Subgroups — C₂₃×#10862 #19158488

Order 5888.

982 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#107	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#340	C₂	1	2	yes	maximal
4	C₂₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#107	—	2	4	no	—
10	C₂×C₂₃×#107	—	2	4	no	—
11	C₂×C₂₃×#107	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₂₃	—	2	4	no	—
13	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
14	C₂×C₂₃×#95	—	2	4	no	—
15	C₂₃×#340	—	2	4	no	—
16	C₂³×C₄×C₂₃	—	2	4	no	—
17	C₂³×C₄×C₂₃	D₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂²×C₈×C₂₃	D₄	1	4	yes	—
26	C₂²×C₈×C₂₃	D₄	1	2	yes	—
27	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
39	C₂₃×#107	—	2	16	no	—
40	C₄×C₈×C₂₃	—	2	8	no	—
41	C₂₃×#95	—	4	8	no	—
42	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₃	—	2	8	no	—
46	C₂²×C₄×C₂₃	—	4	4	no	—
47	C₂²×C₄×C₂₃	—	4	4	no	—
48	C₂²×C₄×C₂₃	—	4	8	no	—
49	C₂²×C₄×C₂₃	—	4	8	no	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₂₃	D₈	1	4	yes	—
52	C₂²×C₄×C₂₃	Q₁₆	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	#41	1	2	yes	—
62	C₂²×C₄×C₂₃	#41	1	2	yes	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	2	4	no	—
67	C₂×C₈×C₂₃	C₄²	1	4	yes	—
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	8	yes	—
69	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
73	C₄⋊C₄×C₂₃	—	2	8	no	—
74	C₄⋊C₄×C₂₃	—	2	8	no	—
75	C₄⋊C₄×C₂₃	—	4	4	no	—
76	C₄⋊C₄×C₂₃	—	4	4	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	2	no	—
79	C₄²×C₂₃	—	2	4	no	—
80	#10862	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₂₃	C₄×D₄	1	2	yes	—
83	C₂³×C₂₃	#121	1	2	yes	—
84	C₂³×C₂₃	#127	1	1	yes	characteristic
85	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂³×C₂₃	C₂×D₈	1	2	yes	—
87	C₂³×C₂₃	C₂×Q₁₆	1	2	yes	—
88	C₂³×C₂₃	#135	1	4	yes	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	4	8	no	—
94	C₂×C₄×C₂₃	—	4	16	no	—
95	C₂×C₄×C₂₃	—	8	8	no	—
96	C₂×C₄×C₂₃	—	8	4	no	—
97	C₂×C₄×C₂₃	—	8	4	no	—
98	C₂×C₄×C₂₃	#107	1	8	yes	complemented
99	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₂₃	#121	1	2	yes	—
111	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	#124	1	2	yes	—
114	C₂×C₄×C₂₃	#125	1	2	yes	—
115	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₈×C₂₃	—	2	4	no	—
118	C₈×C₂₃	#95	1	8	yes	complemented
119	C₂²×#107	C₂×C₂₃	1	2	yes	—
120	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
121	C₂×#340	C₂×C₂₃	1	2	yes	—
122	C₄⋊#95	C₂×C₂₃	1	2	yes	—
123	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₂₃	#380	1	1	yes	characteristic
125	C₂²×C₂₃	#384	1	1	yes	characteristic
126	C₂²×C₂₃	#389	1	2	yes	—
127	C₂²×C₂₃	#390	1	2	yes	—
128	C₂²×C₂₃	C₂×#107	1	2	yes	—
129	C₂²×C₂₃	#427	1	2	yes	—
130	C₂²×C₂₃	C₄×D₈	1	4	yes	—
131	C₂²×C₂₃	C₄×Q₁₆	1	4	yes	—
132	C₂²×C₂₃	#466	1	4	yes	—
133	C₂²×C₂₃	#467	1	4	yes	—
134	C₂²×C₂₃	#493	1	1	yes	characteristic
135	C₂²×C₂₃	#494	1	1	yes	characteristic
136	C₂²×C₂₃	#495	1	2	yes	—
137	C₂²×C₂₃	#499	1	2	yes	—
138	C₂²×C₂₃	#500	1	2	yes	—
139	C₄×C₂₃	—	2	8	no	—
140	C₄×C₂₃	—	8	8	no	—
141	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
142	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₂₃	#380	1	1	yes	characteristic
144	C₄×C₂₃	#382	1	1	yes	characteristic
145	C₄×C₂₃	#389	1	2	yes	—
146	C₄×C₂₃	#391	1	2	yes	—
147	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
150	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#107	—	2	4	no	—
152	C₂×#107	—	2	4	no	—
153	C₂×#107	C₄×C₂₃	1	8	yes	complemented
154	C₂×C₄×C₈	—	2	4	no	—
155	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
156	C₂×#95	—	2	4	no	—
157	#340	—	2	4	no	—
158	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
159	C₂×C₂₃	C₄×#107	1	2	yes	—
160	C₂×C₂₃	#1808	1	2	yes	—
161	C₂×C₂₃	#1845	1	1	yes	characteristic
162	C₂×C₂₃	#1846	1	1	yes	characteristic
163	C₂×C₂₃	#1942	1	4	yes	—
164	C₂×C₂₃	#1969	1	2	yes	—
165	C₂×C₂₃	#1970	1	2	yes	—
166	C₂³×C₄	—	2	4	no	—
167	C₂³×C₄	D₄×C₂₃	1	2	yes	—
168	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
172	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
174	C₂²×C₈	D₄×C₂₃	1	2	yes	—
175	C₂²×C₈	D₄×C₂₃	1	4	yes	—
176	C₂²×C₈	D₄×C₂₃	1	2	yes	—
177	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
178	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
188	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
189	#107	—	2	16	no	—
190	C₄×C₈	—	2	8	no	—
191	#95	—	4	8	no	—
192	C₂₃	#10862	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	4	8	no	—
201	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
202	C₂²×C₄	D₈×C₂₃	1	4	yes	—
203	C₂²×C₄	Q₁₆×C₂₃	1	4	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	#41×C₂₃	1	2	yes	—
213	C₂²×C₄	#41×C₂₃	1	2	yes	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₄²×C₂₃	1	4	yes	—
219	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
220	C₂×C₈	C₂²⋊C₄×C₂₃	1	8	yes	—
221	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
222	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
223	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	4	no	—
231	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂³	C₄×D₄×C₂₃	1	2	yes	—
233	C₂³	C₂₃×#121	1	2	yes	—
234	C₂³	C₂₃×#127	1	1	yes	characteristic
235	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
236	C₂³	C₂×D₈×C₂₃	1	2	yes	—
237	C₂³	C₂×Q₁₆×C₂₃	1	2	yes	—
238	C₂³	C₂₃×#135	1	4	yes	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	4	16	no	—
245	C₂×C₄	—	8	8	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₂₃×#107	1	8	yes	complemented
249	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
258	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
259	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
260	C₂×C₄	C₂₃×#121	1	2	yes	—
261	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
262	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
263	C₂×C₄	C₂₃×#124	1	2	yes	—
264	C₂×C₄	C₂₃×#125	1	2	yes	—
265	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
266	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	C₂₃×#95	1	8	yes	complemented
269	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
270	C₂²	C₂₃×#380	1	1	yes	characteristic
271	C₂²	C₂₃×#384	1	1	yes	characteristic
272	C₂²	C₂₃×#389	1	2	yes	—
273	C₂²	C₂₃×#390	1	2	yes	—
274	C₂²	C₂×C₂₃×#107	1	2	yes	—
275	C₂²	C₂₃×#427	1	2	yes	—
276	C₂²	C₄×D₈×C₂₃	1	4	yes	—
277	C₂²	C₄×Q₁₆×C₂₃	1	4	yes	—
278	C₂²	C₂₃×#466	1	4	yes	—
279	C₂²	C₂₃×#467	1	4	yes	—
280	C₂²	C₂₃×#493	1	1	yes	characteristic
281	C₂²	C₂₃×#494	1	1	yes	characteristic
282	C₂²	C₂₃×#495	1	2	yes	—
283	C₂²	C₂₃×#499	1	2	yes	—
284	C₂²	C₂₃×#500	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	8	8	no	—
287	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
288	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
289	C₄	C₂₃×#380	1	1	yes	characteristic
290	C₄	C₂₃×#382	1	1	yes	characteristic
291	C₄	C₂₃×#389	1	2	yes	—
292	C₄	C₂₃×#391	1	2	yes	—
293	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
294	C₂	C₄×C₂₃×#107	1	2	yes	—
295	C₂	C₂₃×#1808	1	2	yes	—
296	C₂	C₂₃×#1845	1	1	yes	characteristic
297	C₂	C₂₃×#1846	1	1	yes	characteristic
298	C₂	C₂₃×#1942	1	4	yes	—
299	C₂	C₂₃×#1969	1	2	yes	—
300	C₂	C₂₃×#1970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂×C₂₃×#107	C₄	semidirect
13. C₂×C₄×C₈×C₂₃	C₄	semidirect
80. #10862	C₂₃	direct (N × H)
98. C₂×C₄×C₂₃	#107	semidirect
118. C₈×C₂₃	#95	semidirect
153. C₂×#107	C₄×C₂₃	semidirect
155. C₂×C₄×C₈	C₄×C₂₃	semidirect
192. C₂₃	#10862	direct (N × H)
248. C₂×C₄	C₂₃×#107	semidirect
268. C₈	C₂₃×#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#107	2
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#340	2
C₂₃×C₄⋊#95	2
#10862	23

Subgroups — C23×#10862 #19158488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10862 #19158488