Subgroups — C₂₃×#23669 #19171295

Order 5888.

874 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2316	C₂	1	2	yes	maximal
4	C₂₃×#2088	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2087	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2086	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1736	C₂	1	2	yes	maximal
8	C₂×C₂₃×#340	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#117	C₂²	1	2	yes	—
12	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#427	—	2	4	no	—
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#106	C₂²	1	2	yes	—
18	C₂₃×#424	—	2	4	no	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂₃×#389	—	2	2	no	—
22	C₂₃×#389	—	2	2	no	—
23	C₂₃×#389	C₂²	1	4	yes	—
24	C₂₃×#389	C₂²	1	4	yes	—
25	C₂₃×#380	—	2	4	no	—
26	C₂₃×#340	C₂²	1	2	yes	—
27	C₂₃×#340	C₂²	1	2	yes	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂³×C₄×C₂₃	—	2	2	no	—
31	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	—	2	4	no	—
34	C₂²×C₈×C₂₃	—	2	4	no	—
35	C₂²×C₈×C₂₃	D₄	1	4	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#117	—	2	2	no	—
40	C₂₃×#117	—	2	2	no	—
41	C₂₃×#117	—	2	4	no	—
42	C₂₃×#117	—	2	4	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₂₃	Q₈	1	8	yes	—
50	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
55	C₂×C₄²×C₂₃	—	2	2	no	—
56	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
57	C₂₃×#107	—	2	2	no	—
58	C₂₃×#107	—	2	2	no	—
59	C₂₃×#106	—	2	4	no	—
60	C₂₃×C₄⋊C₈	—	2	4	no	—
61	C₂₃×C₄⋊C₈	—	2	4	no	—
62	C₂₃×#95	—	2	4	no	—
63	C₂₃×#95	—	2	4	no	—
64	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	—	2	4	no	—
66	C₂²×C₄×C₂₃	—	2	8	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	2	2	no	—
69	C₂²×C₄×C₂₃	—	8	2	no	—
70	C₂²×C₄×C₂₃	—	8	2	no	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₂₃	#41	1	2	yes	—
77	C₂²×C₄×C₂₃	#41	1	4	yes	—
78	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₂₃	—	4	4	no	—
80	C₂×C₈×C₂₃	—	2	8	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	4	no	—
85	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	8	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	8	4	no	—
96	C₄⋊C₄×C₂₃	—	8	4	no	—
97	C₄⋊C₄×C₂₃	C₂×Q₈	1	8	yes	—
98	C₂²⋊C₄×C₂₃	—	2	4	no	—
99	C₂²⋊C₄×C₂₃	—	4	4	no	—
100	C₄²×C₂₃	—	2	2	no	—
101	C₄²×C₂₃	—	2	2	no	—
102	C₄²×C₂₃	—	4	4	no	—
103	#23669	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	1	no	—
105	C₂³×C₂₃	—	2	2	no	—
106	C₂³×C₂₃	—	2	1	no	—
107	C₂³×C₂₃	#121	1	2	yes	—
108	C₂³×C₂₃	C₂²⋊Q₈	1	4	yes	—
109	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	8	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	8	8	no	—
125	C₂×C₄×C₂₃	#121	1	2	yes	—
126	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₂₃	#123	1	4	yes	—
128	C₂×C₄×C₂₃	#125	1	8	yes	—
129	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
130	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	—	4	2	no	—
137	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
138	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
139	#2316	C₂×C₂₃	1	2	yes	—
140	#2088	C₂×C₂₃	1	4	yes	complemented
141	#2087	C₂×C₂₃	1	2	yes	complemented
142	#2086	C₂×C₂₃	1	2	yes	complemented
143	#1736	C₂×C₂₃	1	2	yes	—
144	C₂×#340	C₂×C₂₃	1	1	yes	characteristic
145	C₂²×C₂₃	—	2	4	no	—
146	C₂²×C₂₃	—	2	4	no	—
147	C₂²×C₂₃	—	2	2	no	—
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	#398	1	4	yes	—
150	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
151	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	2	yes	—
152	C₂²×C₂₃	#576	1	1	yes	characteristic
153	C₂²×C₂₃	#577	1	2	yes	—
154	C₂²×C₂₃	#578	1	1	yes	characteristic
155	C₄×C₂₃	—	4	8	no	—
156	C₄×C₂₃	—	4	2	no	—
157	C₄×C₂₃	—	4	2	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	—	8	2	no	—
160	C₄×C₂₃	#398	1	4	yes	—
161	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
162	C₄×C₂₃	C₂×#125	1	2	yes	—
163	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
164	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#117	C₂²×C₂₃	1	2	yes	—
167	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
168	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
169	#427	—	2	4	no	—
170	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×#106	C₂²×C₂₃	1	2	yes	—
173	#424	—	2	4	no	—
174	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
176	#389	—	2	2	no	—
177	#389	—	2	2	no	—
178	#389	C₂²×C₂₃	1	4	yes	—
179	#389	C₂²×C₂₃	1	4	yes	—
180	#380	—	2	4	no	—
181	#340	C₂²×C₂₃	1	2	yes	—
182	#340	C₂²×C₂₃	1	4	yes	—
183	#340	C₂²×C₂₃	1	4	yes	—
184	#340	C₂²×C₂₃	1	2	yes	—
185	C₂×C₂₃	—	2	4	no	—
186	C₂×C₂₃	C₂×#398	1	1	yes	characteristic
187	C₂×C₂₃	#3060	1	1	yes	characteristic
188	C₂×C₂₃	#3061	1	1	yes	characteristic
189	C₂×C₂₃	#3081	1	2	yes	—
190	C₂×C₂₃	#3082	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	D₄×C₂₃	1	4	yes	—
197	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	4	no	—
203	#117	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	Q₈×C₂₃	1	8	yes	—
211	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#106	—	2	4	no	—
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	2	4	no	—
223	#95	—	2	4	no	—
224	#95	—	2	4	no	—
225	C₂₃	#23669	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	—	8	2	no	—
233	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
237	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
238	C₂²×C₄	#41×C₂₃	1	2	yes	—
239	C₂²×C₄	#41×C₂₃	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₄⋊C₄	—	8	4	no	—
259	C₄⋊C₄	C₂×Q₈×C₂₃	1	8	yes	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	4	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂₃×#121	1	2	yes	—
269	C₂³	C₂₃×C₂²⋊Q₈	1	4	yes	—
270	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	C₂₃×#121	1	2	yes	—
287	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
288	C₂×C₄	C₂₃×#123	1	4	yes	—
289	C₂×C₄	C₂₃×#125	1	8	yes	—
290	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
291	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
293	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
294	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₂₃×#398	1	4	yes	—
303	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
304	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	2	yes	—
305	C₂²	C₂₃×#576	1	1	yes	characteristic
306	C₂²	C₂₃×#577	1	2	yes	—
307	C₂²	C₂₃×#578	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	8	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₂₃×#398	1	4	yes	—
314	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
315	C₄	C₂×C₂₃×#125	1	2	yes	—
316	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
317	C₂	—	2	4	no	—
318	C₂	C₂×C₂₃×#398	1	1	yes	characteristic
319	C₂	C₂₃×#3060	1	1	yes	characteristic
320	C₂	C₂₃×#3061	1	1	yes	characteristic
321	C₂	C₂₃×#3081	1	2	yes	—
322	C₂	C₂₃×#3082	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#2088	C₂	semidirect
5. C₂₃×#2087	C₂	semidirect
6. C₂₃×#2086	C₂	semidirect
103. #23669	C₂₃	direct (N × H)
140. #2088	C₂×C₂₃	semidirect
141. #2087	C₂×C₂₃	semidirect
142. #2086	C₂×C₂₃	semidirect
225. C₂₃	#23669	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×#427	2
C₂×C₂₃×#424	2
C₂₃×#2316	2
C₂₃×#2088	2
C₂₃×#2087	2
C₂₃×#2086	2
C₂₃×#1736	2
C₂×C₂₃×#340	2
#23669	23

Subgroups — C23×#23669 #19171295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23669 #19171295