Subgroups — C₂₃×#18045 #19165671

Order 5888.

648 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2970	C₂	1	2	yes	maximal, complemented
2	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1593	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1585	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1580	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1488	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₄	1	1	yes	characteristic
10	C₄×#41×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#446	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₂₃	C₄	1	2	yes	—
15	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic
17	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#435	—	2	4	no	—
19	C₈×D₄×C₂₃	C₂²	1	4	yes	—
20	C₂₃×#432	—	2	2	no	—
21	C₂₃×#432	C₂²	1	2	yes	—
22	C₂₃×#431	C₂²	1	1	yes	characteristic
23	C₂₃×#427	C₄	1	1	yes	characteristic
24	C₂₃×#416	C₄	1	2	yes	—
25	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#97	—	2	2	no	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#335	C₂²	1	2	yes	—
29	C₂₃×#334	C₂²	1	2	yes	—
30	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
32	C₂₃×#322	C₂²	1	2	yes	—
33	C₂₃×#322	C₂²	1	2	yes	—
34	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
35	C₂₃×#135	—	2	4	no	—
36	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
40	C₂²×C₈×C₂₃	—	2	4	no	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
45	C₄×D₄×C₂₃	—	2	2	no	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₄×D₄×C₂₃	C₂³	1	2	yes	—
49	C₂₃×#117	—	2	2	no	—
50	C₂₃×#117	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
54	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₂₃×#106	C₂×C₄	1	2	yes	—
56	C₂₃×C₄⋊C₈	—	2	4	no	—
57	C₂₃×C₄⋊C₈	—	2	1	no	—
58	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
59	C₂₃×#103	C₂×C₄	1	2	yes	—
60	C₂₃×#103	C₂×C₄	1	2	yes	—
61	C₂₃×#102	C₂×C₄	1	2	yes	—
62	C₂₃×#102	C₂×C₄	1	2	yes	—
63	C₂₃×C₂²⋊C₈	—	2	2	no	—
64	C₂₃×C₂²⋊C₈	—	2	1	no	—
65	C₂₃×C₂²⋊C₈	—	2	4	no	—
66	C₂₃×#97	—	2	4	no	—
67	C₂₃×#97	—	2	2	no	—
68	C₂₃×#97	D₄	1	2	yes	—
69	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂³	1	2	yes	—
72	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₂₃	C₂³	1	2	yes	—
75	#41×C₂₃	—	2	4	no	—
76	#41×C₂₃	—	4	2	no	—
77	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₂₃	—	2	2	no	—
79	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	—	2	2	no	—
83	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
86	Q₁₆×C₂₃	—	2	2	no	—
87	SD₁₆×C₂₃	—	2	4	no	—
88	D₈×C₂₃	M₁₆	1	4	yes	complemented
89	M₁₆×C₂₃	—	4	2	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	4	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	—	4	8	no	—
95	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
104	C₂²⋊C₄×C₂₃	—	2	2	no	—
105	C₂²⋊C₄×C₂₃	—	2	2	no	—
106	C₂²⋊C₄×C₂₃	—	2	2	no	—
107	C₄²×C₂₃	—	2	2	no	—
108	C₄²×C₂₃	—	2	2	no	—
109	C₄²×C₂₃	—	2	2	no	—
110	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₂₃	#41	1	1	yes	characteristic
113	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
114	#18045	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₂₃	—	2	2	no	—
116	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₂₃	—	2	2	no	—
118	Q₈×C₂₃	—	2	2	no	—
119	D₄×C₂₃	—	2	2	no	—
120	D₄×C₂₃	—	4	4	no	—
121	D₄×C₂₃	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	—	4	4	no	—
131	C₂×C₄×C₂₃	—	8	2	no	—
132	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₂₃	—	2	1	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	4	no	—
142	C₈×C₂₃	—	4	2	no	—
143	C₈×C₂₃	—	4	4	no	—
144	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
145	#2970	C₂×C₂₃	1	2	yes	complemented
146	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
147	#1593	C₂×C₂₃	1	2	yes	complemented
148	#1585	C₂×C₂₃	1	4	yes	complemented
149	#1580	C₂×C₂₃	1	2	yes	complemented
150	#1568	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#1488	C₂×C₂₃	1	2	yes	complemented
152	#1451	C₂×C₂₃	1	1	yes	characteristic, complemented
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	—	2	4	no	—
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	2	1	no	—
157	C₂²×C₂₃	—	4	2	no	—
158	C₂²×C₂₃	—	4	2	no	—
159	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	4	2	no	—
165	C₄×C₂₃	—	4	2	no	—
166	C₄×C₂₃	—	4	2	no	—
167	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
168	C₄×C₂₃	#568	1	1	yes	characteristic
169	C₄×C₂₃	#575	1	2	yes	—
170	C₂×#135	C₄×C₂₃	1	1	yes	characteristic
171	C₄×#41	C₂²×C₂₃	1	2	yes	—
172	#446	C₂²×C₂₃	1	4	yes	—
173	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
174	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
175	C₄×SD₁₆	C₄×C₂₃	1	2	yes	—
176	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
177	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic
178	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
179	#435	—	2	4	no	—
180	C₈×D₄	C₂²×C₂₃	1	4	yes	—
181	#432	—	2	2	no	—
182	#432	C₂²×C₂₃	1	2	yes	—
183	#431	C₂²×C₂₃	1	1	yes	characteristic
184	#427	C₄×C₂₃	1	1	yes	characteristic
185	#416	C₄×C₂₃	1	2	yes	—
186	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×#97	—	2	2	no	—
188	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
189	#335	C₂²×C₂₃	1	2	yes	—
190	#334	C₂²×C₂₃	1	2	yes	—
191	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
192	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
193	#322	C₂²×C₂₃	1	2	yes	—
194	#322	C₂²×C₂₃	1	2	yes	—
195	C₂×C₂₃	—	4	1	no	—
196	C₂×C₂₃	—	4	4	no	—
197	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
198	C₂×C₂₃	#2943	1	1	yes	characteristic
199	C₂×C₂₃	#2957	1	1	yes	characteristic
200	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
201	#135	—	2	4	no	—
202	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
203	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
204	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
210	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
214	C₄×D₄	C₂³×C₂₃	1	2	yes	—
215	#117	—	2	2	no	—
216	#117	C₂×C₄×C₂₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
219	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
220	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
221	#106	C₂×C₄×C₂₃	1	2	yes	—
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
225	#103	C₂×C₄×C₂₃	1	2	yes	—
226	#103	C₂×C₄×C₂₃	1	2	yes	—
227	#102	C₂×C₄×C₂₃	1	2	yes	—
228	#102	C₂×C₄×C₂₃	1	2	yes	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	C₂²⋊C₈	—	2	4	no	—
232	#97	—	2	4	no	—
233	#97	—	2	2	no	—
234	#97	D₄×C₂₃	1	2	yes	—
235	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
236	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₂₃	1	2	yes	—
238	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₂₃	1	2	yes	—
241	C₂₃	#18045	1	1	yes	characteristic, direct product factor, Sylow
242	#41	—	2	4	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	SD₁₆	—	2	4	no	—
255	D₈	M₁₆×C₂₃	1	4	yes	complemented
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	8	no	—
262	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
270	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₄²	#41×C₂₃	1	1	yes	characteristic
280	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
281	C₂³	—	2	2	no	—
282	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	2	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	4	4	no	—
287	D₄	C₂×M₁₆×C₂₃	1	4	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
301	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	4	no	—
310	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
326	C₄	C₂₃×#568	1	1	yes	characteristic
327	C₄	C₂₃×#575	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	4	4	no	—
330	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
331	C₂	C₂₃×#2943	1	1	yes	characteristic
332	C₂	C₂₃×#2957	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	C₂×C₄×D₄
prime radical	C₂×C₂₃×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2970	C₂	semidirect
3. C₂₃×#1593	C₂	semidirect
4. C₂₃×#1585	C₂	semidirect
5. C₂₃×#1580	C₂	semidirect
6. C₂₃×#1568	C₂	semidirect
7. C₂₃×#1488	C₂	semidirect
8. C₂₃×#1451	C₂	semidirect
88. D₈×C₂₃	M₁₆	semidirect
114. #18045	C₂₃	direct (N × H)
145. #2970	C₂×C₂₃	semidirect
147. #1593	C₂×C₂₃	semidirect
148. #1585	C₂×C₂₃	semidirect
149. #1580	C₂×C₂₃	semidirect
150. #1568	C₂×C₂₃	semidirect
151. #1488	C₂×C₂₃	semidirect
152. #1451	C₂×C₂₃	semidirect
241. C₂₃	#18045	direct (N × H)
255. D₈	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2970	2
C₄×C₂₃×#135	2
C₂₃×#1593	2
C₂₃×#1585	2
C₂₃×#1580	2
C₂₃×#1568	2
C₂₃×#1488	2
C₂₃×#1451	2
#18045	23

Subgroups — C23×#18045 #19165671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18045 #19165671