Subgroups — C₂₃×#20561 #19168187

Order 5888.

990 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#435	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2290	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1750	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#436	C₄	1	8	yes	complemented
24	C₂₃×#435	C₄	1	8	yes	complemented
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#388	C₄	1	2	yes	—
40	C₂₃×#387	C₄	1	2	yes	—
41	C₂₃×#386	C₄	1	2	yes	—
42	C₂₃×#385	C₄	1	2	yes	—
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₂₃×#376	C₂²	1	1	yes	characteristic
47	C₂₃×#336	C₂²	1	1	yes	characteristic
48	C₂₃×#336	C₂²	1	1	yes	characteristic
49	C₂₃×#336	C₂²	1	1	yes	characteristic
50	C₂₃×#336	C₂²	1	1	yes	characteristic
51	C₂₃×#336	C₂²	1	1	yes	characteristic
52	C₂₃×#336	C₂²	1	1	yes	characteristic
53	C₂₃×#336	C₂²	1	1	yes	characteristic
54	C₂₃×#336	C₂²	1	1	yes	characteristic
55	C₂₃×#336	C₂²	1	1	yes	characteristic
56	C₂₃×#336	C₂²	1	1	yes	characteristic
57	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₂₃	—	4	1	no	—
62	C₂×M₁₆×C₂₃	—	4	1	no	—
63	C₂×M₁₆×C₂₃	—	4	1	no	—
64	C₂×M₁₆×C₂₃	—	4	1	no	—
65	C₂×M₁₆×C₂₃	—	4	1	no	—
66	C₂×M₁₆×C₂₃	—	4	1	no	—
67	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
68	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
69	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
70	C₂²×C₈×C₂₃	—	4	1	no	—
71	C₂²×C₈×C₂₃	—	4	1	no	—
72	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
82	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
83	C₂₃×#117	—	4	1	no	—
84	C₂₃×#117	—	4	1	no	—
85	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
86	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₂₃	—	4	1	no	—
94	C₂×C₄²×C₂₃	—	4	1	no	—
95	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
102	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
103	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
104	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
105	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
106	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
107	C₂₃×#97	—	4	1	no	—
108	C₂₃×#97	—	4	1	no	—
109	C₂₃×#97	C₂×C₄	1	4	yes	complemented
110	C₄×C₈×C₂₃	—	4	1	no	—
111	C₄×C₈×C₂₃	—	4	1	no	—
112	C₄×C₈×C₂₃	—	4	1	no	—
113	C₄×C₈×C₂₃	—	4	1	no	—
114	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
115	C₂₃×#95	C₂×C₄	1	2	yes	—
116	C₂₃×#95	C₂×C₄	1	2	yes	—
117	C₂₃×#95	C₂×C₄	1	2	yes	—
118	C₂₃×#95	C₂×C₄	1	2	yes	—
119	C₂₃×#95	C₂×C₄	1	2	yes	—
120	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
121	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄×C₂₃	—	4	2	no	—
123	C₂×D₄×C₂₃	C₄²	1	4	yes	—
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	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	4	2	no	—
134	C₂²×C₄×C₂₃	—	4	2	no	—
135	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
148	M₁₆×C₂₃	—	4	4	no	—
149	M₁₆×C₂₃	—	4	4	no	—
150	M₁₆×C₂₃	—	4	4	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	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	1	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	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	—	4	4	no	—
168	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
169	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
170	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
171	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
172	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
173	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	4	2	no	—
178	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
179	C₂²⋊C₄×C₂₃	—	2	4	no	—
180	C₂²⋊C₄×C₂₃	—	4	2	no	—
181	C₂²⋊C₄×C₂₃	—	4	2	no	—
182	C₂²⋊C₄×C₂₃	C₄²	1	4	yes	—
183	C₂²⋊C₄×C₂₃	C₄²	1	4	yes	—
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	—	2	1	no	—
189	C₄²×C₂₃	—	2	1	no	—
190	C₄²×C₂₃	—	2	1	no	—
191	C₄²×C₂₃	—	2	1	no	—
192	C₄²×C₂₃	—	4	2	no	—
193	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
194	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
195	#20561	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₂₃	—	2	1	no	—
197	C₂³×C₂₃	—	2	1	no	—
198	C₂³×C₂₃	—	2	1	no	—
199	C₂³×C₂₃	—	2	1	no	—
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	2	2	no	—
202	C₂³×C₂₃	—	2	2	no	—
203	C₂³×C₂₃	—	2	2	no	—
204	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
205	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
206	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
207	D₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	1	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	2	no	—
214	C₂×C₄×C₂₃	—	2	2	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	4	8	no	—
234	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
235	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
236	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
237	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
238	C₂×C₄×C₂₃	#131	1	4	yes	—
239	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
241	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
245	C₈×C₂₃	—	2	2	no	—
246	C₈×C₂₃	—	2	2	no	—
247	C₈×C₂₃	—	2	2	no	—
248	C₈×C₂₃	—	2	2	no	—
249	C₈×C₂₃	—	2	2	no	—
250	C₈×C₂₃	—	2	2	no	—
251	C₈×C₂₃	—	2	2	no	—
252	C₈×C₂₃	—	2	2	no	—
253	C₂×#436	C₂×C₂₃	1	1	yes	characteristic
254	C₂×#435	C₂×C₂₃	1	1	yes	characteristic
255	#2290	C₂×C₂₃	1	1	yes	characteristic
256	#1771	C₂×C₂₃	1	1	yes	characteristic, complemented
257	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#1768	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#1753	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1753	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1753	C₂×C₂₃	1	1	yes	characteristic, complemented
262	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1750	C₂×C₂₃	1	1	yes	characteristic, complemented
264	C₄×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1746	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1729	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1728	C₂×C₂₃	1	1	yes	characteristic, complemented
268	C₂²×C₂₃	—	2	2	no	—
269	C₂²×C₂₃	—	2	2	no	—
270	C₂²×C₂₃	—	2	2	no	—
271	C₂²×C₂₃	—	2	2	no	—
272	C₂²×C₂₃	—	2	2	no	—
273	C₂²×C₂₃	—	2	2	no	—
274	C₂²×C₂₃	—	4	2	no	—
275	C₂²×C₂₃	—	4	2	no	—
276	C₂²×C₂₃	—	4	2	no	—
277	C₂²×C₂₃	—	4	2	no	—
278	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
279	C₂²×C₂₃	#518	1	1	yes	characteristic
280	C₂²×C₂₃	#520	1	1	yes	characteristic
281	C₂²×C₂₃	C₂×#131	1	2	yes	—
282	C₂²×C₂₃	#568	1	2	yes	—
283	C₄×C₂₃	—	2	2	no	—
284	C₄×C₂₃	—	2	2	no	—
285	C₄×C₂₃	—	2	4	no	—
286	C₄×C₂₃	—	2	2	no	—
287	C₄×C₂₃	—	2	2	no	—
288	C₄×C₂₃	—	4	2	no	—
289	C₄×C₂₃	—	4	2	no	—
290	C₄×C₂₃	—	4	2	no	—
291	C₄×C₂₃	#518	1	1	yes	characteristic
292	C₄×C₂₃	#518	1	1	yes	characteristic
293	C₄×C₂₃	#520	1	1	yes	characteristic
294	C₄×C₂₃	#520	1	1	yes	characteristic
295	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
296	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
297	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
298	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
301	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
302	#436	C₄×C₂₃	1	8	yes	complemented
303	#435	C₄×C₂₃	1	8	yes	complemented
304	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
318	#388	C₄×C₂₃	1	2	yes	—
319	#387	C₄×C₂₃	1	2	yes	—
320	#386	C₄×C₂₃	1	2	yes	—
321	#385	C₄×C₂₃	1	2	yes	—
322	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
323	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
324	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
325	#376	C₂²×C₂₃	1	1	yes	characteristic
326	#336	C₂²×C₂₃	1	1	yes	characteristic
327	#336	C₂²×C₂₃	1	1	yes	characteristic
328	#336	C₂²×C₂₃	1	1	yes	characteristic
329	#336	C₂²×C₂₃	1	1	yes	characteristic
330	#336	C₂²×C₂₃	1	1	yes	characteristic
331	#336	C₂²×C₂₃	1	1	yes	characteristic
332	#336	C₂²×C₂₃	1	1	yes	characteristic
333	#336	C₂²×C₂₃	1	1	yes	characteristic
334	#336	C₂²×C₂₃	1	1	yes	characteristic
335	#336	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×C₂₃	—	4	2	no	—
337	C₂×C₂₃	—	4	2	no	—
338	C₂×C₂₃	#2275	1	1	yes	characteristic
339	C₂×C₂₃	#2873	1	1	yes	characteristic
340	C₂×C₂₃	C₄×#131	1	1	yes	characteristic
341	C₂×C₂₃	#2976	1	2	yes	—
342	C₂×C₂₃	#2993	1	2	yes	—
343	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
344	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
345	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
347	C₂×M₁₆	—	4	1	no	—
348	C₂×M₁₆	—	4	1	no	—
349	C₂×M₁₆	—	4	1	no	—
350	C₂×M₁₆	—	4	1	no	—
351	C₂×M₁₆	—	4	1	no	—
352	C₂×M₁₆	—	4	1	no	—
353	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
354	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
355	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
356	C₂²×C₈	—	4	1	no	—
357	C₂²×C₈	—	4	1	no	—
358	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
359	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
361	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
368	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
369	#117	—	4	1	no	—
370	#117	—	4	1	no	—
371	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
372	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
373	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
375	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
376	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
379	C₂×C₄²	—	4	1	no	—
380	C₂×C₄²	—	4	1	no	—
381	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
382	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
387	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
388	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
389	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
390	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
391	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
392	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
393	#97	—	4	1	no	—
394	#97	—	4	1	no	—
395	#97	C₂×C₄×C₂₃	1	4	yes	complemented
396	C₄×C₈	—	4	1	no	—
397	C₄×C₈	—	4	1	no	—
398	C₄×C₈	—	4	1	no	—
399	C₄×C₈	—	4	1	no	—
400	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
401	#95	C₂×C₄×C₂₃	1	2	yes	—
402	#95	C₂×C₄×C₂₃	1	2	yes	—
403	#95	C₂×C₄×C₂₃	1	2	yes	—
404	#95	C₂×C₄×C₂₃	1	2	yes	—
405	#95	C₂×C₄×C₂₃	1	2	yes	—
406	C₂₃	#20561	1	1	yes	characteristic, direct product factor, Sylow
407	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
408	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
409	C₂×D₄	—	4	2	no	—
410	C₂×D₄	C₄²×C₂₃	1	4	yes	—
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	4	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
423	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
424	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
425	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
428	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
430	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
431	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
432	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
433	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
435	M₁₆	—	4	4	no	—
436	M₁₆	—	4	4	no	—
437	M₁₆	—	4	4	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	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	4	4	no	—
455	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
456	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
457	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
458	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
459	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
460	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
461	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
462	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	4	2	no	—
465	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	4	2	no	—
468	C₂²⋊C₄	—	4	2	no	—
469	C₂²⋊C₄	C₄²×C₂₃	1	4	yes	—
470	C₂²⋊C₄	C₄²×C₂₃	1	4	yes	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	1	no	—
474	C₄²	—	2	1	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	4	2	no	—
480	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
481	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	1	no	—
490	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
491	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
492	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
493	D₄	—	4	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	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	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
521	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
522	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
523	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
524	C₂×C₄	C₂₃×#131	1	4	yes	—
525	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
526	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
527	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
528	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
529	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
530	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
531	C₈	—	2	2	no	—
532	C₈	—	2	2	no	—
533	C₈	—	2	2	no	—
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₈	—	2	2	no	—
537	C₈	—	2	2	no	—
538	C₈	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
550	C₂²	C₂₃×#518	1	1	yes	characteristic
551	C₂²	C₂₃×#520	1	1	yes	characteristic
552	C₂²	C₂×C₂₃×#131	1	2	yes	—
553	C₂²	C₂₃×#568	1	2	yes	—
554	C₄	—	2	4	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	2	no	—
562	C₄	C₂₃×#518	1	1	yes	characteristic
563	C₄	C₂₃×#518	1	1	yes	characteristic
564	C₄	C₂₃×#520	1	1	yes	characteristic
565	C₄	C₂₃×#520	1	1	yes	characteristic
566	C₂	—	4	2	no	—
567	C₂	—	4	2	no	—
568	C₂	C₂₃×#2275	1	1	yes	characteristic
569	C₂	C₂₃×#2873	1	1	yes	characteristic
570	C₂	C₄×C₂₃×#131	1	1	yes	characteristic
571	C₂	C₂₃×#2976	1	2	yes	—
572	C₂	C₂₃×#2993	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1771	C₂	semidirect
5. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1768	C₂	semidirect
7. C₂₃×#1753	C₂	semidirect
8. C₂₃×#1753	C₂	semidirect
9. C₂₃×#1753	C₂	semidirect
10. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
11. C₂₃×#1750	C₂	semidirect
12. C₄×C₂₃×C₂²⋊C₈	C₂	semidirect
13. C₂₃×#1746	C₂	semidirect
14. C₂₃×#1729	C₂	semidirect
15. C₂₃×#1728	C₂	semidirect
23. C₂₃×#436	C₄	semidirect
24. C₂₃×#435	C₄	semidirect
69. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
72. C₂²×C₈×C₂₃	C₂×C₄	semidirect
101. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
102. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
105. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
106. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
109. C₂₃×#97	C₂×C₄	semidirect
114. C₄×C₈×C₂₃	C₂×C₄	semidirect
195. #20561	C₂₃	direct (N × H)
256. #1771	C₂×C₂₃	semidirect
257. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
258. #1768	C₂×C₂₃	semidirect
259. #1753	C₂×C₂₃	semidirect
260. #1753	C₂×C₂₃	semidirect
261. #1753	C₂×C₂₃	semidirect
262. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
263. #1750	C₂×C₂₃	semidirect
264. C₄×C₂²⋊C₈	C₂×C₂₃	semidirect
265. #1746	C₂×C₂₃	semidirect
266. #1729	C₂×C₂₃	semidirect
267. #1728	C₂×C₂₃	semidirect
302. #436	C₄×C₂₃	semidirect
303. #435	C₄×C₂₃	semidirect
355. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
358. C₂²×C₈	C₂×C₄×C₂₃	semidirect
387. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
388. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
391. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
392. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
395. #97	C₂×C₄×C₂₃	semidirect
400. C₄×C₈	C₂×C₄×C₂₃	semidirect
406. C₂₃	#20561	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#436	2
C₂×C₂₃×#435	2
C₂₃×#2290	2
C₂₃×#1771	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1768	2
C₂₃×#1753	2
C₈×C₂²⋊C₄×C₂₃	2
C₂₃×#1750	2
C₄×C₂₃×C₂²⋊C₈	2
C₂₃×#1746	2
C₂₃×#1729	2
C₂₃×#1728	2
#20561	23

Subgroups — C23×#20561 #19168187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20561 #19168187