Subgroups — C₂₃×#8013 #19155639

Order 5888.

762 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×#431	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1294	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₂₃	—	2	1	no	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#431	—	2	2	no	—
8	C₂₃×#431	C₄	1	4	yes	complemented
9	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	—	2	1	no	—
12	C₂₃×#383	—	2	1	no	—
13	C₂₃×#382	C₄	1	2	yes	complemented
14	C₂₃×#381	—	2	1	no	—
15	C₂₃×#380	C₂²	1	2	yes	—
16	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	complemented
17	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
18	C₂₃×#376	—	2	2	no	—
19	C₂₃×#376	—	2	1	no	—
20	C₂₃×#376	C₄	1	4	yes	complemented
21	C₂×C₂₃×#95	—	2	1	no	—
22	C₂×C₂₃×#95	C₂²	1	2	yes	—
23	C₂³×C₄×C₂₃	—	2	2	no	—
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄²×C₂₃	—	2	8	no	—
31	C₂×C₄²×C₂₃	—	2	4	no	—
32	C₂×C₄²×C₂₃	—	2	2	no	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
41	C₂₃×C₄⋊C₈	—	2	4	no	—
42	C₂₃×C₂²⋊C₈	—	2	2	no	—
43	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₂₃	—	2	4	no	—
45	C₂₃×#95	—	2	4	no	—
46	C₂₃×#95	—	2	2	no	—
47	C₂₃×#95	—	2	2	no	—
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	C₂×C₄	1	4	yes	—
52	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	8	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	4	4	no	—
60	C₂²×C₄×C₂₃	—	4	4	no	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₄⋊C₄×C₂₃	—	4	4	no	—
78	C₂²⋊C₄×C₂₃	—	4	4	no	—
79	C₄²×C₂₃	—	2	16	no	—
80	C₄²×C₂₃	—	4	4	no	—
81	C₄²×C₂₃	C₄²	1	2	yes	—
82	C₄²×C₂₃	C₄²	1	2	yes	—
83	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
89	#8013	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₂₃	#135	1	4	yes	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	4	16	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	#95	1	2	yes	—
117	C₂×C₄×C₂₃	#95	1	2	yes	—
118	C₂×C₄×C₂₃	#95	1	2	yes	—
119	C₂×C₄×C₂₃	#95	1	2	yes	—
120	C₂×C₄×C₂₃	#104	1	4	yes	—
121	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈×C₂₃	—	4	4	no	—
126	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
127	#2283	C₂×C₂₃	1	1	yes	characteristic
128	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
129	#1294	C₂×C₂₃	1	4	yes	—
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	2	8	no	—
135	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
136	C₂²×C₂₃	#410	1	2	yes	—
137	C₂²×C₂₃	#416	1	2	yes	—
138	C₂²×C₂₃	#416	1	2	yes	—
139	C₂²×C₂₃	C₂×#104	1	2	yes	—
140	C₂²×C₂₃	#427	1	2	yes	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	4	8	no	—
149	C₄×C₂₃	—	8	4	no	—
150	C₄×C₂₃	#339	1	4	yes	—
151	C₂²×C₄²	—	2	1	no	—
152	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
153	#431	—	2	2	no	—
154	#431	C₄×C₂₃	1	4	yes	complemented
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄×C₈	—	2	1	no	—
158	#383	—	2	1	no	—
159	#382	C₄×C₂₃	1	2	yes	complemented
160	#381	—	2	1	no	—
161	#380	C₂²×C₂₃	1	2	yes	—
162	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	complemented
163	C₄×C₂²⋊C₄	—	2	1	no	—
164	#376	—	2	2	no	—
165	#376	—	2	1	no	—
166	#376	C₄×C₂₃	1	4	yes	complemented
167	C₂×#95	—	2	1	no	—
168	C₂×#95	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	—	2	2	no	—
171	C₂×C₂₃	#1500	1	4	yes	—
172	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
173	C₂×C₂₃	#1735	1	1	yes	characteristic
174	C₂×C₂₃	#1740	1	1	yes	characteristic
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	—	2	8	no	—
183	C₂×C₄²	—	2	4	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₂₃	1	2	yes	—
191	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
193	C₄⋊C₈	—	2	4	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
196	C₄×C₈	—	2	4	no	—
197	#95	—	2	4	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	C₂×C₄×C₂₃	1	4	yes	—
204	C₂₃	#8013	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
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	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	2	16	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₄²×C₂₃	1	2	yes	—
235	C₄²	C₄²×C₂₃	1	2	yes	—
236	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
237	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
238	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
239	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
240	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
241	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂³	C₂₃×#135	1	4	yes	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	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	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₂₃×#95	1	2	yes	—
269	C₂×C₄	C₂₃×#95	1	2	yes	—
270	C₂×C₄	C₂₃×#95	1	2	yes	—
271	C₂×C₄	C₂₃×#95	1	2	yes	—
272	C₂×C₄	C₂₃×#104	1	4	yes	—
273	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
277	C₈	—	4	4	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	8	no	—
283	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
284	C₂²	C₂₃×#410	1	2	yes	—
285	C₂²	C₂₃×#416	1	2	yes	—
286	C₂²	C₂₃×#416	1	2	yes	—
287	C₂²	C₂×C₂₃×#104	1	2	yes	—
288	C₂²	C₂₃×#427	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	4	8	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂₃×#339	1	4	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₂₃×#1500	1	4	yes	—
302	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
303	C₂	C₂₃×#1735	1	1	yes	characteristic
304	C₂	C₂₃×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂₃×#431	C₄	semidirect
13. C₂₃×#382	C₄	semidirect
16. C₄×C₄⋊C₄×C₂₃	C₄	semidirect
20. C₂₃×#376	C₄	semidirect
89. #8013	C₂₃	direct (N × H)
154. #431	C₄×C₂₃	semidirect
159. #382	C₄×C₂₃	semidirect
162. C₄×C₄⋊C₄	C₄×C₂₃	semidirect
166. #376	C₄×C₂₃	semidirect
204. C₂₃	#8013	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×#431	2
C₂₃×#2283	2
C₂×C₂₃×#376	2
C₂₃×#1294	2
#8013	23

Subgroups — C23×#8013 #19155639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8013 #19155639