Subgroups — C₂₃×#20469 #19168095

Order 5888.

1002 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1768	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1749	C₂	1	2	yes	maximal
6	C₄×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal
7	C₂₃×#1746	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1726	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#432	C₄	1	8	yes	complemented
12	C₂₃×#431	C₄	1	8	yes	complemented
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	4	yes	—
19	C₂×C₂₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#376	C₄	1	2	yes	—
23	C₂₃×#376	C₄	1	2	yes	—
24	C₂₃×#376	C₂²	1	1	yes	characteristic
25	C₂₃×#376	C₂²	1	1	yes	characteristic
26	C₂₃×#376	C₂²	1	2	yes	—
27	C₂×C₂₃×#95	C₄	1	2	yes	—
28	C₂×C₂₃×#95	C₄	1	2	yes	—
29	C₂₃×#336	C₂²	1	4	yes	—
30	C₂₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₂₃	—	2	8	no	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
48	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
49	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
50	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
53	C₂₃×#97	—	2	8	no	—
54	C₂₃×#97	C₂×C₄	1	8	yes	complemented
55	C₄×C₈×C₂₃	—	2	8	no	—
56	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
57	C₂₃×#95	C₂×C₄	1	2	yes	—
58	C₂₃×#95	C₂×C₄	1	2	yes	—
59	C₂₃×#95	C₂×C₄	1	4	yes	—
60	C₂₃×#95	C₂×C₄	1	4	yes	—
61	C₂₃×#95	C₂×C₄	1	2	yes	—
62	C₂₃×#95	C₂×C₄	1	2	yes	—
63	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	2	4	no	—
65	C₂²×C₄×C₂₃	—	2	4	no	—
66	C₂²×C₄×C₂₃	—	2	8	no	—
67	C₂²×C₄×C₂₃	—	2	8	no	—
68	C₂²×C₄×C₂₃	—	4	2	no	—
69	C₂²×C₄×C₂₃	—	4	4	no	—
70	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
71	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	4	no	—
85	C₂×C₈×C₂₃	—	2	4	no	—
86	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
89	C₄²×C₂₃	—	2	8	no	—
90	C₄²×C₂₃	—	2	4	no	—
91	C₄²×C₂₃	C₄²	1	4	yes	—
92	C₄²×C₂₃	C₄²	1	4	yes	—
93	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₂₃	#41	1	8	yes	—
98	#20469	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	—	2	2	no	—
100	C₂³×C₂₃	—	2	2	no	—
101	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
102	C₂³×C₂₃	#131	1	4	yes	—
103	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	8	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	16	no	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₂₃	#117	1	8	yes	—
125	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
128	C₈×C₂₃	—	2	4	no	—
129	C₈×C₂₃	—	2	8	no	—
130	C₈×C₂₃	—	2	4	no	—
131	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
132	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
133	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
134	#1768	C₂×C₂₃	1	4	yes	complemented
135	#1749	C₂×C₂₃	1	2	yes	—
136	C₄×C₂²⋊C₈	C₂×C₂₃	1	2	yes	—
137	#1746	C₂×C₂₃	1	2	yes	complemented
138	#1726	C₂×C₂₃	1	2	yes	complemented
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	2	8	no	—
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	#405	1	4	yes	—
143	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₂₃	C₂×#117	1	2	yes	—
145	C₂²×C₂₃	C₂×#131	1	2	yes	—
146	C₂²×C₂₃	#568	1	2	yes	—
147	C₄×C₂₃	—	2	2	no	—
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	2	2	no	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	2	8	no	—
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	4	4	no	—
154	C₄×C₂₃	C₄×#41	1	2	yes	—
155	C₄×C₂₃	C₄×#41	1	2	yes	—
156	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
157	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
158	#432	C₄×C₂₃	1	8	yes	complemented
159	#431	C₄×C₂₃	1	8	yes	complemented
160	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
165	C₂×#97	C₂²×C₂₃	1	4	yes	—
166	C₂×#97	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
168	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
169	#376	C₄×C₂₃	1	2	yes	—
170	#376	C₄×C₂₃	1	2	yes	—
171	#376	C₂²×C₂₃	1	1	yes	characteristic
172	#376	C₂²×C₂₃	1	1	yes	characteristic
173	#376	C₂²×C₂₃	1	2	yes	—
174	C₂×#95	C₄×C₂₃	1	2	yes	—
175	C₂×#95	C₄×C₂₃	1	2	yes	—
176	#336	C₂²×C₂₃	1	4	yes	—
177	#336	C₂²×C₂₃	1	4	yes	—
178	C₂×C₂₃	—	2	4	no	—
179	C₂×C₂₃	C₄×#117	1	1	yes	characteristic
180	C₂×C₂₃	C₂×#405	1	1	yes	characteristic
181	C₂×C₂₃	#2873	1	1	yes	characteristic
182	C₂×C₂₃	#2924	1	4	yes	—
183	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
184	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
189	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
190	C₂×C₄²	—	2	8	no	—
191	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
192	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
196	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
200	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
201	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
202	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
203	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
204	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
205	#97	—	2	8	no	—
206	#97	C₂×C₄×C₂₃	1	8	yes	complemented
207	C₄×C₈	—	2	8	no	—
208	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
209	#95	C₂×C₄×C₂₃	1	2	yes	—
210	#95	C₂×C₄×C₂₃	1	2	yes	—
211	#95	C₂×C₄×C₂₃	1	4	yes	—
212	#95	C₂×C₄×C₂₃	1	4	yes	—
213	#95	C₂×C₄×C₂₃	1	2	yes	—
214	#95	C₂×C₄×C₂₃	1	2	yes	—
215	C₂₃	#20469	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
224	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₄²×C₂₃	1	4	yes	—
245	C₄²	C₄²×C₂₃	1	4	yes	—
246	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
247	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
248	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
249	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
250	C₄²	#41×C₂₃	1	8	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
254	C₂³	C₂₃×#131	1	4	yes	—
255	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	16	no	—
271	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
274	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
276	C₂×C₄	C₂₃×#117	1	8	yes	—
277	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	8	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	C₂₃×#405	1	4	yes	—
287	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
288	C₂²	C₂×C₂₃×#117	1	2	yes	—
289	C₂²	C₂×C₂₃×#131	1	2	yes	—
290	C₂²	C₂₃×#568	1	2	yes	—
291	C₄	—	2	8	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	C₄×#41×C₂₃	1	2	yes	—
299	C₄	C₄×#41×C₂₃	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	C₄×C₂₃×#117	1	1	yes	characteristic
302	C₂	C₂×C₂₃×#405	1	1	yes	characteristic
303	C₂	C₂₃×#2873	1	1	yes	characteristic
304	C₂	C₂₃×#2924	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×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
4. C₂₃×#1768	C₂	semidirect
7. C₂₃×#1746	C₂	semidirect
8. C₂₃×#1726	C₂	semidirect
11. C₂₃×#432	C₄	semidirect
12. C₂₃×#431	C₄	semidirect
48. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
49. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
54. C₂₃×#97	C₂×C₄	semidirect
56. C₄×C₈×C₂₃	C₂×C₄	semidirect
98. #20469	C₂₃	direct (N × H)
134. #1768	C₂×C₂₃	semidirect
137. #1746	C₂×C₂₃	semidirect
138. #1726	C₂×C₂₃	semidirect
158. #432	C₄×C₂₃	semidirect
159. #431	C₄×C₂₃	semidirect
200. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
201. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
206. #97	C₂×C₄×C₂₃	semidirect
208. C₄×C₈	C₂×C₄×C₂₃	semidirect
215. C₂₃	#20469	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×#432	2
C₂×C₂₃×#431	2
C₂×C₂₃×#376	2
C₂₃×#1768	2
C₂₃×#1749	2
C₄×C₂₃×C₂²⋊C₈	2
C₂₃×#1746	2
C₂₃×#1726	2
#20469	23

Subgroups — C23×#20469 #19168095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20469 #19168095