Subgroups — C₂₃×#8878 #19156504

Order 5888.

432 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2932	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1502	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1480	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	—	2	1	no	—
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂₃×#435	—	2	2	no	—
11	C₂₃×#435	—	2	2	no	—
12	C₂₃×#431	C₂²	1	1	yes	characteristic
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂₃×#423	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂₃×#408	—	2	1	no	—
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#336	—	2	1	no	—
21	C₂₃×#335	—	2	2	no	—
22	C₂₃×#327	C₄	1	2	yes	—
23	C₂₃×#327	C₄	1	2	yes	—
24	C₂₃×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₂₃	—	2	1	no	—
26	C₂₃×#131	—	4	2	no	—
27	C₂₃×#131	—	4	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
35	C₂₃×#127	Q₈	1	1	yes	characteristic
36	C₂₃×#124	—	2	1	no	—
37	C₂₃×#121	—	2	1	no	—
38	C₂₃×#121	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₂₃	—	2	1	no	—
40	C₄×D₄×C₂₃	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
42	C₂×C₄²×C₂₃	—	2	1	no	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	—	2	1	no	—
45	C₂₃×C₄⋊C₈	—	2	1	no	—
46	C₂₃×C₄⋊C₈	—	2	1	no	—
47	C₂₃×C₄⋊C₈	—	2	1	no	—
48	C₂₃×C₄⋊C₈	—	2	1	no	—
49	C₂₃×C₄⋊C₈	—	4	2	no	—
50	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
51	C₂₃×C₂²⋊C₈	—	2	4	no	—
52	C₂₃×C₂²⋊C₈	—	2	1	no	—
53	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₂₃×#97	—	2	1	no	—
55	C₂₃×#97	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	1	no	—
57	C₄×C₈×C₂₃	—	2	1	no	—
58	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
59	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
60	#41×C₂₃	—	4	1	no	—
61	#41×C₂₃	—	4	1	no	—
62	#41×C₂₃	—	4	1	no	—
63	#41×C₂₃	—	4	1	no	—
64	C₂×Q₈×C₂₃	—	2	1	no	—
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	2	1	no	—
67	C₂×D₄×C₂₃	—	2	1	no	—
68	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	—	2	1	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	1	no	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
75	M₁₆×C₂₃	—	4	4	no	—
76	M₁₆×C₂₃	—	4	1	no	—
77	M₁₆×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	1	no	—
85	C₂×C₈×C₂₃	—	4	4	no	—
86	C₂×C₈×C₂₃	—	4	1	no	—
87	C₂×C₈×C₂₃	—	4	1	no	—
88	C₂×C₈×C₂₃	—	4	1	no	—
89	C₂×C₈×C₂₃	—	4	1	no	—
90	C₂×C₈×C₂₃	D₈	1	2	yes	—
91	C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
94	C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	4	1	no	—
97	C₄²×C₂₃	—	4	1	no	—
98	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₂₃	#41	1	1	yes	characteristic
102	#8878	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	—	2	2	no	—
104	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₂₃	—	4	1	no	—
106	Q₈×C₂₃	—	4	1	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	1	no	—
111	D₄×C₂₃	—	4	1	no	—
112	D₄×C₂₃	—	8	1	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	1	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	1	no	—
122	C₂×C₄×C₂₃	—	4	1	no	—
123	C₂×C₄×C₂₃	—	4	1	no	—
124	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	4	2	no	—
136	C₈×C₂₃	—	4	2	no	—
137	C₈×C₂₃	—	4	1	no	—
138	C₈×C₂₃	—	8	1	no	—
139	C₈×C₂₃	—	8	1	no	—
140	#2932	C₂×C₂₃	1	1	yes	characteristic
141	#1940	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#1571	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#1502	C₂×C₂₃	1	1	yes	characteristic
144	#1480	C₂×C₂₃	1	1	yes	characteristic
145	#1303	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#1281	C₂×C₂₃	1	1	yes	characteristic, complemented
147	C₂²×C₂₃	—	2	1	no	—
148	C₂²×C₂₃	—	2	1	no	—
149	C₂²×C₂₃	—	4	2	no	—
150	C₂²×C₂₃	—	4	2	no	—
151	C₂²×C₂₃	—	4	2	no	—
152	C₂²×C₂₃	#385	1	1	yes	characteristic
153	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
154	C₂²×C₂₃	#440	1	1	yes	characteristic
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	4	2	no	—
160	C₄×C₂₃	—	4	2	no	—
161	C₄×C₂₃	—	8	1	no	—
162	C₄×C₂₃	#447	1	1	yes	characteristic
163	C₄×C₂₃	#452	1	1	yes	characteristic
164	C₄×C₂₃	#474	1	1	yes	characteristic
165	C₄×C₂₃	#478	1	1	yes	characteristic
166	C₂×#131	—	2	1	no	—
167	#532	C₂²×C₂₃	1	1	yes	characteristic
168	#435	—	2	2	no	—
169	#435	—	2	2	no	—
170	#431	C₂²×C₂₃	1	1	yes	characteristic
171	#431	C₂²×C₂₃	1	1	yes	characteristic
172	#423	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₄⋊C₈	—	2	1	no	—
174	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
175	#408	—	2	1	no	—
176	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
177	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
178	#336	—	2	1	no	—
179	#335	—	2	2	no	—
180	#327	C₄×C₂₃	1	2	yes	—
181	#327	C₄×C₂₃	1	2	yes	—
182	D₄⋊C₈	—	2	2	no	—
183	C₂×C₂₃	—	2	1	no	—
184	C₂×C₂₃	—	2	1	no	—
185	C₂×C₂₃	—	8	2	no	—
186	C₂×C₂₃	#1861	1	1	yes	characteristic
187	C₂×C₂₃	#1862	1	1	yes	characteristic
188	C₂×C₂₃	#1868	1	1	yes	characteristic
189	C₂×#41	—	2	1	no	—
190	#131	—	4	2	no	—
191	#131	—	4	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
198	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
199	#127	Q₈×C₂₃	1	1	yes	characteristic
200	#124	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	C₂×C₄×C₂₃	1	2	yes	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₄×C₂₃	1	2	yes	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	4	2	no	—
214	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
218	#97	—	2	1	no	—
219	#97	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
223	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
224	C₂₃	#8878	1	1	yes	characteristic, direct product factor, Sylow
225	#41	—	4	1	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	#41	—	4	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	D₈×C₂₃	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
266	C₄²	#41×C₂₃	1	1	yes	characteristic
267	C₂³	—	2	2	no	—
268	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
269	Q₈	—	4	1	no	—
270	Q₈	—	4	1	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	1	no	—
276	D₄	—	8	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
291	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
294	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
296	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂₃×#385	1	1	yes	characteristic
310	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
311	C₂²	C₂₃×#440	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	C₂₃×#447	1	1	yes	characteristic
320	C₄	C₂₃×#452	1	1	yes	characteristic
321	C₄	C₂₃×#474	1	1	yes	characteristic
322	C₄	C₂₃×#478	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	8	2	no	—
326	C₂	C₂₃×#1861	1	1	yes	characteristic
327	C₂	C₂₃×#1862	1	1	yes	characteristic
328	C₂	C₂₃×#1868	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#385
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1940	C₂	semidirect
3. C₂₃×#1571	C₂	semidirect
6. C₂₃×#1303	C₂	semidirect
7. C₂₃×#1281	C₂	semidirect
102. #8878	C₂₃	direct (N × H)
141. #1940	C₂×C₂₃	semidirect
142. #1571	C₂×C₂₃	semidirect
145. #1303	C₂×C₂₃	semidirect
146. #1281	C₂×C₂₃	semidirect
224. C₂₃	#8878	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×#2932	2
C₂₃×#1940	2
C₂₃×#1571	2
C₂₃×#1502	2
C₂₃×#1480	2
C₂₃×#1303	2
C₂₃×#1281	2
#8878	23

Subgroups — C23×#8878 #19156504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8878 #19156504