Subgroups — C₂₃×#10292 #19157918

Order 5888.

640 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2858	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#533	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#124	—	2	2	no	—
12	C₂₃×#500	—	2	1	no	—
13	C₂₃×#492	—	2	1	no	—
14	C₂₃×#487	—	2	1	no	—
15	C₂₃×#486	—	2	1	no	—
16	C₂₃×#484	—	2	2	no	—
17	C₂₃×#477	—	2	2	no	—
18	C₂₃×#460	—	2	2	no	—
19	C₂₃×#450	—	2	2	no	—
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂₃×#431	C₂²	1	1	yes	characteristic
22	C₂₃×#423	C₂²	1	1	yes	characteristic
23	C₂₃×#390	—	2	2	no	—
24	C₂₃×#390	—	2	2	no	—
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#339	—	2	2	no	—
27	C₂₃×#332	—	2	2	no	—
28	C₂₃×#331	—	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	4	no	—
34	C₂×M₁₆×C₂₃	—	2	1	no	—
35	C₂₃×C₄⋊Q₈	—	2	1	no	—
36	C₂₃×C₄⋊Q₈	—	2	1	no	—
37	C₂₃×C₄⋊Q₈	—	2	1	no	—
38	C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₂₃×#127	—	2	1	no	—
42	C₂₃×#127	—	2	1	no	—
43	C₂₃×#127	—	2	1	no	—
44	C₂₃×#127	—	2	1	no	—
45	C₂₃×#125	—	2	1	no	—
46	C₂₃×#124	—	4	8	no	—
47	C₂₃×C₂²⋊Q₈	—	2	1	no	—
48	C₂₃×C₂²⋊Q₈	—	2	1	no	—
49	C₄×Q₈×C₂₃	—	2	1	no	—
50	C₄×Q₈×C₂₃	—	2	1	no	—
51	C₂₃×#117	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
55	C₂×C₄²×C₂₃	—	2	1	no	—
56	C₂×C₄²×C₂₃	—	2	1	no	—
57	C₂×C₄²×C₂₃	D₄	1	2	yes	—
58	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#107	—	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×C₄⋊C₈	D₄	1	2	yes	—
64	C₂₃×#103	—	2	4	no	—
65	C₂₃×#102	—	2	4	no	—
66	C₂₃×C₂²⋊C₈	—	2	1	no	—
67	C₂₃×C₂²⋊C₈	—	2	1	no	—
68	C₄×C₈×C₂₃	—	2	1	no	—
69	C₄×C₈×C₂₃	—	2	1	no	—
70	C₂⁴×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	2	1	no	—
72	C₂×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	4	4	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	4	4	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	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	2	2	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	C₂×D₄	1	1	yes	characteristic
91	SD₁₆×C₂₃	—	8	4	no	—
92	M₁₆×C₂₃	—	4	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	4	1	no	—
96	C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₄⋊C₄×C₂₃	—	4	1	no	—
103	C₄⋊C₄×C₂₃	—	4	1	no	—
104	C₄⋊C₄×C₂₃	—	4	1	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₂²⋊C₄×C₂₃	—	4	16	no	—
107	C₂²⋊C₄×C₂₃	—	4	1	no	—
108	C₄²×C₂₃	—	2	4	no	—
109	C₄²×C₂₃	—	2	2	no	—
110	C₄²×C₂₃	—	2	2	no	—
111	C₄²×C₂₃	—	4	2	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	#41	1	1	yes	characteristic
117	#10292	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₂₃	—	4	4	no	—
119	C₂³×C₂₃	—	4	4	no	—
120	C₂³×C₂₃	—	4	2	no	—
121	C₂³×C₂₃	—	4	4	no	—
122	C₂³×C₂₃	#120	1	1	yes	characteristic
123	Q₈×C₂₃	—	4	2	no	—
124	Q₈×C₂₃	—	4	2	no	—
125	Q₈×C₂₃	—	8	2	no	—
126	Q₈×C₂₃	—	8	2	no	—
127	D₄×C₂₃	—	4	4	no	—
128	D₄×C₂₃	—	8	2	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	—	2	4	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	4	4	no	—
140	C₂×C₄×C₂₃	—	4	1	no	—
141	C₂×C₄×C₂₃	—	4	1	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
149	C₂×C₄×C₂₃	#135	1	2	yes	—
150	C₂×C₄×C₂₃	#137	1	2	yes	—
151	C₈×C₂₃	—	4	2	no	—
152	C₈×C₂₃	—	4	2	no	—
153	C₈×C₂₃	—	8	1	no	—
154	#3230	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#3208	C₂×C₂₃	1	1	yes	characteristic
156	#2858	C₂×C₂₃	1	1	yes	characteristic
157	#1994	C₂×C₂₃	1	1	yes	characteristic, complemented
158	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
159	#1545	C₂×C₂₃	1	1	yes	characteristic
160	#1276	C₂×C₂₃	1	1	yes	characteristic, complemented
161	C₂²×C₂₃	—	2	1	no	—
162	C₂²×C₂₃	—	2	1	no	—
163	C₂²×C₂₃	—	8	8	no	—
164	C₂²×C₂₃	—	8	2	no	—
165	C₂²×C₂₃	—	8	2	no	—
166	C₂²×C₂₃	—	8	2	no	—
167	C₂²×C₂₃	#392	1	1	yes	characteristic
168	C₂²×C₂₃	#452	1	2	yes	—
169	C₄×C₂₃	—	2	1	no	—
170	C₄×C₂₃	—	2	1	no	—
171	C₄×C₂₃	—	4	4	no	—
172	C₄×C₂₃	—	8	2	no	—
173	C₄×C₂₃	—	8	1	no	—
174	C₄×C₂₃	—	8	1	no	—
175	C₄×C₂₃	#453	1	2	yes	—
176	C₄×C₂₃	#454	1	2	yes	—
177	C₄×C₂₃	#464	1	2	yes	—
178	C₄×C₂₃	#495	1	1	yes	characteristic
179	C₄×C₂₃	#497	1	1	yes	characteristic
180	#533	C₂²×C₂₃	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
183	C₂×#124	—	2	2	no	—
184	#500	—	2	1	no	—
185	#492	—	2	1	no	—
186	#487	—	2	1	no	—
187	#486	—	2	1	no	—
188	#484	—	2	2	no	—
189	#477	—	2	2	no	—
190	#460	—	2	2	no	—
191	#450	—	2	2	no	—
192	#431	C₂²×C₂₃	1	1	yes	characteristic
193	#431	C₂²×C₂₃	1	1	yes	characteristic
194	#423	C₂²×C₂₃	1	1	yes	characteristic
195	#390	—	2	2	no	—
196	#390	—	2	2	no	—
197	C₄³	C₂²×C₂₃	1	1	yes	characteristic
198	#339	—	2	2	no	—
199	#332	—	2	2	no	—
200	#331	—	2	2	no	—
201	C₂×C₂₃	—	2	2	no	—
202	C₂×C₂₃	—	16	2	no	—
203	C₂×C₂₃	#2002	1	1	yes	characteristic
204	C₂×C₂₃	#2003	1	1	yes	characteristic
205	C₂×C₂₃	#2015	1	1	yes	characteristic
206	C₂²×Q₈	—	2	1	no	—
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	D₄×C₂₃	1	2	yes	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₄⋊Q₈	—	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	1	yes	characteristic
217	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#127	—	2	1	no	—
222	#125	—	2	1	no	—
223	#124	—	4	8	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	D₄×C₂₃	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	8	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	D₄×C₂₃	1	2	yes	—
235	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
236	#107	—	2	4	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
240	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
241	#103	—	2	4	no	—
242	#102	—	2	4	no	—
243	C₂²⋊C₈	—	2	1	no	—
244	C₂²⋊C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	C₂₃	#10292	1	1	yes	characteristic, direct product factor, Sylow
248	C₂⁴	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	4	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
269	SD₁₆	—	8	4	no	—
270	M₁₆	—	4	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	16	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	4	2	no	—
290	C₄²	—	4	1	no	—
291	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
294	C₄²	#41×C₂₃	1	1	yes	characteristic
295	C₂³	—	4	4	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	4	no	—
299	C₂³	C₂₃×#120	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	2	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	8	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
321	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
322	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
323	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
324	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
325	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
326	C₂×C₄	C₂₃×#135	1	2	yes	—
327	C₂×C₄	C₂₃×#137	1	2	yes	—
328	C₈	—	4	2	no	—
329	C₈	—	4	2	no	—
330	C₈	—	8	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	8	8	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₂₃×#392	1	1	yes	characteristic
338	C₂²	C₂₃×#452	1	2	yes	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	4	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	C₂₃×#453	1	2	yes	—
346	C₄	C₂₃×#454	1	2	yes	—
347	C₄	C₂₃×#464	1	2	yes	—
348	C₄	C₂₃×#495	1	1	yes	characteristic
349	C₄	C₂₃×#497	1	1	yes	characteristic
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₂₃×#2002	1	1	yes	characteristic
353	C₂	C₂₃×#2003	1	1	yes	characteristic
354	C₂	C₂₃×#2015	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
1. C₂₃×#3230	C₂	semidirect
4. C₂₃×#1994	C₂	semidirect
5. C₂₃×#1549	C₂	semidirect
7. C₂₃×#1276	C₂	semidirect
117. #10292	C₂₃	direct (N × H)
154. #3230	C₂×C₂₃	semidirect
157. #1994	C₂×C₂₃	semidirect
158. #1549	C₂×C₂₃	semidirect
160. #1276	C₂×C₂₃	semidirect
247. C₂₃	#10292	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×#3230	2
C₂₃×#3208	2
C₂₃×#2858	2
C₂₃×#1994	2
C₂₃×#1549	2
C₂₃×#1545	2
C₂₃×#1276	2
#10292	23

Subgroups — C23×#10292 #19157918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10292 #19157918