Subgroups — C₂₃×#9851 #19157477

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1962	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1492	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#533	C₂²	1	1	yes	characteristic
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂₃×#498	C₄	1	1	yes	characteristic, complemented
11	C₂₃×#492	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#487	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#486	C₄	1	1	yes	characteristic, complemented
14	C₂₃×#465	—	2	2	no	—
15	C₄×SD₁₆×C₂₃	—	2	2	no	—
16	C₂₃×#423	C₂²	1	1	yes	characteristic
17	C₂₃×#423	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#104	—	2	2	no	—
19	C₂×C₂₃×#97	—	2	2	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₄³×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#339	—	2	2	no	—
24	C₂₃×#335	—	2	2	no	—
25	C₂₃×Q₈⋊C₈	—	2	2	no	—
26	C₂₃×D₄⋊C₈	—	2	2	no	—
27	C₂×#41×C₂₃	—	2	1	no	—
28	C₂×SD₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	D₄	1	2	yes	—
33	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
37	C₂₃×#125	—	2	1	no	—
38	C₂₃×#124	—	2	1	no	—
39	C₂₃×C₂²⋊Q₈	—	2	1	no	—
40	C₂₃×C₂²⋊Q₈	—	2	1	no	—
41	C₂₃×#121	—	2	1	no	—
42	C₂₃×#121	—	2	1	no	—
43	C₄×Q₈×C₂₃	—	2	1	no	—
44	C₄×Q₈×C₂₃	—	2	1	no	—
45	C₄×D₄×C₂₃	—	2	1	no	—
46	C₄×D₄×C₂₃	—	2	1	no	—
47	C₂₃×#117	—	2	1	no	—
48	C₂×C₄²×C₂₃	—	2	1	no	—
49	C₂×C₄²×C₂₃	—	2	1	no	—
50	C₂×C₄²×C₂₃	D₄	1	2	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#106	—	2	2	no	—
53	C₂₃×C₄⋊C₈	—	2	1	no	—
54	C₂₃×C₄⋊C₈	—	2	1	no	—
55	C₂₃×C₄⋊C₈	—	2	1	no	—
56	C₂₃×C₄⋊C₈	—	2	1	no	—
57	C₂₃×#104	—	4	4	no	—
58	C₂₃×#103	—	2	2	no	—
59	C₂₃×#102	—	2	2	no	—
60	C₂₃×#97	—	2	8	no	—
61	C₄×C₈×C₂₃	—	2	1	no	—
62	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	D₄	1	2	yes	—
66	#41×C₂₃	—	4	2	no	—
67	#41×C₂₃	—	4	2	no	—
68	C₂×Q₈×C₂₃	—	2	1	no	—
69	C₂×Q₈×C₂₃	—	2	1	no	—
70	C₂×Q₈×C₂₃	—	2	1	no	—
71	C₂×D₄×C₂₃	—	2	1	no	—
72	C₂×D₄×C₂₃	—	2	1	no	—
73	C₂×D₄×C₂₃	—	2	1	no	—
74	C₂×D₄×C₂₃	—	4	1	no	—
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	4	1	no	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
80	SD₁₆×C₂₃	—	4	4	no	—
81	M₁₆×C₂₃	—	4	2	no	—
82	M₁₆×C₂₃	—	4	2	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	4	no	—
86	C₂×C₈×C₂₃	—	4	1	no	—
87	C₂×C₈×C₂₃	—	4	1	no	—
88	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₄⋊C₄×C₂₃	—	4	1	no	—
97	C₄⋊C₄×C₂₃	—	4	1	no	—
98	C₂²⋊C₄×C₂₃	—	4	1	no	—
99	C₂²⋊C₄×C₂₃	—	4	1	no	—
100	C₂²⋊C₄×C₂₃	—	4	1	no	—
101	C₄²×C₂₃	—	2	4	no	—
102	C₄²×C₂₃	—	2	2	no	—
103	C₄²×C₂₃	—	2	2	no	—
104	C₄²×C₂₃	—	4	2	no	—
105	C₄²×C₂₃	—	4	1	no	—
106	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₂₃	#41	1	1	yes	characteristic
110	#9851	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₂₃	—	4	1	no	—
112	C₂³×C₂₃	#121	1	1	yes	characteristic
113	Q₈×C₂₃	—	4	2	no	—
114	Q₈×C₂₃	—	4	2	no	—
115	Q₈×C₂₃	—	4	2	no	—
116	D₄×C₂₃	—	4	2	no	—
117	D₄×C₂₃	—	4	2	no	—
118	D₄×C₂₃	—	4	2	no	—
119	D₄×C₂₃	—	8	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	2	4	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	1	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	8	1	no	—
129	C₂×C₄×C₂₃	—	8	1	no	—
130	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₂₃	#137	1	2	yes	—
138	C₈×C₂₃	—	2	4	no	—
139	C₈×C₂₃	—	2	2	no	—
140	C₈×C₂₃	—	8	1	no	—
141	C₈×C₂₃	—	8	1	no	—
142	C₈×C₂₃	#104	1	4	yes	complemented
143	#3167	C₂×C₂₃	1	1	yes	characteristic
144	#1994	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#1962	C₂×C₂₃	1	1	yes	characteristic
146	C₄×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#1492	C₂×C₂₃	1	1	yes	characteristic
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	8	1	no	—
153	C₂²×C₂₃	—	8	1	no	—
154	C₂²×C₂₃	—	8	1	no	—
155	C₂²×C₂₃	#390	1	1	yes	characteristic
156	C₂²×C₂₃	#468	1	2	yes	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	4	4	no	—
160	C₄×C₂₃	—	8	1	no	—
161	C₄×C₂₃	—	8	1	no	—
162	C₄×C₂₃	—	8	1	no	—
163	C₄×C₂₃	C₂×#104	1	2	yes	—
164	C₄×C₂₃	#440	1	2	yes	—
165	C₄×C₂₃	#444	1	2	yes	—
166	C₄×C₂₃	#496	1	1	yes	characteristic
167	C₄×C₂₃	#497	1	1	yes	characteristic
168	#533	C₂²×C₂₃	1	1	yes	characteristic
169	#532	C₂²×C₂₃	1	1	yes	characteristic
170	#498	C₄×C₂₃	1	1	yes	characteristic, complemented
171	#492	C₄×C₂₃	1	1	yes	characteristic, complemented
172	#487	C₄×C₂₃	1	1	yes	characteristic, complemented
173	#486	C₄×C₂₃	1	1	yes	characteristic, complemented
174	#465	—	2	2	no	—
175	C₄×SD₁₆	—	2	2	no	—
176	#423	C₂²×C₂₃	1	1	yes	characteristic
177	#423	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×#104	—	2	2	no	—
179	C₂×#97	—	2	2	no	—
180	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
182	C₄³	C₂²×C₂₃	1	1	yes	characteristic
183	#339	—	2	2	no	—
184	#335	—	2	2	no	—
185	Q₈⋊C₈	—	2	2	no	—
186	D₄⋊C₈	—	2	2	no	—
187	C₂×C₂₃	—	2	2	no	—
188	C₂×C₂₃	—	16	1	no	—
189	C₂×C₂₃	#1962	1	1	yes	characteristic
190	C₂×C₂₃	#1973	1	1	yes	characteristic
191	C₂×C₂₃	#1975	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	D₄×C₂₃	1	2	yes	—
198	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
201	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
202	#125	—	2	1	no	—
203	#124	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	#121	—	2	1	no	—
207	#121	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	#117	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	D₄×C₂₃	1	2	yes	—
216	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
217	#106	—	2	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	#104	—	4	4	no	—
223	#103	—	2	2	no	—
224	#102	—	2	2	no	—
225	#97	—	2	8	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
228	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
229	C₄×C₈	D₄×C₂₃	1	2	yes	—
230	C₄×C₈	D₄×C₂₃	1	2	yes	—
231	C₂₃	#9851	1	1	yes	characteristic, direct product factor, Sylow
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
246	SD₁₆	—	4	4	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
255	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₄²	#41×C₂₃	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	C₂₃×#121	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
298	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
299	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
300	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
301	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
302	C₂×C₄	C₂₃×#137	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	8	1	no	—
306	C₈	—	8	1	no	—
307	C₈	C₂₃×#104	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₂₃×#390	1	1	yes	characteristic
314	C₂²	C₂₃×#468	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	1	no	—
321	C₄	C₂×C₂₃×#104	1	2	yes	—
322	C₄	C₂₃×#440	1	2	yes	—
323	C₄	C₂₃×#444	1	2	yes	—
324	C₄	C₂₃×#496	1	1	yes	characteristic
325	C₄	C₂₃×#497	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	C₂₃×#1962	1	1	yes	characteristic
329	C₂	C₂₃×#1973	1	1	yes	characteristic
330	C₂	C₂₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₄² ▷ 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×#1994	C₂	semidirect
4. C₄×C₂₃×#97	C₂	semidirect
5. C₂₃×#1567	C₂	semidirect
6. C₂₃×#1493	C₂	semidirect
10. C₂₃×#498	C₄	semidirect
11. C₂₃×#492	C₄	semidirect
12. C₂₃×#487	C₄	semidirect
13. C₂₃×#486	C₄	semidirect
110. #9851	C₂₃	direct (N × H)
142. C₈×C₂₃	#104	semidirect
144. #1994	C₂×C₂₃	semidirect
146. C₄×#97	C₂×C₂₃	semidirect
147. #1567	C₂×C₂₃	semidirect
148. #1493	C₂×C₂₃	semidirect
170. #498	C₄×C₂₃	semidirect
171. #492	C₄×C₂₃	semidirect
172. #487	C₄×C₂₃	semidirect
173. #486	C₄×C₂₃	semidirect
231. C₂₃	#9851	direct (N × H)
307. C₈	C₂₃×#104	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3167	2
C₂₃×#1994	2
C₂₃×#1962	2
C₄×C₂₃×#97	2
C₂₃×#1567	2
C₂₃×#1493	2
C₂₃×#1492	2
#9851	23

Subgroups — C23×#9851 #19157477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9851 #19157477