Subgroups — C₂₃×#17986 #19165612

Order 5888.

578 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3172	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2966	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1588	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1587	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1584	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1582	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1497	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1495	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#529	C₂²	1	1	yes	characteristic
14	C₂₃×#501	C₄	1	1	yes	characteristic
15	C₂₃×#497	C₄	1	1	yes	characteristic
16	C₂₃×#491	C₄	1	1	yes	characteristic
17	C₂₃×#489	C₄	1	1	yes	characteristic
18	C₂₃×#470	C₄	1	2	yes	—
19	C₂₃×#468	C₄	1	2	yes	—
20	C₂₃×#446	—	2	1	no	—
21	C₂₃×#446	C₂²	1	2	yes	—
22	C₂₃×#446	C₂²	1	1	yes	characteristic
23	C₈×Q₈×C₂₃	—	2	1	no	—
24	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
27	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
28	C₂₃×#436	C₂²	1	1	yes	characteristic
29	C₂₃×#435	—	2	1	no	—
30	C₈×D₄×C₂₃	—	2	1	no	—
31	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#433	—	2	1	no	—
33	C₂₃×#432	C₂²	1	1	yes	characteristic
34	C₂₃×#431	C₂²	1	1	yes	characteristic
35	C₂₃×#424	—	2	1	no	—
36	C₂₃×#423	C₂²	1	1	yes	characteristic
37	C₂₃×#405	—	2	1	no	—
38	C₂₃×#405	—	2	1	no	—
39	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#335	C₂²	1	2	yes	—
42	C₂₃×#334	C₂²	1	2	yes	—
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
47	C₂₃×#322	C₂²	1	2	yes	—
48	C₂₃×#322	C₂²	1	2	yes	—
49	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₂₃	—	2	1	no	—
55	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
57	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
59	C₂₃×#126	—	2	1	no	—
60	C₂₃×#125	—	2	1	no	—
61	C₂₃×#125	C₂×C₄	1	1	yes	characteristic
62	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
63	C₂₃×#123	—	2	1	no	—
64	C₂₃×C₂²⋊Q₈	—	2	1	no	—
65	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
66	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₂₃×#121	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₂₃	—	2	1	no	—
69	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
70	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₄×D₄×C₂₃	—	2	1	no	—
72	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂₃×#117	—	2	1	no	—
74	C₂₃×#117	—	2	1	no	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×#107	C₂×C₄	1	2	yes	—
77	C₂₃×#106	C₂×C₄	1	2	yes	—
78	C₂₃×C₄⋊C₈	—	2	2	no	—
79	C₂₃×C₄⋊C₈	—	2	2	no	—
80	C₂₃×C₄⋊C₈	—	2	1	no	—
81	C₂₃×C₄⋊C₈	—	2	1	no	—
82	C₂₃×C₄⋊C₈	—	2	1	no	—
83	C₂₃×C₄⋊C₈	—	2	1	no	—
84	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
85	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₂₃×#103	C₂×C₄	1	2	yes	—
88	C₂₃×#103	C₂×C₄	1	2	yes	—
89	C₂₃×#103	C₂×C₄	1	2	yes	—
90	C₂₃×#102	C₂×C₄	1	2	yes	—
91	C₂₃×C₂²⋊C₈	—	2	1	no	—
92	C₂₃×C₂²⋊C₈	—	2	1	no	—
93	C₂₃×C₂²⋊C₈	—	2	1	no	—
94	C₂₃×C₂²⋊C₈	—	2	1	no	—
95	C₂₃×#97	—	2	1	no	—
96	C₂₃×#97	—	2	2	no	—
97	C₂₃×#97	—	2	1	no	—
98	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
99	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
100	C₄×C₈×C₂₃	—	2	1	no	—
101	C₄×C₈×C₂₃	—	2	2	no	—
102	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
103	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
104	C₄×C₈×C₂₃	C₂³	1	2	yes	—
105	C₄×C₈×C₂₃	C₂³	1	2	yes	—
106	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₂₃	—	2	1	no	—
112	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	—	2	1	no	—
114	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
117	Q₁₆×C₂₃	—	4	2	no	—
118	SD₁₆×C₂₃	—	4	2	no	—
119	M₁₆×C₂₃	—	4	2	no	—
120	M₁₆×C₂₃	—	4	2	no	—
121	M₁₆×C₂₃	—	4	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	—	2	1	no	—
125	C₂×C₈×C₂₃	—	2	1	no	—
126	C₂×C₈×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	4	1	no	—
128	C₂×C₈×C₂₃	—	4	2	no	—
129	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	2	1	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
145	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²⋊C₄×C₂₃	—	2	2	no	—
148	C₂²⋊C₄×C₂₃	—	2	1	no	—
149	C₂²⋊C₄×C₂₃	—	2	1	no	—
150	C₂²⋊C₄×C₂₃	—	2	1	no	—
151	C₂²⋊C₄×C₂₃	—	2	1	no	—
152	C₂²⋊C₄×C₂₃	—	2	1	no	—
153	C₄²×C₂₃	—	2	1	no	—
154	C₄²×C₂₃	—	2	2	no	—
155	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₂₃	#41	1	1	yes	characteristic
158	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
159	#17986	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₂₃	—	2	1	no	—
161	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
162	Q₈×C₂₃	—	2	2	no	—
163	Q₈×C₂₃	—	2	2	no	—
164	Q₈×C₂₃	—	2	2	no	—
165	Q₈×C₂₃	—	4	2	no	—
166	D₄×C₂₃	—	2	2	no	—
167	D₄×C₂₃	—	4	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	2	1	no	—
171	C₂×C₄×C₂₃	—	2	1	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	1	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	4	1	no	—
177	C₂×C₄×C₂₃	—	4	1	no	—
178	C₂×C₄×C₂₃	—	4	1	no	—
179	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
185	C₈×C₂₃	—	2	2	no	—
186	C₈×C₂₃	—	2	2	no	—
187	C₈×C₂₃	—	2	1	no	—
188	C₈×C₂₃	—	2	1	no	—
189	C₈×C₂₃	—	2	1	no	—
190	C₈×C₂₃	—	4	2	no	—
191	C₈×C₂₃	—	4	1	no	—
192	C₈×C₂₃	—	4	2	no	—
193	C₈×C₂₃	—	4	1	no	—
194	C₈×C₂₃	#137	1	2	yes	—
195	#3172	C₂×C₂₃	1	1	yes	characteristic
196	#2966	C₂×C₂₃	1	1	yes	characteristic
197	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
198	#1588	C₂×C₂₃	1	2	yes	complemented
199	#1587	C₂×C₂₃	1	2	yes	complemented
200	#1584	C₂×C₂₃	1	2	yes	complemented
201	#1582	C₂×C₂₃	1	2	yes	complemented
202	#1569	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#1497	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#1495	C₂×C₂₃	1	1	yes	characteristic
205	#1451	C₂×C₂₃	1	1	yes	characteristic, complemented
206	C₂²×C₂₃	—	2	1	no	—
207	C₂²×C₂₃	—	2	1	no	—
208	C₂²×C₂₃	—	2	1	no	—
209	C₂²×C₂₃	—	4	1	no	—
210	C₂²×C₂₃	—	4	1	no	—
211	C₂²×C₂₃	—	4	1	no	—
212	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₄×C₂₃	—	2	1	no	—
214	C₄×C₂₃	—	2	1	no	—
215	C₄×C₂₃	—	2	1	no	—
216	C₄×C₂₃	—	2	1	no	—
217	C₄×C₂₃	—	4	1	no	—
218	C₄×C₂₃	—	4	2	no	—
219	C₄×C₂₃	—	4	1	no	—
220	C₄×C₂₃	—	4	1	no	—
221	C₄×C₂₃	—	4	2	no	—
222	C₄×C₂₃	#568	1	2	yes	—
223	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
224	C₄×C₂₃	#575	1	1	yes	characteristic
225	#533	C₂²×C₂₃	1	1	yes	characteristic
226	#529	C₂²×C₂₃	1	1	yes	characteristic
227	#501	C₄×C₂₃	1	1	yes	characteristic
228	#497	C₄×C₂₃	1	1	yes	characteristic
229	#491	C₄×C₂₃	1	1	yes	characteristic
230	#489	C₄×C₂₃	1	1	yes	characteristic
231	#470	C₄×C₂₃	1	2	yes	—
232	#468	C₄×C₂₃	1	2	yes	—
233	#446	—	2	1	no	—
234	#446	C₂²×C₂₃	1	2	yes	—
235	#446	C₂²×C₂₃	1	1	yes	characteristic
236	C₈×Q₈	—	2	1	no	—
237	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
238	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
239	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
240	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
241	#436	C₂²×C₂₃	1	1	yes	characteristic
242	#435	—	2	1	no	—
243	C₈×D₄	—	2	1	no	—
244	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
245	#433	—	2	1	no	—
246	#432	C₂²×C₂₃	1	1	yes	characteristic
247	#431	C₂²×C₂₃	1	1	yes	characteristic
248	#424	—	2	1	no	—
249	#423	C₂²×C₂₃	1	1	yes	characteristic
250	#405	—	2	1	no	—
251	#405	—	2	1	no	—
252	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
254	#335	C₂²×C₂₃	1	2	yes	—
255	#334	C₂²×C₂₃	1	2	yes	—
256	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
257	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
258	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
259	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
260	#322	C₂²×C₂₃	1	2	yes	—
261	#322	C₂²×C₂₃	1	2	yes	—
262	C₂×C₂₃	—	4	1	no	—
263	C₂×C₂₃	—	8	1	no	—
264	C₂×C₂₃	#2932	1	1	yes	characteristic
265	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
266	C₂×C₂₃	#2955	1	1	yes	characteristic
267	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
268	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
269	C₂×M₁₆	—	2	1	no	—
270	C₂×M₁₆	—	2	1	no	—
271	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
272	C₂²×C₈	—	2	1	no	—
273	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
274	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
275	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
276	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
277	#126	—	2	1	no	—
278	#125	—	2	1	no	—
279	#125	C₂×C₄×C₂₃	1	1	yes	characteristic
280	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
281	#123	—	2	1	no	—
282	C₂²⋊Q₈	—	2	1	no	—
283	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
284	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
285	#121	C₂×C₄×C₂₃	1	1	yes	characteristic
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
288	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
289	C₄×D₄	—	2	1	no	—
290	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
291	#117	—	2	1	no	—
292	#117	—	2	1	no	—
293	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
294	#107	C₂×C₄×C₂₃	1	2	yes	—
295	#106	C₂×C₄×C₂₃	1	2	yes	—
296	C₄⋊C₈	—	2	2	no	—
297	C₄⋊C₈	—	2	2	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	—	2	1	no	—
301	C₄⋊C₈	—	2	1	no	—
302	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
303	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
304	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
305	#103	C₂×C₄×C₂₃	1	2	yes	—
306	#103	C₂×C₄×C₂₃	1	2	yes	—
307	#103	C₂×C₄×C₂₃	1	2	yes	—
308	#102	C₂×C₄×C₂₃	1	2	yes	—
309	C₂²⋊C₈	—	2	1	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₂²⋊C₈	—	2	1	no	—
313	#97	—	2	1	no	—
314	#97	—	2	2	no	—
315	#97	—	2	1	no	—
316	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
317	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
321	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
322	C₄×C₈	C₂³×C₂₃	1	2	yes	—
323	C₄×C₈	C₂³×C₂₃	1	2	yes	—
324	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
326	C₂₃	#17986	1	1	yes	characteristic, direct product factor, Sylow
327	C₂×Q₈	—	2	1	no	—
328	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
329	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
335	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
336	Q₁₆	—	4	2	no	—
337	SD₁₆	—	4	2	no	—
338	M₁₆	—	4	2	no	—
339	M₁₆	—	4	2	no	—
340	M₁₆	—	4	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	4	1	no	—
347	C₂×C₈	—	4	2	no	—
348	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
349	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
350	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
351	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
364	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
365	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	2	no	—
374	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
375	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₄²	#41×C₂₃	1	1	yes	characteristic
377	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
378	C₂³	—	2	1	no	—
379	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
380	Q₈	—	2	2	no	—
381	Q₈	—	2	2	no	—
382	Q₈	—	2	2	no	—
383	Q₈	—	4	2	no	—
384	D₄	—	2	2	no	—
385	D₄	—	4	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
398	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
399	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
400	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
403	C₈	—	2	2	no	—
404	C₈	—	2	2	no	—
405	C₈	—	2	1	no	—
406	C₈	—	2	1	no	—
407	C₈	—	2	1	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	1	no	—
411	C₈	—	4	2	no	—
412	C₈	C₂₃×#137	1	2	yes	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	4	1	no	—
417	C₂²	—	4	1	no	—
418	C₂²	—	4	1	no	—
419	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	2	1	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	2	no	—
426	C₄	—	4	1	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	2	no	—
429	C₄	C₂₃×#568	1	2	yes	—
430	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
431	C₄	C₂₃×#575	1	1	yes	characteristic
432	C₂	—	4	1	no	—
433	C₂	—	8	1	no	—
434	C₂	C₂₃×#2932	1	1	yes	characteristic
435	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
436	C₂	C₂₃×#2955	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×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2960	C₂	semidirect
4. C₂₃×#1588	C₂	semidirect
5. C₂₃×#1587	C₂	semidirect
6. C₂₃×#1584	C₂	semidirect
7. C₂₃×#1582	C₂	semidirect
8. C₂₃×#1569	C₂	semidirect
9. C₂₃×#1497	C₂	semidirect
11. C₂₃×#1451	C₂	semidirect
159. #17986	C₂₃	direct (N × H)
197. #2960	C₂×C₂₃	semidirect
198. #1588	C₂×C₂₃	semidirect
199. #1587	C₂×C₂₃	semidirect
200. #1584	C₂×C₂₃	semidirect
201. #1582	C₂×C₂₃	semidirect
202. #1569	C₂×C₂₃	semidirect
203. #1497	C₂×C₂₃	semidirect
205. #1451	C₂×C₂₃	semidirect
326. C₂₃	#17986	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×#3172	2
C₂₃×#2966	2
C₂₃×#2960	2
C₂₃×#1588	2
C₂₃×#1587	2
C₂₃×#1584	2
C₂₃×#1582	2
C₂₃×#1569	2
C₂₃×#1497	2
C₂₃×#1495	2
C₂₃×#1451	2
#17986	23

Subgroups — C23×#17986 #19165612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17986 #19165612