Subgroups — C₂₃×#61886 #19209512

Order 5888.

1796 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3522	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3396	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3394	C₂	1	3	yes	maximal, complemented
5	C₂₃×#3392	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#3391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3363	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3362	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3361	C₂	1	3	yes	maximal, complemented
10	C₂₃×#3360	C₂	1	3	yes	maximal, complemented
11	C₂₃×#3212	C₂	1	3	yes	maximal, complemented
12	C₂₃×#3207	C₂	1	3	yes	maximal, complemented
13	C₂₃×#3106	C₂	1	3	yes	maximal, complemented
14	C₂₃×#3102	C₂	1	3	yes	maximal, complemented
15	C₂₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#559	C₂²	1	3	yes	—
17	C₂₃×#559	C₂²	1	3	yes	—
18	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#556	C₂²	1	3	yes	—
21	C₂₃×#554	C₂²	1	3	yes	—
22	C₂₃×#553	—	2	3	no	—
23	C₂₃×#550	C₂²	1	1	yes	characteristic
24	C₂₃×#550	C₂²	1	1	yes	characteristic
25	C₂₃×#548	—	2	1	no	—
26	C₂₃×#548	—	2	1	no	—
27	C₂₃×#547	—	2	3	no	—
28	C₂₃×#542	—	2	3	no	—
29	C₂₃×#532	C₂²	1	3	yes	—
30	C₂₃×#532	C₂²	1	3	yes	—
31	C₂₃×#529	—	2	3	no	—
32	C₂₃×#529	C₂²	1	3	yes	—
33	C₂₃×#520	—	2	1	no	—
34	C₄×#41×C₂₃	—	2	1	no	—
35	C₄×#41×C₂₃	C₂²	1	3	yes	—
36	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#495	C₂²	1	6	yes	—
38	C₂₃×#493	C₂²	1	3	yes	—
39	C₂₃×#492	C₂²	1	3	yes	—
40	C₂₃×#488	C₂²	1	6	yes	—
41	C₂₃×#487	C₂²	1	3	yes	—
42	C₂₃×#486	C₂²	1	3	yes	—
43	C₂₃×#485	C₂²	1	1	yes	characteristic
44	C₂₃×#482	C₂²	1	3	yes	—
45	C₂₃×#479	C₂²	1	3	yes	—
46	C₂₃×#478	C₂²	1	3	yes	—
47	C₂₃×#477	C₂²	1	3	yes	—
48	C₂₃×#476	C₂²	1	3	yes	—
49	C₂₃×#464	C₂²	1	3	yes	—
50	C₂₃×#463	C₂²	1	6	yes	—
51	C₂₃×#463	C₂²	1	6	yes	—
52	C₂₃×#463	C₂²	1	3	yes	—
53	C₂₃×#462	C₂²	1	3	yes	—
54	C₂₃×#461	C₂²	1	3	yes	—
55	C₂₃×#460	C₂²	1	3	yes	—
56	C₂₃×#460	C₂²	1	3	yes	—
57	C₂₃×#459	C₂²	1	3	yes	—
58	C₂₃×#459	C₂²	1	3	yes	—
59	C₂₃×#452	C₂²	1	3	yes	—
60	C₂₃×#449	C₂²	1	3	yes	—
61	C₂₃×#449	C₂²	1	6	yes	—
62	C₈×Q₈×C₂₃	C₂²	1	3	yes	—
63	C₈×Q₈×C₂₃	C₂²	1	3	yes	—
64	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
65	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
66	C₄×Q₁₆×C₂₃	C₂²	1	3	yes	—
67	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
68	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
69	C₄×SD₁₆×C₂₃	C₂²	1	6	yes	—
70	C₄×D₈×C₂₃	C₂²	1	3	yes	—
71	C₄×D₈×C₂₃	C₂²	1	6	yes	—
72	C₂₃×#431	C₂²	1	3	yes	—
73	C₂₃×#431	C₂²	1	3	yes	—
74	C₂×#41×C₂₃	—	2	1	no	—
75	C₂×#41×C₂₃	—	2	1	no	—
76	C₂×#41×C₂₃	—	2	3	no	—
77	C₂×#41×C₂₃	C₂³	1	3	yes	—
78	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
79	C₂×Q₁₆×C₂₃	C₂³	1	3	yes	—
80	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
81	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
82	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
83	C₂×D₈×C₂₃	C₂³	1	6	yes	—
84	C₂×D₈×C₂₃	C₂³	1	3	yes	—
85	C₂₃×C₄⋊Q₈	—	2	3	no	—
86	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
87	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
88	C₂₃×#127	C₂³	1	3	yes	—
89	C₂₃×#127	C₂³	1	3	yes	—
90	C₂₃×#126	—	2	6	no	—
91	C₂₃×#126	—	2	3	no	—
92	C₂₃×#125	—	2	3	no	—
93	C₂₃×#125	C₂³	1	3	yes	—
94	C₂₃×#124	—	2	3	no	—
95	C₂₃×#124	—	2	3	no	—
96	C₂₃×#124	—	2	3	no	—
97	C₂₃×#124	C₂³	1	6	yes	—
98	C₂₃×#124	C₂³	1	3	yes	—
99	C₂₃×#123	—	2	3	no	—
100	C₂₃×#123	—	2	6	no	—
101	C₂₃×#123	—	2	6	no	—
102	C₂₃×C₂²⋊Q₈	—	2	3	no	—
103	C₂₃×C₂²⋊Q₈	—	2	3	no	—
104	C₂₃×C₂²⋊Q₈	—	2	6	no	—
105	C₂₃×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
106	C₂₃×#121	—	2	3	no	—
107	C₂₃×#121	—	2	3	no	—
108	C₂₃×#121	—	2	3	no	—
109	C₂₃×#121	—	2	3	no	—
110	C₂₃×#121	—	2	3	no	—
111	C₂₃×#121	C₂³	1	3	yes	—
112	C₄×Q₈×C₂₃	—	2	1	no	—
113	C₄×Q₈×C₂₃	—	2	1	no	—
114	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
115	C₄×Q₈×C₂₃	D₄	1	3	yes	—
116	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
117	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
118	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
119	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
120	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
121	C₄×D₄×C₂₃	—	2	3	no	—
122	C₄×D₄×C₂₃	—	2	3	no	—
123	C₄×D₄×C₂₃	—	2	3	no	—
124	C₄×D₄×C₂₃	—	2	3	no	—
125	C₄×D₄×C₂₃	—	2	3	no	—
126	C₄×D₄×C₂₃	—	2	3	no	—
127	C₄×D₄×C₂₃	—	2	3	no	—
128	C₄×D₄×C₂₃	—	2	3	no	—
129	C₄×D₄×C₂₃	C₂³	1	3	yes	—
130	C₄×D₄×C₂₃	C₂³	1	3	yes	—
131	C₄×D₄×C₂₃	C₂³	1	6	yes	—
132	C₄×D₄×C₂₃	C₂³	1	3	yes	—
133	C₂₃×#117	—	2	3	no	—
134	C₂₃×#117	—	2	3	no	—
135	C₂₃×#117	—	2	3	no	—
136	C₂₃×#117	C₂³	1	1	yes	characteristic
137	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
138	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
139	C₂×C₄²×C₂₃	—	2	3	no	—
140	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
141	C₂₃×#107	C₂³	1	1	yes	characteristic
142	C₂₃×#107	C₂³	1	3	yes	—
143	C₂₃×#106	C₂³	1	3	yes	—
144	C₂₃×#106	C₂³	1	1	yes	characteristic
145	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
146	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
147	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
148	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
149	C₂₃×#103	C₂³	1	1	yes	characteristic
150	C₂₃×#103	C₂³	1	6	yes	—
151	C₂₃×#103	C₂³	1	3	yes	—
152	C₂₃×#103	C₂³	1	1	yes	characteristic
153	C₂₃×#103	C₂³	1	3	yes	—
154	C₂₃×#102	C₂³	1	3	yes	—
155	C₂₃×#102	C₂³	1	3	yes	—
156	C₂₃×#102	C₂³	1	6	yes	—
157	C₂₃×#102	C₂³	1	3	yes	—
158	C₂₃×#102	C₂³	1	3	yes	—
159	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₂²⋊C₈	C₂³	1	3	yes	—
161	C₄×C₈×C₂₃	C₂³	1	6	yes	—
162	C₄×C₈×C₂₃	C₂³	1	3	yes	—
163	C₄×C₈×C₂₃	C₂³	1	3	yes	—
164	#41×C₂₃	—	4	3	no	—
165	#41×C₂₃	—	4	3	no	—
166	#41×C₂₃	—	4	4	no	—
167	#41×C₂₃	—	4	4	no	—
168	#41×C₂₃	—	8	6	no	—
169	C₂×Q₈×C₂₃	—	2	4	no	—
170	C₂×Q₈×C₂₃	—	2	3	no	—
171	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
172	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
174	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
175	C₂×D₄×C₂₃	—	2	3	no	—
176	C₂×D₄×C₂₃	—	2	3	no	—
177	C₂×D₄×C₂₃	—	2	3	no	—
178	C₂×D₄×C₂₃	—	2	3	no	—
179	C₂×D₄×C₂₃	—	2	3	no	—
180	C₂×D₄×C₂₃	—	2	3	no	—
181	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
182	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
183	C₂²×C₄×C₂₃	—	2	3	no	—
184	C₂²×C₄×C₂₃	—	2	3	no	—
185	C₂²×C₄×C₂₃	—	2	3	no	—
186	C₂²×C₄×C₂₃	—	2	3	no	—
187	C₂²×C₄×C₂₃	—	2	6	no	—
188	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
189	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
190	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
191	Q₁₆×C₂₃	—	4	3	no	—
192	SD₁₆×C₂₃	—	4	3	no	—
193	SD₁₆×C₂₃	—	4	6	no	—
194	SD₁₆×C₂₃	—	4	3	no	—
195	D₈×C₂₃	—	4	3	no	—
196	D₈×C₂₃	—	4	6	no	—
197	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
198	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
199	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
200	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
201	C₄⋊C₄×C₂₃	—	2	3	no	—
202	C₄⋊C₄×C₂₃	—	2	3	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	3	no	—
205	C₄⋊C₄×C₂₃	—	2	3	no	—
206	C₄⋊C₄×C₂₃	—	2	3	no	—
207	C₄⋊C₄×C₂₃	—	2	3	no	—
208	C₄⋊C₄×C₂₃	—	2	3	no	—
209	C₄⋊C₄×C₂₃	—	2	3	no	—
210	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
211	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
212	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
213	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
214	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
215	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
216	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
217	C₂²⋊C₄×C₂₃	—	2	1	no	—
218	C₂²⋊C₄×C₂₃	—	2	1	no	—
219	C₂²⋊C₄×C₂₃	—	2	6	no	—
220	C₂²⋊C₄×C₂₃	—	2	3	no	—
221	C₂²⋊C₄×C₂₃	—	2	3	no	—
222	C₂²⋊C₄×C₂₃	—	2	3	no	—
223	C₂²⋊C₄×C₂₃	—	2	3	no	—
224	C₂²⋊C₄×C₂₃	—	2	6	no	—
225	C₂²⋊C₄×C₂₃	—	2	3	no	—
226	C₂²⋊C₄×C₂₃	—	2	3	no	—
227	C₂²⋊C₄×C₂₃	—	2	6	no	—
228	C₄²×C₂₃	—	2	1	no	—
229	C₄²×C₂₃	—	2	3	no	—
230	C₄²×C₂₃	—	2	3	no	—
231	C₄²×C₂₃	—	2	3	no	—
232	C₄²×C₂₃	C₂×D₄	1	3	yes	—
233	C₄²×C₂₃	C₂×D₄	1	3	yes	—
234	C₄²×C₂₃	C₂⁴	1	3	yes	—
235	C₄²×C₂₃	C₂⁴	1	3	yes	—
236	#61886	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂³×C₂₃	—	2	3	no	—
238	C₂³×C₂₃	—	2	3	no	—
239	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
240	Q₈×C₂₃	—	2	6	no	—
241	Q₈×C₂₃	—	2	1	no	—
242	Q₈×C₂₃	—	2	3	no	—
243	Q₈×C₂₃	—	2	3	no	—
244	Q₈×C₂₃	—	2	1	no	—
245	Q₈×C₂₃	—	8	3	no	—
246	Q₈×C₂₃	#135	1	4	yes	—
247	D₄×C₂₃	—	2	3	no	—
248	D₄×C₂₃	—	2	3	no	—
249	D₄×C₂₃	—	2	3	no	—
250	D₄×C₂₃	—	2	3	no	—
251	D₄×C₂₃	—	4	6	no	—
252	D₄×C₂₃	—	4	6	no	—
253	D₄×C₂₃	—	4	6	no	—
254	D₄×C₂₃	—	4	6	no	—
255	D₄×C₂₃	—	8	3	no	—
256	D₄×C₂₃	—	8	3	no	—
257	C₂×C₄×C₂₃	—	2	6	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	3	no	—
264	C₂×C₄×C₂₃	—	2	3	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	4	3	no	—
267	C₂×C₄×C₂₃	—	4	6	no	—
268	C₂×C₄×C₂₃	—	4	6	no	—
269	C₂×C₄×C₂₃	—	4	6	no	—
270	C₂×C₄×C₂₃	—	8	3	no	—
271	C₂×C₄×C₂₃	—	8	6	no	—
272	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
274	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
276	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
277	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
278	C₈×C₂₃	—	2	1	no	—
279	C₈×C₂₃	—	2	3	no	—
280	C₈×C₂₃	—	2	3	no	—
281	C₈×C₂₃	—	2	1	no	—
282	#3522	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#3480	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#3396	C₂×C₂₃	1	3	yes	complemented
285	#3394	C₂×C₂₃	1	3	yes	complemented
286	#3392	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#3391	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#3363	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#3362	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#3361	C₂×C₂₃	1	3	yes	complemented
291	#3360	C₂×C₂₃	1	3	yes	complemented
292	#3212	C₂×C₂₃	1	3	yes	complemented
293	#3207	C₂×C₂₃	1	3	yes	complemented
294	#3106	C₂×C₂₃	1	3	yes	complemented
295	#3102	C₂×C₂₃	1	3	yes	complemented
296	#2982	C₂×C₂₃	1	1	yes	characteristic, complemented
297	C₂²×C₂₃	—	2	1	no	—
298	C₂²×C₂₃	—	2	1	no	—
299	C₂²×C₂₃	—	2	1	no	—
300	C₂²×C₂₃	—	4	3	no	—
301	C₂²×C₂₃	—	4	3	no	—
302	C₂²×C₂₃	—	4	3	no	—
303	C₂²×C₂₃	—	4	3	no	—
304	C₂²×C₂₃	—	4	3	no	—
305	C₂²×C₂₃	—	4	3	no	—
306	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
307	C₄×C₂₃	—	2	3	no	—
308	C₄×C₂₃	—	2	3	no	—
309	C₄×C₂₃	—	2	1	no	—
310	C₄×C₂₃	—	2	3	no	—
311	C₄×C₂₃	—	4	1	no	—
312	C₄×C₂₃	—	4	3	no	—
313	C₄×C₂₃	—	4	3	no	—
314	C₄×C₂₃	—	4	1	no	—
315	C₄×C₂₃	—	4	1	no	—
316	C₄×C₂₃	—	4	1	no	—
317	C₄×C₂₃	C₂×#135	1	6	yes	—
318	C₄×C₂₃	#585	1	1	yes	characteristic
319	C₄×C₂₃	#585	1	1	yes	characteristic
320	#559	C₂²×C₂₃	1	3	yes	—
321	#559	C₂²×C₂₃	1	3	yes	—
322	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
323	#557	C₂²×C₂₃	1	1	yes	characteristic
324	#556	C₂²×C₂₃	1	3	yes	—
325	#554	C₂²×C₂₃	1	3	yes	—
326	#553	—	2	3	no	—
327	#550	C₂²×C₂₃	1	1	yes	characteristic
328	#550	C₂²×C₂₃	1	1	yes	characteristic
329	#548	—	2	1	no	—
330	#548	—	2	1	no	—
331	#547	—	2	3	no	—
332	#542	—	2	3	no	—
333	#532	C₂²×C₂₃	1	3	yes	—
334	#532	C₂²×C₂₃	1	3	yes	—
335	#529	—	2	3	no	—
336	#529	C₂²×C₂₃	1	3	yes	—
337	#520	—	2	1	no	—
338	C₄×#41	—	2	1	no	—
339	C₄×#41	C₂²×C₂₃	1	3	yes	—
340	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
341	#495	C₂²×C₂₃	1	6	yes	—
342	#493	C₂²×C₂₃	1	3	yes	—
343	#492	C₂²×C₂₃	1	3	yes	—
344	#488	C₂²×C₂₃	1	6	yes	—
345	#487	C₂²×C₂₃	1	3	yes	—
346	#486	C₂²×C₂₃	1	3	yes	—
347	#485	C₂²×C₂₃	1	1	yes	characteristic
348	#482	C₂²×C₂₃	1	3	yes	—
349	#479	C₂²×C₂₃	1	3	yes	—
350	#478	C₂²×C₂₃	1	3	yes	—
351	#477	C₂²×C₂₃	1	3	yes	—
352	#476	C₂²×C₂₃	1	3	yes	—
353	#464	C₂²×C₂₃	1	3	yes	—
354	#463	C₂²×C₂₃	1	3	yes	—
355	#463	C₂²×C₂₃	1	6	yes	—
356	#463	C₂²×C₂₃	1	6	yes	—
357	#462	C₂²×C₂₃	1	3	yes	—
358	#461	C₂²×C₂₃	1	3	yes	—
359	#460	C₂²×C₂₃	1	3	yes	—
360	#460	C₂²×C₂₃	1	3	yes	—
361	#459	C₂²×C₂₃	1	3	yes	—
362	#459	C₂²×C₂₃	1	3	yes	—
363	#452	C₂²×C₂₃	1	3	yes	—
364	#449	C₂²×C₂₃	1	3	yes	—
365	#449	C₂²×C₂₃	1	6	yes	—
366	C₈×Q₈	C₂²×C₂₃	1	3	yes	—
367	C₈×Q₈	C₂²×C₂₃	1	3	yes	—
368	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
369	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
370	C₄×Q₁₆	C₂²×C₂₃	1	3	yes	—
371	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
372	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
373	C₄×SD₁₆	C₂²×C₂₃	1	6	yes	—
374	C₄×D₈	C₂²×C₂₃	1	3	yes	—
375	C₄×D₈	C₂²×C₂₃	1	6	yes	—
376	#431	C₂²×C₂₃	1	3	yes	—
377	#431	C₂²×C₂₃	1	3	yes	—
378	C₂×C₂₃	—	4	1	no	—
379	C₂×C₂₃	—	8	3	no	—
380	C₂×C₂₃	—	8	3	no	—
381	C₂×C₂₃	#3487	1	1	yes	characteristic
382	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
383	C₂×C₂₃	#3585	1	1	yes	characteristic
384	C₂×#41	—	2	1	no	—
385	C₂×#41	—	2	1	no	—
386	C₂×#41	—	2	3	no	—
387	C₂×#41	C₂³×C₂₃	1	3	yes	—
388	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
389	C₂×Q₁₆	C₂³×C₂₃	1	3	yes	—
390	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
391	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
392	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
393	C₂×D₈	C₂³×C₂₃	1	6	yes	—
394	C₂×D₈	C₂³×C₂₃	1	3	yes	—
395	C₄⋊Q₈	—	2	3	no	—
396	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
397	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
398	#127	C₂³×C₂₃	1	3	yes	—
399	#127	C₂³×C₂₃	1	3	yes	—
400	#126	—	2	6	no	—
401	#126	—	2	3	no	—
402	#125	—	2	3	no	—
403	#125	C₂³×C₂₃	1	3	yes	—
404	#124	—	2	3	no	—
405	#124	—	2	3	no	—
406	#124	—	2	3	no	—
407	#124	C₂³×C₂₃	1	3	yes	—
408	#124	C₂³×C₂₃	1	6	yes	—
409	#123	—	2	3	no	—
410	#123	—	2	6	no	—
411	#123	—	2	6	no	—
412	C₂²⋊Q₈	—	2	3	no	—
413	C₂²⋊Q₈	—	2	3	no	—
414	C₂²⋊Q₈	—	2	6	no	—
415	C₂²⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
416	#121	—	2	3	no	—
417	#121	—	2	3	no	—
418	#121	—	2	3	no	—
419	#121	—	2	3	no	—
420	#121	—	2	3	no	—
421	#121	C₂³×C₂₃	1	3	yes	—
422	C₄×Q₈	—	2	1	no	—
423	C₄×Q₈	—	2	1	no	—
424	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
425	C₄×Q₈	D₄×C₂₃	1	3	yes	—
426	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
427	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
428	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
429	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
431	C₄×D₄	—	2	3	no	—
432	C₄×D₄	—	2	3	no	—
433	C₄×D₄	—	2	3	no	—
434	C₄×D₄	—	2	3	no	—
435	C₄×D₄	—	2	3	no	—
436	C₄×D₄	—	2	3	no	—
437	C₄×D₄	—	2	3	no	—
438	C₄×D₄	—	2	3	no	—
439	C₄×D₄	C₂³×C₂₃	1	3	yes	—
440	C₄×D₄	C₂³×C₂₃	1	3	yes	—
441	C₄×D₄	C₂³×C₂₃	1	3	yes	—
442	C₄×D₄	C₂³×C₂₃	1	6	yes	—
443	#117	—	2	3	no	—
444	#117	—	2	3	no	—
445	#117	—	2	3	no	—
446	#117	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×C₄⋊C₄	—	2	3	no	—
448	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
449	C₂×C₄²	—	2	3	no	—
450	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
451	#107	C₂³×C₂₃	1	1	yes	characteristic
452	#107	C₂³×C₂₃	1	3	yes	—
453	#106	C₂³×C₂₃	1	3	yes	—
454	#106	C₂³×C₂₃	1	1	yes	characteristic
455	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
456	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
457	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
458	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
459	#103	C₂³×C₂₃	1	1	yes	characteristic
460	#103	C₂³×C₂₃	1	6	yes	—
461	#103	C₂³×C₂₃	1	3	yes	—
462	#103	C₂³×C₂₃	1	1	yes	characteristic
463	#103	C₂³×C₂₃	1	3	yes	—
464	#102	C₂³×C₂₃	1	3	yes	—
465	#102	C₂³×C₂₃	1	3	yes	—
466	#102	C₂³×C₂₃	1	6	yes	—
467	#102	C₂³×C₂₃	1	3	yes	—
468	#102	C₂³×C₂₃	1	3	yes	—
469	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
470	C₂²⋊C₈	C₂³×C₂₃	1	3	yes	—
471	C₄×C₈	C₂³×C₂₃	1	6	yes	—
472	C₄×C₈	C₂³×C₂₃	1	3	yes	—
473	C₄×C₈	C₂³×C₂₃	1	3	yes	—
474	C₂₃	#61886	1	1	yes	characteristic, direct product factor, Sylow
475	#41	—	4	3	no	—
476	#41	—	4	3	no	—
477	#41	—	4	4	no	—
478	#41	—	4	4	no	—
479	#41	—	8	6	no	—
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	3	no	—
482	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
483	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
485	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
486	C₂×D₄	—	2	3	no	—
487	C₂×D₄	—	2	3	no	—
488	C₂×D₄	—	2	3	no	—
489	C₂×D₄	—	2	3	no	—
490	C₂×D₄	—	2	3	no	—
491	C₂×D₄	—	2	3	no	—
492	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
493	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
494	C₂²×C₄	—	2	3	no	—
495	C₂²×C₄	—	2	3	no	—
496	C₂²×C₄	—	2	3	no	—
497	C₂²×C₄	—	2	3	no	—
498	C₂²×C₄	—	2	6	no	—
499	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
500	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
501	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
502	Q₁₆	—	4	3	no	—
503	SD₁₆	—	4	3	no	—
504	SD₁₆	—	4	6	no	—
505	SD₁₆	—	4	3	no	—
506	D₈	—	4	3	no	—
507	D₈	—	4	6	no	—
508	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
509	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
510	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
511	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
512	C₄⋊C₄	—	2	3	no	—
513	C₄⋊C₄	—	2	3	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	3	no	—
516	C₄⋊C₄	—	2	3	no	—
517	C₄⋊C₄	—	2	3	no	—
518	C₄⋊C₄	—	2	3	no	—
519	C₄⋊C₄	—	2	3	no	—
520	C₄⋊C₄	—	2	3	no	—
521	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
522	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
523	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
524	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
525	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
526	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
527	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
528	C₂²⋊C₄	—	2	1	no	—
529	C₂²⋊C₄	—	2	1	no	—
530	C₂²⋊C₄	—	2	6	no	—
531	C₂²⋊C₄	—	2	3	no	—
532	C₂²⋊C₄	—	2	3	no	—
533	C₂²⋊C₄	—	2	3	no	—
534	C₂²⋊C₄	—	2	3	no	—
535	C₂²⋊C₄	—	2	6	no	—
536	C₂²⋊C₄	—	2	3	no	—
537	C₂²⋊C₄	—	2	3	no	—
538	C₂²⋊C₄	—	2	6	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	3	no	—
541	C₄²	—	2	3	no	—
542	C₄²	—	2	3	no	—
543	C₄²	C₂×D₄×C₂₃	1	3	yes	—
544	C₄²	C₂×D₄×C₂₃	1	3	yes	—
545	C₄²	C₂⁴×C₂₃	1	3	yes	—
546	C₄²	C₂⁴×C₂₃	1	3	yes	—
547	C₂³	—	2	3	no	—
548	C₂³	—	2	3	no	—
549	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
550	Q₈	—	2	6	no	—
551	Q₈	—	2	1	no	—
552	Q₈	—	2	3	no	—
553	Q₈	—	2	3	no	—
554	Q₈	—	2	1	no	—
555	Q₈	—	8	3	no	—
556	Q₈	C₂₃×#135	1	4	yes	—
557	D₄	—	2	3	no	—
558	D₄	—	2	3	no	—
559	D₄	—	2	3	no	—
560	D₄	—	2	3	no	—
561	D₄	—	4	6	no	—
562	D₄	—	4	6	no	—
563	D₄	—	4	6	no	—
564	D₄	—	4	6	no	—
565	D₄	—	8	3	no	—
566	D₄	—	8	3	no	—
567	C₂×C₄	—	2	6	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	3	no	—
574	C₂×C₄	—	2	3	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	4	3	no	—
577	C₂×C₄	—	4	6	no	—
578	C₂×C₄	—	4	6	no	—
579	C₂×C₄	—	4	6	no	—
580	C₂×C₄	—	8	6	no	—
581	C₂×C₄	—	8	3	no	—
582	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
584	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
585	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
586	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
587	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
588	C₈	—	2	1	no	—
589	C₈	—	2	3	no	—
590	C₈	—	2	3	no	—
591	C₈	—	2	1	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	4	3	no	—
596	C₂²	—	4	3	no	—
597	C₂²	—	4	3	no	—
598	C₂²	—	4	3	no	—
599	C₂²	—	4	3	no	—
600	C₂²	—	4	3	no	—
601	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
602	C₄	—	2	1	no	—
603	C₄	—	2	3	no	—
604	C₄	—	2	3	no	—
605	C₄	—	2	3	no	—
606	C₄	—	4	3	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	1	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	1	no	—
611	C₄	—	4	3	no	—
612	C₄	C₂×C₂₃×#135	1	6	yes	—
613	C₄	C₂₃×#585	1	1	yes	characteristic
614	C₄	C₂₃×#585	1	1	yes	characteristic
615	C₂	—	4	1	no	—
616	C₂	—	8	3	no	—
617	C₂	—	8	3	no	—
618	C₂	C₂₃×#3487	1	1	yes	characteristic
619	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
620	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3522	C₂	semidirect
2. C₂₃×#3480	C₂	semidirect
3. C₂₃×#3396	C₂	semidirect
4. C₂₃×#3394	C₂	semidirect
5. C₂₃×#3392	C₂	semidirect
6. C₂₃×#3391	C₂	semidirect
7. C₂₃×#3363	C₂	semidirect
8. C₂₃×#3362	C₂	semidirect
9. C₂₃×#3361	C₂	semidirect
10. C₂₃×#3360	C₂	semidirect
11. C₂₃×#3212	C₂	semidirect
12. C₂₃×#3207	C₂	semidirect
13. C₂₃×#3106	C₂	semidirect
14. C₂₃×#3102	C₂	semidirect
15. C₂₃×#2982	C₂	semidirect
236. #61886	C₂₃	direct (N × H)
282. #3522	C₂×C₂₃	semidirect
283. #3480	C₂×C₂₃	semidirect
284. #3396	C₂×C₂₃	semidirect
285. #3394	C₂×C₂₃	semidirect
286. #3392	C₂×C₂₃	semidirect
287. #3391	C₂×C₂₃	semidirect
288. #3363	C₂×C₂₃	semidirect
289. #3362	C₂×C₂₃	semidirect
290. #3361	C₂×C₂₃	semidirect
291. #3360	C₂×C₂₃	semidirect
292. #3212	C₂×C₂₃	semidirect
293. #3207	C₂×C₂₃	semidirect
294. #3106	C₂×C₂₃	semidirect
295. #3102	C₂×C₂₃	semidirect
296. #2982	C₂×C₂₃	semidirect
474. C₂₃	#61886	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×#3522	2
C₂₃×#3480	2
C₂₃×#3396	2
C₂₃×#3394	2
C₂₃×#3392	2
C₂₃×#3391	2
C₂₃×#3363	2
C₂₃×#3362	2
C₂₃×#3361	2
C₂₃×#3360	2
C₂₃×#3212	2
C₂₃×#3207	2
C₂₃×#3106	2
C₂₃×#3102	2
C₂₃×#2982	2
#61886	23

Subgroups — C23×#61886 #19209512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61886 #19209512