Subgroups — C₂₃×#60627 #19208253

Order 5888.

1998 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3142	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3139	C₂	1	3	yes	maximal, complemented
5	C₂₃×#3137	C₂	1	12	yes	maximal, complemented
6	C₂₃×#3135	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#491	C₂	1	3	yes	maximal, complemented
8	C₂₃×#2898	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
11	C₂×C₂₃×#131	C₂²	1	2	yes	—
12	C₂₃×#557	—	2	2	no	—
13	C₂₃×#554	—	2	6	no	—
14	D₄×Q₈×C₂₃	—	2	2	no	—
15	C₂₃×#547	—	2	6	no	—
16	C₂₃×#544	C₂²	1	6	yes	—
17	C₂₃×#536	C₂²	1	6	yes	—
18	C₂₃×#533	C₂²	1	3	yes	—
19	C₂₃×#532	C₂²	1	3	yes	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
21	C₂×C₂₃×#125	C₂²	1	3	yes	—
22	C₂₃×#520	C₂²	1	2	yes	—
23	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#491	C₂²	1	3	yes	—
25	C₂₃×#491	C₂²	1	12	yes	—
26	C₂₃×#491	C₂²	1	6	yes	—
27	C₂₃×#491	C₂²	1	3	yes	—
28	C₂₃×#490	C₂²	1	1	yes	characteristic
29	C₂₃×#490	C₂²	1	3	yes	—
30	C₂₃×#489	C₂²	1	12	yes	—
31	C₂₃×#488	C₂²	1	12	yes	—
32	C₂₃×#487	C₂²	1	12	yes	—
33	C₂₃×#419	C₂²	1	12	yes	—
34	C₂₃×#417	C₂²	1	12	yes	—
35	C₂₃×#416	C₂²	1	12	yes	—
36	C₂×C₂₃×#103	C₂²	1	12	yes	—
37	C₂₃×#405	C₂²	1	6	yes	—
38	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
39	C₂×C₂₃×#97	C₂²	1	3	yes	—
40	C₂₃×2₋¹⁺⁴	—	2	16	no	—
41	C₂×#41×C₂₃	—	2	6	no	—
42	C₂×#41×C₂₃	C₂³	1	6	yes	—
43	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₂₃	—	2	2	no	—
45	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
46	C₂₃×#131	—	2	8	no	—
47	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
49	C₂₃×C₄⋊Q₈	—	2	6	no	—
50	C₂₃×C₄⋊Q₈	—	2	6	no	—
51	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
52	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
53	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
54	C₂₃×#127	C₂³	1	1	yes	characteristic
55	C₂₃×#125	—	2	6	no	—
56	C₂₃×#125	—	2	6	no	—
57	C₂₃×#125	C₂³	1	6	yes	—
58	C₂₃×#125	C₂³	1	3	yes	—
59	C₂₃×#125	C₂³	1	1	yes	characteristic
60	C₂₃×#124	C₂³	1	6	yes	—
61	C₂₃×#123	—	2	12	no	—
62	C₂₃×C₂²⋊Q₈	—	2	6	no	—
63	C₂₃×C₂²⋊Q₈	—	2	12	no	—
64	C₂₃×C₂²⋊Q₈	—	2	12	no	—
65	C₂₃×#121	—	2	6	no	—
66	C₄×Q₈×C₂₃	—	2	2	no	—
67	C₄×Q₈×C₂₃	—	2	6	no	—
68	C₄×Q₈×C₂₃	D₄	1	2	yes	—
69	C₄×D₄×C₂₃	—	2	6	no	—
70	C₄×D₄×C₂₃	—	2	6	no	—
71	C₄×D₄×C₂₃	D₄	1	6	yes	—
72	C₂₃×#117	C₂³	1	6	yes	—
73	C₂₃×#117	C₂³	1	3	yes	—
74	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
76	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
77	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
78	C₂₃×#103	C₂³	1	12	yes	—
79	C₂₃×#103	C₂³	1	24	yes	—
80	C₂₃×#103	C₂³	1	12	yes	—
81	C₂₃×#102	C₂³	1	4	yes	—
82	C₂₃×#102	C₂³	1	12	yes	—
83	C₂₃×#97	C₂³	1	1	yes	characteristic
84	C₂₃×#97	C₂³	1	3	yes	—
85	C₂₃×#97	C₂³	1	6	yes	—
86	C₄×C₈×C₂₃	C₂³	1	6	yes	—
87	#41×C₂₃	—	2	24	no	—
88	#41×C₂₃	—	4	24	no	—
89	#41×C₂₃	#41	1	8	yes	—
90	C₂×Q₈×C₂₃	—	2	6	no	—
91	C₂×Q₈×C₂₃	—	2	6	no	—
92	C₂×Q₈×C₂₃	—	2	24	no	—
93	C₂×Q₈×C₂₃	—	4	8	no	—
94	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₂₃	C₂⁴	1	6	yes	—
96	C₂×D₄×C₂₃	—	2	6	no	—
97	C₂×D₄×C₂₃	—	2	2	no	—
98	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
99	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
100	C₂²×C₄×C₂₃	—	2	6	no	—
101	C₂²×C₄×C₂₃	—	2	6	no	—
102	C₂²×C₄×C₂₃	—	2	6	no	—
103	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
104	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
105	M₁₆×C₂₃	—	2	12	no	—
106	C₂×C₈×C₂₃	—	2	12	no	—
107	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
108	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
109	C₄⋊C₄×C₂₃	—	2	6	no	—
110	C₄⋊C₄×C₂₃	—	2	6	no	—
111	C₄⋊C₄×C₂₃	—	2	12	no	—
112	C₄⋊C₄×C₂₃	—	2	12	no	—
113	C₄⋊C₄×C₂₃	—	2	6	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	6	no	—
116	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
117	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
118	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
119	C₂²⋊C₄×C₂₃	—	2	6	no	—
120	C₂²⋊C₄×C₂₃	—	2	12	no	—
121	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
122	C₄²×C₂₃	—	2	6	no	—
123	C₄²×C₂₃	C₂×D₄	1	6	yes	—
124	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
125	C₄²×C₂₃	C₂⁴	1	3	yes	—
126	#60627	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₂₃	—	2	2	no	—
128	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
129	Q₈×C₂₃	—	2	12	no	—
130	Q₈×C₂₃	—	4	12	no	—
131	Q₈×C₂₃	—	8	6	no	—
132	Q₈×C₂₃	#124	1	4	yes	—
133	D₄×C₂₃	—	2	4	no	—
134	D₄×C₂₃	—	4	4	no	—
135	D₄×C₂₃	—	4	12	no	—
136	D₄×C₂₃	#124	1	12	yes	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	6	no	—
139	C₂×C₄×C₂₃	—	2	6	no	—
140	C₂×C₄×C₂₃	—	4	6	no	—
141	C₂×C₄×C₂₃	—	4	12	no	—
142	C₂×C₄×C₂₃	—	4	12	no	—
143	C₂×C₄×C₂₃	—	8	6	no	—
144	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
146	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
147	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
148	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
149	C₈×C₂₃	—	2	2	no	—
150	C₈×C₂₃	—	2	6	no	—
151	#3504	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#3499	C₂×C₂₃	1	1	yes	characteristic
153	#3142	C₂×C₂₃	1	3	yes	complemented
154	#3139	C₂×C₂₃	1	3	yes	complemented
155	#3137	C₂×C₂₃	1	12	yes	complemented
156	#3135	C₂×C₂₃	1	3	yes	complemented
157	C₂×#491	C₂×C₂₃	1	3	yes	complemented
158	#2898	C₂×C₂₃	1	4	yes	complemented
159	#2875	C₂×C₂₃	1	1	yes	characteristic, complemented
160	C₂²×C₂₃	—	2	6	no	—
161	C₂²×C₂₃	—	4	4	no	—
162	C₂²×C₂₃	—	4	2	no	—
163	C₂²×C₂₃	C₂×#124	1	6	yes	—
164	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
165	C₄×C₂₃	—	2	2	no	—
166	C₄×C₂₃	—	2	6	no	—
167	C₄×C₂₃	—	4	6	no	—
168	C₄×C₂₃	—	4	2	no	—
169	C₄×C₂₃	—	4	6	no	—
170	C₄×C₂₃	C₂×#124	1	6	yes	—
171	C₄×C₂₃	C₂²×#41	1	2	yes	—
172	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
173	C₂×#131	C₂²×C₂₃	1	2	yes	—
174	#557	—	2	2	no	—
175	#554	—	2	6	no	—
176	D₄×Q₈	—	2	2	no	—
177	#547	—	2	6	no	—
178	#544	C₂²×C₂₃	1	6	yes	—
179	#536	C₂²×C₂₃	1	6	yes	—
180	#533	C₂²×C₂₃	1	3	yes	—
181	#532	C₂²×C₂₃	1	3	yes	—
182	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
183	C₂×#125	C₂²×C₂₃	1	3	yes	—
184	#520	C₂²×C₂₃	1	2	yes	—
185	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
186	#491	C₂²×C₂₃	1	3	yes	—
187	#491	C₂²×C₂₃	1	12	yes	—
188	#491	C₂²×C₂₃	1	6	yes	—
189	#491	C₂²×C₂₃	1	3	yes	—
190	#490	C₂²×C₂₃	1	1	yes	characteristic
191	#490	C₂²×C₂₃	1	3	yes	—
192	#489	C₂²×C₂₃	1	12	yes	—
193	#488	C₂²×C₂₃	1	12	yes	—
194	#487	C₂²×C₂₃	1	12	yes	—
195	#419	C₂²×C₂₃	1	12	yes	—
196	#417	C₂²×C₂₃	1	12	yes	—
197	#416	C₂²×C₂₃	1	12	yes	—
198	C₂×#103	C₂²×C₂₃	1	12	yes	—
199	#405	C₂²×C₂₃	1	6	yes	—
200	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
201	C₂×#97	C₂²×C₂₃	1	3	yes	—
202	C₂×C₂₃	—	2	6	no	—
203	C₂×C₂₃	—	8	2	no	—
204	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
205	C₂×C₂₃	#3586	1	2	yes	—
206	2₋¹⁺⁴	—	2	16	no	—
207	C₂×#41	—	2	6	no	—
208	C₂×#41	C₂³×C₂₃	1	6	yes	—
209	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
210	C₂²×Q₈	—	2	2	no	—
211	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
212	#131	—	2	8	no	—
213	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
214	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
215	C₄⋊Q₈	—	2	6	no	—
216	C₄⋊Q₈	—	2	6	no	—
217	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
218	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
219	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
220	#127	C₂³×C₂₃	1	1	yes	characteristic
221	#125	—	2	6	no	—
222	#125	—	2	6	no	—
223	#125	C₂³×C₂₃	1	3	yes	—
224	#125	C₂³×C₂₃	1	6	yes	—
225	#125	C₂³×C₂₃	1	1	yes	characteristic
226	#124	C₂³×C₂₃	1	6	yes	—
227	#123	—	2	12	no	—
228	C₂²⋊Q₈	—	2	6	no	—
229	C₂²⋊Q₈	—	2	12	no	—
230	C₂²⋊Q₈	—	2	12	no	—
231	#121	—	2	6	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	6	no	—
234	C₄×Q₈	D₄×C₂₃	1	2	yes	—
235	C₄×D₄	—	2	6	no	—
236	C₄×D₄	—	2	6	no	—
237	C₄×D₄	D₄×C₂₃	1	6	yes	—
238	#117	C₂³×C₂₃	1	6	yes	—
239	#117	C₂³×C₂₃	1	3	yes	—
240	C₂×C₄⋊C₄	—	2	6	no	—
241	C₂×C₄⋊C₄	—	2	6	no	—
242	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
243	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
244	#103	C₂³×C₂₃	1	12	yes	—
245	#103	C₂³×C₂₃	1	24	yes	—
246	#103	C₂³×C₂₃	1	12	yes	—
247	#102	C₂³×C₂₃	1	4	yes	—
248	#102	C₂³×C₂₃	1	12	yes	—
249	#97	C₂³×C₂₃	1	1	yes	characteristic
250	#97	C₂³×C₂₃	1	3	yes	—
251	#97	C₂³×C₂₃	1	6	yes	—
252	C₄×C₈	C₂³×C₂₃	1	6	yes	—
253	C₂₃	#60627	1	1	yes	characteristic, direct product factor, Sylow
254	#41	—	2	24	no	—
255	#41	—	4	24	no	—
256	#41	#41×C₂₃	1	8	yes	—
257	C₂×Q₈	—	2	6	no	—
258	C₂×Q₈	—	2	6	no	—
259	C₂×Q₈	—	2	24	no	—
260	C₂×Q₈	—	4	8	no	—
261	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
262	C₂×Q₈	C₂⁴×C₂₃	1	6	yes	—
263	C₂×D₄	—	2	6	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
266	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
267	C₂²×C₄	—	2	6	no	—
268	C₂²×C₄	—	2	6	no	—
269	C₂²×C₄	—	2	6	no	—
270	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
271	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
272	M₁₆	—	2	12	no	—
273	C₂×C₈	—	2	12	no	—
274	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
275	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
276	C₄⋊C₄	—	2	6	no	—
277	C₄⋊C₄	—	2	6	no	—
278	C₄⋊C₄	—	2	12	no	—
279	C₄⋊C₄	—	2	12	no	—
280	C₄⋊C₄	—	2	6	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	6	no	—
283	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
284	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
285	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
286	C₂²⋊C₄	—	2	6	no	—
287	C₂²⋊C₄	—	2	12	no	—
288	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
289	C₄²	—	2	6	no	—
290	C₄²	C₂×D₄×C₂₃	1	6	yes	—
291	C₄²	C₂⁴×C₂₃	1	3	yes	—
292	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
293	C₂³	—	2	2	no	—
294	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
295	Q₈	—	2	12	no	—
296	Q₈	—	4	12	no	—
297	Q₈	—	8	6	no	—
298	Q₈	C₂₃×#124	1	4	yes	—
299	D₄	—	2	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	12	no	—
302	D₄	C₂₃×#124	1	12	yes	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	6	no	—
305	C₂×C₄	—	2	6	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	12	no	—
309	C₂×C₄	—	8	6	no	—
310	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
311	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
312	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
313	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
314	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	6	no	—
317	C₂²	—	2	6	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	4	no	—
320	C₂²	C₂×C₂₃×#124	1	6	yes	—
321	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	6	no	—
324	C₄	—	4	6	no	—
325	C₄	—	4	6	no	—
326	C₄	—	4	2	no	—
327	C₄	C₂×C₂₃×#124	1	6	yes	—
328	C₄	C₂²×#41×C₂₃	1	2	yes	—
329	C₂	—	2	6	no	—
330	C₂	—	8	2	no	—
331	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
332	C₂	C₂₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3504	C₂	semidirect
3. C₂₃×#3142	C₂	semidirect
4. C₂₃×#3139	C₂	semidirect
5. C₂₃×#3137	C₂	semidirect
6. C₂₃×#3135	C₂	semidirect
7. C₂×C₂₃×#491	C₂	semidirect
8. C₂₃×#2898	C₂	semidirect
9. C₂₃×#2875	C₂	semidirect
126. #60627	C₂₃	direct (N × H)
151. #3504	C₂×C₂₃	semidirect
153. #3142	C₂×C₂₃	semidirect
154. #3139	C₂×C₂₃	semidirect
155. #3137	C₂×C₂₃	semidirect
156. #3135	C₂×C₂₃	semidirect
157. C₂×#491	C₂×C₂₃	semidirect
158. #2898	C₂×C₂₃	semidirect
159. #2875	C₂×C₂₃	semidirect
253. C₂₃	#60627	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3504	2
C₂₃×#3499	2
C₂₃×#3142	2
C₂₃×#3139	2
C₂₃×#3137	2
C₂₃×#3135	2
C₂×C₂₃×#491	2
C₂₃×#2898	2
C₂₃×#2875	2
#60627	23

Subgroups — C23×#60627 #19208253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60627 #19208253