Subgroups — C₂₃×#20544 #19168170

Order 5888.

906 conjugacy classes of subgroups.

The subgroup lattice has 294 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	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1771	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1768	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1752	C₂	1	2	yes	maximal
7	C₈×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#432	C₄	1	8	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	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×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	4	yes	—
18	C₂×C₂₃×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
20	C₂₃×#383	C₄	1	2	yes	—
21	C₂₃×#382	C₄	1	2	yes	—
22	C₂₃×#381	C₄	1	2	yes	—
23	C₂₃×#380	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#376	C₂²	1	2	yes	—
27	C₂₃×#336	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	4	yes	—
29	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	—	4	2	no	—
33	C₂²×C₈×C₂₃	—	4	2	no	—
34	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
51	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
52	C₂₃×#97	—	4	4	no	—
53	C₂₃×#97	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₂₃	—	4	4	no	—
55	C₂₃×#95	C₂×C₄	1	4	yes	—
56	C₂₃×#95	C₂×C₄	1	4	yes	—
57	C₂₃×#95	C₂×C₄	1	4	yes	—
58	C₂₃×#95	C₂×C₄	1	4	yes	—
59	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
65	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	4	4	no	—
80	C₂×C₈×C₂₃	—	4	4	no	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
84	C₄⋊C₄×C₂₃	—	4	4	no	—
85	C₂²⋊C₄×C₂₃	—	2	8	no	—
86	C₄²×C₂₃	—	2	4	no	—
87	C₄²×C₂₃	—	2	4	no	—
88	C₄²×C₂₃	C₄²	1	4	yes	—
89	C₄²×C₂₃	C₄²	1	4	yes	—
90	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
94	#20544	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
98	C₂³×C₂₃	#131	1	4	yes	—
99	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	8	no	—
110	C₂×C₄×C₂₃	—	2	8	no	—
111	C₂×C₄×C₂₃	—	2	8	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
121	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₂₃	—	2	4	no	—
123	C₈×C₂₃	—	2	4	no	—
124	C₈×C₂₃	—	2	8	no	—
125	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
126	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
127	#2283	C₂×C₂₃	1	1	yes	characteristic
128	#1771	C₂×C₂₃	1	4	yes	complemented
129	#1768	C₂×C₂₃	1	4	yes	complemented
130	#1752	C₂×C₂₃	1	2	yes	—
131	C₈×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	8	no	—
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	#405	1	4	yes	—
136	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
137	C₂²×C₂₃	#520	1	2	yes	—
138	C₂²×C₂₃	C₂×#131	1	2	yes	—
139	C₂²×C₂₃	#568	1	2	yes	—
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	4	8	no	—
146	C₄×C₂₃	#518	1	1	yes	characteristic
147	C₄×C₂₃	#519	1	1	yes	characteristic
148	C₄×C₂₃	#520	1	2	yes	—
149	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
152	#432	C₄×C₂₃	1	8	yes	complemented
153	C₂×C₄⋊C₈	C₄×C₂₃	1	8	yes	complemented
154	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
158	C₂×#97	C₂²×C₂₃	1	4	yes	—
159	C₂×#97	C₂²×C₂₃	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
161	#383	C₄×C₂₃	1	2	yes	—
162	#382	C₄×C₂₃	1	2	yes	—
163	#381	C₄×C₂₃	1	2	yes	—
164	#380	C₄×C₂₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
166	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
167	#376	C₂²×C₂₃	1	2	yes	—
168	#336	C₂²×C₂₃	1	4	yes	—
169	#336	C₂²×C₂₃	1	4	yes	—
170	C₂×C₂₃	—	2	4	no	—
171	C₂×C₂₃	#2274	1	1	yes	characteristic
172	C₂×C₂₃	C₂×#405	1	1	yes	characteristic
173	C₂×C₂₃	#2873	1	1	yes	characteristic
174	C₂×C₂₃	#2992	1	4	yes	—
175	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	—	4	2	no	—
180	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
182	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
183	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
187	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
191	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
192	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
194	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
195	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
196	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
197	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
198	#97	—	4	4	no	—
199	#97	C₂×C₄×C₂₃	1	8	yes	—
200	C₄×C₈	—	4	4	no	—
201	#95	C₂×C₄×C₂₃	1	4	yes	—
202	#95	C₂×C₄×C₂₃	1	4	yes	—
203	#95	C₂×C₄×C₂₃	1	4	yes	—
204	#95	C₂×C₄×C₂₃	1	4	yes	—
205	C₂₃	#20544	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
212	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
214	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
229	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
231	C₄⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	C₄²×C₂₃	1	4	yes	—
236	C₄²	C₄²×C₂₃	1	4	yes	—
237	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
238	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
239	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
240	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
244	C₂³	C₂₃×#131	1	4	yes	—
245	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
260	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
262	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
263	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
264	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
267	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	2	8	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₂₃×#405	1	4	yes	—
275	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
276	C₂²	C₂₃×#520	1	2	yes	—
277	C₂²	C₂×C₂₃×#131	1	2	yes	—
278	C₂²	C₂₃×#568	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	8	no	—
285	C₄	C₂₃×#518	1	1	yes	characteristic
286	C₄	C₂₃×#519	1	1	yes	characteristic
287	C₄	C₂₃×#520	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	C₂₃×#2274	1	1	yes	characteristic
290	C₂	C₂×C₂₃×#405	1	1	yes	characteristic
291	C₂	C₂₃×#2873	1	1	yes	characteristic
292	C₂	C₂₃×#2992	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×#1771	C₂	semidirect
5. C₂₃×#1768	C₂	semidirect
11. C₂₃×#432	C₄	semidirect
12. C₂×C₂₃×C₄⋊C₈	C₄	semidirect
94. #20544	C₂₃	direct (N × H)
128. #1771	C₂×C₂₃	semidirect
129. #1768	C₂×C₂₃	semidirect
152. #432	C₄×C₂₃	semidirect
153. C₂×C₄⋊C₈	C₄×C₂₃	semidirect
205. C₂₃	#20544	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
C₂²×C₂₃×C₄⋊C₈	2
C₂₃×#2283	2
C₂₃×#1771	2
C₂₃×#1768	2
C₂₃×#1752	2
C₈×C₂²⋊C₄×C₂₃	2
#20544	23

Subgroups — C23×#20544 #19168170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20544 #19168170