Subgroups — C₂₃×#9858 #19157484

Order 5888.

428 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3131	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1492	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1492	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#533	C₂²	1	1	yes	characteristic
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂₃×#488	—	2	2	no	—
11	C₂₃×#487	C₄	1	1	yes	characteristic
12	C₂₃×#487	C₄	1	1	yes	characteristic
13	C₂₃×#486	C₄	1	1	yes	characteristic
14	C₂₃×#486	C₄	1	1	yes	characteristic
15	C₂×C₂₃×#108	—	2	2	no	—
16	C₂₃×#423	C₂²	1	1	yes	characteristic
17	C₂₃×#423	C₂²	1	1	yes	characteristic
18	C₂₃×#416	—	2	2	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×Q₈⋊C₈	—	2	2	no	—
23	C₂₃×Q₈⋊C₈	—	2	2	no	—
24	C₂₃×D₄⋊C₈	—	2	2	no	—
25	C₂₃×D₄⋊C₈	—	2	2	no	—
26	C₂₃×#322	—	2	2	no	—
27	C₂×#41×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂²×C₈×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	C₂³	1	1	yes	characteristic
49	C₂₃×#108	—	4	4	no	—
50	C₂₃×C₄⋊C₈	—	2	1	no	—
51	C₂₃×C₄⋊C₈	—	2	1	no	—
52	C₂₃×C₄⋊C₈	—	2	1	no	—
53	C₂₃×C₄⋊C₈	—	2	1	no	—
54	C₂₃×#103	—	2	2	no	—
55	C₂₃×#103	—	2	2	no	—
56	C₂₃×#102	—	2	2	no	—
57	C₂₃×#102	—	2	2	no	—
58	C₄×C₈×C₂₃	—	2	1	no	—
59	C₄×C₈×C₂₃	—	2	1	no	—
60	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₂₃	D₄	1	2	yes	—
63	C₄×C₈×C₂₃	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	#41×C₂₃	—	4	2	no	—
66	#41×C₂₃	—	4	2	no	—
67	C₂×Q₈×C₂₃	—	2	1	no	—
68	C₂×Q₈×C₂₃	—	2	1	no	—
69	C₂×Q₈×C₂₃	—	2	1	no	—
70	C₂×D₄×C₂₃	—	2	1	no	—
71	C₂×D₄×C₂₃	—	2	1	no	—
72	C₂×D₄×C₂₃	—	2	1	no	—
73	C₂×D₄×C₂₃	—	4	1	no	—
74	C₂²×C₄×C₂₃	—	4	1	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
78	M₁₆×C₂₃	—	4	2	no	—
79	M₁₆×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	4	2	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	—	4	1	no	—
102	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
106	#9858	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₃	—	4	1	no	—
108	C₂³×C₂₃	#124	1	1	yes	characteristic
109	Q₈×C₂₃	—	4	2	no	—
110	Q₈×C₂₃	—	4	2	no	—
111	Q₈×C₂₃	—	4	2	no	—
112	D₄×C₂₃	—	4	2	no	—
113	D₄×C₂₃	—	4	2	no	—
114	D₄×C₂₃	—	4	2	no	—
115	D₄×C₂₃	—	8	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	—	4	1	no	—
120	C₂×C₄×C₂₃	—	4	1	no	—
121	C₂×C₄×C₂₃	—	8	1	no	—
122	C₂×C₄×C₂₃	—	8	1	no	—
123	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
129	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₂₃	#137	1	2	yes	—
131	C₈×C₂₃	—	2	4	no	—
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	2	2	no	—
134	C₈×C₂₃	—	4	2	no	—
135	C₈×C₂₃	—	8	1	no	—
136	C₈×C₂₃	—	8	1	no	—
137	#3131	C₂×C₂₃	1	1	yes	characteristic
138	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#1492	C₂×C₂₃	1	1	yes	characteristic
142	#1492	C₂×C₂₃	1	1	yes	characteristic
143	C₂×#322	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	8	1	no	—
147	C₂²×C₂₃	—	8	1	no	—
148	C₂²×C₂₃	—	8	1	no	—
149	C₂²×C₂₃	#390	1	1	yes	characteristic
150	C₂²×C₂₃	#489	1	2	yes	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	8	1	no	—
154	C₄×C₂₃	—	8	1	no	—
155	C₄×C₂₃	—	8	1	no	—
156	C₄×C₂₃	#417	1	2	yes	—
157	C₄×C₂₃	#443	1	2	yes	—
158	C₄×C₂₃	#444	1	2	yes	—
159	C₄×C₂₃	#496	1	1	yes	characteristic
160	C₄×C₂₃	#497	1	1	yes	characteristic
161	#533	C₂²×C₂₃	1	1	yes	characteristic
162	#532	C₂²×C₂₃	1	1	yes	characteristic
163	#488	—	2	2	no	—
164	#487	C₄×C₂₃	1	1	yes	characteristic
165	#487	C₄×C₂₃	1	1	yes	characteristic
166	#486	C₄×C₂₃	1	1	yes	characteristic
167	#486	C₄×C₂₃	1	1	yes	characteristic
168	C₂×#108	—	2	2	no	—
169	#423	C₂²×C₂₃	1	1	yes	characteristic
170	#423	C₂²×C₂₃	1	1	yes	characteristic
171	#416	—	2	2	no	—
172	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
175	Q₈⋊C₈	—	2	2	no	—
176	Q₈⋊C₈	—	2	2	no	—
177	D₄⋊C₈	—	2	2	no	—
178	D₄⋊C₈	—	2	2	no	—
179	#322	—	2	2	no	—
180	C₂×C₂₃	—	2	2	no	—
181	C₂×C₂₃	—	16	1	no	—
182	C₂×C₂₃	#1959	1	1	yes	characteristic
183	C₂×C₂₃	#1974	1	1	yes	characteristic
184	C₂×C₂₃	#1975	1	1	yes	characteristic
185	C₂×#41	—	2	1	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	D₄×C₂₃	1	2	yes	—
191	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
194	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
195	#125	—	2	1	no	—
196	#124	—	2	1	no	—
197	C₂²⋊Q₈	—	2	1	no	—
198	C₂²⋊Q₈	—	2	1	no	—
199	#121	—	2	1	no	—
200	#121	—	2	1	no	—
201	C₄×Q₈	—	2	1	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	—	2	1	no	—
205	#117	—	2	1	no	—
206	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
207	#108	—	4	4	no	—
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	#103	—	2	2	no	—
213	#103	—	2	2	no	—
214	#102	—	2	2	no	—
215	#102	—	2	2	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
219	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
220	C₄×C₈	D₄×C₂₃	1	2	yes	—
221	C₄×C₈	D₄×C₂₃	1	2	yes	—
222	C₄×C₈	D₄×C₂₃	1	2	yes	—
223	C₂₃	#9858	1	1	yes	characteristic, direct product factor, Sylow
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
248	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₂³	—	4	1	no	—
266	C₂³	C₂₃×#124	1	1	yes	characteristic
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	8	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
284	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
285	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
286	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
287	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
288	C₂×C₄	C₂₃×#137	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	—	8	1	no	—
299	C₂²	—	8	1	no	—
300	C₂²	C₂₃×#390	1	1	yes	characteristic
301	C₂²	C₂₃×#489	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	C₂₃×#417	1	2	yes	—
308	C₄	C₂₃×#443	1	2	yes	—
309	C₄	C₂₃×#444	1	2	yes	—
310	C₄	C₂₃×#496	1	1	yes	characteristic
311	C₄	C₂₃×#497	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	C₂₃×#1959	1	1	yes	characteristic
315	C₂	C₂₃×#1974	1	1	yes	characteristic
316	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×#1848	C₂	semidirect
3. C₂₃×#1493	C₂	semidirect
4. C₂₃×#1493	C₂	semidirect
7. C₂×C₂₃×#322	C₂	semidirect
106. #9858	C₂₃	direct (N × H)
138. #1848	C₂×C₂₃	semidirect
139. #1493	C₂×C₂₃	semidirect
140. #1493	C₂×C₂₃	semidirect
143. C₂×#322	C₂×C₂₃	semidirect
223. C₂₃	#9858	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×#3131	2
C₂₃×#1848	2
C₂₃×#1493	2
C₂₃×#1492	2
C₂×C₂₃×#322	2
#9858	23

Subgroups — C23×#9858 #19157484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9858 #19157484