Subgroups — C₂₃×#17824 #19165450

Order 5888.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2989	C₂	1	2	yes	maximal
2	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1574	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1573	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1572	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1570	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1569	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#446	—	2	2	no	—
12	C₂₃×#436	—	2	2	no	—
13	C₂₃×#435	—	2	2	no	—
14	C₂₃×#435	—	2	2	no	—
15	C₂₃×#433	C₄	1	4	yes	—
16	C₂₃×#432	C₂²	1	2	yes	—
17	C₂₃×#432	C₂²	1	2	yes	—
18	C₂₃×#431	C₂²	1	2	yes	—
19	C₂₃×#431	C₂²	1	2	yes	—
20	C₂₃×#431	C₂²	1	2	yes	—
21	C₂₃×#424	C₄	1	2	yes	—
22	C₂₃×#423	C₂²	1	2	yes	—
23	C₂₃×#423	C₂²	1	2	yes	—
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂₃×#408	C₄	1	2	yes	—
26	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#335	C₂²	1	2	yes	—
29	C₂₃×#335	C₂²	1	2	yes	—
30	C₂₃×#335	C₂²	1	2	yes	—
31	C₂₃×#334	C₂²	1	2	yes	—
32	C₂₃×#334	C₂²	1	1	yes	characteristic
33	C₂₃×#334	C₂²	1	1	yes	characteristic
34	C₂₃×#334	C₂²	1	1	yes	characteristic
35	C₂₃×#334	C₂²	1	1	yes	characteristic
36	C₂₃×#334	C₂²	1	2	yes	—
37	C₂₃×#334	C₂²	1	2	yes	—
38	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₂₃	—	2	2	no	—
40	C₂×M₁₆×C₂₃	—	2	2	no	—
41	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₂₃	—	2	2	no	—
43	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₂₃	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
47	C₄×D₄×C₂₃	D₄	1	2	yes	—
48	C₄×D₄×C₂₃	Q₈	1	1	yes	characteristic
49	C₄×D₄×C₂₃	Q₈	1	2	yes	—
50	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
51	C₂₃×#117	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
54	C₂₃×C₄⋊C₈	—	2	2	no	—
55	C₂₃×C₄⋊C₈	—	2	2	no	—
56	C₂₃×C₄⋊C₈	—	2	2	no	—
57	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
58	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
59	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
60	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
61	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
62	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
63	C₂₃×C₂²⋊C₈	—	2	2	no	—
64	C₂₃×C₂²⋊C₈	—	2	2	no	—
65	C₂₃×C₂²⋊C₈	—	2	2	no	—
66	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
67	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₂₃×#97	—	2	2	no	—
69	C₂₃×#97	—	2	2	no	—
70	C₂₃×#97	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₂₃	—	2	2	no	—
72	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₂₃	C₂³	1	2	yes	—
76	#41×C₂₃	—	4	2	no	—
77	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
86	M₁₆×C₂₃	—	4	2	no	—
87	M₁₆×C₂₃	—	8	2	no	—
88	M₁₆×C₂₃	—	8	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	4	2	no	—
94	C₂×C₈×C₂₃	—	8	2	no	—
95	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
103	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
105	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₂₃	C₂×D₄	1	2	yes	—
108	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
110	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
111	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
112	#17824	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
115	Q₈×C₂₃	—	4	1	no	—
116	D₄×C₂₃	—	4	1	no	—
117	D₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	4	1	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	—	2	4	no	—
137	C₈×C₂₃	—	4	2	no	—
138	C₈×C₂₃	—	4	2	no	—
139	C₈×C₂₃	—	4	2	no	—
140	C₈×C₂₃	—	4	2	no	—
141	#2989	C₂×C₂₃	1	2	yes	—
142	#2922	C₂×C₂₃	1	1	yes	characteristic
143	#1574	C₂×C₂₃	1	2	yes	complemented
144	#1573	C₂×C₂₃	1	2	yes	complemented
145	#1572	C₂×C₂₃	1	2	yes	complemented
146	#1570	C₂×C₂₃	1	2	yes	complemented
147	#1569	C₂×C₂₃	1	2	yes	complemented
148	#1568	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#1566	C₂×C₂₃	1	1	yes	characteristic, complemented
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	2	no	—
152	C₂²×C₂₃	—	2	2	no	—
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	2	2	no	—
156	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	2	2	no	—
159	C₄×C₂₃	—	2	2	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	2	no	—
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	#568	1	2	yes	—
165	C₄×C₂₃	#577	1	2	yes	—
166	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
167	#446	—	2	2	no	—
168	#436	—	2	2	no	—
169	#435	—	2	2	no	—
170	#435	—	2	2	no	—
171	#433	C₄×C₂₃	1	4	yes	—
172	#432	C₂²×C₂₃	1	2	yes	—
173	#432	C₂²×C₂₃	1	2	yes	—
174	#431	C₂²×C₂₃	1	2	yes	—
175	#431	C₂²×C₂₃	1	2	yes	—
176	#431	C₂²×C₂₃	1	2	yes	—
177	#424	C₄×C₂₃	1	2	yes	—
178	#423	C₂²×C₂₃	1	2	yes	—
179	#423	C₂²×C₂₃	1	2	yes	—
180	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
181	#408	C₄×C₂₃	1	2	yes	—
182	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
184	#335	C₂²×C₂₃	1	2	yes	—
185	#335	C₂²×C₂₃	1	2	yes	—
186	#335	C₂²×C₂₃	1	2	yes	—
187	#334	C₂²×C₂₃	1	2	yes	—
188	#334	C₂²×C₂₃	1	1	yes	characteristic
189	#334	C₂²×C₂₃	1	1	yes	characteristic
190	#334	C₂²×C₂₃	1	1	yes	characteristic
191	#334	C₂²×C₂₃	1	1	yes	characteristic
192	#334	C₂²×C₂₃	1	2	yes	—
193	#334	C₂²×C₂₃	1	2	yes	—
194	C₂×C₂₃	—	4	1	no	—
195	C₂×C₂₃	—	4	2	no	—
196	C₂×C₂₃	#2905	1	1	yes	characteristic
197	C₂×C₂₃	#2912	1	1	yes	characteristic
198	C₂×C₂₃	#2916	1	1	yes	characteristic
199	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
205	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
206	C₄×Q₈	Q₈×C₂₃	1	1	yes	characteristic
207	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
208	C₄×D₄	D₄×C₂₃	1	2	yes	—
209	C₄×D₄	Q₈×C₂₃	1	1	yes	characteristic
210	C₄×D₄	Q₈×C₂₃	1	2	yes	—
211	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
212	#117	C₂×C₄×C₂₃	1	2	yes	—
213	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
219	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
220	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
221	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
222	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
223	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
228	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
229	#97	—	2	2	no	—
230	#97	—	2	2	no	—
231	#97	C₂×C₄×C₂₃	1	2	yes	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
234	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₂₃	1	2	yes	—
237	C₂₃	#17824	1	1	yes	characteristic, direct product factor, Sylow
238	#41	—	4	2	no	—
239	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂×D₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	8	2	no	—
250	M₁₆	—	8	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	8	2	no	—
257	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
258	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
259	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
260	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
261	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
262	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
264	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
265	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
266	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
267	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
268	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
269	C₄²	C₂×D₄×C₂₃	1	2	yes	—
270	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
271	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
272	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
273	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
274	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
275	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	D₄	—	4	1	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	C₂₃×#568	1	2	yes	—
317	C₄	C₂₃×#577	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	—	4	1	no	—
320	C₂	C₂₃×#2905	1	1	yes	characteristic
321	C₂	C₂₃×#2912	1	1	yes	characteristic
322	C₂	C₂₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1574	C₂	semidirect
4. C₂₃×#1573	C₂	semidirect
5. C₂₃×#1572	C₂	semidirect
6. C₂₃×#1570	C₂	semidirect
7. C₂₃×#1569	C₂	semidirect
8. C₂₃×#1568	C₂	semidirect
9. C₂₃×#1566	C₂	semidirect
112. #17824	C₂₃	direct (N × H)
143. #1574	C₂×C₂₃	semidirect
144. #1573	C₂×C₂₃	semidirect
145. #1572	C₂×C₂₃	semidirect
146. #1570	C₂×C₂₃	semidirect
147. #1569	C₂×C₂₃	semidirect
148. #1568	C₂×C₂₃	semidirect
149. #1566	C₂×C₂₃	semidirect
237. C₂₃	#17824	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×#2989	2
C₂₃×#2922	2
C₂₃×#1574	2
C₂₃×#1573	2
C₂₃×#1572	2
C₂₃×#1570	2
C₂₃×#1569	2
C₂₃×#1568	2
C₂₃×#1566	2
#17824	23

Subgroups — C23×#17824 #19165450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17824 #19165450