Subgroups — C₂₃×#20518 #19168144

Order 5888.

838 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	2	yes	maximal
2	C₂₃×#2295	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1771	C₂	1	2	yes	maximal
4	C₂₃×#1771	C₂	1	2	yes	maximal
5	C₂₃×#1770	C₂	1	2	yes	maximal
6	C₂₃×#1768	C₂	1	2	yes	maximal
7	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1726	C₂	1	2	yes	maximal
9	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#446	C₄	1	16	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	2	yes	—
19	C₂×C₂₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#391	C₄	1	2	yes	—
25	C₂₃×#389	C₄	1	2	yes	—
26	C₂₃×#387	C₄	1	4	yes	—
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#376	C₂²	1	2	yes	—
30	C₄³×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#336	C₂²	1	1	yes	characteristic
32	C₂₃×#336	C₂²	1	1	yes	characteristic
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₂₃×#97	—	2	4	no	—
56	C₂₃×#97	—	2	4	no	—
57	C₂₃×#97	—	2	4	no	—
58	C₂₃×#97	C₂×C₄	1	8	yes	—
59	C₂₃×#97	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₂₃	—	2	4	no	—
61	C₄×C₈×C₂₃	—	2	2	no	—
62	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₂₃	Q₈	1	4	yes	complemented
64	C₂₃×#95	C₂×C₄	1	4	yes	—
65	C₂₃×#95	C₂×C₄	1	4	yes	—
66	C₂₃×#95	C₂×C₄	1	2	yes	—
67	C₂₃×#95	C₂×C₄	1	4	yes	—
68	C₂×Q₈×C₂₃	—	4	2	no	—
69	C₂×Q₈×C₂₃	C₄²	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	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	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₂₃	#41	1	4	yes	—
90	C₄⋊C₄×C₂₃	—	4	2	no	—
91	C₄⋊C₄×C₂₃	—	4	4	no	—
92	C₄⋊C₄×C₂₃	C₄²	1	8	yes	—
93	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	4	2	no	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
100	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₂₃	#41	1	2	yes	—
104	#20518	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	4	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
127	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₂₃	—	2	4	no	—
129	C₈×C₂₃	—	2	4	no	—
130	C₈×C₂₃	—	2	4	no	—
131	C₈×C₂₃	C₄×Q₈	1	8	yes	—
132	C₂×#446	C₂×C₂₃	1	2	yes	—
133	#2295	C₂×C₂₃	1	1	yes	characteristic
134	#1771	C₂×C₂₃	1	2	yes	—
135	#1771	C₂×C₂₃	1	2	yes	—
136	#1770	C₂×C₂₃	1	2	yes	—
137	#1768	C₂×C₂₃	1	2	yes	—
138	#1767	C₂×C₂₃	1	1	yes	characteristic
139	#1726	C₂×C₂₃	1	2	yes	—
140	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
141	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
142	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
144	C₂²×C₂₃	#568	1	2	yes	—
145	C₂²×C₂₃	#568	1	2	yes	—
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	2	4	no	—
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	2	4	no	—
150	C₄×C₂₃	—	2	4	no	—
151	C₄×C₂₃	—	4	2	no	—
152	C₄×C₂₃	—	4	4	no	—
153	C₄×C₂₃	C₂×C₄×Q₈	1	2	yes	—
154	C₄×C₂₃	C₄×#41	1	2	yes	—
155	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
156	#446	C₄×C₂₃	1	16	yes	complemented
157	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
160	C₂×#97	C₂²×C₂₃	1	2	yes	—
161	C₂×#97	C₂²×C₂₃	1	2	yes	—
162	C₂×#97	C₂²×C₂₃	1	2	yes	—
163	C₂×#97	C₂²×C₂₃	1	2	yes	—
164	C₂×#97	C₂²×C₂₃	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	#391	C₄×C₂₃	1	2	yes	—
170	#389	C₄×C₂₃	1	2	yes	—
171	#387	C₄×C₂₃	1	4	yes	—
172	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
173	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
174	#376	C₂²×C₂₃	1	2	yes	—
175	C₄³	C₂²×C₂₃	1	1	yes	characteristic
176	#336	C₂²×C₂₃	1	1	yes	characteristic
177	#336	C₂²×C₂₃	1	1	yes	characteristic
178	#336	C₂²×C₂₃	1	2	yes	—
179	#336	C₂²×C₂₃	1	2	yes	—
180	C₂×C₂₃	C₄²×Q₈	1	1	yes	characteristic
181	C₂×C₂₃	#2873	1	1	yes	characteristic
182	C₂×C₂₃	#2875	1	1	yes	characteristic
183	C₂×C₂₃	#2961	1	2	yes	—
184	C₂×C₂₃	#2967	1	2	yes	—
185	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
190	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
203	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
204	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
205	#97	—	2	4	no	—
206	#97	—	2	4	no	—
207	#97	—	2	4	no	—
208	#97	C₂×C₄×C₂₃	1	8	yes	—
209	#97	C₂×C₄×C₂₃	1	8	yes	—
210	C₄×C₈	—	2	4	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
213	C₄×C₈	Q₈×C₂₃	1	4	yes	complemented
214	#95	C₂×C₄×C₂₃	1	4	yes	—
215	#95	C₂×C₄×C₂₃	1	4	yes	—
216	#95	C₂×C₄×C₂₃	1	2	yes	—
217	#95	C₂×C₄×C₂₃	1	4	yes	—
218	C₂₃	#20518	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	C₄²×C₂₃	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
237	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
238	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
239	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
240	C₂×C₈	#41×C₂₃	1	4	yes	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	C₄²×C₂₃	1	8	yes	—
244	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
251	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
252	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
253	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
254	C₄²	#41×C₂₃	1	2	yes	—
255	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
270	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
272	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
274	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
275	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
277	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	C₄×Q₈×C₂₃	1	8	yes	—
282	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
283	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
284	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
285	C₂²	C₂₃×#568	1	2	yes	—
286	C₂²	C₂₃×#568	1	2	yes	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₄×Q₈×C₂₃	1	2	yes	—
295	C₄	C₄×#41×C₂₃	1	2	yes	—
296	C₂	C₄²×Q₈×C₂₃	1	1	yes	characteristic
297	C₂	C₂₃×#2873	1	1	yes	characteristic
298	C₂	C₂₃×#2875	1	1	yes	characteristic
299	C₂	C₂₃×#2961	1	2	yes	—
300	C₂	C₂₃×#2967	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×#446	C₄	semidirect
63. C₄×C₈×C₂₃	Q₈	semidirect
104. #20518	C₂₃	direct (N × H)
156. #446	C₄×C₂₃	semidirect
213. C₄×C₈	Q₈×C₂₃	semidirect
218. C₂₃	#20518	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#446	2
C₂₃×#2295	2
C₂₃×#1771	2
C₂₃×#1770	2
C₂₃×#1768	2
C₂₃×#1767	2
C₂₃×#1726	2
C₄×C₂₃×#97	2
#20518	23

Subgroups — C23×#20518 #19168144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20518 #19168144