Subgroups — C₂₃×#60625 #19208251

Order 5888.

2066 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3500	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3141	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3140	C₂	1	3	yes	maximal, complemented
5	C₂₃×#3137	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3136	C₂	1	6	yes	maximal, complemented
7	C₂₃×#3135	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×#489	C₂	1	3	yes	maximal, complemented
9	C₂₃×#2898	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2897	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#131	C₂²	1	2	yes	—
15	Q₈²×C₂₃	—	2	1	no	—
16	C₂₃×#557	—	2	1	no	—
17	C₂₃×#554	—	2	3	no	—
18	D₄×Q₈×C₂₃	—	2	3	no	—
19	C₂₃×#548	—	2	1	no	—
20	C₂₃×#548	—	2	1	no	—
21	C₂₃×#546	—	2	3	no	—
22	C₂₃×#546	—	2	3	no	—
23	C₂₃×#544	C₂²	1	6	yes	complemented
24	C₂₃×#536	C₂²	1	3	yes	—
25	C₂₃×#535	C₂²	1	3	yes	—
26	C₂₃×#533	C₂²	1	3	yes	complemented
27	C₂₃×#532	C₂²	1	3	yes	—
28	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
29	C₂×C₂₃×#124	C₂²	1	3	yes	—
30	C₂₃×#520	C₂²	1	2	yes	—
31	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#491	C₂²	1	6	yes	complemented
33	C₂₃×#490	C₂²	1	6	yes	complemented
34	C₂₃×#489	C₂²	1	3	yes	complemented
35	C₂₃×#489	C₂²	1	12	yes	complemented
36	C₂₃×#489	C₂²	1	6	yes	complemented
37	C₂₃×#489	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#489	C₂²	1	3	yes	complemented
39	C₂₃×#489	C₂²	1	3	yes	complemented
40	C₂₃×#488	C₂²	1	12	yes	complemented
41	C₂₃×#487	C₂²	1	6	yes	complemented
42	C₂₃×#486	C₂²	1	6	yes	complemented
43	C₂₃×#419	C₂²	1	6	yes	complemented
44	C₂₃×#418	C₂²	1	6	yes	—
45	C₂₃×#417	C₂²	1	6	yes	—
46	C₂₃×#417	C₂²	1	6	yes	complemented
47	C₂₃×#416	C₂²	1	6	yes	—
48	C₂₃×#416	C₂²	1	6	yes	complemented
49	C₂×C₂₃×#103	C₂²	1	6	yes	complemented
50	C₂×C₂₃×#102	C₂²	1	6	yes	—
51	C₂₃×#405	C₂²	1	6	yes	complemented
52	C₄×M₁₆×C₂₃	C₂²	1	3	yes	complemented
53	C₂×C₂₃×#97	C₂²	1	3	yes	complemented
54	C₂₃×2₋¹⁺⁴	—	2	8	no	—
55	C₂₃×H(5,𝔽₂)	—	2	8	no	—
56	C₂×#41×C₂₃	—	2	1	no	—
57	C₂×#41×C₂₃	—	2	3	no	—
58	C₂×#41×C₂₃	C₂³	1	3	yes	—
59	C₂×#41×C₂₃	C₂³	1	3	yes	—
60	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×Q₈×C₂₃	—	2	3	no	—
63	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
64	C₂²×D₄×C₂₃	—	2	3	no	—
65	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
66	C₂₃×#131	—	2	8	no	—
67	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
69	C₂₃×C₄⋊Q₈	—	2	6	no	—
70	C₂₃×C₄⋊Q₈	—	2	3	no	—
71	C₂₃×C₄⋊Q₈	—	2	3	no	—
72	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×#127	C₂³	1	3	yes	—
77	C₂₃×#125	—	2	6	no	—
78	C₂₃×#125	C₂³	1	6	yes	—
79	C₂₃×#124	—	2	3	no	—
80	C₂₃×#124	—	2	3	no	—
81	C₂₃×#124	C₂³	1	6	yes	—
82	C₂₃×#124	C₂³	1	3	yes	—
83	C₂₃×#124	C₂³	1	1	yes	characteristic
84	C₂₃×#123	—	2	6	no	—
85	C₂₃×C₂²⋊Q₈	—	2	3	no	—
86	C₂₃×C₂²⋊Q₈	—	2	3	no	—
87	C₂₃×C₂²⋊Q₈	—	2	3	no	—
88	C₂₃×C₂²⋊Q₈	—	2	3	no	—
89	C₂₃×C₂²⋊Q₈	—	2	12	no	—
90	C₂₃×#121	—	2	3	no	—
91	C₂₃×#121	—	2	3	no	—
92	C₂₃×#121	—	2	3	no	—
93	C₂₃×#121	—	2	3	no	—
94	C₂₃×#120	—	2	6	no	—
95	C₄×Q₈×C₂₃	—	2	2	no	—
96	C₄×Q₈×C₂₃	—	2	3	no	—
97	C₄×Q₈×C₂₃	—	2	3	no	—
98	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
100	C₄×D₄×C₂₃	—	2	6	no	—
101	C₄×D₄×C₂₃	—	2	3	no	—
102	C₄×D₄×C₂₃	—	2	3	no	—
103	C₄×D₄×C₂₃	D₄	1	3	yes	—
104	C₄×D₄×C₂₃	D₄	1	3	yes	—
105	C₂₃×#117	C₂³	1	6	yes	—
106	C₂₃×#117	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
111	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
112	C₂₃×#103	C₂³	1	6	yes	—
113	C₂₃×#103	C₂³	1	12	yes	—
114	C₂₃×#103	C₂³	1	2	yes	—
115	C₂₃×#103	C₂³	1	6	yes	—
116	C₂₃×#103	C₂³	1	6	yes	—
117	C₂₃×#102	C₂³	1	6	yes	—
118	C₂₃×#102	C₂³	1	12	yes	—
119	C₂₃×#102	C₂³	1	6	yes	—
120	C₂₃×#102	C₂³	1	2	yes	—
121	C₂₃×#102	C₂³	1	6	yes	—
122	C₂₃×#97	C₂³	1	1	yes	characteristic
123	C₂₃×#97	C₂³	1	3	yes	—
124	C₂₃×#97	C₂³	1	6	yes	—
125	C₄×C₈×C₂₃	C₂³	1	6	yes	—
126	C₂⁴×C₂₃	—	2	3	no	—
127	#41×C₂₃	—	2	12	no	—
128	#41×C₂₃	—	2	12	no	—
129	#41×C₂₃	—	4	4	no	—
130	#41×C₂₃	—	4	12	no	—
131	#41×C₂₃	#41	1	8	yes	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	3	no	—
134	C₂×Q₈×C₂₃	—	2	3	no	—
135	C₂×Q₈×C₂₃	—	2	6	no	—
136	C₂×Q₈×C₂₃	—	2	12	no	—
137	C₂×Q₈×C₂₃	—	4	4	no	—
138	C₂×Q₈×C₂₃	—	8	6	no	—
139	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
141	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
142	C₂×D₄×C₂₃	—	2	3	no	—
143	C₂×D₄×C₂₃	—	2	3	no	—
144	C₂×D₄×C₂₃	—	2	3	no	—
145	C₂×D₄×C₂₃	—	2	3	no	—
146	C₂×D₄×C₂₃	—	2	3	no	—
147	C₂×D₄×C₂₃	—	2	12	no	—
148	C₂×D₄×C₂₃	—	4	12	no	—
149	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
151	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	—	2	3	no	—
153	C₂²×C₄×C₂₃	—	2	6	no	—
154	C₂²×C₄×C₂₃	—	2	3	no	—
155	C₂²×C₄×C₂₃	—	2	3	no	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
159	M₁₆×C₂₃	—	2	12	no	—
160	C₂×C₈×C₂₃	—	2	12	no	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
163	C₄⋊C₄×C₂₃	—	2	3	no	—
164	C₄⋊C₄×C₂₃	—	2	6	no	—
165	C₄⋊C₄×C₂₃	—	2	6	no	—
166	C₄⋊C₄×C₂₃	—	2	3	no	—
167	C₄⋊C₄×C₂₃	—	2	3	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	6	no	—
170	C₄⋊C₄×C₂₃	—	2	6	no	—
171	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
172	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
174	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
175	C₂²⋊C₄×C₂₃	—	2	6	no	—
176	C₂²⋊C₄×C₂₃	—	2	3	no	—
177	C₂²⋊C₄×C₂₃	—	2	3	no	—
178	C₂²⋊C₄×C₂₃	—	2	6	no	—
179	C₂²⋊C₄×C₂₃	—	2	6	no	—
180	C₂²⋊C₄×C₂₃	—	2	1	no	—
181	C₂²⋊C₄×C₂₃	—	2	1	no	—
182	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
183	C₄²×C₂₃	—	2	6	no	—
184	C₄²×C₂₃	C₂×D₄	1	3	yes	—
185	C₄²×C₂₃	C₂×D₄	1	3	yes	—
186	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
187	C₄²×C₂₃	C₂⁴	1	3	yes	—
188	#60625	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₂₃	—	2	3	no	—
190	C₂³×C₂₃	—	2	1	no	—
191	C₂³×C₂₃	—	4	6	no	—
192	C₂³×C₂₃	—	8	3	no	—
193	C₂³×C₂₃	—	8	3	no	—
194	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
195	Q₈×C₂₃	—	2	6	no	—
196	Q₈×C₂₃	—	2	2	no	—
197	Q₈×C₂₃	—	4	2	no	—
198	Q₈×C₂₃	—	4	6	no	—
199	Q₈×C₂₃	—	4	6	no	—
200	Q₈×C₂₃	—	8	6	no	—
201	Q₈×C₂₃	#124	1	4	yes	—
202	D₄×C₂₃	—	2	6	no	—
203	D₄×C₂₃	—	2	2	no	—
204	D₄×C₂₃	—	4	6	no	—
205	D₄×C₂₃	—	4	6	no	—
206	D₄×C₂₃	—	4	6	no	—
207	D₄×C₂₃	—	8	3	no	—
208	D₄×C₂₃	—	8	3	no	—
209	D₄×C₂₃	#124	1	12	yes	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	6	no	—
212	C₂×C₄×C₂₃	—	2	3	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	4	3	no	—
215	C₂×C₄×C₂₃	—	4	3	no	—
216	C₂×C₄×C₂₃	—	4	6	no	—
217	C₂×C₄×C₂₃	—	4	6	no	—
218	C₂×C₄×C₂₃	—	4	6	no	—
219	C₂×C₄×C₂₃	—	8	6	no	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
226	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
227	C₈×C₂₃	—	2	2	no	—
228	C₈×C₂₃	—	2	6	no	—
229	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#3500	C₂×C₂₃	1	1	yes	characteristic
231	#3141	C₂×C₂₃	1	3	yes	complemented
232	#3140	C₂×C₂₃	1	3	yes	complemented
233	#3137	C₂×C₂₃	1	6	yes	complemented
234	#3136	C₂×C₂₃	1	6	yes	complemented
235	#3135	C₂×C₂₃	1	3	yes	complemented
236	C₂×#489	C₂×C₂₃	1	3	yes	complemented
237	#2898	C₂×C₂₃	1	2	yes	complemented
238	#2897	C₂×C₂₃	1	2	yes	complemented
239	#2875	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂²×C₂₃	—	2	3	no	—
241	C₂²×C₂₃	—	2	3	no	—
242	C₂²×C₂₃	—	4	1	no	—
243	C₂²×C₂₃	—	4	3	no	—
244	C₂²×C₂₃	—	4	1	no	—
245	C₂²×C₂₃	—	4	3	no	—
246	C₂²×C₂₃	—	4	1	no	—
247	C₂²×C₂₃	—	4	3	no	—
248	C₂²×C₂₃	—	16	6	no	—
249	C₂²×C₂₃	C₂×#124	1	6	yes	—
250	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
251	C₄×C₂₃	—	2	1	no	—
252	C₄×C₂₃	—	2	3	no	—
253	C₄×C₂₃	—	2	1	no	—
254	C₄×C₂₃	—	2	3	no	—
255	C₄×C₂₃	—	4	6	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	4	1	no	—
258	C₄×C₂₃	—	4	3	no	—
259	C₄×C₂₃	C₂×#124	1	6	yes	—
260	C₄×C₂₃	C₂²×#41	1	2	yes	—
261	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×#131	C₂²×C₂₃	1	2	yes	—
264	Q₈²	—	2	1	no	—
265	#557	—	2	1	no	—
266	#554	—	2	3	no	—
267	D₄×Q₈	—	2	3	no	—
268	#548	—	2	1	no	—
269	#548	—	2	1	no	—
270	#546	—	2	3	no	—
271	#546	—	2	3	no	—
272	#544	C₂²×C₂₃	1	6	yes	complemented
273	#536	C₂²×C₂₃	1	3	yes	—
274	#535	C₂²×C₂₃	1	3	yes	—
275	#533	C₂²×C₂₃	1	3	yes	complemented
276	#532	C₂²×C₂₃	1	3	yes	—
277	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	complemented
278	C₂×#124	C₂²×C₂₃	1	3	yes	—
279	#520	C₂²×C₂₃	1	2	yes	—
280	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
281	#491	C₂²×C₂₃	1	6	yes	complemented
282	#490	C₂²×C₂₃	1	6	yes	complemented
283	#489	C₂²×C₂₃	1	3	yes	complemented
284	#489	C₂²×C₂₃	1	12	yes	complemented
285	#489	C₂²×C₂₃	1	6	yes	complemented
286	#489	C₂²×C₂₃	1	1	yes	characteristic, complemented
287	#489	C₂²×C₂₃	1	3	yes	complemented
288	#489	C₂²×C₂₃	1	3	yes	complemented
289	#488	C₂²×C₂₃	1	12	yes	complemented
290	#487	C₂²×C₂₃	1	6	yes	complemented
291	#486	C₂²×C₂₃	1	6	yes	complemented
292	#419	C₂²×C₂₃	1	6	yes	complemented
293	#418	C₂²×C₂₃	1	6	yes	—
294	#417	C₂²×C₂₃	1	6	yes	—
295	#417	C₂²×C₂₃	1	6	yes	complemented
296	#416	C₂²×C₂₃	1	6	yes	—
297	#416	C₂²×C₂₃	1	6	yes	complemented
298	C₂×#103	C₂²×C₂₃	1	6	yes	complemented
299	C₂×#102	C₂²×C₂₃	1	6	yes	—
300	#405	C₂²×C₂₃	1	6	yes	complemented
301	C₄×M₁₆	C₂²×C₂₃	1	3	yes	complemented
302	C₂×#97	C₂²×C₂₃	1	3	yes	complemented
303	C₂×C₂₃	—	2	6	no	—
304	C₂×C₂₃	—	8	1	no	—
305	C₂×C₂₃	—	8	3	no	—
306	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
307	C₂×C₂₃	#3585	1	1	yes	characteristic
308	C₂×C₂₃	#3586	1	1	yes	characteristic
309	2₋¹⁺⁴	—	2	8	no	—
310	H(5,𝔽₂)	—	2	8	no	—
311	C₂×#41	—	2	1	no	—
312	C₂×#41	—	2	3	no	—
313	C₂×#41	C₂³×C₂₃	1	3	yes	—
314	C₂×#41	C₂³×C₂₃	1	3	yes	—
315	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
316	C₂²×Q₈	—	2	1	no	—
317	C₂²×Q₈	—	2	3	no	—
318	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
319	C₂²×D₄	—	2	3	no	—
320	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
321	#131	—	2	8	no	—
322	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
323	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
324	C₄⋊Q₈	—	2	6	no	—
325	C₄⋊Q₈	—	2	3	no	—
326	C₄⋊Q₈	—	2	3	no	—
327	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
328	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
329	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
330	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
331	#127	C₂³×C₂₃	1	3	yes	—
332	#125	—	2	6	no	—
333	#125	C₂³×C₂₃	1	6	yes	—
334	#124	—	2	3	no	—
335	#124	—	2	3	no	—
336	#124	C₂³×C₂₃	1	3	yes	—
337	#124	C₂³×C₂₃	1	6	yes	—
338	#124	C₂³×C₂₃	1	1	yes	characteristic
339	#123	—	2	6	no	—
340	C₂²⋊Q₈	—	2	3	no	—
341	C₂²⋊Q₈	—	2	3	no	—
342	C₂²⋊Q₈	—	2	3	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	—	2	12	no	—
345	#121	—	2	3	no	—
346	#121	—	2	3	no	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#120	—	2	6	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	3	no	—
352	C₄×Q₈	—	2	3	no	—
353	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
354	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
355	C₄×D₄	—	2	6	no	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	D₄×C₂₃	1	3	yes	—
359	C₄×D₄	D₄×C₂₃	1	3	yes	—
360	#117	C₂³×C₂₃	1	6	yes	—
361	#117	C₂³×C₂₃	1	3	yes	—
362	C₂×C₄⋊C₄	—	2	3	no	—
363	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
364	C₂×C₂²⋊C₄	—	2	3	no	—
365	C₂×C₂²⋊C₄	—	2	3	no	—
366	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
367	#103	C₂³×C₂₃	1	6	yes	—
368	#103	C₂³×C₂₃	1	12	yes	—
369	#103	C₂³×C₂₃	1	2	yes	—
370	#103	C₂³×C₂₃	1	6	yes	—
371	#103	C₂³×C₂₃	1	6	yes	—
372	#102	C₂³×C₂₃	1	6	yes	—
373	#102	C₂³×C₂₃	1	12	yes	—
374	#102	C₂³×C₂₃	1	6	yes	—
375	#102	C₂³×C₂₃	1	2	yes	—
376	#102	C₂³×C₂₃	1	6	yes	—
377	#97	C₂³×C₂₃	1	1	yes	characteristic
378	#97	C₂³×C₂₃	1	3	yes	—
379	#97	C₂³×C₂₃	1	6	yes	—
380	C₄×C₈	C₂³×C₂₃	1	6	yes	—
381	C₂₃	#60625	1	1	yes	characteristic, direct product factor, Sylow
382	C₂⁴	—	2	3	no	—
383	#41	—	2	12	no	—
384	#41	—	2	12	no	—
385	#41	—	4	4	no	—
386	#41	—	4	12	no	—
387	#41	#41×C₂₃	1	8	yes	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	2	3	no	—
391	C₂×Q₈	—	2	6	no	—
392	C₂×Q₈	—	2	12	no	—
393	C₂×Q₈	—	4	4	no	—
394	C₂×Q₈	—	8	6	no	—
395	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
397	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	3	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	—	2	3	no	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	12	no	—
404	C₂×D₄	—	4	12	no	—
405	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
406	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
407	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	2	3	no	—
412	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
413	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
414	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
415	M₁₆	—	2	12	no	—
416	C₂×C₈	—	2	12	no	—
417	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
418	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
419	C₄⋊C₄	—	2	3	no	—
420	C₄⋊C₄	—	2	6	no	—
421	C₄⋊C₄	—	2	6	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	6	no	—
426	C₄⋊C₄	—	2	6	no	—
427	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
428	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
429	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
430	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
431	C₂²⋊C₄	—	2	6	no	—
432	C₂²⋊C₄	—	2	3	no	—
433	C₂²⋊C₄	—	2	3	no	—
434	C₂²⋊C₄	—	2	6	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
439	C₄²	—	2	6	no	—
440	C₄²	C₂×D₄×C₂₃	1	3	yes	—
441	C₄²	C₂×D₄×C₂₃	1	3	yes	—
442	C₄²	C₂⁴×C₂₃	1	3	yes	—
443	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
444	C₂³	—	2	3	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	4	6	no	—
447	C₂³	—	8	3	no	—
448	C₂³	—	8	3	no	—
449	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
450	Q₈	—	2	6	no	—
451	Q₈	—	2	2	no	—
452	Q₈	—	4	6	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	6	no	—
455	Q₈	—	8	6	no	—
456	Q₈	C₂₃×#124	1	4	yes	—
457	D₄	—	2	6	no	—
458	D₄	—	2	2	no	—
459	D₄	—	4	6	no	—
460	D₄	—	4	6	no	—
461	D₄	—	4	6	no	—
462	D₄	—	8	3	no	—
463	D₄	—	8	3	no	—
464	D₄	C₂₃×#124	1	12	yes	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	6	no	—
467	C₂×C₄	—	2	3	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	3	no	—
470	C₂×C₄	—	4	3	no	—
471	C₂×C₄	—	4	6	no	—
472	C₂×C₄	—	4	6	no	—
473	C₂×C₄	—	4	6	no	—
474	C₂×C₄	—	8	6	no	—
475	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
478	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
479	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
480	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
481	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	6	no	—
484	C₂²	—	2	3	no	—
485	C₂²	—	2	3	no	—
486	C₂²	—	4	1	no	—
487	C₂²	—	4	3	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	4	3	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	3	no	—
492	C₂²	—	16	6	no	—
493	C₂²	C₂×C₂₃×#124	1	6	yes	—
494	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
495	C₄	—	2	1	no	—
496	C₄	—	2	3	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	3	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	3	no	—
501	C₄	—	4	6	no	—
502	C₄	—	4	2	no	—
503	C₄	C₂×C₂₃×#124	1	6	yes	—
504	C₄	C₂²×#41×C₂₃	1	2	yes	—
505	C₂	—	2	6	no	—
506	C₂	—	8	3	no	—
507	C₂	—	8	1	no	—
508	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
509	C₂	C₂₃×#3585	1	1	yes	characteristic
510	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3500	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×#3503	C₂	semidirect
3. C₂₃×#3141	C₂	semidirect
4. C₂₃×#3140	C₂	semidirect
5. C₂₃×#3137	C₂	semidirect
6. C₂₃×#3136	C₂	semidirect
7. C₂₃×#3135	C₂	semidirect
8. C₂×C₂₃×#489	C₂	semidirect
9. C₂₃×#2898	C₂	semidirect
10. C₂₃×#2897	C₂	semidirect
11. C₂₃×#2875	C₂	semidirect
23. C₂₃×#544	C₂²	semidirect
26. C₂₃×#533	C₂²	semidirect
28. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
32. C₂₃×#491	C₂²	semidirect
33. C₂₃×#490	C₂²	semidirect
34. C₂₃×#489	C₂²	semidirect
35. C₂₃×#489	C₂²	semidirect
36. C₂₃×#489	C₂²	semidirect
37. C₂₃×#489	C₂²	semidirect
38. C₂₃×#489	C₂²	semidirect
39. C₂₃×#489	C₂²	semidirect
40. C₂₃×#488	C₂²	semidirect
41. C₂₃×#487	C₂²	semidirect
42. C₂₃×#486	C₂²	semidirect
43. C₂₃×#419	C₂²	semidirect
46. C₂₃×#417	C₂²	semidirect
48. C₂₃×#416	C₂²	semidirect
49. C₂×C₂₃×#103	C₂²	semidirect
51. C₂₃×#405	C₂²	semidirect
52. C₄×M₁₆×C₂₃	C₂²	semidirect
53. C₂×C₂₃×#97	C₂²	semidirect
188. #60625	C₂₃	direct (N × H)
229. #3503	C₂×C₂₃	semidirect
231. #3141	C₂×C₂₃	semidirect
232. #3140	C₂×C₂₃	semidirect
233. #3137	C₂×C₂₃	semidirect
234. #3136	C₂×C₂₃	semidirect
235. #3135	C₂×C₂₃	semidirect
236. C₂×#489	C₂×C₂₃	semidirect
237. #2898	C₂×C₂₃	semidirect
238. #2897	C₂×C₂₃	semidirect
239. #2875	C₂×C₂₃	semidirect
272. #544	C₂²×C₂₃	semidirect
275. #533	C₂²×C₂₃	semidirect
277. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
281. #491	C₂²×C₂₃	semidirect
282. #490	C₂²×C₂₃	semidirect
283. #489	C₂²×C₂₃	semidirect
284. #489	C₂²×C₂₃	semidirect
285. #489	C₂²×C₂₃	semidirect
286. #489	C₂²×C₂₃	semidirect
287. #489	C₂²×C₂₃	semidirect
288. #489	C₂²×C₂₃	semidirect
289. #488	C₂²×C₂₃	semidirect
290. #487	C₂²×C₂₃	semidirect
291. #486	C₂²×C₂₃	semidirect
292. #419	C₂²×C₂₃	semidirect
295. #417	C₂²×C₂₃	semidirect
297. #416	C₂²×C₂₃	semidirect
298. C₂×#103	C₂²×C₂₃	semidirect
300. #405	C₂²×C₂₃	semidirect
301. C₄×M₁₆	C₂²×C₂₃	semidirect
302. C₂×#97	C₂²×C₂₃	semidirect
381. C₂₃	#60625	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×#3503	2
C₂₃×#3500	2
C₂₃×#3141	2
C₂₃×#3140	2
C₂₃×#3137	2
C₂₃×#3136	2
C₂₃×#3135	2
C₂×C₂₃×#489	2
C₂₃×#2898	2
C₂₃×#2897	2
C₂₃×#2875	2
#60625	23

Subgroups — C23×#60625 #19208251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60625 #19208251