Subgroups — C₂₃×#17918 #19165544

Order 5888.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3105	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2993	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2972	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1586	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1482	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#484	C₄	1	1	yes	characteristic
19	C₂₃×#481	C₄	1	1	yes	characteristic
20	C₂₃×#478	C₄	1	1	yes	characteristic
21	C₂₃×#478	C₄	1	1	yes	characteristic
22	C₂₃×#463	C₄	1	1	yes	characteristic
23	C₂₃×#463	C₄	1	1	yes	characteristic
24	C₂₃×#450	C₄	1	1	yes	characteristic
25	C₂₃×#447	C₄	1	1	yes	characteristic
26	C₂₃×#446	C₂²	1	1	yes	characteristic, complemented
27	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
28	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
29	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#436	—	2	1	no	—
33	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#435	—	2	1	no	—
35	C₈×D₄×C₂₃	—	2	1	no	—
36	C₈×D₄×C₂₃	—	2	1	no	—
37	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#433	—	2	1	no	—
43	C₂₃×#431	C₂²	1	1	yes	characteristic
44	C₂₃×#431	C₂²	1	1	yes	characteristic
45	C₂₃×#431	C₂²	1	1	yes	characteristic
46	C₂₃×#431	C₂²	1	1	yes	characteristic
47	C₂₃×#424	—	2	1	no	—
48	C₂₃×#423	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
50	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
51	C₂₃×#335	C₂²	1	1	yes	characteristic, complemented
52	C₂₃×#335	C₂²	1	1	yes	characteristic, complemented
53	C₂₃×#334	C₂²	1	1	yes	characteristic, complemented
54	C₂₃×#334	C₂²	1	1	yes	characteristic, complemented
55	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
62	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
64	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
65	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
66	C₈²×C₂₃	C₂²	1	1	yes	characteristic, complemented
67	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₂₃	—	2	1	no	—
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂²×C₈×C₂₃	—	2	1	no	—
79	C₂²×C₈×C₂₃	—	2	1	no	—
80	C₂₃×#126	—	2	1	no	—
81	C₂₃×#125	C₂×C₄	1	1	yes	characteristic
82	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
83	C₂₃×#123	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
85	C₂₃×#121	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₄×D₄×C₂₃	—	2	1	no	—
88	C₄×D₄×C₂₃	—	2	1	no	—
89	C₄×D₄×C₂₃	—	2	1	no	—
90	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂₃×#117	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
100	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
101	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
102	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
103	C₂₃×C₄⋊C₈	—	2	1	no	—
104	C₂₃×C₄⋊C₈	—	2	1	no	—
105	C₂₃×C₄⋊C₈	—	2	1	no	—
106	C₂₃×C₄⋊C₈	—	2	1	no	—
107	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
108	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
109	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
116	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
117	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
118	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
119	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
120	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
121	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
122	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
123	C₂₃×C₂²⋊C₈	—	2	1	no	—
124	C₂₃×C₂²⋊C₈	—	2	1	no	—
125	C₂₃×C₂²⋊C₈	—	2	1	no	—
126	C₂₃×C₂²⋊C₈	—	2	1	no	—
127	C₂₃×C₂²⋊C₈	—	2	1	no	—
128	C₂₃×C₂²⋊C₈	—	2	1	no	—
129	C₂₃×C₂²⋊C₈	—	2	1	no	—
130	C₂₃×C₂²⋊C₈	—	2	1	no	—
131	C₂₃×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
132	C₂₃×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
133	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
134	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
135	C₂₃×#97	—	2	1	no	—
136	C₄×C₈×C₂₃	—	2	1	no	—
137	C₄×C₈×C₂₃	—	2	1	no	—
138	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₂⁴×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	8	1	no	—
160	C₂²×C₄×C₂₃	—	8	1	no	—
161	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
164	SD₁₆×C₂₃	—	2	2	no	—
165	SD₁₆×C₂₃	—	2	2	no	—
166	D₈×C₂₃	—	2	2	no	—
167	D₈×C₂₃	—	2	2	no	—
168	M₁₆×C₂₃	—	8	1	no	—
169	M₁₆×C₂₃	—	8	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	8	1	no	—
177	C₂×C₈×C₂₃	—	8	1	no	—
178	C₂×C₈×C₂₃	—	8	1	no	—
179	C₂×C₈×C₂₃	—	8	1	no	—
180	C₂×C₈×C₂₃	—	8	1	no	—
181	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	2	1	no	—
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	2	1	no	—
202	C₂²⋊C₄×C₂₃	—	2	1	no	—
203	C₂²⋊C₄×C₂₃	—	2	1	no	—
204	C₂²⋊C₄×C₂₃	—	2	1	no	—
205	C₂²⋊C₄×C₂₃	—	2	1	no	—
206	C₂²⋊C₄×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₂₃	#41	1	1	yes	characteristic
211	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
212	#17918	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₂₃	—	2	4	no	—
214	C₂³×C₂₃	—	2	1	no	—
215	C₂³×C₂₃	—	2	1	no	—
216	C₂³×C₂₃	—	2	1	no	—
217	C₂³×C₂₃	—	8	1	no	—
218	C₂³×C₂₃	—	8	1	no	—
219	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
220	Q₈×C₂₃	—	2	1	no	—
221	Q₈×C₂₃	—	2	1	no	—
222	D₄×C₂₃	—	2	4	no	—
223	D₄×C₂₃	—	2	1	no	—
224	D₄×C₂₃	—	2	1	no	—
225	D₄×C₂₃	—	2	1	no	—
226	D₄×C₂₃	—	2	1	no	—
227	D₄×C₂₃	—	8	1	no	—
228	D₄×C₂₃	#131	1	2	yes	—
229	D₄×C₂₃	#131	1	2	yes	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	8	1	no	—
247	C₂×C₄×C₂₃	—	8	1	no	—
248	C₂×C₄×C₂₃	—	8	1	no	—
249	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	2	1	no	—
257	C₈×C₂₃	—	2	1	no	—
258	C₈×C₂₃	—	2	1	no	—
259	C₈×C₂₃	—	2	1	no	—
260	C₈×C₂₃	—	2	1	no	—
261	C₈×C₂₃	—	2	1	no	—
262	C₈×C₂₃	—	2	1	no	—
263	C₈×C₂₃	—	4	1	no	—
264	C₈×C₂₃	—	4	1	no	—
265	C₈×C₂₃	—	4	1	no	—
266	C₈×C₂₃	—	4	1	no	—
267	C₈×C₂₃	—	4	1	no	—
268	C₈×C₂₃	—	4	1	no	—
269	#3105	C₂×C₂₃	1	1	yes	characteristic
270	#2993	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#2972	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1586	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1585	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1580	C₂×C₂₃	1	1	yes	characteristic, complemented
277	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
279	C₈×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1574	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1486	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1482	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1454	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂²×C₂₃	—	2	1	no	—
285	C₂²×C₂₃	—	2	2	no	—
286	C₂²×C₂₃	—	2	2	no	—
287	C₂²×C₂₃	—	2	1	no	—
288	C₂²×C₂₃	—	2	1	no	—
289	C₂²×C₂₃	—	4	1	no	—
290	C₂²×C₂₃	—	4	1	no	—
291	C₂²×C₂₃	—	4	1	no	—
292	C₂²×C₂₃	—	4	1	no	—
293	C₂²×C₂₃	—	4	1	no	—
294	C₂²×C₂₃	—	4	1	no	—
295	C₂²×C₂₃	—	4	1	no	—
296	C₂²×C₂₃	—	8	4	no	—
297	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
298	C₄×C₂₃	—	2	1	no	—
299	C₄×C₂₃	—	2	1	no	—
300	C₄×C₂₃	—	2	1	no	—
301	C₄×C₂₃	—	2	1	no	—
302	C₄×C₂₃	—	4	1	no	—
303	C₄×C₂₃	—	4	1	no	—
304	C₄×C₂₃	—	4	1	no	—
305	C₄×C₂₃	—	4	1	no	—
306	C₄×C₂₃	—	4	1	no	—
307	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
308	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
309	C₄×C₂₃	#576	1	1	yes	characteristic
310	C₄×C₂₃	#577	1	1	yes	characteristic
311	#529	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
313	#484	C₄×C₂₃	1	1	yes	characteristic
314	#481	C₄×C₂₃	1	1	yes	characteristic
315	#478	C₄×C₂₃	1	1	yes	characteristic
316	#478	C₄×C₂₃	1	1	yes	characteristic
317	#463	C₄×C₂₃	1	1	yes	characteristic
318	#463	C₄×C₂₃	1	1	yes	characteristic
319	#450	C₄×C₂₃	1	1	yes	characteristic
320	#447	C₄×C₂₃	1	1	yes	characteristic
321	#446	C₂²×C₂₃	1	1	yes	characteristic, complemented
322	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
323	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
324	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
325	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
326	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
327	#436	—	2	1	no	—
328	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
329	#435	—	2	1	no	—
330	C₈×D₄	—	2	1	no	—
331	C₈×D₄	—	2	1	no	—
332	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
333	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
334	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
335	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
337	#433	—	2	1	no	—
338	#431	C₂²×C₂₃	1	1	yes	characteristic
339	#431	C₂²×C₂₃	1	1	yes	characteristic
340	#431	C₂²×C₂₃	1	1	yes	characteristic
341	#431	C₂²×C₂₃	1	1	yes	characteristic
342	#424	—	2	1	no	—
343	#423	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	—	2	1	no	—
345	C₂×C₂²⋊C₈	—	2	1	no	—
346	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
347	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
348	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
349	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
350	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
351	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
352	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
353	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
354	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
355	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
356	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
357	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
358	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
359	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
360	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
361	C₈²	C₂²×C₂₃	1	1	yes	characteristic, complemented
362	C₂×C₂₃	—	4	1	no	—
363	C₂×C₂₃	—	4	2	no	—
364	C₂×C₂₃	—	4	2	no	—
365	C₂×C₂₃	—	8	1	no	—
366	C₂×C₂₃	#2929	1	1	yes	characteristic
367	C₂×C₂₃	#2946	1	1	yes	characteristic
368	C₂×C₂₃	#2957	1	1	yes	characteristic
369	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
370	C₂³×C₄	—	2	1	no	—
371	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
372	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
373	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
374	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
375	C₂×M₁₆	—	2	1	no	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂²×C₈	—	2	1	no	—
378	C₂²×C₈	—	2	1	no	—
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	—	2	1	no	—
381	C₂²×C₈	—	2	1	no	—
382	#126	—	2	1	no	—
383	#125	C₂×C₄×C₂₃	1	1	yes	characteristic
384	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
385	#123	—	2	1	no	—
386	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
387	#121	C₂×C₄×C₂₃	1	1	yes	characteristic
388	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
393	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
394	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
395	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
396	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
397	#117	—	2	1	no	—
398	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
401	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
402	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
403	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
404	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	—	2	1	no	—
407	C₄⋊C₈	—	2	1	no	—
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
410	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
411	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
412	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
413	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
414	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
415	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
416	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
417	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
418	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
419	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
420	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
421	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
422	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
423	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
424	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
425	C₂²⋊C₈	—	2	1	no	—
426	C₂²⋊C₈	—	2	1	no	—
427	C₂²⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	1	no	—
429	C₂²⋊C₈	—	2	1	no	—
430	C₂²⋊C₈	—	2	1	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₂²⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
434	C₂²⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
435	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
436	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
437	#97	—	2	1	no	—
438	C₄×C₈	—	2	1	no	—
439	C₄×C₈	—	2	1	no	—
440	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
441	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
442	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
443	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
444	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
446	C₂₃	#17918	1	1	yes	characteristic, direct product factor, Sylow
447	C₂⁴	—	2	1	no	—
448	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
454	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
455	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	8	1	no	—
463	C₂²×C₄	—	8	1	no	—
464	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
467	SD₁₆	—	2	2	no	—
468	SD₁₆	—	2	2	no	—
469	D₈	—	2	2	no	—
470	D₈	—	2	2	no	—
471	M₁₆	—	8	1	no	—
472	M₁₆	—	8	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	8	1	no	—
480	C₂×C₈	—	8	1	no	—
481	C₂×C₈	—	8	1	no	—
482	C₂×C₈	—	8	1	no	—
483	C₂×C₈	—	8	1	no	—
484	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
485	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
486	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
487	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
488	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
498	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
499	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
500	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
501	C₂²⋊C₄	—	2	1	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₄²	#41×C₂₃	1	1	yes	characteristic
514	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
515	C₂³	—	2	4	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	8	1	no	—
520	C₂³	—	8	1	no	—
521	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
522	Q₈	—	2	1	no	—
523	Q₈	—	2	1	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	1	no	—
526	D₄	—	2	1	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	8	1	no	—
530	D₄	C₂₃×#131	1	2	yes	—
531	D₄	C₂₃×#131	1	2	yes	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	8	1	no	—
549	C₂×C₄	—	8	1	no	—
550	C₂×C₄	—	8	1	no	—
551	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
552	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
553	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
554	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
555	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
556	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
557	C₈	—	2	1	no	—
558	C₈	—	2	1	no	—
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	2	1	no	—
562	C₈	—	2	1	no	—
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	8	4	no	—
584	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	1	no	—
594	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
595	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
596	C₄	C₂₃×#576	1	1	yes	characteristic
597	C₄	C₂₃×#577	1	1	yes	characteristic
598	C₂	—	4	1	no	—
599	C₂	—	4	2	no	—
600	C₂	—	4	2	no	—
601	C₂	—	8	1	no	—
602	C₂	C₂₃×#2929	1	1	yes	characteristic
603	C₂	C₂₃×#2946	1	1	yes	characteristic
604	C₂	C₂₃×#2957	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×#447	C₄
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×#2993	C₂	semidirect
3. C₂₃×#2972	C₂	semidirect
4. C₂₃×#1586	C₂	semidirect
5. C₂₃×#1585	C₂	semidirect
6. C₂₃×#1582	C₂	semidirect
7. C₂₃×#1581	C₂	semidirect
8. C₂₃×#1580	C₂	semidirect
9. C₂₃×SD₁₆⋊C₈	C₂	semidirect
10. C₈×SD₁₆×C₂₃	C₂	semidirect
11. C₈×D₈×C₂₃	C₂	semidirect
12. C₂₃×#1574	C₂	semidirect
13. C₂₃×#1486	C₂	semidirect
14. C₂₃×#1482	C₂	semidirect
15. C₂₃×#1454	C₂	semidirect
26. C₂₃×#446	C₂²	semidirect
27. C₈×Q₈×C₂₃	C₂²	semidirect
33. C₂₃×#436	C₂²	semidirect
41. C₈×D₄×C₂₃	C₂²	semidirect
51. C₂₃×#335	C₂²	semidirect
52. C₂₃×#335	C₂²	semidirect
53. C₂₃×#334	C₂²	semidirect
54. C₂₃×#334	C₂²	semidirect
55. C₂₃×Q₈⋊C₈	C₂²	semidirect
56. C₂₃×Q₈⋊C₈	C₂²	semidirect
61. C₂₃×D₄⋊C₈	C₂²	semidirect
62. C₂₃×D₄⋊C₈	C₂²	semidirect
63. C₂₃×#322	C₂²	semidirect
64. C₂₃×#322	C₂²	semidirect
65. C₂₃×#322	C₂²	semidirect
66. C₈²×C₂₃	C₂²	semidirect
212. #17918	C₂₃	direct (N × H)
270. #2993	C₂×C₂₃	semidirect
271. #2972	C₂×C₂₃	semidirect
272. #1586	C₂×C₂₃	semidirect
273. #1585	C₂×C₂₃	semidirect
274. #1582	C₂×C₂₃	semidirect
275. #1581	C₂×C₂₃	semidirect
276. #1580	C₂×C₂₃	semidirect
277. SD₁₆⋊C₈	C₂×C₂₃	semidirect
278. C₈×SD₁₆	C₂×C₂₃	semidirect
279. C₈×D₈	C₂×C₂₃	semidirect
280. #1574	C₂×C₂₃	semidirect
281. #1486	C₂×C₂₃	semidirect
282. #1482	C₂×C₂₃	semidirect
283. #1454	C₂×C₂₃	semidirect
321. #446	C₂²×C₂₃	semidirect
322. C₈×Q₈	C₂²×C₂₃	semidirect
328. #436	C₂²×C₂₃	semidirect
336. C₈×D₄	C₂²×C₂₃	semidirect
346. #335	C₂²×C₂₃	semidirect
347. #335	C₂²×C₂₃	semidirect
348. #334	C₂²×C₂₃	semidirect
349. #334	C₂²×C₂₃	semidirect
350. Q₈⋊C₈	C₂²×C₂₃	semidirect
351. Q₈⋊C₈	C₂²×C₂₃	semidirect
356. D₄⋊C₈	C₂²×C₂₃	semidirect
357. D₄⋊C₈	C₂²×C₂₃	semidirect
358. #322	C₂²×C₂₃	semidirect
359. #322	C₂²×C₂₃	semidirect
360. #322	C₂²×C₂₃	semidirect
361. C₈²	C₂²×C₂₃	semidirect
446. C₂₃	#17918	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×#3105	2
C₂₃×#2993	2
C₂₃×#2972	2
C₂₃×#1586	2
C₂₃×#1585	2
C₂₃×#1582	2
C₂₃×#1581	2
C₂₃×#1580	2
C₂₃×SD₁₆⋊C₈	2
C₈×SD₁₆×C₂₃	2
C₈×D₈×C₂₃	2
C₂₃×#1574	2
C₂₃×#1486	2
C₂₃×#1482	2
C₂₃×#1454	2
#17918	23

Subgroups — C23×#17918 #19165544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17918 #19165544