Subgroups — C₂₃×#18085 #19165711

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2979	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1590	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1588	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1585	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1484	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂₃×#498	C₄	1	1	yes	characteristic
15	C₂₃×#492	C₄	1	1	yes	characteristic
16	C₂₃×#487	C₄	1	1	yes	characteristic
17	C₂₃×#486	C₄	1	1	yes	characteristic
18	C₂₃×#465	C₄	1	2	yes	—
19	C₂₃×#465	C₄	1	2	yes	—
20	C₂₃×#446	—	2	1	no	—
21	C₂₃×#446	C₂²	1	2	yes	—
22	C₈×Q₈×C₂₃	—	2	1	no	—
23	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#436	—	2	1	no	—
27	C₂₃×#436	C₂²	1	2	yes	—
28	C₈×D₄×C₂₃	—	2	1	no	—
29	C₈×D₄×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#433	—	2	2	no	—
31	C₂₃×#431	C₂²	1	1	yes	characteristic
32	C₂₃×#431	C₂²	1	1	yes	characteristic
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#408	—	2	2	no	—
35	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#335	C₂²	1	1	yes	characteristic
38	C₂₃×#335	C₂²	1	1	yes	characteristic
39	C₂₃×#334	C₂²	1	1	yes	characteristic
40	C₂₃×#334	C₂²	1	1	yes	characteristic
41	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₂₃×#322	C₂²	1	2	yes	—
46	C₈²×C₂₃	C₂²	1	2	yes	—
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₂₃	—	2	1	no	—
51	C₂×M₁₆×C₂₃	—	2	2	no	—
52	C₂²×C₈×C₂₃	—	2	2	no	—
53	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
56	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
59	C₂₃×#125	—	2	1	no	—
60	C₂₃×#124	—	2	1	no	—
61	C₂₃×C₂²⋊Q₈	—	2	1	no	—
62	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
63	C₂₃×#121	—	2	1	no	—
64	C₂₃×#121	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₂₃	—	2	1	no	—
66	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
67	C₄×D₄×C₂₃	—	2	1	no	—
68	C₄×D₄×C₂₃	C₂³	1	2	yes	—
69	C₂₃×#117	—	2	1	no	—
70	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×#106	C₂×C₄	1	2	yes	—
72	C₂₃×#106	C₂×C₄	1	2	yes	—
73	C₂₃×C₄⋊C₈	—	2	2	no	—
74	C₂₃×C₄⋊C₈	—	2	2	no	—
75	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
77	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₂₃×#103	C₂×C₄	1	2	yes	—
82	C₂₃×#103	C₂×C₄	1	2	yes	—
83	C₂₃×#102	C₂×C₄	1	2	yes	—
84	C₂₃×#102	C₂×C₄	1	2	yes	—
85	C₂₃×C₂²⋊C₈	—	2	1	no	—
86	C₂₃×C₂²⋊C₈	—	2	1	no	—
87	C₂₃×C₂²⋊C₈	—	2	2	no	—
88	C₂₃×C₂²⋊C₈	—	2	2	no	—
89	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
90	C₂₃×#97	—	2	2	no	—
91	C₂₃×#97	—	2	1	no	—
92	C₄×C₈×C₂₃	—	2	2	no	—
93	C₄×C₈×C₂₃	—	2	1	no	—
94	C₄×C₈×C₂₃	—	2	1	no	—
95	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₂₃	C₂³	1	2	yes	—
98	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₂₃	C₂³	1	2	yes	—
101	#41×C₂₃	—	8	1	no	—
102	#41×C₂₃	—	8	1	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₂₃	—	2	2	no	—
107	C₂×D₄×C₂₃	—	2	1	no	—
108	C₂×D₄×C₂₃	—	2	1	no	—
109	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₂₃	—	2	2	no	—
111	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
114	SD₁₆×C₂₃	—	2	4	no	—
115	SD₁₆×C₂₃	—	4	2	no	—
116	M₁₆×C₂₃	—	2	4	no	—
117	M₁₆×C₂₃	—	8	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	—	2	2	no	—
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	—	2	1	no	—
122	C₂×C₈×C₂₃	—	2	1	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	4	1	no	—
125	C₂×C₈×C₂₃	—	8	2	no	—
126	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	1	no	—
134	C₄⋊C₄×C₂₃	—	2	1	no	—
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	2	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
140	C₂²⋊C₄×C₂₃	—	2	2	no	—
141	C₂²⋊C₄×C₂₃	—	2	2	no	—
142	C₂²⋊C₄×C₂₃	—	2	2	no	—
143	C₄²×C₂₃	—	2	2	no	—
144	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₂₃	#41	1	1	yes	characteristic
147	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
148	#18085	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₂₃	—	2	2	no	—
150	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
151	Q₈×C₂₃	—	2	2	no	—
152	Q₈×C₂₃	—	2	2	no	—
153	Q₈×C₂₃	—	4	2	no	—
154	Q₈×C₂₃	—	8	1	no	—
155	D₄×C₂₃	—	2	2	no	—
156	D₄×C₂₃	—	2	2	no	—
157	D₄×C₂₃	—	4	2	no	—
158	D₄×C₂₃	—	8	1	no	—
159	D₄×C₂₃	—	8	2	no	—
160	C₂×C₄×C₂₃	—	2	1	no	—
161	C₂×C₄×C₂₃	—	2	1	no	—
162	C₂×C₄×C₂₃	—	2	1	no	—
163	C₂×C₄×C₂₃	—	2	1	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	4	2	no	—
169	C₂×C₄×C₂₃	—	4	2	no	—
170	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
176	C₈×C₂₃	—	2	4	no	—
177	C₈×C₂₃	—	2	1	no	—
178	C₈×C₂₃	—	2	1	no	—
179	C₈×C₂₃	—	2	1	no	—
180	C₈×C₂₃	—	2	2	no	—
181	C₈×C₂₃	—	2	2	no	—
182	C₈×C₂₃	—	4	1	no	—
183	C₈×C₂₃	—	4	1	no	—
184	C₈×C₂₃	—	4	1	no	—
185	C₈×C₂₃	—	4	1	no	—
186	C₈×C₂₃	#131	1	2	yes	—
187	#3167	C₂×C₂₃	1	1	yes	characteristic
188	#2986	C₂×C₂₃	1	1	yes	characteristic, complemented
189	#2979	C₂×C₂₃	1	1	yes	characteristic
190	#1590	C₂×C₂₃	1	2	yes	complemented
191	#1588	C₂×C₂₃	1	2	yes	complemented
192	#1585	C₂×C₂₃	1	2	yes	complemented
193	C₈×SD₁₆	C₂×C₂₃	1	2	yes	complemented
194	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
195	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
196	#1484	C₂×C₂₃	1	1	yes	characteristic
197	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
198	C₂²×C₂₃	—	2	1	no	—
199	C₂²×C₂₃	—	2	1	no	—
200	C₂²×C₂₃	—	2	1	no	—
201	C₂²×C₂₃	—	4	2	no	—
202	C₂²×C₂₃	—	4	2	no	—
203	C₂²×C₂₃	—	4	2	no	—
204	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
205	C₄×C₂₃	—	2	1	no	—
206	C₄×C₂₃	—	2	1	no	—
207	C₄×C₂₃	—	2	1	no	—
208	C₄×C₂₃	—	4	2	no	—
209	C₄×C₂₃	—	4	2	no	—
210	C₄×C₂₃	—	4	2	no	—
211	C₄×C₂₃	#443	1	2	yes	—
212	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
213	C₄×C₂₃	#568	1	1	yes	characteristic
214	C₄×C₂₃	#577	1	1	yes	characteristic
215	C₄×C₂₃	#577	1	1	yes	characteristic
216	#533	C₂²×C₂₃	1	1	yes	characteristic
217	#532	C₂²×C₂₃	1	1	yes	characteristic
218	#498	C₄×C₂₃	1	1	yes	characteristic
219	#492	C₄×C₂₃	1	1	yes	characteristic
220	#487	C₄×C₂₃	1	1	yes	characteristic
221	#486	C₄×C₂₃	1	1	yes	characteristic
222	#465	C₄×C₂₃	1	2	yes	—
223	#465	C₄×C₂₃	1	2	yes	—
224	#446	—	2	1	no	—
225	#446	C₂²×C₂₃	1	2	yes	—
226	C₈×Q₈	—	2	1	no	—
227	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
228	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
229	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
230	#436	—	2	1	no	—
231	#436	C₂²×C₂₃	1	2	yes	—
232	C₈×D₄	—	2	1	no	—
233	C₈×D₄	C₂²×C₂₃	1	2	yes	—
234	#433	—	2	2	no	—
235	#431	C₂²×C₂₃	1	1	yes	characteristic
236	#431	C₂²×C₂₃	1	1	yes	characteristic
237	#431	C₂²×C₂₃	1	1	yes	characteristic
238	#408	—	2	2	no	—
239	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
241	#335	C₂²×C₂₃	1	1	yes	characteristic
242	#335	C₂²×C₂₃	1	1	yes	characteristic
243	#334	C₂²×C₂₃	1	1	yes	characteristic
244	#334	C₂²×C₂₃	1	1	yes	characteristic
245	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
246	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
247	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
248	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
249	#322	C₂²×C₂₃	1	2	yes	—
250	C₈²	C₂²×C₂₃	1	2	yes	—
251	C₂×C₂₃	—	4	1	no	—
252	C₂×C₂₃	—	8	2	no	—
253	C₂×C₂₃	#2935	1	1	yes	characteristic
254	C₂×C₂₃	#2949	1	1	yes	characteristic
255	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
256	C₂×#41	—	2	1	no	—
257	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
258	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
259	C₂×M₁₆	—	2	1	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
263	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
264	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
265	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
266	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
267	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
268	#125	—	2	1	no	—
269	#124	—	2	1	no	—
270	C₂²⋊Q₈	—	2	1	no	—
271	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
272	#121	—	2	1	no	—
273	#121	C₂×C₄×C₂₃	1	2	yes	—
274	C₄×Q₈	—	2	1	no	—
275	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
276	C₄×D₄	—	2	1	no	—
277	C₄×D₄	C₂³×C₂₃	1	2	yes	—
278	#117	—	2	1	no	—
279	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
280	#106	C₂×C₄×C₂₃	1	2	yes	—
281	#106	C₂×C₄×C₂₃	1	2	yes	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
285	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
286	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
289	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
290	#103	C₂×C₄×C₂₃	1	2	yes	—
291	#103	C₂×C₄×C₂₃	1	2	yes	—
292	#102	C₂×C₄×C₂₃	1	2	yes	—
293	#102	C₂×C₄×C₂₃	1	2	yes	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
299	#97	—	2	2	no	—
300	#97	—	2	1	no	—
301	C₄×C₈	—	2	2	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
305	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₄×C₈	C₂³×C₂₃	1	2	yes	—
307	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
308	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
309	C₄×C₈	C₂³×C₂₃	1	2	yes	—
310	C₂₃	#18085	1	1	yes	characteristic, direct product factor, Sylow
311	#41	—	8	1	no	—
312	#41	—	8	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
323	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
324	SD₁₆	—	2	4	no	—
325	SD₁₆	—	4	2	no	—
326	M₁₆	—	2	4	no	—
327	M₁₆	—	8	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	8	2	no	—
336	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
337	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
339	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
349	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
356	C₄²	#41×C₂₃	1	1	yes	characteristic
357	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
358	C₂³	—	2	2	no	—
359	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	Q₈	—	2	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	8	1	no	—
364	D₄	—	2	2	no	—
365	D₄	—	2	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	1	no	—
368	D₄	—	8	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
380	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
381	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
382	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	2	1	no	—
387	C₈	—	2	1	no	—
388	C₈	—	2	1	no	—
389	C₈	—	2	2	no	—
390	C₈	—	2	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	1	no	—
395	C₈	C₂₃×#131	1	2	yes	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	C₂₃×#443	1	2	yes	—
410	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
411	C₄	C₂₃×#568	1	1	yes	characteristic
412	C₄	C₂₃×#577	1	1	yes	characteristic
413	C₄	C₂₃×#577	1	1	yes	characteristic
414	C₂	—	4	1	no	—
415	C₂	—	8	2	no	—
416	C₂	C₂₃×#2935	1	1	yes	characteristic
417	C₂	C₂₃×#2949	1	1	yes	characteristic
418	C₂	C₂×C₂₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2986	C₂	semidirect
4. C₂₃×#1590	C₂	semidirect
5. C₂₃×#1588	C₂	semidirect
6. C₂₃×#1585	C₂	semidirect
7. C₈×SD₁₆×C₂₃	C₂	semidirect
8. C₂₃×#1564	C₂	semidirect
9. C₂₃×#1485	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
148. #18085	C₂₃	direct (N × H)
188. #2986	C₂×C₂₃	semidirect
190. #1590	C₂×C₂₃	semidirect
191. #1588	C₂×C₂₃	semidirect
192. #1585	C₂×C₂₃	semidirect
193. C₈×SD₁₆	C₂×C₂₃	semidirect
194. #1564	C₂×C₂₃	semidirect
195. #1485	C₂×C₂₃	semidirect
197. C₈×M₁₆	C₂×C₂₃	semidirect
310. C₂₃	#18085	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×#3167	2
C₂₃×#2986	2
C₂₃×#2979	2
C₂₃×#1590	2
C₂₃×#1588	2
C₂₃×#1585	2
C₈×SD₁₆×C₂₃	2
C₂₃×#1564	2
C₂₃×#1485	2
C₂₃×#1484	2
C₈×M₁₆×C₂₃	2
#18085	23

Subgroups — C23×#18085 #19165711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18085 #19165711