Subgroups — C₂₃×#19413 #19167039

Order 5888.

628 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3210	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3115	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1682	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1679	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1670	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1669	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1665	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1572	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂₃×#532	C₂²	1	1	yes	characteristic
18	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#494	C₂²	1	1	yes	characteristic
20	C₂₃×#492	C₂²	1	1	yes	characteristic
21	C₂₃×#487	C₂²	1	1	yes	characteristic
22	C₂₃×#486	C₂²	1	1	yes	characteristic
23	C₂₃×#485	—	2	1	no	—
24	C₂₃×#482	—	2	1	no	—
25	C₂₃×#475	C₂²	1	1	yes	characteristic
26	C₂₃×#475	C₂²	1	1	yes	characteristic
27	C₂₃×#474	C₂²	1	1	yes	characteristic
28	C₂₃×#474	C₂²	1	1	yes	characteristic
29	C₂₃×#464	—	2	1	no	—
30	C₂₃×#464	—	2	1	no	—
31	C₂₃×#462	C₂²	1	1	yes	characteristic
32	C₂₃×#461	C₂²	1	1	yes	characteristic
33	C₂₃×#460	C₂²	1	1	yes	characteristic
34	C₂₃×#459	C₂²	1	1	yes	characteristic
35	C₂₃×#452	—	2	1	no	—
36	C₂₃×#452	—	2	1	no	—
37	C₂₃×#452	—	2	1	no	—
38	C₂₃×#449	—	2	1	no	—
39	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#442	—	2	1	no	—
42	C₂₃×#440	—	2	1	no	—
43	C₄×Q₁₆×C₂₃	—	2	1	no	—
44	C₄×SD₁₆×C₂₃	—	2	1	no	—
45	C₈×D₄×C₂₃	—	2	1	no	—
46	C₈×D₄×C₂₃	—	2	1	no	—
47	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#432	C₂²	1	1	yes	characteristic
50	C₂₃×#431	—	2	1	no	—
51	C₂₃×#431	C₂²	1	1	yes	characteristic
52	C₂₃×#431	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
54	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
55	C₂₃×#335	C₂²	1	1	yes	characteristic
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#334	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×#332	C₂²	1	1	yes	characteristic
64	C₂₃×#332	C₂²	1	1	yes	characteristic
65	C₂₃×#331	C₂²	1	1	yes	characteristic
66	C₂₃×#331	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₂₃	—	2	1	no	—
70	C₂×Q₁₆×C₂₃	—	2	1	no	—
71	C₂×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×D₈×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂²×C₈×C₂₃	—	2	1	no	—
79	C₂²×C₈×C₂₃	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×#127	C₂³	1	1	yes	characteristic
84	C₂₃×#125	—	2	1	no	—
85	C₂₃×#124	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	—	2	1	no	—
88	C₂₃×#121	—	2	1	no	—
89	C₂₃×#121	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
93	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
99	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂₃×#117	—	2	1	no	—
101	C₂₃×#117	—	2	1	no	—
102	C₂₃×#117	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	—	2	1	no	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#106	—	2	1	no	—
108	C₂₃×#106	—	2	1	no	—
109	C₂₃×C₄⋊C₈	—	2	1	no	—
110	C₂₃×C₄⋊C₈	—	2	1	no	—
111	C₂₃×C₄⋊C₈	—	2	1	no	—
112	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
113	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
114	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×#103	—	2	1	no	—
121	C₂₃×#103	—	2	1	no	—
122	C₂₃×#103	—	2	1	no	—
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×#102	—	2	1	no	—
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×C₂²⋊C₈	—	2	1	no	—
133	C₂₃×C₂²⋊C₈	—	2	1	no	—
134	C₂₃×C₂²⋊C₈	—	2	1	no	—
135	C₂₃×C₂²⋊C₈	—	2	1	no	—
136	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
137	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
138	C₂₃×#97	—	2	1	no	—
139	C₄×C₈×C₂₃	—	2	1	no	—
140	C₄×C₈×C₂₃	—	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	#41×C₂₃	—	2	2	no	—
145	#41×C₂₃	—	2	2	no	—
146	#41×C₂₃	—	8	1	no	—
147	#41×C₂₃	—	8	1	no	—
148	C₂×Q₈×C₂₃	—	2	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×D₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
157	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	4	1	no	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
165	Q₁₆×C₂₃	—	4	2	no	—
166	Q₁₆×C₂₃	—	8	1	no	—
167	Q₁₆×C₂₃	—	8	1	no	—
168	SD₁₆×C₂₃	—	4	2	no	—
169	SD₁₆×C₂₃	—	4	2	no	—
170	SD₁₆×C₂₃	—	8	1	no	—
171	SD₁₆×C₂₃	—	8	1	no	—
172	D₈×C₂₃	—	4	2	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	8	1	no	—
180	C₂×C₈×C₂₃	—	8	1	no	—
181	C₂×C₈×C₂₃	—	8	1	no	—
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
195	C₄⋊C₄×C₂₃	—	4	1	no	—
196	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	2	1	no	—
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	4	1	no	—
202	C₂²⋊C₄×C₂₃	—	4	1	no	—
203	C₂²⋊C₄×C₂₃	—	4	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	2	1	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	4	1	no	—
208	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₂₃	#41	1	1	yes	characteristic
211	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
212	#19413	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₂₃	—	2	1	no	—
214	C₂³×C₂₃	—	4	1	no	—
215	C₂³×C₂₃	#121	1	1	yes	characteristic
216	Q₈×C₂₃	—	2	1	no	—
217	Q₈×C₂₃	—	4	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	Q₈×C₂₃	—	4	1	no	—
222	Q₈×C₂₃	—	4	1	no	—
223	Q₈×C₂₃	#135	1	2	yes	—
224	D₄×C₂₃	—	2	4	no	—
225	D₄×C₂₃	—	2	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	4	1	no	—
228	D₄×C₂₃	—	4	1	no	—
229	D₄×C₂₃	—	4	1	no	—
230	D₄×C₂₃	—	8	1	no	—
231	D₄×C₂₃	—	8	2	no	—
232	D₄×C₂₃	#135	1	2	yes	—
233	C₂×C₄×C₂₃	—	2	4	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	8	1	no	—
248	C₂×C₄×C₂₃	—	8	2	no	—
249	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
250	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
251	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	2	1	no	—
257	C₈×C₂₃	—	2	1	no	—
258	C₈×C₂₃	—	2	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	C₈×C₂₃	—	4	1	no	—
263	C₈×C₂₃	—	4	1	no	—
264	C₈×C₂₃	—	4	1	no	—
265	#3210	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#3115	C₂×C₂₃	1	1	yes	characteristic
267	#3107	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#2988	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1682	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1681	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1680	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1679	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1670	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1669	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1666	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1665	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1572	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1540	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₂²×C₂₃	—	2	1	no	—
281	C₂²×C₂₃	—	2	1	no	—
282	C₂²×C₂₃	—	2	2	no	—
283	C₂²×C₂₃	—	2	1	no	—
284	C₂²×C₂₃	—	4	1	no	—
285	C₂²×C₂₃	—	4	1	no	—
286	C₂²×C₂₃	—	8	1	no	—
287	C₂²×C₂₃	—	8	1	no	—
288	C₂²×C₂₃	—	8	1	no	—
289	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
290	C₄×C₂₃	—	2	1	no	—
291	C₄×C₂₃	—	2	1	no	—
292	C₄×C₂₃	—	2	2	no	—
293	C₄×C₂₃	—	2	1	no	—
294	C₄×C₂₃	—	2	1	no	—
295	C₄×C₂₃	—	4	1	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	8	1	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	—	8	1	no	—
300	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
301	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
302	C₄×C₂₃	#576	1	1	yes	characteristic
303	C₄×C₂₃	#577	1	1	yes	characteristic
304	#533	C₂²×C₂₃	1	1	yes	characteristic
305	#532	C₂²×C₂₃	1	1	yes	characteristic
306	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
307	#494	C₂²×C₂₃	1	1	yes	characteristic
308	#492	C₂²×C₂₃	1	1	yes	characteristic
309	#487	C₂²×C₂₃	1	1	yes	characteristic
310	#486	C₂²×C₂₃	1	1	yes	characteristic
311	#485	—	2	1	no	—
312	#482	—	2	1	no	—
313	#475	C₂²×C₂₃	1	1	yes	characteristic
314	#475	C₂²×C₂₃	1	1	yes	characteristic
315	#474	C₂²×C₂₃	1	1	yes	characteristic
316	#474	C₂²×C₂₃	1	1	yes	characteristic
317	#464	—	2	1	no	—
318	#464	—	2	1	no	—
319	#462	C₂²×C₂₃	1	1	yes	characteristic
320	#461	C₂²×C₂₃	1	1	yes	characteristic
321	#460	C₂²×C₂₃	1	1	yes	characteristic
322	#459	C₂²×C₂₃	1	1	yes	characteristic
323	#452	—	2	1	no	—
324	#452	—	2	1	no	—
325	#452	—	2	1	no	—
326	#449	—	2	1	no	—
327	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
328	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
329	#442	—	2	1	no	—
330	#440	—	2	1	no	—
331	C₄×Q₁₆	—	2	1	no	—
332	C₄×SD₁₆	—	2	1	no	—
333	C₈×D₄	—	2	1	no	—
334	C₈×D₄	—	2	1	no	—
335	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
337	#432	C₂²×C₂₃	1	1	yes	characteristic
338	#431	—	2	1	no	—
339	#431	C₂²×C₂₃	1	1	yes	characteristic
340	#431	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₈	—	2	1	no	—
342	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	#335	C₂²×C₂₃	1	1	yes	characteristic
344	#335	C₂²×C₂₃	1	1	yes	characteristic
345	#334	C₂²×C₂₃	1	1	yes	characteristic
346	#334	C₂²×C₂₃	1	1	yes	characteristic
347	#332	C₂²×C₂₃	1	1	yes	characteristic
348	#332	C₂²×C₂₃	1	1	yes	characteristic
349	#332	C₂²×C₂₃	1	1	yes	characteristic
350	#332	C₂²×C₂₃	1	1	yes	characteristic
351	#332	C₂²×C₂₃	1	1	yes	characteristic
352	#332	C₂²×C₂₃	1	1	yes	characteristic
353	#331	C₂²×C₂₃	1	1	yes	characteristic
354	#331	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₂₃	—	4	1	no	—
356	C₂×C₂₃	—	4	2	no	—
357	C₂×C₂₃	—	16	1	no	—
358	C₂×C₂₃	#3045	1	1	yes	characteristic
359	C₂×C₂₃	#3050	1	1	yes	characteristic
360	C₂×C₂₃	#3069	1	1	yes	characteristic
361	C₂×#41	—	2	1	no	—
362	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
363	C₂×Q₁₆	—	2	1	no	—
364	C₂×Q₁₆	—	2	1	no	—
365	C₂×Q₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	1	no	—
367	C₂×SD₁₆	—	2	1	no	—
368	C₂×SD₁₆	—	2	1	no	—
369	C₂×SD₁₆	—	2	1	no	—
370	C₂×D₈	—	2	1	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₂²×C₈	—	2	1	no	—
374	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
375	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
376	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
377	#127	C₂³×C₂₃	1	1	yes	characteristic
378	#125	—	2	1	no	—
379	#124	—	2	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	1	no	—
382	#121	—	2	1	no	—
383	#121	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
387	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
393	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	D₄×C₂₃	1	1	yes	characteristic
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
399	#107	—	2	1	no	—
400	#107	—	2	1	no	—
401	#106	—	2	1	no	—
402	#106	—	2	1	no	—
403	C₄⋊C₈	—	2	1	no	—
404	C₄⋊C₈	—	2	1	no	—
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
407	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
408	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
412	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
413	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
414	#103	—	2	1	no	—
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	2	1	no	—
418	#103	—	2	1	no	—
419	#103	—	2	1	no	—
420	#102	—	2	1	no	—
421	#102	—	2	1	no	—
422	#102	—	2	1	no	—
423	#102	—	2	1	no	—
424	#102	—	2	1	no	—
425	#102	—	2	1	no	—
426	C₂²⋊C₈	—	2	1	no	—
427	C₂²⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	1	no	—
429	C₂²⋊C₈	—	2	1	no	—
430	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
431	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
432	#97	—	2	1	no	—
433	C₄×C₈	—	2	1	no	—
434	C₄×C₈	—	2	1	no	—
435	C₄×C₈	—	2	1	no	—
436	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
437	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
438	C₂₃	#19413	1	1	yes	characteristic, direct product factor, Sylow
439	#41	—	2	2	no	—
440	#41	—	2	2	no	—
441	#41	—	8	1	no	—
442	#41	—	8	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
460	Q₁₆	—	4	2	no	—
461	Q₁₆	—	8	1	no	—
462	Q₁₆	—	8	1	no	—
463	SD₁₆	—	4	2	no	—
464	SD₁₆	—	4	2	no	—
465	SD₁₆	—	8	1	no	—
466	SD₁₆	—	8	1	no	—
467	D₈	—	4	2	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	8	1	no	—
475	C₂×C₈	—	8	1	no	—
476	C₂×C₈	—	8	1	no	—
477	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	1	no	—
495	C₂²⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	4	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	4	1	no	—
503	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₄²	#41×C₂₃	1	1	yes	characteristic
506	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
507	C₂³	—	2	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	C₂₃×#121	1	1	yes	characteristic
510	Q₈	—	2	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	4	1	no	—
517	Q₈	C₂₃×#135	1	2	yes	—
518	D₄	—	2	4	no	—
519	D₄	—	2	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	8	1	no	—
525	D₄	—	8	2	no	—
526	D₄	C₂₃×#135	1	2	yes	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	8	1	no	—
542	C₂×C₄	—	8	2	no	—
543	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
544	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
545	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
548	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
549	C₈	—	2	1	no	—
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	2	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	8	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	—	8	1	no	—
568	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	1	no	—
573	C₄	—	2	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	1	no	—
576	C₄	—	8	1	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
580	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
581	C₄	C₂₃×#576	1	1	yes	characteristic
582	C₄	C₂₃×#577	1	1	yes	characteristic
583	C₂	—	4	1	no	—
584	C₂	—	4	2	no	—
585	C₂	—	16	1	no	—
586	C₂	C₂₃×#3045	1	1	yes	characteristic
587	C₂	C₂₃×#3050	1	1	yes	characteristic
588	C₂	C₂₃×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3115	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3210	C₂	semidirect
3. C₂₃×#3107	C₂	semidirect
4. C₂₃×#2988	C₂	semidirect
5. C₂₃×#1682	C₂	semidirect
6. C₂₃×#1681	C₂	semidirect
7. C₂₃×#1680	C₂	semidirect
8. C₂₃×#1679	C₂	semidirect
9. C₂₃×#1670	C₂	semidirect
10. C₂₃×#1669	C₂	semidirect
11. C₂₃×#1666	C₂	semidirect
12. C₂₃×#1665	C₂	semidirect
13. C₂₃×#1572	C₂	semidirect
14. C₂₃×#1552	C₂	semidirect
15. C₂₃×#1540	C₂	semidirect
212. #19413	C₂₃	direct (N × H)
265. #3210	C₂×C₂₃	semidirect
267. #3107	C₂×C₂₃	semidirect
268. #2988	C₂×C₂₃	semidirect
269. #1682	C₂×C₂₃	semidirect
270. #1681	C₂×C₂₃	semidirect
271. #1680	C₂×C₂₃	semidirect
272. #1679	C₂×C₂₃	semidirect
273. #1670	C₂×C₂₃	semidirect
274. #1669	C₂×C₂₃	semidirect
275. #1666	C₂×C₂₃	semidirect
276. #1665	C₂×C₂₃	semidirect
277. #1572	C₂×C₂₃	semidirect
278. #1552	C₂×C₂₃	semidirect
279. #1540	C₂×C₂₃	semidirect
438. C₂₃	#19413	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×#3210	2
C₂₃×#3115	2
C₂₃×#3107	2
C₂₃×#2988	2
C₂₃×#1682	2
C₂₃×#1681	2
C₂₃×#1680	2
C₂₃×#1679	2
C₂₃×#1670	2
C₂₃×#1669	2
C₂₃×#1666	2
C₂₃×#1665	2
C₂₃×#1572	2
C₂₃×#1552	2
C₂₃×#1540	2
#19413	23

Subgroups — C23×#19413 #19167039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19413 #19167039