	yes	—
174	C₄×C₂₃	#471	1	2	yes	—
175	C₄×C₂₃	#492	1	1	yes	characteristic
176	C₄×C₂₃	#497	1	1	yes	characteristic
177	C₂²×SD₁₆	—	2	2	no	—
178	#533	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
180	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
181	#501	—	2	2	no	—
182	#500	—	2	1	no	—
183	#498	—	2	1	no	—
184	#492	—	2	2	no	—
185	#492	—	2	1	no	—
186	#492	—	2	2	no	—
187	#486	—	2	1	no	—
188	#460	—	2	2	no	—
189	#450	—	2	2	no	—
190	#431	C₂²×C₂₃	1	1	yes	characteristic
191	#428	—	2	2	no	—
192	#423	C₂²×C₂₃	1	1	yes	characteristic
193	C₂×#100	—	2	2	no	—
194	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
196	#332	—	2	2	no	—
197	Q₈⋊C₈	—	2	2	no	—
198	C₂×C₂₃	—	2	2	no	—
199	C₂×C₂₃	—	16	2	no	—
200	C₂×C₂₃	#2002	1	1	yes	characteristic
201	C₂×C₂₃	#2013	1	1	yes	characteristic
202	C₂×C₂₃	#2019	1	1	yes	characteristic
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	D₄×C₂₃	1	2	yes	—
207	C₂×SD₁₆	—	2	8	no	—
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×SD₁₆	—	4	8	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
212	C₂²×C₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
217	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
218	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#125	—	2	1	no	—
224	C₂²⋊Q₈	—	2	1	no	—
225	C₂²⋊Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	#117	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
232	#107	—	2	4	no	—
233	#106	—	2	4	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
237	#102	—	2	4	no	—
238	#100	—	4	4	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	#97	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	D₄×C₂₃	1	2	yes	—
245	C₂₃	#10153	1	1	yes	characteristic, direct product factor, Sylow
246	C₂⁴	—	2	2	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
266	SD₁₆	—	4	16	no	—
267	SD₁₆	—	8	4	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	SD₁₆×C₂₃	1	4	yes	complemented
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
292	C₂³	—	4	2	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	C₂₃×#121	1	1	yes	characteristic
297	Q₈	—	4	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
315	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
316	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
317	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
319	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
320	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
321	C₂×C₄	C₂₃×#137	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	4	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	16	4	no	—
332	C₂²	C₂₃×#392	1	1	yes	characteristic
333	C₂²	C₂₃×#461	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	C₂₃×#448	1	2	yes	—
340	C₄	C₂₃×#453	1	2	yes	—
341	C₄	C₂₃×#471	1	2	yes	—
342	C₄	C₂₃×#492	1	1	yes	characteristic
343	C₄	C₂₃×#497	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₂₃×#2002	1	1	yes	characteristic
347	C₂	C₂₃×#2013	1	1	yes	characteristic
348	C₂	C₂₃×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3165	C₂	semidirect
5. C₂₃×#1549	C₂	semidirect
6. C₂₃×#1493	C₂	semidirect
7. C₂₃×#1278	C₂	semidirect
95. M₁₆×C₂₃	SD₁₆	semidirect
117. #10153	C₂₃	direct (N × H)
153. #3165	C₂×C₂₃	semidirect
156. #1549	C₂×C₂₃	semidirect
157. #1493	C₂×C₂₃	semidirect
158. #1278	C₂×C₂₃	semidirect
245. C₂₃	#10153	direct (N × H)
270. M₁₆	SD₁₆×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3208	2
C₂₃×#3165	2
C₂×C₂₃×#492	2
C₂₃×#1965	2
C₂₃×#1549	2
C₂₃×#1493	2
C₂₃×#1278	2
#10153	23

Subgroups — C23×#10153 #19157779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10153 #19157779