Subgroups — C₂₃×#19852 #19167478

Order 5888.

660 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3399	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3361	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3284	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1715	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1692	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1677	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1631	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1625	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#135	—	2	1	no	—
17	C₂₃×#559	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	C₂₃×#550	C₂²	1	1	yes	characteristic
20	C₂₃×#547	C₂²	1	1	yes	characteristic
21	C₂₃×#498	C₂²	1	1	yes	characteristic
22	C₂₃×#496	—	2	1	no	—
23	C₂₃×#493	C₂²	1	1	yes	characteristic
24	C₂₃×#492	C₂²	1	1	yes	characteristic
25	C₂₃×#492	C₂²	1	1	yes	characteristic
26	C₂₃×#488	—	2	1	no	—
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#486	C₂²	1	1	yes	characteristic
29	C₂₃×#483	—	2	1	no	—
30	C₂₃×#481	—	2	1	no	—
31	C₂₃×#478	—	2	1	no	—
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#476	C₂²	1	1	yes	characteristic
34	C₂₃×#475	C₂²	1	1	yes	characteristic
35	C₂₃×#465	—	2	1	no	—
36	C₂₃×#465	—	2	1	no	—
37	C₂₃×#463	—	2	1	no	—
38	C₂₃×#463	—	2	1	no	—
39	C₂₃×#462	C₂²	1	1	yes	characteristic
40	C₂₃×#460	C₂²	1	1	yes	characteristic
41	C₂₃×#459	—	2	1	no	—
42	C₂₃×#459	C₂²	1	1	yes	characteristic
43	C₂₃×#459	C₂²	1	1	yes	characteristic
44	C₂₃×#459	C₂²	1	1	yes	characteristic
45	C₂₃×#452	—	2	1	no	—
46	C₂₃×#449	—	2	1	no	—
47	C₂₃×#446	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#442	—	2	1	no	—
50	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×D₈×C₂₃	—	2	1	no	—
53	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#106	—	2	1	no	—
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#331	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×#41×C₂₃	—	2	1	no	—
69	C₂₃×#135	—	4	4	no	—
70	C₂×Q₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	4	1	no	—
75	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
76	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
77	C₂×D₈×C₂₃	—	2	1	no	—
78	C₂×D₈×C₂₃	—	2	1	no	—
79	C₂×D₈×C₂₃	—	2	1	no	—
80	C₂×D₈×C₂₃	—	2	1	no	—
81	C₂×D₈×C₂₃	—	2	1	no	—
82	C₂²×C₈×C₂₃	—	2	1	no	—
83	C₂²×C₈×C₂₃	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×#127	—	2	1	no	—
87	C₂₃×#127	C₂³	1	1	yes	characteristic
88	C₂₃×#127	C₂³	1	1	yes	characteristic
89	C₂₃×#125	—	2	1	no	—
90	C₂₃×#124	—	2	1	no	—
91	C₂₃×#124	—	2	1	no	—
92	C₂₃×#123	—	4	1	no	—
93	C₂₃×#123	—	4	1	no	—
94	C₂₃×C₂²⋊Q₈	—	2	1	no	—
95	C₂₃×C₂²⋊Q₈	—	2	1	no	—
96	C₂₃×C₂²⋊Q₈	—	4	1	no	—
97	C₂₃×#121	—	2	1	no	—
98	C₂₃×#121	—	2	1	no	—
99	C₂₃×#121	—	4	1	no	—
100	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₄×D₄×C₂₃	—	2	1	no	—
105	C₄×D₄×C₂₃	—	2	1	no	—
106	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂₃×#107	—	2	1	no	—
111	C₂₃×#106	—	2	1	no	—
112	C₂₃×#106	—	2	1	no	—
113	C₂₃×#106	—	2	1	no	—
114	C₂₃×#106	—	2	1	no	—
115	C₂₃×#106	D₄	1	1	yes	characteristic
116	C₂₃×#106	D₄	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	—	2	1	no	—
118	C₂₃×C₄⋊C₈	—	2	1	no	—
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#103	D₄	1	1	yes	characteristic
127	C₂₃×#103	D₄	1	1	yes	characteristic
128	C₂₃×#102	—	2	1	no	—
129	C₂₃×#102	—	2	1	no	—
130	C₂₃×#102	—	2	1	no	—
131	C₂₃×#102	—	2	1	no	—
132	C₂₃×#102	—	2	1	no	—
133	C₂₃×#102	—	2	1	no	—
134	C₂₃×#102	—	2	1	no	—
135	C₂₃×#102	—	4	1	no	—
136	C₂₃×#102	D₄	1	1	yes	characteristic
137	C₂₃×#102	D₄	1	1	yes	characteristic
138	C₂₃×C₂²⋊C₈	—	2	1	no	—
139	C₂₃×C₂²⋊C₈	—	2	1	no	—
140	C₂₃×#97	—	2	1	no	—
141	C₄×C₈×C₂₃	—	2	1	no	—
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	#41×C₂₃	—	4	2	no	—
146	#41×C₂₃	—	4	2	no	—
147	#41×C₂₃	—	8	1	no	—
148	#41×C₂₃	—	8	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	4	1	no	—
160	C₂×D₄×C₂₃	—	4	1	no	—
161	C₂×D₄×C₂₃	—	4	1	no	—
162	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	4	1	no	—
167	C₂²×C₄×C₂₃	—	4	1	no	—
168	C₂²×C₄×C₂₃	—	4	1	no	—
169	C₂²×C₄×C₂₃	—	4	1	no	—
170	Q₁₆×C₂₃	—	4	2	no	—
171	SD₁₆×C₂₃	—	2	2	no	—
172	SD₁₆×C₂₃	—	2	2	no	—
173	SD₁₆×C₂₃	—	4	2	no	—
174	SD₁₆×C₂₃	—	8	1	no	—
175	SD₁₆×C₂₃	—	8	1	no	—
176	SD₁₆×C₂₃	—	8	2	no	—
177	D₈×C₂₃	—	4	2	no	—
178	D₈×C₂₃	—	4	2	no	—
179	D₈×C₂₃	—	4	2	no	—
180	D₈×C₂₃	—	8	1	no	—
181	D₈×C₂₃	—	8	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	—	4	2	no	—
188	C₂×C₈×C₂₃	—	4	2	no	—
189	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	4	1	no	—
200	C₄⋊C₄×C₂₃	—	4	1	no	—
201	C₄⋊C₄×C₂₃	—	4	1	no	—
202	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₂₃	—	2	1	no	—
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	4	1	no	—
209	C₂²⋊C₄×C₂₃	—	4	1	no	—
210	C₂²⋊C₄×C₂₃	—	4	1	no	—
211	C₂²⋊C₄×C₂₃	—	4	1	no	—
212	C₂²⋊C₄×C₂₃	—	4	1	no	—
213	C₂²⋊C₄×C₂₃	—	4	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
217	#19852	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₂₃	—	2	1	no	—
219	C₂³×C₂₃	—	2	1	no	—
220	C₂³×C₂₃	—	4	1	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	Q₈×C₂₃	—	2	1	no	—
223	Q₈×C₂₃	—	2	1	no	—
224	Q₈×C₂₃	—	2	1	no	—
225	Q₈×C₂₃	—	2	1	no	—
226	Q₈×C₂₃	—	4	1	no	—
227	Q₈×C₂₃	—	4	1	no	—
228	Q₈×C₂₃	—	8	1	no	—
229	D₄×C₂₃	—	2	1	no	—
230	D₄×C₂₃	—	2	1	no	—
231	D₄×C₂₃	—	2	1	no	—
232	D₄×C₂₃	—	4	1	no	—
233	D₄×C₂₃	—	4	1	no	—
234	D₄×C₂₃	—	4	1	no	—
235	D₄×C₂₃	—	4	1	no	—
236	D₄×C₂₃	—	4	1	no	—
237	D₄×C₂₃	—	4	1	no	—
238	D₄×C₂₃	—	4	1	no	—
239	D₄×C₂₃	—	4	1	no	—
240	D₄×C₂₃	—	8	1	no	—
241	D₄×C₂₃	—	8	1	no	—
242	D₄×C₂₃	—	8	2	no	—
243	D₄×C₂₃	—	8	2	no	—
244	D₄×C₂₃	—	16	1	no	—
245	D₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	8	2	no	—
259	C₂×C₄×C₂₃	—	8	2	no	—
260	C₂×C₄×C₂₃	—	16	1	no	—
261	C₂×C₄×C₂₃	—	16	1	no	—
262	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
265	C₈×C₂₃	—	2	1	no	—
266	C₈×C₂₃	—	2	1	no	—
267	C₈×C₂₃	—	2	1	no	—
268	C₈×C₂₃	—	4	1	no	—
269	C₈×C₂₃	—	4	1	no	—
270	C₈×C₂₃	—	4	1	no	—
271	C₈×C₂₃	—	4	1	no	—
272	C₈×C₂₃	—	4	1	no	—
273	C₈×C₂₃	#135	1	2	yes	complemented
274	#3399	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#3361	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#3335	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#3284	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1715	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1692	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1677	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1667	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1666	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1631	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#1625	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
288	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
289	C₂²×C₂₃	—	2	2	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	—	8	1	no	—
296	C₂²×C₂₃	—	8	1	no	—
297	C₂²×C₂₃	—	8	1	no	—
298	C₂²×C₂₃	—	8	1	no	—
299	C₂²×C₂₃	—	8	1	no	—
300	C₂²×C₂₃	—	8	1	no	—
301	C₂²×C₂₃	D₄²	1	1	yes	characteristic
302	C₄×C₂₃	—	2	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	—	4	1	no	—
308	C₄×C₂₃	—	4	1	no	—
309	C₄×C₂₃	—	8	1	no	—
310	C₄×C₂₃	—	8	1	no	—
311	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
312	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₄×C₂₃	#577	1	1	yes	characteristic
314	C₄×C₂₃	#577	1	1	yes	characteristic
315	C₂×#135	—	2	1	no	—
316	#559	C₂²×C₂₃	1	1	yes	characteristic
317	#554	C₂²×C₂₃	1	1	yes	characteristic
318	#550	C₂²×C₂₃	1	1	yes	characteristic
319	#547	C₂²×C₂₃	1	1	yes	characteristic
320	#498	C₂²×C₂₃	1	1	yes	characteristic
321	#496	—	2	1	no	—
322	#493	C₂²×C₂₃	1	1	yes	characteristic
323	#492	C₂²×C₂₃	1	1	yes	characteristic
324	#492	C₂²×C₂₃	1	1	yes	characteristic
325	#488	—	2	1	no	—
326	#486	C₂²×C₂₃	1	1	yes	characteristic
327	#486	C₂²×C₂₃	1	1	yes	characteristic
328	#483	—	2	1	no	—
329	#481	—	2	1	no	—
330	#478	—	2	1	no	—
331	#477	C₂²×C₂₃	1	1	yes	characteristic
332	#476	C₂²×C₂₃	1	1	yes	characteristic
333	#475	C₂²×C₂₃	1	1	yes	characteristic
334	#465	—	2	1	no	—
335	#465	—	2	1	no	—
336	#463	—	2	1	no	—
337	#463	—	2	1	no	—
338	#462	C₂²×C₂₃	1	1	yes	characteristic
339	#460	C₂²×C₂₃	1	1	yes	characteristic
340	#459	—	2	1	no	—
341	#459	C₂²×C₂₃	1	1	yes	characteristic
342	#459	C₂²×C₂₃	1	1	yes	characteristic
343	#459	C₂²×C₂₃	1	1	yes	characteristic
344	#452	—	2	1	no	—
345	#449	—	2	1	no	—
346	#446	C₂²×C₂₃	1	1	yes	characteristic
347	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
348	#442	—	2	1	no	—
349	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
350	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
351	C₄×D₈	—	2	1	no	—
352	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
353	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#106	—	2	1	no	—
355	#335	C₂²×C₂₃	1	1	yes	characteristic
356	#334	C₂²×C₂₃	1	1	yes	characteristic
357	#333	C₂²×C₂₃	1	1	yes	characteristic
358	#332	C₂²×C₂₃	1	1	yes	characteristic
359	#331	C₂²×C₂₃	1	1	yes	characteristic
360	#331	C₂²×C₂₃	1	1	yes	characteristic
361	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
362	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
364	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
365	#322	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×C₂₃	—	4	2	no	—
367	C₂×C₂₃	—	8	1	no	—
368	C₂×C₂₃	—	16	1	no	—
369	C₂×C₂₃	—	16	1	no	—
370	C₂×C₂₃	#3265	1	1	yes	characteristic
371	C₂×C₂₃	#3284	1	1	yes	characteristic
372	C₂×C₂₃	#3288	1	1	yes	characteristic
373	C₂×#41	—	2	1	no	—
374	C₂×#41	—	2	1	no	—
375	#135	—	4	4	no	—
376	C₂×Q₁₆	—	2	1	no	—
377	C₂×SD₁₆	—	2	1	no	—
378	C₂×SD₁₆	—	2	1	no	—
379	C₂×SD₁₆	—	2	1	no	—
380	C₂×SD₁₆	—	4	1	no	—
381	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
382	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
383	C₂×D₈	—	2	1	no	—
384	C₂×D₈	—	2	1	no	—
385	C₂×D₈	—	2	1	no	—
386	C₂×D₈	—	2	1	no	—
387	C₂×D₈	—	2	1	no	—
388	C₂²×C₈	—	2	1	no	—
389	C₂²×C₈	—	2	1	no	—
390	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
391	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
392	#127	—	2	1	no	—
393	#127	C₂³×C₂₃	1	1	yes	characteristic
394	#127	C₂³×C₂₃	1	1	yes	characteristic
395	#125	—	2	1	no	—
396	#124	—	2	1	no	—
397	#124	—	2	1	no	—
398	#123	—	4	1	no	—
399	#123	—	4	1	no	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	1	no	—
402	C₂²⋊Q₈	—	4	1	no	—
403	#121	—	2	1	no	—
404	#121	—	2	1	no	—
405	#121	—	4	1	no	—
406	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
413	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	#107	—	2	1	no	—
417	#106	—	2	1	no	—
418	#106	—	2	1	no	—
419	#106	—	2	1	no	—
420	#106	—	2	1	no	—
421	#106	D₄×C₂₃	1	1	yes	characteristic
422	#106	D₄×C₂₃	1	1	yes	characteristic
423	C₄⋊C₈	—	2	1	no	—
424	C₄⋊C₈	—	2	1	no	—
425	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
426	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
428	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
429	#103	—	2	1	no	—
430	#103	—	2	1	no	—
431	#103	—	2	1	no	—
432	#103	D₄×C₂₃	1	1	yes	characteristic
433	#103	D₄×C₂₃	1	1	yes	characteristic
434	#102	—	2	1	no	—
435	#102	—	2	1	no	—
436	#102	—	2	1	no	—
437	#102	—	2	1	no	—
438	#102	—	2	1	no	—
439	#102	—	2	1	no	—
440	#102	—	2	1	no	—
441	#102	—	4	1	no	—
442	#102	D₄×C₂₃	1	1	yes	characteristic
443	#102	D₄×C₂₃	1	1	yes	characteristic
444	C₂²⋊C₈	—	2	1	no	—
445	C₂²⋊C₈	—	2	1	no	—
446	#97	—	2	1	no	—
447	C₄×C₈	—	2	1	no	—
448	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
451	C₂₃	#19852	1	1	yes	characteristic, direct product factor, Sylow
452	#41	—	4	2	no	—
453	#41	—	4	2	no	—
454	#41	—	8	1	no	—
455	#41	—	8	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	4	1	no	—
467	C₂×D₄	—	4	1	no	—
468	C₂×D₄	—	4	1	no	—
469	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	4	1	no	—
474	C₂²×C₄	—	4	1	no	—
475	C₂²×C₄	—	4	1	no	—
476	C₂²×C₄	—	4	1	no	—
477	Q₁₆	—	4	2	no	—
478	SD₁₆	—	2	2	no	—
479	SD₁₆	—	2	2	no	—
480	SD₁₆	—	4	2	no	—
481	SD₁₆	—	8	1	no	—
482	SD₁₆	—	8	1	no	—
483	SD₁₆	—	8	2	no	—
484	D₈	—	4	2	no	—
485	D₈	—	4	2	no	—
486	D₈	—	4	2	no	—
487	D₈	—	8	1	no	—
488	D₈	—	8	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	4	2	no	—
495	C₂×C₈	—	4	2	no	—
496	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	4	1	no	—
507	C₄⋊C₄	—	4	1	no	—
508	C₄⋊C₄	—	4	1	no	—
509	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
510	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
511	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
512	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₂²⋊C₄	—	4	1	no	—
516	C₂²⋊C₄	—	4	1	no	—
517	C₂²⋊C₄	—	4	1	no	—
518	C₂²⋊C₄	—	4	1	no	—
519	C₂²⋊C₄	—	4	1	no	—
520	C₂²⋊C₄	—	4	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
524	C₂³	—	2	1	no	—
525	C₂³	—	2	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	—	4	1	no	—
528	Q₈	—	2	1	no	—
529	Q₈	—	2	1	no	—
530	Q₈	—	2	1	no	—
531	Q₈	—	2	1	no	—
532	Q₈	—	4	1	no	—
533	Q₈	—	4	1	no	—
534	Q₈	—	8	1	no	—
535	D₄	—	2	1	no	—
536	D₄	—	2	1	no	—
537	D₄	—	2	1	no	—
538	D₄	—	4	1	no	—
539	D₄	—	4	1	no	—
540	D₄	—	4	1	no	—
541	D₄	—	4	1	no	—
542	D₄	—	4	1	no	—
543	D₄	—	4	1	no	—
544	D₄	—	4	1	no	—
545	D₄	—	4	1	no	—
546	D₄	—	8	1	no	—
547	D₄	—	8	1	no	—
548	D₄	—	8	2	no	—
549	D₄	—	8	2	no	—
550	D₄	—	16	1	no	—
551	D₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	8	2	no	—
565	C₂×C₄	—	8	2	no	—
566	C₂×C₄	—	16	1	no	—
567	C₂×C₄	—	16	1	no	—
568	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
569	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
570	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
571	C₈	—	2	1	no	—
572	C₈	—	2	1	no	—
573	C₈	—	2	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₈	—	4	1	no	—
579	C₈	C₂₃×#135	1	2	yes	complemented
580	C₂²	—	2	2	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	8	1	no	—
587	C₂²	—	8	1	no	—
588	C₂²	—	8	1	no	—
589	C₂²	—	8	1	no	—
590	C₂²	—	8	1	no	—
591	C₂²	—	8	1	no	—
592	C₂²	D₄²×C₂₃	1	1	yes	characteristic
593	C₄	—	2	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	8	1	no	—
601	C₄	—	8	1	no	—
602	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
603	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
604	C₄	C₂₃×#577	1	1	yes	characteristic
605	C₄	C₂₃×#577	1	1	yes	characteristic
606	C₂	—	4	2	no	—
607	C₂	—	8	1	no	—
608	C₂	—	16	1	no	—
609	C₂	—	16	1	no	—
610	C₂	C₂₃×#3265	1	1	yes	characteristic
611	C₂	C₂₃×#3284	1	1	yes	characteristic
612	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3399	C₂	semidirect
2. C₂₃×#3361	C₂	semidirect
3. C₂₃×#3335	C₂	semidirect
4. C₂₃×#3284	C₂	semidirect
5. C₂₃×#1715	C₂	semidirect
6. C₂₃×#1692	C₂	semidirect
7. C₂₃×#1677	C₂	semidirect
8. C₂₃×#1667	C₂	semidirect
9. C₂₃×#1666	C₂	semidirect
10. C₂₃×#1640	C₂	semidirect
11. C₂₃×#1631	C₂	semidirect
12. C₂₃×#1625	C₂	semidirect
13. C₂₃×#1622	C₂	semidirect
14. C₂₃×#1582	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
217. #19852	C₂₃	direct (N × H)
273. C₈×C₂₃	#135	semidirect
274. #3399	C₂×C₂₃	semidirect
275. #3361	C₂×C₂₃	semidirect
276. #3335	C₂×C₂₃	semidirect
277. #3284	C₂×C₂₃	semidirect
278. #1715	C₂×C₂₃	semidirect
279. #1692	C₂×C₂₃	semidirect
280. #1677	C₂×C₂₃	semidirect
281. #1667	C₂×C₂₃	semidirect
282. #1666	C₂×C₂₃	semidirect
283. #1640	C₂×C₂₃	semidirect
284. #1631	C₂×C₂₃	semidirect
285. #1625	C₂×C₂₃	semidirect
286. #1622	C₂×C₂₃	semidirect
287. #1582	C₂×C₂₃	semidirect
288. SD₁₆⋊C₈	C₂×C₂₃	semidirect
451. C₂₃	#19852	direct (N × H)
579. C₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3399	2
C₂₃×#3361	2
C₂₃×#3335	2
C₂₃×#3284	2
C₂₃×#1715	2
C₂₃×#1692	2
C₂₃×#1677	2
C₂₃×#1667	2
C₂₃×#1666	2
C₂₃×#1640	2
C₂₃×#1631	2
C₂₃×#1625	2
C₂₃×#1622	2
C₂₃×#1582	2
C₂₃×SD₁₆⋊C₈	2
#19852	23

Subgroups — C23×#19852 #19167478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19852 #19167478