Subgroups — C₂₃×#18632 #19166258

Order 5888.

938 conjugacy classes of subgroups.

The subgroup lattice has 506 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
2	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3039	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1645	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1643	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#137	—	2	1	no	—
11	Q₈²×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#557	—	2	1	no	—
13	C₂₃×#554	—	2	2	no	—
14	C₂₃×#554	C₂²	1	2	yes	—
15	D₄×Q₈×C₂₃	—	2	2	no	—
16	D₄×Q₈×C₂₃	—	2	1	no	—
17	C₂₃×#544	—	2	2	no	—
18	C₂₃×#544	—	2	1	no	—
19	C₂₃×#533	—	2	1	no	—
20	C₂₃×#533	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂₃×#520	—	2	1	no	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#498	—	2	1	no	—
27	C₂₃×#487	—	2	1	no	—
28	C₂₃×#487	—	2	2	no	—
29	C₂₃×#487	C₂²	1	4	yes	—
30	C₂₃×#483	—	2	2	no	—
31	C₂₃×#475	C₂²	1	4	yes	—
32	C₂₃×#462	—	2	1	no	—
33	C₂₃×#462	C₂²	1	2	yes	—
34	C₂₃×#461	—	2	1	no	—
35	C₂₃×#461	C₂²	1	2	yes	—
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#423	C₂²	1	1	yes	characteristic
38	C₂₃×#419	—	2	1	no	—
39	C₂×C₂₃×#103	—	2	2	no	—
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#333	C₂²	1	4	yes	—
42	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₂₃	—	2	2	no	—
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
49	C₂₃×#137	—	4	4	no	—
50	C₂×Q₁₆×C₂₃	—	2	2	no	—
51	C₂×SD₁₆×C₂₃	—	2	2	no	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂²×C₈×C₂₃	—	2	1	no	—
54	C₂₃×C₄⋊Q₈	—	2	2	no	—
55	C₂₃×C₄⋊Q₈	—	2	2	no	—
56	C₂₃×C₄⋊Q₈	—	2	4	no	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	—	2	2	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	—	2	1	no	—
64	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
65	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
67	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₂₃×#125	—	2	4	no	—
70	C₂₃×#125	—	2	2	no	—
71	C₂₃×#125	—	2	2	no	—
72	C₂₃×#125	—	2	1	no	—
73	C₂₃×C₂²⋊Q₈	—	2	2	no	—
74	C₂₃×C₂²⋊Q₈	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	—	2	4	no	—
76	C₂₃×C₂²⋊Q₈	—	2	4	no	—
77	C₂₃×C₂²⋊Q₈	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	—	2	1	no	—
79	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	2	2	no	—
83	C₄×Q₈×C₂₃	—	2	1	no	—
84	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
86	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₄×D₄×C₂₃	—	2	4	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₄×D₄×C₂₃	—	2	1	no	—
90	C₄×D₄×C₂₃	—	2	1	no	—
91	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
92	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×#117	—	2	2	no	—
94	C₂₃×#117	—	2	1	no	—
95	C₂₃×#117	—	2	1	no	—
96	C₂₃×#117	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄²×C₂₃	—	2	1	no	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×#106	—	2	4	no	—
106	C₂₃×C₄⋊C₈	—	2	1	no	—
107	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
109	C₂₃×#103	—	2	8	no	—
110	C₂₃×#103	—	2	4	no	—
111	C₂₃×#103	—	2	2	no	—
112	C₂₃×#103	—	2	2	no	—
113	C₂₃×C₂²⋊C₈	—	2	1	no	—
114	C₄×C₈×C₂₃	—	2	1	no	—
115	C₄×C₈×C₂₃	—	2	1	no	—
116	C₄×C₈×C₂₃	C₂³	1	2	yes	—
117	#41×C₂₃	—	2	2	no	—
118	#41×C₂₃	SD₁₆	1	4	yes	—
119	C₂×Q₈×C₂₃	—	2	4	no	—
120	C₂×Q₈×C₂₃	—	2	2	no	—
121	C₂×Q₈×C₂₃	—	2	1	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×Q₈×C₂₃	—	4	1	no	—
125	C₂×Q₈×C₂₃	—	4	2	no	—
126	C₂×Q₈×C₂₃	—	4	2	no	—
127	C₂×Q₈×C₂₃	—	8	4	no	—
128	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
129	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
131	C₂×D₄×C₂₃	—	2	1	no	—
132	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	4	1	no	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	Q₁₆×C₂₃	—	4	4	no	—
143	SD₁₆×C₂₃	—	4	4	no	—
144	M₁₆×C₂₃	—	4	2	no	—
145	C₂×C₈×C₂₃	—	2	2	no	—
146	C₂×C₈×C₂₃	—	2	2	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	4	2	no	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	—	2	2	no	—
161	C₄⋊C₄×C₂₃	—	2	2	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	—	4	1	no	—
167	C₄⋊C₄×C₂₃	—	4	1	no	—
168	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²⋊C₄×C₂₃	—	2	4	no	—
172	C₂²⋊C₄×C₂₃	—	2	2	no	—
173	C₂²⋊C₄×C₂₃	—	2	1	no	—
174	C₂²⋊C₄×C₂₃	—	2	1	no	—
175	C₂²⋊C₄×C₂₃	—	2	1	no	—
176	C₄²×C₂₃	—	2	4	no	—
177	C₄²×C₂₃	—	2	1	no	—
178	C₄²×C₂₃	—	2	1	no	—
179	C₄²×C₂₃	—	2	1	no	—
180	C₄²×C₂₃	—	2	2	no	—
181	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
185	#18632	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₂₃	—	2	1	no	—
187	C₂³×C₂₃	#120	1	1	yes	characteristic
188	Q₈×C₂₃	—	2	1	no	—
189	Q₈×C₂₃	—	2	4	no	—
190	Q₈×C₂₃	—	4	4	no	—
191	Q₈×C₂₃	—	4	4	no	—
192	Q₈×C₂₃	—	4	2	no	—
193	Q₈×C₂₃	—	4	2	no	—
194	Q₈×C₂₃	—	4	2	no	—
195	Q₈×C₂₃	—	8	4	no	—
196	Q₈×C₂₃	—	8	2	no	—
197	Q₈×C₂₃	C₂×SD₁₆	1	2	yes	—
198	D₄×C₂₃	—	2	4	no	—
199	D₄×C₂₃	—	2	1	no	—
200	D₄×C₂₃	C₂×SD₁₆	1	2	yes	—
201	C₂×C₄×C₂₃	—	2	1	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	1	no	—
207	C₂×C₄×C₂₃	—	2	1	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	4	2	no	—
212	C₂×C₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	8	2	no	—
214	C₂×C₄×C₂₃	—	8	4	no	—
215	C₂×C₄×C₂₃	—	8	2	no	—
216	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
217	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
218	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
220	C₂×C₄×C₂₃	#137	1	2	yes	—
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
224	C₈×C₂₃	—	4	2	no	—
225	C₈×C₂₃	—	4	2	no	—
226	C₈×C₂₃	—	4	2	no	—
227	#3503	C₂×C₂₃	1	1	yes	characteristic
228	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#3039	C₂×C₂₃	1	1	yes	characteristic
231	#1645	C₂×C₂₃	1	4	yes	complemented
232	#1643	C₂×C₂₃	1	4	yes	complemented
233	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#1476	C₂×C₂₃	1	1	yes	characteristic
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	4	1	no	—
240	C₂²×C₂₃	—	4	1	no	—
241	C₂²×C₂₃	#448	1	2	yes	—
242	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
243	C₄×C₂₃	—	2	2	no	—
244	C₄×C₂₃	—	2	1	no	—
245	C₄×C₂₃	—	2	1	no	—
246	C₄×C₂₃	—	4	1	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	4	1	no	—
249	C₄×C₂₃	—	4	4	no	—
250	C₄×C₂₃	—	4	2	no	—
251	C₄×C₂₃	—	8	2	no	—
252	C₄×C₂₃	#448	1	2	yes	—
253	C₄×C₂₃	#455	1	2	yes	—
254	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
255	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
256	C₄×C₂₃	#578	1	2	yes	—
257	C₂×#137	—	2	1	no	—
258	Q₈²	C₂²×C₂₃	1	2	yes	—
259	#557	—	2	1	no	—
260	#554	—	2	2	no	—
261	#554	C₂²×C₂₃	1	2	yes	—
262	D₄×Q₈	—	2	2	no	—
263	D₄×Q₈	—	2	1	no	—
264	#544	—	2	2	no	—
265	#544	—	2	1	no	—
266	#533	—	2	1	no	—
267	#533	C₂²×C₂₃	1	1	yes	characteristic
268	C₂×C₄⋊Q₈	—	2	1	no	—
269	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	#520	—	2	1	no	—
272	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
273	#498	—	2	1	no	—
274	#487	—	2	2	no	—
275	#487	—	2	1	no	—
276	#487	C₂²×C₂₃	1	4	yes	—
277	#483	—	2	2	no	—
278	#475	C₂²×C₂₃	1	4	yes	—
279	#462	—	2	1	no	—
280	#462	C₂²×C₂₃	1	2	yes	—
281	#461	—	2	1	no	—
282	#461	C₂²×C₂₃	1	2	yes	—
283	#431	C₂²×C₂₃	1	1	yes	characteristic
284	#423	C₂²×C₂₃	1	1	yes	characteristic
285	#419	—	2	1	no	—
286	C₂×#103	—	2	2	no	—
287	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
288	#333	C₂²×C₂₃	1	4	yes	—
289	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
290	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
291	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
292	C₂×C₂₃	—	2	2	no	—
293	C₂×C₂₃	—	4	2	no	—
294	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
295	C₂×C₂₃	#3011	1	1	yes	characteristic
296	C₂×C₂₃	C₂×#455	1	1	yes	characteristic
297	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
298	C₂²×Q₈	—	2	2	no	—
299	C₂²×Q₈	—	2	1	no	—
300	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
301	#137	—	4	4	no	—
302	C₂×Q₁₆	—	2	2	no	—
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	1	no	—
305	C₂²×C₈	—	2	1	no	—
306	C₄⋊Q₈	—	2	2	no	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	—	2	4	no	—
309	C₄⋊Q₈	—	2	2	no	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	1	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
317	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
318	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
319	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
320	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
321	#125	—	2	2	no	—
322	#125	—	2	4	no	—
323	#125	—	2	2	no	—
324	#125	—	2	1	no	—
325	C₂²⋊Q₈	—	2	2	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	2	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	1	no	—
331	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	1	no	—
336	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
337	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
338	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
339	C₄×D₄	—	2	4	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
344	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
345	#117	—	2	2	no	—
346	#117	—	2	1	no	—
347	#117	—	2	1	no	—
348	#117	D₄×C₂₃	1	1	yes	characteristic
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄²	—	2	1	no	—
356	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
357	#106	—	2	4	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
360	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
361	#103	—	2	8	no	—
362	#103	—	2	4	no	—
363	#103	—	2	2	no	—
364	#103	—	2	2	no	—
365	C₂²⋊C₈	—	2	1	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	—	2	1	no	—
368	C₄×C₈	C₂³×C₂₃	1	2	yes	—
369	C₂₃	#18632	1	1	yes	characteristic, direct product factor, Sylow
370	#41	—	2	2	no	—
371	#41	SD₁₆×C₂₃	1	4	yes	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	4	4	no	—
377	C₂×Q₈	—	4	1	no	—
378	C₂×Q₈	—	4	2	no	—
379	C₂×Q₈	—	4	2	no	—
380	C₂×Q₈	—	8	4	no	—
381	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
382	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
383	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	4	1	no	—
392	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	Q₁₆	—	4	4	no	—
396	SD₁₆	—	4	4	no	—
397	M₁₆	—	4	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	2	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	4	1	no	—
420	C₄⋊C₄	—	4	1	no	—
421	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
422	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	1	no	—
427	C₂²⋊C₄	—	2	1	no	—
428	C₂²⋊C₄	—	2	1	no	—
429	C₄²	—	2	4	no	—
430	C₄²	—	2	1	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	2	no	—
434	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
438	C₂³	—	2	1	no	—
439	C₂³	C₂₃×#120	1	1	yes	characteristic
440	Q₈	—	2	1	no	—
441	Q₈	—	2	4	no	—
442	Q₈	—	4	4	no	—
443	Q₈	—	4	4	no	—
444	Q₈	—	4	2	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	8	4	no	—
448	Q₈	—	8	2	no	—
449	Q₈	C₂×SD₁₆×C₂₃	1	2	yes	—
450	D₄	—	2	4	no	—
451	D₄	—	2	1	no	—
452	D₄	C₂×SD₁₆×C₂₃	1	2	yes	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	—	8	4	no	—
467	C₂×C₄	—	8	2	no	—
468	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
469	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
470	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
471	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
472	C₂×C₄	C₂₃×#137	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	C₈	—	4	2	no	—
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	4	1	no	—
483	C₂²	—	4	1	no	—
484	C₂²	C₂₃×#448	1	2	yes	—
485	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	4	no	—
493	C₄	—	4	2	no	—
494	C₄	—	8	2	no	—
495	C₄	C₂₃×#448	1	2	yes	—
496	C₄	C₂₃×#455	1	2	yes	—
497	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
498	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
499	C₄	C₂₃×#578	1	2	yes	—
500	C₂	—	2	2	no	—
501	C₂	—	4	2	no	—
502	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
503	C₂	C₂₃×#3011	1	1	yes	characteristic
504	C₂	C₂×C₂₃×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3229	C₂	semidirect
3. C₂×C₂₃×#487	C₂	semidirect
5. C₂₃×#1645	C₂	semidirect
6. C₂₃×#1643	C₂	semidirect
7. C₂₃×#1548	C₂	semidirect
8. C₂₃×#1493	C₂	semidirect
185. #18632	C₂₃	direct (N × H)
228. #3229	C₂×C₂₃	semidirect
229. C₂×#487	C₂×C₂₃	semidirect
231. #1645	C₂×C₂₃	semidirect
232. #1643	C₂×C₂₃	semidirect
233. #1548	C₂×C₂₃	semidirect
234. #1493	C₂×C₂₃	semidirect
369. C₂₃	#18632	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×#3229	2
C₂×C₂₃×#487	2
C₂₃×#3039	2
C₂₃×#1645	2
C₂₃×#1643	2
C₂₃×#1548	2
C₂₃×#1493	2
C₂₃×#1476	2
#18632	23

Subgroups — C23×#18632 #19166258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18632 #19166258