Subgroups — C₂₃×#18081 #19165707

Order 5888.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 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×#2994	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2986	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1583	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1482	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1481	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂₃×#529	C₂²	1	1	yes	characteristic
18	C₂₃×#501	C₄	1	1	yes	characteristic
19	C₂₃×#497	C₄	1	1	yes	characteristic
20	C₂₃×#491	C₄	1	1	yes	characteristic
21	C₂₃×#489	C₄	1	1	yes	characteristic
22	C₂₃×#470	C₄	1	1	yes	characteristic
23	C₂₃×#470	C₄	1	1	yes	characteristic
24	C₂₃×#468	C₄	1	1	yes	characteristic
25	C₂₃×#468	C₄	1	1	yes	characteristic
26	C₂₃×#446	—	2	1	no	—
27	C₂₃×#446	C₂²	1	1	yes	characteristic
28	C₂₃×#446	C₂²	1	1	yes	characteristic
29	C₂₃×#446	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₂₃	—	2	1	no	—
31	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#436	C₂²	1	1	yes	characteristic
39	C₂₃×#435	—	2	1	no	—
40	C₈×D₄×C₂₃	—	2	1	no	—
41	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#433	—	2	1	no	—
43	C₂₃×#433	—	2	1	no	—
44	C₂₃×#433	—	2	1	no	—
45	C₂₃×#431	C₂²	1	1	yes	characteristic
46	C₂₃×#431	C₂²	1	1	yes	characteristic
47	C₂₃×#431	C₂²	1	1	yes	characteristic
48	C₂₃×#424	—	2	1	no	—
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
51	C₂₃×#335	C₂²	1	1	yes	characteristic
52	C₂₃×#335	C₂²	1	1	yes	characteristic
53	C₂₃×#334	C₂²	1	1	yes	characteristic
54	C₂₃×#334	C₂²	1	1	yes	characteristic
55	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×#322	C₂²	1	1	yes	characteristic
64	C₂₃×#322	C₂²	1	1	yes	characteristic
65	C₂₃×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₂₃	C₂²	1	1	yes	characteristic
67	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×M₁₆×C₂₃	—	2	1	no	—
72	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
78	C₂₃×#126	—	2	1	no	—
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×#125	C₂×C₄	1	1	yes	characteristic
81	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
82	C₂₃×#123	—	2	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
85	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
86	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
87	C₂₃×#121	C₂×C₄	1	1	yes	characteristic
88	C₄×Q₈×C₂₃	—	2	1	no	—
89	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
90	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
91	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₄×D₄×C₂₃	—	2	1	no	—
93	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂₃×#117	—	2	1	no	—
95	C₂₃×#117	—	2	1	no	—
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
98	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
99	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
100	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
101	C₂₃×C₄⋊C₈	—	2	1	no	—
102	C₂₃×C₄⋊C₈	—	2	1	no	—
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	—	2	1	no	—
108	C₂₃×C₄⋊C₈	—	2	1	no	—
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×#103	C₂×C₄	1	1	yes	characteristic
118	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
119	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
120	C₂₃×#103	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	D₄	1	1	yes	characteristic
128	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
129	C₂₃×#97	—	2	1	no	—
130	C₂₃×#97	—	2	1	no	—
131	C₂₃×#97	—	2	1	no	—
132	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
133	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
134	C₄×C₈×C₂₃	—	2	1	no	—
135	C₄×C₈×C₂₃	—	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	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
154	Q₁₆×C₂₃	—	2	2	no	—
155	Q₁₆×C₂₃	—	4	1	no	—
156	SD₁₆×C₂₃	—	2	2	no	—
157	SD₁₆×C₂₃	—	4	1	no	—
158	M₁₆×C₂₃	—	2	2	no	—
159	M₁₆×C₂₃	—	4	1	no	—
160	M₁₆×C₂₃	—	8	1	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	8	1	no	—
168	C₂×C₈×C₂₃	—	8	1	no	—
169	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
192	C₂²⋊C₄×C₂₃	—	2	1	no	—
193	C₂²⋊C₄×C₂₃	—	2	1	no	—
194	C₂²⋊C₄×C₂₃	—	2	1	no	—
195	C₂²⋊C₄×C₂₃	—	2	1	no	—
196	C₂²⋊C₄×C₂₃	—	2	1	no	—
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₄²×C₂₃	—	2	1	no	—
200	C₄²×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	2	1	no	—
202	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
203	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₂₃	#41	1	1	yes	characteristic
205	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
206	#18081	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂³×C₂₃	—	2	1	no	—
208	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
209	Q₈×C₂₃	—	2	1	no	—
210	Q₈×C₂₃	—	2	1	no	—
211	Q₈×C₂₃	—	2	1	no	—
212	Q₈×C₂₃	—	2	1	no	—
213	Q₈×C₂₃	—	2	1	no	—
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	4	2	no	—
216	D₄×C₂₃	—	2	1	no	—
217	D₄×C₂₃	—	2	1	no	—
218	D₄×C₂₃	—	4	2	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
236	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
239	C₈×C₂₃	—	2	1	no	—
240	C₈×C₂₃	—	2	1	no	—
241	C₈×C₂₃	—	2	1	no	—
242	C₈×C₂₃	—	2	1	no	—
243	C₈×C₂₃	—	2	1	no	—
244	C₈×C₂₃	—	2	1	no	—
245	C₈×C₂₃	—	2	1	no	—
246	C₈×C₂₃	—	4	1	no	—
247	C₈×C₂₃	—	4	1	no	—
248	C₈×C₂₃	—	4	1	no	—
249	C₈×C₂₃	—	4	1	no	—
250	C₈×C₂₃	—	4	1	no	—
251	C₈×C₂₃	—	4	1	no	—
252	C₈×C₂₃	#131	1	2	yes	—
253	#3172	C₂×C₂₃	1	1	yes	characteristic
254	#2994	C₂×C₂₃	1	1	yes	characteristic
255	#2986	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#1592	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#1591	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#1587	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#1585	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1584	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1583	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1579	C₂×C₂₃	1	1	yes	characteristic, complemented
263	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1482	C₂×C₂₃	1	1	yes	characteristic
266	#1481	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1453	C₂×C₂₃	1	1	yes	characteristic, complemented
268	C₂²×C₂₃	—	2	1	no	—
269	C₂²×C₂₃	—	2	1	no	—
270	C₂²×C₂₃	—	2	1	no	—
271	C₂²×C₂₃	—	4	1	no	—
272	C₂²×C₂₃	—	4	1	no	—
273	C₂²×C₂₃	—	4	1	no	—
274	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
275	C₄×C₂₃	—	2	1	no	—
276	C₄×C₂₃	—	2	1	no	—
277	C₄×C₂₃	—	2	1	no	—
278	C₄×C₂₃	—	2	1	no	—
279	C₄×C₂₃	—	4	1	no	—
280	C₄×C₂₃	—	4	1	no	—
281	C₄×C₂₃	—	4	1	no	—
282	C₄×C₂₃	—	4	1	no	—
283	C₄×C₂₃	—	4	1	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
287	C₄×C₂₃	#568	1	1	yes	characteristic
288	C₄×C₂₃	#577	1	1	yes	characteristic
289	C₄×C₂₃	#578	1	1	yes	characteristic
290	#533	C₂²×C₂₃	1	1	yes	characteristic
291	#529	C₂²×C₂₃	1	1	yes	characteristic
292	#501	C₄×C₂₃	1	1	yes	characteristic
293	#497	C₄×C₂₃	1	1	yes	characteristic
294	#491	C₄×C₂₃	1	1	yes	characteristic
295	#489	C₄×C₂₃	1	1	yes	characteristic
296	#470	C₄×C₂₃	1	1	yes	characteristic
297	#470	C₄×C₂₃	1	1	yes	characteristic
298	#468	C₄×C₂₃	1	1	yes	characteristic
299	#468	C₄×C₂₃	1	1	yes	characteristic
300	#446	—	2	1	no	—
301	#446	C₂²×C₂₃	1	1	yes	characteristic
302	#446	C₂²×C₂₃	1	1	yes	characteristic
303	#446	C₂²×C₂₃	1	1	yes	characteristic
304	C₈×Q₈	—	2	1	no	—
305	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
306	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
307	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
309	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
310	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
311	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
312	#436	C₂²×C₂₃	1	1	yes	characteristic
313	#435	—	2	1	no	—
314	C₈×D₄	—	2	1	no	—
315	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
316	#433	—	2	1	no	—
317	#433	—	2	1	no	—
318	#433	—	2	1	no	—
319	#431	C₂²×C₂₃	1	1	yes	characteristic
320	#431	C₂²×C₂₃	1	1	yes	characteristic
321	#431	C₂²×C₂₃	1	1	yes	characteristic
322	#424	—	2	1	no	—
323	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
325	#335	C₂²×C₂₃	1	1	yes	characteristic
326	#335	C₂²×C₂₃	1	1	yes	characteristic
327	#334	C₂²×C₂₃	1	1	yes	characteristic
328	#334	C₂²×C₂₃	1	1	yes	characteristic
329	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
330	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
331	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
332	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
335	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
336	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
337	#322	C₂²×C₂₃	1	1	yes	characteristic
338	#322	C₂²×C₂₃	1	1	yes	characteristic
339	#322	C₂²×C₂₃	1	1	yes	characteristic
340	C₈²	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₂₃	—	4	1	no	—
342	C₂×C₂₃	—	8	1	no	—
343	C₂×C₂₃	#2935	1	1	yes	characteristic
344	C₂×C₂₃	#2947	1	1	yes	characteristic
345	C₂×C₂₃	#2955	1	1	yes	characteristic
346	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
347	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
348	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
349	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
352	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
353	C₂²×C₈	—	2	1	no	—
354	C₂²×C₈	—	2	1	no	—
355	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
357	#126	—	2	1	no	—
358	#125	—	2	1	no	—
359	#125	C₂×C₄×C₂₃	1	1	yes	characteristic
360	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
361	#123	—	2	1	no	—
362	C₂²⋊Q₈	—	2	1	no	—
363	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
364	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
365	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
366	#121	C₂×C₄×C₂₃	1	1	yes	characteristic
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
370	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
373	#117	—	2	1	no	—
374	#117	—	2	1	no	—
375	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
376	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
377	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
378	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
379	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	—	2	1	no	—
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
389	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
390	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
391	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
392	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
393	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
394	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
395	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
396	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
397	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
398	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
399	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
400	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
401	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
402	C₂²⋊C₈	—	2	1	no	—
403	C₂²⋊C₈	—	2	1	no	—
404	C₂²⋊C₈	—	2	1	no	—
405	C₂²⋊C₈	—	2	1	no	—
406	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
407	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
408	#97	—	2	1	no	—
409	#97	—	2	1	no	—
410	#97	—	2	1	no	—
411	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
412	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
413	C₄×C₈	—	2	1	no	—
414	C₄×C₈	—	2	1	no	—
415	C₄×C₈	—	2	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
418	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
419	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
423	C₂₃	#18081	1	1	yes	characteristic, direct product factor, Sylow
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
426	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
427	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
434	Q₁₆	—	2	2	no	—
435	Q₁₆	—	4	1	no	—
436	SD₁₆	—	2	2	no	—
437	SD₁₆	—	4	1	no	—
438	M₁₆	—	2	2	no	—
439	M₁₆	—	4	1	no	—
440	M₁₆	—	8	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	8	1	no	—
448	C₂×C₈	—	8	1	no	—
449	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
450	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
451	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
452	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
453	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
469	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
470	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
471	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
483	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₄²	#41×C₂₃	1	1	yes	characteristic
485	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
486	C₂³	—	2	1	no	—
487	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
488	Q₈	—	2	1	no	—
489	Q₈	—	2	1	no	—
490	Q₈	—	2	1	no	—
491	Q₈	—	2	1	no	—
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	4	2	no	—
495	D₄	—	2	1	no	—
496	D₄	—	2	1	no	—
497	D₄	—	4	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
513	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
514	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
515	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
516	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
517	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
518	C₈	—	2	1	no	—
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	2	1	no	—
522	C₈	—	2	1	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	1	no	—
525	C₈	—	4	1	no	—
526	C₈	—	4	1	no	—
527	C₈	—	4	1	no	—
528	C₈	—	4	1	no	—
529	C₈	—	4	1	no	—
530	C₈	—	4	1	no	—
531	C₈	C₂₃×#131	1	2	yes	—
532	C₂²	—	2	1	no	—
533	C₂²	—	2	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	1	no	—
538	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	4	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
551	C₄	C₂₃×#568	1	1	yes	characteristic
552	C₄	C₂₃×#577	1	1	yes	characteristic
553	C₄	C₂₃×#578	1	1	yes	characteristic
554	C₂	—	4	1	no	—
555	C₂	—	8	1	no	—
556	C₂	C₂₃×#2935	1	1	yes	characteristic
557	C₂	C₂₃×#2947	1	1	yes	characteristic
558	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 ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2986	C₂	semidirect
4. C₂₃×#1592	C₂	semidirect
5. C₂₃×#1591	C₂	semidirect
6. C₂₃×#1587	C₂	semidirect
7. C₂₃×#1585	C₂	semidirect
8. C₂₃×#1584	C₂	semidirect
9. C₂₃×#1583	C₂	semidirect
10. C₂₃×#1579	C₂	semidirect
11. C₈×SD₁₆×C₂₃	C₂	semidirect
12. C₂₃×#1565	C₂	semidirect
14. C₂₃×#1481	C₂	semidirect
15. C₂₃×#1453	C₂	semidirect
206. #18081	C₂₃	direct (N × H)
255. #2986	C₂×C₂₃	semidirect
256. #1592	C₂×C₂₃	semidirect
257. #1591	C₂×C₂₃	semidirect
258. #1587	C₂×C₂₃	semidirect
259. #1585	C₂×C₂₃	semidirect
260. #1584	C₂×C₂₃	semidirect
261. #1583	C₂×C₂₃	semidirect
262. #1579	C₂×C₂₃	semidirect
263. C₈×SD₁₆	C₂×C₂₃	semidirect
264. #1565	C₂×C₂₃	semidirect
266. #1481	C₂×C₂₃	semidirect
267. #1453	C₂×C₂₃	semidirect
423. C₂₃	#18081	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×#2994	2
C₂₃×#2986	2
C₂₃×#1592	2
C₂₃×#1591	2
C₂₃×#1587	2
C₂₃×#1585	2
C₂₃×#1584	2
C₂₃×#1583	2
C₂₃×#1579	2
C₈×SD₁₆×C₂₃	2
C₂₃×#1565	2
C₂₃×#1482	2
C₂₃×#1481	2
C₂₃×#1453	2
#18081	23

Subgroups — C23×#18081 #19165707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18081 #19165707