Subgroups — C₂₃×#18676 #19166302

Order 5888.

912 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3246	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3037	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1653	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1645	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1626	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1623	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#137	—	2	1	no	—
13	Q₈²×C₂₃	—	2	1	no	—
14	Q₈²×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#557	—	2	1	no	—
16	C₂₃×#557	—	2	2	no	—
17	C₂₃×#557	—	2	2	no	—
18	C₂₃×#557	C₂²	1	2	yes	—
19	C₂₃×#544	—	2	2	no	—
20	C₂₃×#544	—	2	1	no	—
21	C₂₃×#533	—	2	2	no	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#533	C₂²	1	1	yes	characteristic
24	C₂₃×#532	C₂²	1	1	yes	characteristic
25	C₂₃×#519	—	2	1	no	—
26	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#498	C₂²	1	1	yes	characteristic
28	C₂₃×#492	C₂²	1	1	yes	characteristic
29	C₂₃×#487	C₂²	1	1	yes	characteristic
30	C₂₃×#486	C₂²	1	1	yes	characteristic
31	C₂₃×#476	—	2	1	no	—
32	C₂₃×#475	C₂²	1	2	yes	—
33	C₂₃×#465	—	2	2	no	—
34	C₂₃×#465	—	2	2	no	—
35	C₂₃×#464	—	2	2	no	—
36	C₂₃×#462	C₂²	1	2	yes	—
37	C₂₃×#461	—	2	1	no	—
38	C₂₃×#460	C₂²	1	2	yes	—
39	C₂₃×#460	C₂²	1	2	yes	—
40	C₄×SD₁₆×C₂₃	—	2	2	no	—
41	C₂₃×#423	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂₃×#417	—	2	1	no	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#333	C₂²	1	2	yes	—
46	C₂₃×#332	C₂²	1	2	yes	—
47	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₂₃	—	2	1	no	—
52	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
53	C₂₃×#137	—	4	4	no	—
54	C₂×Q₁₆×C₂₃	—	2	2	no	—
55	C₂×SD₁₆×C₂₃	—	2	2	no	—
56	C₂×SD₁₆×C₂₃	—	2	2	no	—
57	C₂×SD₁₆×C₂₃	—	2	2	no	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	1	no	—
60	C₂²×C₈×C₂₃	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	4	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	—	2	4	no	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
71	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₂₃×#127	C₂³	1	1	yes	characteristic
73	C₂₃×#125	—	2	4	no	—
74	C₂₃×#125	—	2	4	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	4	no	—
77	C₂₃×#125	—	2	2	no	—
78	C₂₃×#125	—	2	1	no	—
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×#124	—	2	1	no	—
81	C₂₃×C₂²⋊Q₈	—	2	4	no	—
82	C₂₃×C₂²⋊Q₈	—	2	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
85	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
86	C₂₃×#121	—	2	1	no	—
87	C₂₃×#121	—	2	1	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	4	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	—	2	2	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	—	2	2	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×Q₈×C₂₃	—	2	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×Q₈×C₂₃	D₄	1	2	yes	—
100	C₄×Q₈×C₂₃	D₄	1	2	yes	—
101	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
102	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₄×D₄×C₂₃	—	2	1	no	—
105	C₂₃×#117	—	2	4	no	—
106	C₂₃×#117	—	2	1	no	—
107	C₂₃×#117	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	—	2	1	no	—
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×C₄⋊C₈	—	2	1	no	—
115	C₂₃×C₄⋊C₈	—	2	1	no	—
116	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#102	—	2	2	no	—
124	C₄×C₈×C₂₃	—	2	1	no	—
125	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
127	#41×C₂₃	—	4	2	no	—
128	#41×C₂₃	—	4	2	no	—
129	C₂×Q₈×C₂₃	—	2	4	no	—
130	C₂×Q₈×C₂₃	—	2	1	no	—
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	—	2	2	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	SD₁₆	1	4	yes	complemented
136	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
139	C₂×D₄×C₂₃	—	2	1	no	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	1	no	—
142	C₂×D₄×C₂₃	—	4	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	4	1	no	—
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	Q₁₆×C₂₃	—	4	4	no	—
150	SD₁₆×C₂₃	—	4	4	no	—
151	SD₁₆×C₂₃	—	4	4	no	—
152	SD₁₆×C₂₃	—	8	2	no	—
153	M₁₆×C₂₃	—	4	2	no	—
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	2	no	—
158	C₂×C₈×C₂₃	—	4	2	no	—
159	C₂×C₈×C₂₃	—	8	1	no	—
160	C₄⋊C₄×C₂₃	—	2	4	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	4	no	—
164	C₄⋊C₄×C₂₃	—	2	4	no	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	2	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
182	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	—	2	2	no	—
185	C₂²⋊C₄×C₂₃	—	4	1	no	—
186	C₂²⋊C₄×C₂₃	—	4	1	no	—
187	C₄²×C₂₃	—	2	4	no	—
188	C₄²×C₂₃	—	2	4	no	—
189	C₄²×C₂₃	—	2	2	no	—
190	C₄²×C₂₃	—	2	2	no	—
191	C₄²×C₂₃	—	2	2	no	—
192	C₄²×C₂₃	—	2	2	no	—
193	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
197	#18676	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₂₃	—	4	1	no	—
199	C₂³×C₂₃	#120	1	1	yes	characteristic
200	Q₈×C₂₃	—	2	2	no	—
201	Q₈×C₂₃	—	2	4	no	—
202	Q₈×C₂₃	—	2	2	no	—
203	Q₈×C₂₃	—	2	2	no	—
204	Q₈×C₂₃	—	4	2	no	—
205	Q₈×C₂₃	—	4	1	no	—
206	Q₈×C₂₃	—	4	1	no	—
207	Q₈×C₂₃	—	4	1	no	—
208	Q₈×C₂₃	—	4	1	no	—
209	Q₈×C₂₃	—	8	1	no	—
210	Q₈×C₂₃	—	8	4	no	—
211	Q₈×C₂₃	C₂×SD₁₆	1	4	yes	—
212	D₄×C₂₃	—	4	2	no	—
213	D₄×C₂₃	—	4	1	no	—
214	D₄×C₂₃	—	4	1	no	—
215	D₄×C₂₃	—	4	1	no	—
216	D₄×C₂₃	—	4	1	no	—
217	D₄×C₂₃	—	8	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	8	1	no	—
230	C₂×C₄×C₂₃	—	8	2	no	—
231	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
232	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
233	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
235	C₂×C₄×C₂₃	#137	1	2	yes	—
236	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
239	C₈×C₂₃	—	2	2	no	—
240	C₈×C₂₃	—	4	1	no	—
241	C₈×C₂₃	—	4	2	no	—
242	C₈×C₂₃	—	4	2	no	—
243	#3502	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#3246	C₂×C₂₃	1	1	yes	characteristic
245	#3167	C₂×C₂₃	1	1	yes	characteristic
246	#3037	C₂×C₂₃	1	1	yes	characteristic
247	#1653	C₂×C₂₃	1	2	yes	complemented
248	#1645	C₂×C₂₃	1	2	yes	complemented
249	#1626	C₂×C₂₃	1	2	yes	complemented
250	#1623	C₂×C₂₃	1	2	yes	complemented
251	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
252	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
254	C₂²×C₂₃	—	2	1	no	—
255	C₂²×C₂₃	—	2	1	no	—
256	C₂²×C₂₃	—	8	1	no	—
257	C₂²×C₂₃	—	8	1	no	—
258	C₂²×C₂₃	—	8	1	no	—
259	C₂²×C₂₃	#448	1	2	yes	—
260	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
261	C₄×C₂₃	—	2	4	no	—
262	C₄×C₂₃	—	2	1	no	—
263	C₄×C₂₃	—	2	1	no	—
264	C₄×C₂₃	—	2	1	no	—
265	C₄×C₂₃	—	4	2	no	—
266	C₄×C₂₃	—	4	2	no	—
267	C₄×C₂₃	—	4	2	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	2	no	—
270	C₄×C₂₃	—	8	1	no	—
271	C₄×C₂₃	#448	1	2	yes	—
272	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
273	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
274	C₄×C₂₃	#577	1	1	yes	characteristic
275	C₄×C₂₃	#578	1	1	yes	characteristic
276	C₂×#137	—	2	1	no	—
277	Q₈²	—	2	1	no	—
278	Q₈²	C₂²×C₂₃	1	2	yes	—
279	#557	—	2	1	no	—
280	#557	—	2	2	no	—
281	#557	—	2	2	no	—
282	#557	C₂²×C₂₃	1	2	yes	—
283	#544	—	2	2	no	—
284	#544	—	2	1	no	—
285	#533	—	2	2	no	—
286	#533	C₂²×C₂₃	1	1	yes	characteristic
287	#533	C₂²×C₂₃	1	1	yes	characteristic
288	#532	C₂²×C₂₃	1	1	yes	characteristic
289	#519	—	2	1	no	—
290	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
291	#498	C₂²×C₂₃	1	1	yes	characteristic
292	#492	C₂²×C₂₃	1	1	yes	characteristic
293	#487	C₂²×C₂₃	1	1	yes	characteristic
294	#486	C₂²×C₂₃	1	1	yes	characteristic
295	#476	—	2	1	no	—
296	#475	C₂²×C₂₃	1	2	yes	—
297	#465	—	2	2	no	—
298	#465	—	2	2	no	—
299	#464	—	2	2	no	—
300	#462	C₂²×C₂₃	1	2	yes	—
301	#461	—	2	1	no	—
302	#460	C₂²×C₂₃	1	2	yes	—
303	#460	C₂²×C₂₃	1	2	yes	—
304	C₄×SD₁₆	—	2	2	no	—
305	#423	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
307	#417	—	2	1	no	—
308	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
309	#333	C₂²×C₂₃	1	2	yes	—
310	#332	C₂²×C₂₃	1	2	yes	—
311	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
312	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
313	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
314	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
315	C₂×C₂₃	—	2	2	no	—
316	C₂×C₂₃	—	16	1	no	—
317	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
318	C₂×C₂₃	#3013	1	1	yes	characteristic
319	C₂×C₂₃	#3018	1	1	yes	characteristic
320	C₂×#41	—	2	1	no	—
321	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
322	#137	—	4	4	no	—
323	C₂×Q₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	2	2	no	—
325	C₂×SD₁₆	—	2	2	no	—
326	C₂×SD₁₆	—	2	2	no	—
327	C₂×M₁₆	—	2	1	no	—
328	C₂²×C₈	—	2	1	no	—
329	C₂²×C₈	—	2	1	no	—
330	C₄⋊Q₈	—	2	4	no	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	—	2	4	no	—
334	C₄⋊Q₈	—	2	2	no	—
335	C₄⋊Q₈	—	2	2	no	—
336	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
337	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
341	#127	C₂³×C₂₃	1	1	yes	characteristic
342	#125	—	2	4	no	—
343	#125	—	2	4	no	—
344	#125	—	2	2	no	—
345	#125	—	2	4	no	—
346	#125	—	2	2	no	—
347	#125	—	2	1	no	—
348	#125	—	2	1	no	—
349	#124	—	2	1	no	—
350	C₂²⋊Q₈	—	2	4	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
354	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
355	#121	—	2	1	no	—
356	#121	—	2	1	no	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	4	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	—	2	2	no	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	D₄×C₂₃	1	2	yes	—
369	C₄×Q₈	D₄×C₂₃	1	2	yes	—
370	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
371	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	—	2	1	no	—
374	#117	—	2	4	no	—
375	#117	—	2	1	no	—
376	#117	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
381	#106	—	2	2	no	—
382	#106	—	2	2	no	—
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
386	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#103	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	C₄×C₈	—	2	1	no	—
394	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
395	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
396	C₂₃	#18676	1	1	yes	characteristic, direct product factor, Sylow
397	#41	—	4	2	no	—
398	#41	—	4	2	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	SD₁₆×C₂₃	1	4	yes	complemented
406	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
407	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
408	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	4	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	4	1	no	—
416	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
418	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
419	Q₁₆	—	4	4	no	—
420	SD₁₆	—	4	4	no	—
421	SD₁₆	—	4	4	no	—
422	SD₁₆	—	8	2	no	—
423	M₁₆	—	4	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	4	2	no	—
429	C₂×C₈	—	8	1	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
451	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
452	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	—	4	1	no	—
456	C₂²⋊C₄	—	4	1	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	4	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
467	C₂³	—	4	1	no	—
468	C₂³	C₂₃×#120	1	1	yes	characteristic
469	Q₈	—	2	2	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	2	2	no	—
472	Q₈	—	2	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	1	no	—
475	Q₈	—	4	1	no	—
476	Q₈	—	4	1	no	—
477	Q₈	—	4	1	no	—
478	Q₈	—	8	1	no	—
479	Q₈	—	8	4	no	—
480	Q₈	C₂×SD₁₆×C₂₃	1	4	yes	—
481	D₄	—	4	2	no	—
482	D₄	—	4	1	no	—
483	D₄	—	4	1	no	—
484	D₄	—	4	1	no	—
485	D₄	—	4	1	no	—
486	D₄	—	8	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	8	1	no	—
499	C₂×C₄	—	8	2	no	—
500	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
501	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
502	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
503	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
504	C₂×C₄	C₂₃×#137	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
508	C₈	—	2	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	2	no	—
511	C₈	—	4	2	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	8	1	no	—
515	C₂²	—	8	1	no	—
516	C₂²	—	8	1	no	—
517	C₂²	C₂₃×#448	1	2	yes	—
518	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
519	C₄	—	2	4	no	—
520	C₄	—	2	1	no	—
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	8	1	no	—
529	C₄	C₂₃×#448	1	2	yes	—
530	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
531	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
532	C₄	C₂₃×#577	1	1	yes	characteristic
533	C₄	C₂₃×#578	1	1	yes	characteristic
534	C₂	—	2	2	no	—
535	C₂	—	16	1	no	—
536	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
537	C₂	C₂₃×#3013	1	1	yes	characteristic
538	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
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
1. C₂₃×#3502	C₂	semidirect
5. C₂₃×#1653	C₂	semidirect
6. C₂₃×#1645	C₂	semidirect
7. C₂₃×#1626	C₂	semidirect
8. C₂₃×#1623	C₂	semidirect
9. C₂₃×#1550	C₂	semidirect
10. C₂₃×#1493	C₂	semidirect
11. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
135. C₂×Q₈×C₂₃	SD₁₆	semidirect
197. #18676	C₂₃	direct (N × H)
243. #3502	C₂×C₂₃	semidirect
247. #1653	C₂×C₂₃	semidirect
248. #1645	C₂×C₂₃	semidirect
249. #1626	C₂×C₂₃	semidirect
250. #1623	C₂×C₂₃	semidirect
251. #1550	C₂×C₂₃	semidirect
252. #1493	C₂×C₂₃	semidirect
253. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
396. C₂₃	#18676	direct (N × H)
405. C₂×Q₈	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3502	2
C₂₃×#3246	2
C₂₃×#3167	2
C₂₃×#3037	2
C₂₃×#1653	2
C₂₃×#1645	2
C₂₃×#1626	2
C₂₃×#1623	2
C₂₃×#1550	2
C₂₃×#1493	2
C₂×C₂₃×Q₈⋊C₈	2
#18676	23

Subgroups — C23×#18676 #19166302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18676 #19166302