Subgroups — C₂₃×#26166 #19173792

Order 5888.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2446	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2096	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1916	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1877	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1877	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1877	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
16	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂₃×#461	—	2	4	no	—
25	C₄×SD₁₆×C₂₃	—	2	4	no	—
26	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#400	—	2	1	no	—
39	C₂₃×#398	—	2	1	no	—
40	C₂₃×#391	C₂²	1	1	yes	characteristic
41	C₂₃×#391	C₂²	1	1	yes	characteristic
42	C₂₃×#391	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#387	—	2	1	no	—
48	C₂₃×#385	—	2	1	no	—
49	C₂₃×#385	—	2	1	no	—
50	C₂₃×#384	C₂²	1	1	yes	characteristic
51	C₂₃×#380	—	2	1	no	—
52	C₄×C₄⋊C₄×C₂₃	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×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂×SD₁₆×C₂₃	—	2	2	no	—
68	C₂×SD₁₆×C₂₃	—	2	2	no	—
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	2	2	no	—
72	C₂×SD₁₆×C₂₃	D₄	1	4	yes	—
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	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	—	4	1	no	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	—	4	1	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	—	2	2	no	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₄×D₄×C₂₃	—	4	1	no	—
86	C₄×D₄×C₂₃	—	4	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
96	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×#106	—	2	2	no	—
111	C₂₃×#106	—	4	1	no	—
112	C₂₃×C₄⋊C₈	—	2	2	no	—
113	C₂₃×#103	—	2	2	no	—
114	C₂₃×#103	—	2	2	no	—
115	C₂₃×#103	—	2	2	no	—
116	C₂₃×#103	—	2	2	no	—
117	C₂₃×#103	—	2	2	no	—
118	C₂₃×#103	—	4	1	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₄×C₈×C₂₃	—	2	2	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	D₄	1	1	yes	characteristic
132	C₂₃×#95	D₄	1	1	yes	characteristic
133	C₂⁴×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₂₃	—	2	1	no	—
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₂₃	#41	1	2	yes	—
151	C₂×D₄×C₂₃	#41	1	2	yes	—
152	C₂²×C₄×C₂₃	—	2	4	no	—
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	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	4	2	no	—
161	C₂²×C₄×C₂₃	—	4	2	no	—
162	C₂²×C₄×C₂₃	—	4	2	no	—
163	C₂²×C₄×C₂₃	—	4	2	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	8	1	no	—
167	C₂²×C₄×C₂₃	—	8	1	no	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
175	SD₁₆×C₂₃	—	2	8	no	—
176	SD₁₆×C₂₃	—	4	4	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
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	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	4	1	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	4	2	no	—
201	C₄⋊C₄×C₂₃	—	4	2	no	—
202	C₄⋊C₄×C₂₃	—	4	2	no	—
203	C₄⋊C₄×C₂₃	—	4	2	no	—
204	C₄⋊C₄×C₂₃	—	4	2	no	—
205	C₄⋊C₄×C₂₃	—	4	2	no	—
206	C₄⋊C₄×C₂₃	—	8	1	no	—
207	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	complemented
208	C₄⋊C₄×C₂₃	#41	1	2	yes	—
209	C₂²⋊C₄×C₂₃	—	2	4	no	—
210	C₂²⋊C₄×C₂₃	—	4	2	no	—
211	C₂²⋊C₄×C₂₃	—	4	2	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	2	1	no	—
217	C₄²×C₂₃	—	2	1	no	—
218	C₄²×C₂₃	—	4	2	no	—
219	#26166	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	4	1	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	—	4	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₂₃	—	2	2	no	—
229	Q₈×C₂₃	—	2	2	no	—
230	Q₈×C₂₃	—	4	1	no	—
231	Q₈×C₂₃	—	4	1	no	—
232	Q₈×C₂₃	—	4	1	no	—
233	Q₈×C₂₃	—	4	1	no	—
234	Q₈×C₂₃	—	4	1	no	—
235	Q₈×C₂₃	—	4	1	no	—
236	D₄×C₂₃	—	2	2	no	—
237	D₄×C₂₃	—	2	2	no	—
238	D₄×C₂₃	—	2	2	no	—
239	D₄×C₂₃	#123	1	4	yes	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	4	8	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	—	8	4	no	—
278	C₂×C₄×C₂₃	—	8	4	no	—
279	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
280	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
281	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
282	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
287	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
288	C₈×C₂₃	—	2	2	no	—
289	C₈×C₂₃	—	4	1	no	—
290	C₈×C₂₃	—	4	1	no	—
291	C₈×C₂₃	—	4	1	no	—
292	C₂×#461	C₂×C₂₃	1	1	yes	characteristic
293	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
294	#2446	C₂×C₂₃	1	1	yes	characteristic, complemented
295	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
296	#2096	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#2088	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#2072	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1916	C₂×C₂₃	1	1	yes	characteristic, complemented
302	#1877	C₂×C₂₃	1	1	yes	characteristic
303	#1877	C₂×C₂₃	1	1	yes	characteristic
304	#1877	C₂×C₂₃	1	1	yes	characteristic
305	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1864	C₂×C₂₃	1	1	yes	characteristic
307	C₂²×C₂₃	—	2	4	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	4	2	no	—
312	C₂²×C₂₃	—	4	2	no	—
313	C₂²×C₂₃	—	4	2	no	—
314	C₂²×C₂₃	#525	1	1	yes	characteristic
315	C₂²×C₂₃	D₄²	1	1	yes	characteristic
316	C₂²×C₂₃	#548	1	1	yes	characteristic
317	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
318	C₂²×C₂₃	#575	1	1	yes	characteristic
319	C₂²×C₂₃	#577	1	1	yes	characteristic
320	C₂²×C₂₃	#578	1	1	yes	characteristic
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	2	2	no	—
323	C₄×C₂₃	—	2	2	no	—
324	C₄×C₂₃	—	4	2	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	2	no	—
327	C₄×C₂₃	—	4	1	no	—
328	C₄×C₂₃	—	4	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	—	8	1	no	—
332	C₄×C₂₃	—	8	1	no	—
333	C₄×C₂₃	—	8	1	no	—
334	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
335	C₄×C₂₃	#539	1	1	yes	characteristic
336	C₄×C₂₃	#547	1	1	yes	characteristic
337	C₄×C₂₃	#553	1	1	yes	characteristic
338	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
339	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
345	C₂²×C₄⋊C₄	—	2	1	no	—
346	#461	—	2	4	no	—
347	C₄×SD₁₆	—	2	4	no	—
348	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
360	#400	—	2	1	no	—
361	#398	—	2	1	no	—
362	#391	C₂²×C₂₃	1	1	yes	characteristic
363	#391	C₂²×C₂₃	1	1	yes	characteristic
364	#391	C₂²×C₂₃	1	1	yes	characteristic
365	#389	C₂²×C₂₃	1	1	yes	characteristic
366	#389	C₂²×C₂₃	1	1	yes	characteristic
367	#389	C₂²×C₂₃	1	1	yes	characteristic
368	#387	—	2	1	no	—
369	#387	—	2	1	no	—
370	#385	—	2	1	no	—
371	#385	—	2	1	no	—
372	#384	C₂²×C₂₃	1	1	yes	characteristic
373	#380	—	2	1	no	—
374	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#340	C₂²×C₂₃	1	1	yes	characteristic
378	#340	C₂²×C₂₃	1	1	yes	characteristic
379	#336	C₂²×C₂₃	1	1	yes	characteristic
380	#336	C₂²×C₂₃	1	1	yes	characteristic
381	#336	C₂²×C₂₃	1	1	yes	characteristic
382	#336	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×C₂₃	—	4	4	no	—
384	C₂×C₂₃	#2443	1	1	yes	characteristic
385	C₂×C₂₃	#3101	1	1	yes	characteristic
386	C₂×C₂₃	#3120	1	1	yes	characteristic
387	C₂×C₂₃	#3269	1	1	yes	characteristic
388	C₂×C₂₃	D₄×SD₁₆	1	1	yes	characteristic
389	C₂×C₂₃	#3359	1	1	yes	characteristic
390	C₂×C₂₃	#3376	1	1	yes	characteristic
391	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
392	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	—	2	1	no	—
396	C₂³×C₄	—	2	1	no	—
397	C₂×SD₁₆	—	2	2	no	—
398	C₂×SD₁₆	—	2	2	no	—
399	C₂×SD₁₆	—	2	2	no	—
400	C₂×SD₁₆	—	2	2	no	—
401	C₂×SD₁₆	—	2	2	no	—
402	C₂×SD₁₆	D₄×C₂₃	1	4	yes	—
403	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₄⋊Q₈	—	2	2	no	—
408	C₄⋊Q₈	—	4	1	no	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×Q₈	—	4	1	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	4	1	no	—
416	C₄×D₄	—	4	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	4	4	no	—
426	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₄²	—	2	1	no	—
437	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
440	#106	—	2	2	no	—
441	#106	—	4	1	no	—
442	C₄⋊C₈	—	2	2	no	—
443	#103	—	2	2	no	—
444	#103	—	2	2	no	—
445	#103	—	2	2	no	—
446	#103	—	2	2	no	—
447	#103	—	2	2	no	—
448	#103	—	4	1	no	—
449	#102	—	2	2	no	—
450	#102	—	2	2	no	—
451	C₄×C₈	—	2	2	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	—	2	1	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	D₄×C₂₃	1	1	yes	characteristic
462	#95	D₄×C₂₃	1	1	yes	characteristic
463	C₂₃	#26166	1	1	yes	characteristic, direct product factor, Sylow
464	C₂⁴	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
481	C₂×D₄	#41×C₂₃	1	2	yes	—
482	C₂×D₄	#41×C₂₃	1	2	yes	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	4	2	no	—
495	C₂²×C₄	—	4	2	no	—
496	C₂²×C₄	—	4	2	no	—
497	C₂²×C₄	—	8	1	no	—
498	C₂²×C₄	—	8	1	no	—
499	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
506	SD₁₆	—	2	8	no	—
507	SD₁₆	—	4	4	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	4	1	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	2	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	—	4	2	no	—
535	C₄⋊C₄	—	4	2	no	—
536	C₄⋊C₄	—	4	2	no	—
537	C₄⋊C₄	—	8	1	no	—
538	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	complemented
539	C₄⋊C₄	#41×C₂₃	1	2	yes	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	4	2	no	—
542	C₂²⋊C₄	—	4	2	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	4	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
558	Q₈	—	2	2	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	4	1	no	—
561	Q₈	—	4	1	no	—
562	Q₈	—	4	1	no	—
563	Q₈	—	4	1	no	—
564	Q₈	—	4	1	no	—
565	Q₈	—	4	1	no	—
566	D₄	—	2	2	no	—
567	D₄	—	2	2	no	—
568	D₄	—	2	2	no	—
569	D₄	C₂₃×#123	1	4	yes	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	8	4	no	—
608	C₂×C₄	—	8	4	no	—
609	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
610	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
611	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
612	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
613	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
614	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
615	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
616	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
617	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
618	C₈	—	2	2	no	—
619	C₈	—	4	1	no	—
620	C₈	—	4	1	no	—
621	C₈	—	4	1	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	C₂₃×#525	1	1	yes	characteristic
630	C₂²	D₄²×C₂₃	1	1	yes	characteristic
631	C₂²	C₂₃×#548	1	1	yes	characteristic
632	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
633	C₂²	C₂₃×#575	1	1	yes	characteristic
634	C₂²	C₂₃×#577	1	1	yes	characteristic
635	C₂²	C₂₃×#578	1	1	yes	characteristic
636	C₄	—	2	2	no	—
637	C₄	—	2	2	no	—
638	C₄	—	2	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	—	8	1	no	—
647	C₄	—	8	1	no	—
648	C₄	—	8	1	no	—
649	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
650	C₄	C₂₃×#539	1	1	yes	characteristic
651	C₄	C₂₃×#547	1	1	yes	characteristic
652	C₄	C₂₃×#553	1	1	yes	characteristic
653	C₂	—	4	4	no	—
654	C₂	C₂₃×#2443	1	1	yes	characteristic
655	C₂	C₂₃×#3101	1	1	yes	characteristic
656	C₂	C₂₃×#3120	1	1	yes	characteristic
657	C₂	C₂₃×#3269	1	1	yes	characteristic
658	C₂	D₄×SD₁₆×C₂₃	1	1	yes	characteristic
659	C₂	C₂₃×#3359	1	1	yes	characteristic
660	C₂	C₂₃×#3376	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2446	C₂	semidirect
5. C₂₃×#2096	C₂	semidirect
6. C₂₃×#2088	C₂	semidirect
7. C₂₃×#2072	C₂	semidirect
8. C₂₃×#2043	C₂	semidirect
9. C₂₃×#1933	C₂	semidirect
10. C₂₃×#1916	C₂	semidirect
14. C₂₃×#1865	C₂	semidirect
207. C₄⋊C₄×C₂₃	SD₁₆	semidirect
219. #26166	C₂₃	direct (N × H)
294. #2446	C₂×C₂₃	semidirect
296. #2096	C₂×C₂₃	semidirect
297. #2088	C₂×C₂₃	semidirect
298. #2072	C₂×C₂₃	semidirect
299. #2043	C₂×C₂₃	semidirect
300. #1933	C₂×C₂₃	semidirect
301. #1916	C₂×C₂₃	semidirect
305. #1865	C₂×C₂₃	semidirect
463. C₂₃	#26166	direct (N × H)
538. C₄⋊C₄	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#461	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#2446	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#2096	2
C₂₃×#2088	2
C₂₃×#2072	2
C₂₃×#2043	2
C₂₃×#1933	2
C₂₃×#1916	2
C₂₃×#1877	2
C₂₃×#1865	2
C₂₃×#1864	2
#26166	23

Subgroups — C23×#26166 #19173792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26166 #19173792