Subgroups — C₂₃×#26019 #19173645

Order 5888.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2442	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2346	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2088	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2088	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1919	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1919	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1825	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1809	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1736	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₃×#400	—	2	1	no	—
28	C₂₃×#398	—	2	1	no	—
29	C₂₃×#389	C₂²	1	1	yes	characteristic
30	C₂₃×#389	C₂²	1	1	yes	characteristic
31	C₂₃×#389	C₂²	1	1	yes	characteristic
32	C₂₃×#389	C₂²	1	1	yes	characteristic
33	C₂₃×#388	—	2	1	no	—
34	C₂₃×#388	—	2	1	no	—
35	C₂₃×#385	—	2	1	no	—
36	C₂₃×#382	—	2	1	no	—
37	C₂₃×#382	C₂²	1	1	yes	characteristic
38	C₂₃×#380	—	2	1	no	—
39	C₂₃×#380	C₂²	1	1	yes	characteristic
40	C₂₃×#380	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#376	—	2	1	no	—
45	C₂₃×#340	C₂²	1	1	yes	characteristic
46	C₂₃×#340	C₂²	1	1	yes	characteristic
47	C₂₃×#340	C₂²	1	1	yes	characteristic
48	C₂₃×#340	C₂²	1	1	yes	characteristic
49	C₂₃×#340	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	—	2	1	no	—
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₂₃	—	2	2	no	—
62	C₂×M₁₆×C₂₃	—	2	2	no	—
63	C₂×M₁₆×C₂₃	—	4	1	no	—
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	—	4	1	no	—
70	C₂₃×#117	—	2	2	no	—
71	C₂₃×#117	Q₈	1	4	yes	—
72	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	—	2	1	no	—
95	C₂×C₄²×C₂₃	—	2	1	no	—
96	C₂×C₄²×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#107	—	4	1	no	—
100	C₂₃×#106	—	2	2	no	—
101	C₂₃×#106	—	2	2	no	—
102	C₂₃×#106	—	4	1	no	—
103	C₂₃×C₄⋊C₈	—	2	2	no	—
104	C₂₃×C₄⋊C₈	—	4	1	no	—
105	C₂₃×C₂²⋊C₈	—	2	2	no	—
106	C₂₃×#95	—	2	1	no	—
107	C₂₃×#95	—	2	1	no	—
108	C₂₃×#95	—	2	1	no	—
109	C₂₃×#95	—	2	1	no	—
110	C₂₃×#95	—	2	1	no	—
111	C₂₃×#95	—	2	1	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	D₄	1	1	yes	characteristic
117	C₂₃×#95	D₄	1	1	yes	characteristic
118	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	—	2	1	no	—
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	4	2	no	—
132	C₂²×C₄×C₂₃	—	4	2	no	—
133	C₂²×C₄×C₂₃	—	8	1	no	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
140	C₂²×C₄×C₂₃	#41	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
142	M₁₆×C₂₃	—	4	4	no	—
143	C₂×C₈×C₂₃	—	2	1	no	—
144	C₂×C₈×C₂₃	—	2	1	no	—
145	C₂×C₈×C₂₃	—	2	1	no	—
146	C₂×C₈×C₂₃	—	2	1	no	—
147	C₂×C₈×C₂₃	—	2	1	no	—
148	C₂×C₈×C₂₃	—	2	1	no	—
149	C₂×C₈×C₂₃	—	2	1	no	—
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	1	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	#41	1	2	yes	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	4	2	no	—
169	C₄⋊C₄×C₂₃	—	4	2	no	—
170	C₄⋊C₄×C₂₃	—	4	2	no	—
171	C₄⋊C₄×C₂₃	—	8	1	no	—
172	C₄⋊C₄×C₂₃	—	8	1	no	—
173	C₄⋊C₄×C₂₃	—	8	1	no	—
174	C₄⋊C₄×C₂₃	—	8	1	no	—
175	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
176	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
177	C₄⋊C₄×C₂₃	#41	1	2	yes	—
178	C₂²⋊C₄×C₂₃	—	2	4	no	—
179	C₂²⋊C₄×C₂₃	—	4	1	no	—
180	C₂²⋊C₄×C₂₃	—	4	1	no	—
181	C₂²⋊C₄×C₂₃	—	4	2	no	—
182	C₂²⋊C₄×C₂₃	—	8	1	no	—
183	C₄²×C₂₃	—	2	4	no	—
184	C₄²×C₂₃	—	2	1	no	—
185	C₄²×C₂₃	—	2	1	no	—
186	C₄²×C₂₃	—	2	1	no	—
187	C₄²×C₂₃	—	2	1	no	—
188	C₄²×C₂₃	—	4	2	no	—
189	C₄²×C₂₃	—	4	2	no	—
190	#26019	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₂₃	—	2	1	no	—
192	C₂³×C₂₃	—	2	1	no	—
193	C₂³×C₂₃	—	2	1	no	—
194	C₂³×C₂₃	—	2	1	no	—
195	C₂³×C₂₃	—	2	1	no	—
196	C₂³×C₂₃	—	2	1	no	—
197	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
198	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₂₃	—	2	1	no	—
200	C₂×C₄×C₂₃	—	2	1	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	1	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	4	1	no	—
221	C₂×C₄×C₂₃	—	4	1	no	—
222	C₂×C₄×C₂₃	—	4	1	no	—
223	C₂×C₄×C₂₃	—	4	1	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	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	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	8	4	no	—
242	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
243	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
247	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
249	C₈×C₂₃	—	2	2	no	—
250	C₈×C₂₃	—	4	1	no	—
251	C₈×C₂₃	—	4	1	no	—
252	C₈×C₂₃	—	4	1	no	—
253	#2442	C₂×C₂₃	1	1	yes	characteristic
254	#2346	C₂×C₂₃	1	1	yes	characteristic
255	#2101	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#2088	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#2088	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#1989	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1919	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1919	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1853	C₂×C₂₃	1	1	yes	characteristic
264	#1851	C₂×C₂₃	1	1	yes	characteristic
265	#1825	C₂×C₂₃	1	1	yes	characteristic
266	#1809	C₂×C₂₃	1	1	yes	characteristic
267	#1736	C₂×C₂₃	1	1	yes	characteristic
268	C₂²×C₂₃	—	2	2	no	—
269	C₂²×C₂₃	—	2	2	no	—
270	C₂²×C₂₃	—	2	2	no	—
271	C₂²×C₂₃	—	4	1	no	—
272	C₂²×C₂₃	—	4	1	no	—
273	C₂²×C₂₃	—	4	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂²×C₂₃	#546	1	1	yes	characteristic
277	C₂²×C₂₃	#547	1	1	yes	characteristic
278	C₂²×C₂₃	#575	1	1	yes	characteristic
279	C₂²×C₂₃	#575	1	1	yes	characteristic
280	C₂²×C₂₃	#577	1	1	yes	characteristic
281	C₂²×C₂₃	#577	1	1	yes	characteristic
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	4	2	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	4	1	no	—
288	C₄×C₂₃	—	4	1	no	—
289	C₄×C₂₃	—	4	2	no	—
290	C₄×C₂₃	—	8	1	no	—
291	C₄×C₂₃	—	8	1	no	—
292	C₄×C₂₃	—	8	1	no	—
293	C₄×C₂₃	—	8	1	no	—
294	C₄×C₂₃	—	8	1	no	—
295	C₄×C₂₃	#533	1	1	yes	characteristic
296	C₄×C₂₃	#544	1	1	yes	characteristic
297	C₄×C₂₃	#551	1	1	yes	characteristic
298	C₄×C₂₃	#552	1	1	yes	characteristic
299	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
310	#400	—	2	1	no	—
311	#398	—	2	1	no	—
312	#389	C₂²×C₂₃	1	1	yes	characteristic
313	#389	C₂²×C₂₃	1	1	yes	characteristic
314	#389	C₂²×C₂₃	1	1	yes	characteristic
315	#389	C₂²×C₂₃	1	1	yes	characteristic
316	#388	—	2	1	no	—
317	#388	—	2	1	no	—
318	#385	—	2	1	no	—
319	#382	—	2	1	no	—
320	#382	C₂²×C₂₃	1	1	yes	characteristic
321	#380	—	2	1	no	—
322	#380	C₂²×C₂₃	1	1	yes	characteristic
323	#380	C₂²×C₂₃	1	1	yes	characteristic
324	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
325	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
326	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
327	#376	—	2	1	no	—
328	#340	C₂²×C₂₃	1	1	yes	characteristic
329	#340	C₂²×C₂₃	1	1	yes	characteristic
330	#340	C₂²×C₂₃	1	1	yes	characteristic
331	#340	C₂²×C₂₃	1	1	yes	characteristic
332	#340	C₂²×C₂₃	1	1	yes	characteristic
333	#340	C₂²×C₂₃	1	1	yes	characteristic
334	#340	C₂²×C₂₃	1	1	yes	characteristic
335	#340	C₂²×C₂₃	1	1	yes	characteristic
336	#340	C₂²×C₂₃	1	1	yes	characteristic
337	#340	C₂²×C₂₃	1	1	yes	characteristic
338	#340	C₂²×C₂₃	1	1	yes	characteristic
339	#340	C₂²×C₂₃	1	1	yes	characteristic
340	#336	C₂²×C₂₃	1	1	yes	characteristic
341	#336	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₂₃	—	4	2	no	—
343	C₂×C₂₃	#2438	1	1	yes	characteristic
344	C₂×C₂₃	#3080	1	1	yes	characteristic
345	C₂×C₂₃	#3084	1	1	yes	characteristic
346	C₂×C₂₃	#3321	1	1	yes	characteristic
347	C₂×C₂₃	#3323	1	1	yes	characteristic
348	C₂×C₂₃	#3348	1	1	yes	characteristic
349	C₂×C₂₃	#3351	1	1	yes	characteristic
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
352	C₂×M₁₆	—	2	2	no	—
353	C₂×M₁₆	—	2	2	no	—
354	C₂×M₁₆	—	4	1	no	—
355	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
356	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
359	#125	—	2	2	no	—
360	#125	—	4	1	no	—
361	#117	—	2	2	no	—
362	#117	Q₈×C₂₃	1	4	yes	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	4	1	no	—
372	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
390	#107	—	4	1	no	—
391	#106	—	2	2	no	—
392	#106	—	2	2	no	—
393	#106	—	4	1	no	—
394	C₄⋊C₈	—	2	2	no	—
395	C₄⋊C₈	—	4	1	no	—
396	C₂²⋊C₈	—	2	2	no	—
397	#95	—	2	1	no	—
398	#95	—	2	1	no	—
399	#95	—	2	1	no	—
400	#95	—	2	1	no	—
401	#95	—	2	1	no	—
402	#95	—	2	1	no	—
403	#95	—	2	1	no	—
404	#95	—	2	1	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	D₄×C₂₃	1	1	yes	characteristic
408	#95	D₄×C₂₃	1	1	yes	characteristic
409	C₂₃	#26019	1	1	yes	characteristic, direct product factor, Sylow
410	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	4	2	no	—
424	C₂²×C₄	—	4	2	no	—
425	C₂²×C₄	—	8	1	no	—
426	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
432	C₂²×C₄	#41×C₂₃	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
434	M₁₆	—	4	4	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	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	#41×C₂₃	1	2	yes	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	4	2	no	—
461	C₄⋊C₄	—	4	2	no	—
462	C₄⋊C₄	—	4	2	no	—
463	C₄⋊C₄	—	8	1	no	—
464	C₄⋊C₄	—	8	1	no	—
465	C₄⋊C₄	—	8	1	no	—
466	C₄⋊C₄	—	8	1	no	—
467	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
468	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
469	C₄⋊C₄	#41×C₂₃	1	2	yes	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	4	1	no	—
472	C₂²⋊C₄	—	4	1	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₂²⋊C₄	—	8	1	no	—
475	C₄²	—	2	4	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	4	2	no	—
481	C₄²	—	4	2	no	—
482	C₂³	—	2	1	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
489	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	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	—	2	1	no	—
497	C₂×C₄	—	2	1	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	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	8	4	no	—
533	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
534	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
535	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
536	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
537	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
538	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
539	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
540	C₈	—	2	2	no	—
541	C₈	—	4	1	no	—
542	C₈	—	4	1	no	—
543	C₈	—	4	1	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
552	C₂²	C₂₃×#546	1	1	yes	characteristic
553	C₂²	C₂₃×#547	1	1	yes	characteristic
554	C₂²	C₂₃×#575	1	1	yes	characteristic
555	C₂²	C₂₃×#575	1	1	yes	characteristic
556	C₂²	C₂₃×#577	1	1	yes	characteristic
557	C₂²	C₂₃×#577	1	1	yes	characteristic
558	C₄	—	2	2	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	8	1	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	—	8	1	no	—
571	C₄	C₂₃×#533	1	1	yes	characteristic
572	C₄	C₂₃×#544	1	1	yes	characteristic
573	C₄	C₂₃×#551	1	1	yes	characteristic
574	C₄	C₂₃×#552	1	1	yes	characteristic
575	C₂	—	4	2	no	—
576	C₂	C₂₃×#2438	1	1	yes	characteristic
577	C₂	C₂₃×#3080	1	1	yes	characteristic
578	C₂	C₂₃×#3084	1	1	yes	characteristic
579	C₂	C₂₃×#3321	1	1	yes	characteristic
580	C₂	C₂₃×#3323	1	1	yes	characteristic
581	C₂	C₂₃×#3348	1	1	yes	characteristic
582	C₂	C₂₃×#3351	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2101	C₂	semidirect
4. C₂₃×#2088	C₂	semidirect
5. C₂₃×#2088	C₂	semidirect
6. C₂₃×#2059	C₂	semidirect
7. C₂₃×#1989	C₂	semidirect
8. C₂₃×#1949	C₂	semidirect
9. C₂₃×#1919	C₂	semidirect
10. C₂₃×#1919	C₂	semidirect
190. #26019	C₂₃	direct (N × H)
255. #2101	C₂×C₂₃	semidirect
256. #2088	C₂×C₂₃	semidirect
257. #2088	C₂×C₂₃	semidirect
258. #2059	C₂×C₂₃	semidirect
259. #1989	C₂×C₂₃	semidirect
260. #1949	C₂×C₂₃	semidirect
261. #1919	C₂×C₂₃	semidirect
262. #1919	C₂×C₂₃	semidirect
409. C₂₃	#26019	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×#2442	2
C₂₃×#2346	2
C₂₃×#2101	2
C₂₃×#2088	2
C₂₃×#2059	2
C₂₃×#1989	2
C₂₃×#1949	2
C₂₃×#1919	2
C₂₃×#1853	2
C₂₃×#1851	2
C₂₃×#1825	2
C₂₃×#1809	2
C₂₃×#1736	2
#26019	23

Subgroups — C23×#26019 #19173645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26019 #19173645