Subgroups — C₂₃×#18042 #19165668

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1593	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1587	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1579	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1487	C₂	1	2	yes	maximal
8	C₂₃×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#446	—	2	4	no	—
12	C₂₃×#446	C₂²	1	4	yes	—
13	C₈×Q₈×C₂₃	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
15	C₄×Q₁₆×C₂₃	C₄	1	2	yes	—
16	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
17	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
22	C₂₃×#423	—	2	2	no	—
23	C₂₃×#423	C₂²	1	2	yes	—
24	C₂×C₂₃×#103	C₄	1	2	yes	—
25	C₄×M₁₆×C₂₃	—	2	2	no	—
26	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#334	C₂²	1	2	yes	—
29	C₂₃×#334	C₂²	1	2	yes	—
30	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
32	C₂₃×#322	C₂²	1	2	yes	—
33	C₂₃×#322	C₂²	1	2	yes	—
34	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₂₃	—	2	4	no	—
36	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₂₃	—	2	2	no	—
40	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₂₃	—	2	2	no	—
44	C₄×Q₈×C₂₃	—	2	2	no	—
45	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₂₃	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	2	yes	—
54	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
56	C₂₃×C₄⋊C₈	—	2	4	no	—
57	C₂₃×C₄⋊C₈	—	2	4	no	—
58	C₂₃×C₄⋊C₈	—	2	2	no	—
59	C₂₃×C₄⋊C₈	—	2	1	no	—
60	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
61	C₂₃×#103	C₂×C₄	1	2	yes	—
62	C₂₃×#103	C₂×C₄	1	2	yes	—
63	C₂₃×#103	C₂×C₄	1	2	yes	—
64	C₂₃×#103	C₂×C₄	1	2	yes	—
65	C₂₃×C₂²⋊C₈	—	2	1	no	—
66	C₂₃×#97	—	2	4	no	—
67	C₂₃×#97	D₄	1	2	yes	—
68	C₄×C₈×C₂₃	—	2	4	no	—
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	C₂×Q₈×C₂₃	—	2	2	no	—
76	C₂×Q₈×C₂₃	—	2	4	no	—
77	C₂×Q₈×C₂₃	—	4	2	no	—
78	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
85	Q₁₆×C₂₃	—	2	4	no	—
86	Q₁₆×C₂₃	—	2	2	no	—
87	Q₁₆×C₂₃	M₁₆	1	4	yes	complemented
88	M₁₆×C₂₃	—	2	4	no	—
89	M₁₆×C₂₃	—	4	4	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	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	4	yes	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
105	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
106	C₄²×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	#18042	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₂₃	—	2	2	no	—
117	Q₈×C₂₃	—	2	2	no	—
118	Q₈×C₂₃	—	2	2	no	—
119	Q₈×C₂₃	—	4	4	no	—
120	Q₈×C₂₃	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₂₃	—	2	2	no	—
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	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	—	8	2	no	—
130	C₂×C₄×C₂₃	C₄×D₄	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	#137	1	4	yes	—
134	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
137	C₈×C₂₃	—	2	1	no	—
138	C₈×C₂₃	—	2	1	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	4	no	—
141	C₈×C₂₃	—	4	2	no	—
142	C₈×C₂₃	—	4	4	no	—
143	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
144	Q₈×M₁₆	C₂×C₂₃	1	2	yes	—
145	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
146	#1593	C₂×C₂₃	1	2	yes	complemented
147	#1587	C₂×C₂₃	1	4	yes	complemented
148	#1579	C₂×C₂₃	1	2	yes	complemented
149	#1566	C₂×C₂₃	1	1	yes	characteristic
150	#1487	C₂×C₂₃	1	2	yes	—
151	#1451	C₂×C₂₃	1	1	yes	characteristic
152	C₂²×C₂₃	—	2	1	no	—
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	#441	1	2	yes	—
155	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	2	4	no	—
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	4	2	no	—
161	C₄×C₂₃	—	4	2	no	—
162	C₄×C₂₃	—	4	2	no	—
163	C₄×C₂₃	#441	1	2	yes	—
164	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
165	C₄×C₂₃	#568	1	1	yes	characteristic
166	C₄×C₂₃	C₂×#137	1	2	yes	—
167	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
169	#446	—	2	4	no	—
170	#446	C₂²×C₂₃	1	4	yes	—
171	C₈×Q₈	C₂²×C₂₃	1	4	yes	—
172	C₄×Q₁₆	C₄×C₂₃	1	2	yes	—
173	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
174	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
175	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
176	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
177	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
178	#431	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
180	#423	—	2	2	no	—
181	#423	C₂²×C₂₃	1	2	yes	—
182	C₂×#103	C₄×C₂₃	1	2	yes	—
183	C₄×M₁₆	—	2	2	no	—
184	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
186	#334	C₂²×C₂₃	1	2	yes	—
187	#334	C₂²×C₂₃	1	2	yes	—
188	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
189	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
190	#322	C₂²×C₂₃	1	2	yes	—
191	#322	C₂²×C₂₃	1	2	yes	—
192	C₂×C₂₃	—	2	2	no	—
193	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
194	C₂×C₂₃	C₂×#441	1	1	yes	characteristic
195	C₂×C₂₃	#2957	1	1	yes	characteristic
196	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
199	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
203	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
208	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
209	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
210	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
215	#107	C₂×C₄×C₂₃	1	2	yes	—
216	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
217	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
223	#103	C₂×C₄×C₂₃	1	2	yes	—
224	#103	C₂×C₄×C₂₃	1	2	yes	—
225	#103	C₂×C₄×C₂₃	1	2	yes	—
226	#103	C₂×C₄×C₂₃	1	2	yes	—
227	C₂²⋊C₈	—	2	1	no	—
228	#97	—	2	4	no	—
229	#97	D₄×C₂₃	1	2	yes	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
232	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
233	C₄×C₈	C₂³×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	#18042	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
242	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
248	Q₁₆	—	2	4	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	M₁₆×C₂₃	1	4	yes	complemented
251	M₁₆	—	2	4	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
268	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₄²	#41×C₂₃	1	1	yes	characteristic
276	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
277	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	4	4	no	—
282	Q₈	C₂×M₁₆×C₂₃	1	4	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂₃×#137	1	4	yes	—
296	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	4	no	—
303	C₈	—	4	2	no	—
304	C₈	—	4	4	no	—
305	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂₃×#441	1	2	yes	—
309	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	4	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	C₂₃×#441	1	2	yes	—
318	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
319	C₄	C₂₃×#568	1	1	yes	characteristic
320	C₄	C₂×C₂₃×#137	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
323	C₂	C₂×C₂₃×#441	1	1	yes	characteristic
324	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	C₂³×C₂₃	C₄×D₄
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
3. C₂₃×#1593	C₂	semidirect
4. C₂₃×#1587	C₂	semidirect
5. C₂₃×#1579	C₂	semidirect
87. Q₁₆×C₂₃	M₁₆	semidirect
114. #18042	C₂₃	direct (N × H)
146. #1593	C₂×C₂₃	semidirect
147. #1587	C₂×C₂₃	semidirect
148. #1579	C₂×C₂₃	semidirect
237. C₂₃	#18042	direct (N × H)
250. Q₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₂₃	2
C₂×C₄×Q₁₆×C₂₃	2
C₂₃×#1593	2
C₂₃×#1587	2
C₂₃×#1579	2
C₂₃×#1566	2
C₂₃×#1487	2
C₂₃×#1451	2
#18042	23

Subgroups — C23×#18042 #19165668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18042 #19165668