Subgroups — C₂₃×#17899 #19165525

Order 5888.

646 conjugacy classes of subgroups.

The subgroup lattice has 336 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, complemented
2	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1594	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1593	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1590	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1477	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1450	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×#436	C₂²	1	4	yes	—
19	C₂₃×#435	—	2	4	no	—
20	C₂₃×#432	—	2	2	no	—
21	C₂₃×#431	C₂²	1	1	yes	characteristic
22	C₂₃×#431	C₂²	1	1	yes	characteristic
23	C₂₃×#427	C₄	1	1	yes	characteristic
24	C₂₃×#423	—	2	2	no	—
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂₃×#416	C₄	1	2	yes	—
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×D₄⋊C₈	C₂²	1	4	yes	—
32	C₂₃×#322	C₂²	1	2	yes	—
33	C₈²×C₂₃	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	—	2	4	no	—
40	C₂²×C₈×C₂₃	—	2	2	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	2	no	—
58	C₂₃×C₄⋊C₈	—	2	1	no	—
59	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
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×#102	C₂×C₄	1	2	yes	—
64	C₂₃×#102	C₂×C₄	1	2	yes	—
65	C₂₃×C₂²⋊C₈	—	2	1	no	—
66	C₂₃×C₂²⋊C₈	—	2	4	no	—
67	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
68	C₂₃×#97	—	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	#41×C₂₃	—	2	4	no	—
76	#41×C₂₃	—	2	4	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	—	2	2	no	—
89	M₁₆×C₂₃	—	8	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	—	8	2	no	—
94	C₂×C₈×C₂₃	M₁₆	1	4	yes	complemented
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	#17899	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	—	2	2	no	—
121	D₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	—	8	2	no	—
131	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₂₃	#131	1	2	yes	—
136	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
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	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
146	#2989	C₂×C₂₃	1	2	yes	complemented
147	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
148	#1594	C₂×C₂₃	1	2	yes	complemented
149	#1593	C₂×C₂₃	1	2	yes	complemented
150	#1590	C₂×C₂₃	1	4	yes	complemented
151	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#1477	C₂×C₂₃	1	2	yes	complemented
153	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	4	2	no	—
157	C₂²×C₂₃	—	4	2	no	—
158	C₂²×C₂₃	—	4	2	no	—
159	C₂²×C₂₃	#436	1	2	yes	—
160	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
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	#436	1	2	yes	—
168	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
169	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₂₃	#577	1	2	yes	—
171	C₂×#135	C₄×C₂₃	1	1	yes	characteristic
172	C₄×#41	C₂²×C₂₃	1	2	yes	—
173	#446	C₂²×C₂₃	1	4	yes	—
174	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
175	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
176	C₄×SD₁₆	C₄×C₂₃	1	2	yes	—
177	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
178	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic
179	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
180	#436	C₂²×C₂₃	1	4	yes	—
181	#435	—	2	4	no	—
182	#432	—	2	2	no	—
183	#431	C₂²×C₂₃	1	1	yes	characteristic
184	#431	C₂²×C₂₃	1	1	yes	characteristic
185	#427	C₄×C₂₃	1	1	yes	characteristic
186	#423	—	2	2	no	—
187	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
188	#416	C₄×C₂₃	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
190	#334	C₂²×C₂₃	1	2	yes	—
191	#334	C₂²×C₂₃	1	2	yes	—
192	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
193	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
194	#322	C₂²×C₂₃	1	2	yes	—
195	C₈²	C₂²×C₂₃	1	2	yes	—
196	C₂×C₂₃	—	2	2	no	—
197	C₂×C₂₃	—	8	2	no	—
198	C₂×C₂₃	C₂×#436	1	1	yes	characteristic
199	C₂×C₂₃	#2952	1	1	yes	characteristic
200	C₂×C₂₃	#2957	1	1	yes	characteristic
201	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
202	#135	—	2	4	no	—
203	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
205	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₂×M₁₆	—	2	4	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
210	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
211	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
215	C₄×D₄	C₂³×C₂₃	1	2	yes	—
216	#117	—	2	2	no	—
217	#117	C₂×C₄×C₂₃	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
220	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
221	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
222	#106	C₂×C₄×C₂₃	1	2	yes	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
227	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
228	#103	C₂×C₄×C₂₃	1	2	yes	—
229	#103	C₂×C₄×C₂₃	1	2	yes	—
230	#102	C₂×C₄×C₂₃	1	2	yes	—
231	#102	C₂×C₄×C₂₃	1	2	yes	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	2	4	no	—
234	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
235	#97	—	2	4	no	—
236	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
237	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₂₃	1	2	yes	—
239	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₂₃	1	2	yes	—
242	C₂₃	#17899	1	1	yes	characteristic, direct product factor, Sylow
243	#41	—	2	4	no	—
244	#41	—	2	4	no	—
245	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	2	4	no	—
256	D₈	—	2	2	no	—
257	M₁₆	—	8	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	8	2	no	—
262	C₂×C₈	M₁₆×C₂₃	1	4	yes	complemented
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	C₂²×C₄×C₂₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
271	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₄²	#41×C₂₃	1	1	yes	characteristic
281	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	4	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	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	C₂×M₁₆×C₂₃	1	2	yes	—
302	C₂×C₄	C₂₃×#131	1	2	yes	—
303	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	4	no	—
311	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
312	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₂₃×#436	1	2	yes	—
319	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	C₂₃×#436	1	2	yes	—
327	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
328	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
329	C₄	C₂₃×#577	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂×C₂₃×#436	1	1	yes	characteristic
333	C₂	C₂₃×#2952	1	1	yes	characteristic
334	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 ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2989	C₂	semidirect
3. C₂₃×#1594	C₂	semidirect
4. C₂₃×#1593	C₂	semidirect
5. C₂₃×#1590	C₂	semidirect
6. C₂×C₂₃×#334	C₂	semidirect
7. C₂₃×#1477	C₂	semidirect
8. C₂₃×#1450	C₂	semidirect
94. C₂×C₈×C₂₃	M₁₆	semidirect
114. #17899	C₂₃	direct (N × H)
146. #2989	C₂×C₂₃	semidirect
148. #1594	C₂×C₂₃	semidirect
149. #1593	C₂×C₂₃	semidirect
150. #1590	C₂×C₂₃	semidirect
151. C₂×#334	C₂×C₂₃	semidirect
152. #1477	C₂×C₂₃	semidirect
153. #1450	C₂×C₂₃	semidirect
242. C₂₃	#17899	direct (N × H)
262. C₂×C₈	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×#2989	2
C₄×C₂₃×#135	2
C₂₃×#1594	2
C₂₃×#1593	2
C₂₃×#1590	2
C₂×C₂₃×#334	2
C₂₃×#1477	2
C₂₃×#1450	2
#17899	23

Subgroups — C23×#17899 #19165525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17899 #19165525