Subgroups — C₂₃×#19925 #19167551

Order 5888.

608 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×#3375	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3362	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1698	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1697	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1620	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1590	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#559	C₂²	1	1	yes	characteristic
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	C₂₃×#550	C₂²	1	1	yes	characteristic
20	C₂₃×#494	C₂²	1	1	yes	characteristic
21	C₂₃×#493	C₂²	1	1	yes	characteristic
22	C₂₃×#489	—	2	1	no	—
23	C₂₃×#488	—	2	1	no	—
24	C₂₃×#486	—	2	1	no	—
25	C₂₃×#486	C₂²	1	1	yes	characteristic
26	C₂₃×#486	C₂²	1	1	yes	characteristic
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#486	C₂²	1	1	yes	characteristic
29	C₂₃×#480	—	2	1	no	—
30	C₂₃×#478	—	2	1	no	—
31	C₂₃×#477	C₂²	1	1	yes	characteristic
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#477	C₂²	1	1	yes	characteristic
34	C₂₃×#476	—	2	1	no	—
35	C₂₃×#476	C₂²	1	1	yes	characteristic
36	C₂₃×#474	—	2	1	no	—
37	C₂₃×#468	—	2	1	no	—
38	C₂₃×#464	—	2	1	no	—
39	C₂₃×#463	—	2	1	no	—
40	C₂₃×#460	—	2	1	no	—
41	C₂₃×#460	C₂²	1	1	yes	characteristic
42	C₂₃×#460	C₂²	1	1	yes	characteristic
43	C₂₃×#460	C₂²	1	1	yes	characteristic
44	C₂₃×#459	C₂²	1	1	yes	characteristic
45	C₂₃×#452	—	2	1	no	—
46	C₂₃×#446	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#440	—	2	1	no	—
50	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×SD₁₆×C₂₃	—	2	1	no	—
52	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
53	C₄×D₈×C₂₃	—	2	1	no	—
54	C₂₃×#436	C₂²	1	1	yes	characteristic
55	C₂₃×#417	—	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×#333	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₈²×C₂₃	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×Q₁₆×C₂₃	—	4	1	no	—
69	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
74	C₂×D₈×C₂₃	—	2	1	no	—
75	C₂×D₈×C₂₃	—	4	1	no	—
76	C₂×M₁₆×C₂₃	—	2	1	no	—
77	C₂₃×C₄⋊Q₈	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	4	1	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×#127	—	2	1	no	—
83	C₂₃×#127	C₂³	1	1	yes	characteristic
84	C₂₃×#127	C₂³	1	1	yes	characteristic
85	C₂₃×#125	—	2	1	no	—
86	C₂₃×#124	—	2	1	no	—
87	C₂₃×#124	—	2	1	no	—
88	C₂₃×#123	—	4	1	no	—
89	C₂₃×C₂²⋊Q₈	—	2	1	no	—
90	C₂₃×C₂²⋊Q₈	—	4	1	no	—
91	C₂₃×#121	—	2	1	no	—
92	C₂₃×#121	—	4	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#107	D₄	1	1	yes	characteristic
108	C₂₃×#106	—	2	1	no	—
109	C₂₃×#106	D₄	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	—	2	1	no	—
111	C₂₃×C₄⋊C₈	—	2	1	no	—
112	C₂₃×C₄⋊C₈	—	2	1	no	—
113	C₂₃×C₄⋊C₈	C₂³	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×#103	—	2	1	no	—
118	C₂₃×#103	—	2	1	no	—
119	C₂₃×#103	—	2	1	no	—
120	C₂₃×#103	D₄	1	1	yes	characteristic
121	C₂₃×#103	D₄	1	1	yes	characteristic
122	C₂₃×#103	D₄	1	1	yes	characteristic
123	C₂₃×#102	—	2	1	no	—
124	C₂₃×#102	—	2	1	no	—
125	C₂₃×#102	—	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×#102	—	2	1	no	—
133	C₂₃×#102	—	2	1	no	—
134	C₂₃×#102	—	2	1	no	—
135	C₂₃×#102	—	2	1	no	—
136	C₂₃×#102	D₄	1	1	yes	characteristic
137	C₂₃×C₂²⋊C₈	—	2	1	no	—
138	C₂₃×#97	—	2	1	no	—
139	C₄×C₈×C₂₃	—	2	1	no	—
140	C₄×C₈×C₂₃	—	2	1	no	—
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	#41×C₂₃	—	4	2	no	—
145	#41×C₂₃	—	8	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	4	1	no	—
149	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	4	1	no	—
159	C₂×D₄×C₂₃	—	4	1	no	—
160	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	4	1	no	—
163	C₂²×C₄×C₂₃	—	4	1	no	—
164	C₂²×C₄×C₂₃	—	4	1	no	—
165	Q₁₆×C₂₃	—	2	2	no	—
166	Q₁₆×C₂₃	—	8	2	no	—
167	SD₁₆×C₂₃	—	4	1	no	—
168	SD₁₆×C₂₃	—	4	2	no	—
169	SD₁₆×C₂₃	—	4	2	no	—
170	SD₁₆×C₂₃	—	8	1	no	—
171	D₈×C₂₃	—	4	2	no	—
172	D₈×C₂₃	—	8	2	no	—
173	M₁₆×C₂₃	—	8	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	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
191	C₄⋊C₄×C₂₃	—	4	1	no	—
192	C₄⋊C₄×C₂₃	—	4	1	no	—
193	C₄⋊C₄×C₂₃	—	4	1	no	—
194	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
200	C₂²⋊C₄×C₂₃	—	4	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	C₂⁴	1	1	yes	characteristic
209	#19925	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₂₃	—	2	1	no	—
211	C₂³×C₂₃	—	4	1	no	—
212	C₂³×C₂₃	—	4	1	no	—
213	Q₈×C₂₃	—	2	1	no	—
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	2	1	no	—
216	Q₈×C₂₃	—	2	1	no	—
217	Q₈×C₂₃	—	2	1	no	—
218	Q₈×C₂₃	—	8	1	no	—
219	Q₈×C₂₃	—	8	1	no	—
220	Q₈×C₂₃	—	16	1	no	—
221	Q₈×C₂₃	#135	1	2	yes	—
222	D₄×C₂₃	—	2	1	no	—
223	D₄×C₂₃	—	2	1	no	—
224	D₄×C₂₃	—	4	1	no	—
225	D₄×C₂₃	—	4	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	—	4	1	no	—
231	D₄×C₂₃	—	4	1	no	—
232	D₄×C₂₃	—	8	2	no	—
233	D₄×C₂₃	—	8	1	no	—
234	D₄×C₂₃	—	8	1	no	—
235	D₄×C₂₃	—	16	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	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	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	2	no	—
248	C₂×C₄×C₂₃	—	16	1	no	—
249	C₂×C₄×C₂₃	—	16	1	no	—
250	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
253	C₈×C₂₃	—	2	2	no	—
254	C₈×C₂₃	—	2	1	no	—
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	2	1	no	—
257	C₈×C₂₃	—	2	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	#3375	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#3362	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#3361	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#3333	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1709	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1698	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1697	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1621	C₂×C₂₃	1	1	yes	characteristic
274	#1620	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1590	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₈×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂²×C₂₃	—	4	1	no	—
278	C₂²×C₂₃	—	4	1	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	8	1	no	—
281	C₂²×C₂₃	—	8	1	no	—
282	C₂²×C₂₃	—	8	1	no	—
283	C₂²×C₂₃	—	8	1	no	—
284	C₂²×C₂₃	—	8	1	no	—
285	C₂²×C₂₃	—	8	1	no	—
286	C₂²×C₂₃	D₄²	1	1	yes	characteristic
287	C₄×C₂₃	—	2	2	no	—
288	C₄×C₂₃	—	2	1	no	—
289	C₄×C₂₃	—	4	1	no	—
290	C₄×C₂₃	—	4	1	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	4	1	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	8	1	no	—
296	C₄×C₂₃	—	8	1	no	—
297	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
298	C₄×C₂₃	#575	1	1	yes	characteristic
299	C₄×C₂₃	#577	1	1	yes	characteristic
300	C₄×C₂₃	#578	1	1	yes	characteristic
301	#559	C₂²×C₂₃	1	1	yes	characteristic
302	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
303	#554	C₂²×C₂₃	1	1	yes	characteristic
304	#550	C₂²×C₂₃	1	1	yes	characteristic
305	#494	C₂²×C₂₃	1	1	yes	characteristic
306	#493	C₂²×C₂₃	1	1	yes	characteristic
307	#489	—	2	1	no	—
308	#488	—	2	1	no	—
309	#486	—	2	1	no	—
310	#486	C₂²×C₂₃	1	1	yes	characteristic
311	#486	C₂²×C₂₃	1	1	yes	characteristic
312	#486	C₂²×C₂₃	1	1	yes	characteristic
313	#486	C₂²×C₂₃	1	1	yes	characteristic
314	#480	—	2	1	no	—
315	#478	—	2	1	no	—
316	#477	C₂²×C₂₃	1	1	yes	characteristic
317	#477	C₂²×C₂₃	1	1	yes	characteristic
318	#477	C₂²×C₂₃	1	1	yes	characteristic
319	#476	—	2	1	no	—
320	#476	C₂²×C₂₃	1	1	yes	characteristic
321	#474	—	2	1	no	—
322	#468	—	2	1	no	—
323	#464	—	2	1	no	—
324	#463	—	2	1	no	—
325	#460	—	2	1	no	—
326	#460	C₂²×C₂₃	1	1	yes	characteristic
327	#460	C₂²×C₂₃	1	1	yes	characteristic
328	#460	C₂²×C₂₃	1	1	yes	characteristic
329	#459	C₂²×C₂₃	1	1	yes	characteristic
330	#452	—	2	1	no	—
331	#446	C₂²×C₂₃	1	1	yes	characteristic
332	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
334	#440	—	2	1	no	—
335	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
336	C₄×SD₁₆	—	2	1	no	—
337	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
338	C₄×D₈	—	2	1	no	—
339	#436	C₂²×C₂₃	1	1	yes	characteristic
340	#417	—	2	1	no	—
341	#335	C₂²×C₂₃	1	1	yes	characteristic
342	#334	C₂²×C₂₃	1	1	yes	characteristic
343	#333	C₂²×C₂₃	1	1	yes	characteristic
344	#333	C₂²×C₂₃	1	1	yes	characteristic
345	#331	C₂²×C₂₃	1	1	yes	characteristic
346	#331	C₂²×C₂₃	1	1	yes	characteristic
347	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
349	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
351	C₈²	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×C₂₃	—	8	1	no	—
353	C₂×C₂₃	—	16	1	no	—
354	C₂×C₂₃	—	16	1	no	—
355	C₂×C₂₃	#3277	1	1	yes	characteristic
356	C₂×C₂₃	#3282	1	1	yes	characteristic
357	C₂×C₂₃	#3292	1	1	yes	characteristic
358	C₂×#41	—	2	1	no	—
359	C₂×Q₁₆	—	4	1	no	—
360	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
361	C₂×SD₁₆	—	2	1	no	—
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
365	C₂×D₈	—	2	1	no	—
366	C₂×D₈	—	4	1	no	—
367	C₂×M₁₆	—	2	1	no	—
368	C₄⋊Q₈	—	2	1	no	—
369	C₄⋊Q₈	—	2	1	no	—
370	C₄⋊Q₈	—	4	1	no	—
371	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
373	#127	—	2	1	no	—
374	#127	C₂³×C₂₃	1	1	yes	characteristic
375	#127	C₂³×C₂₃	1	1	yes	characteristic
376	#125	—	2	1	no	—
377	#124	—	2	1	no	—
378	#124	—	2	1	no	—
379	#123	—	4	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	4	1	no	—
382	#121	—	2	1	no	—
383	#121	—	4	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
394	C₂×C₄⋊C₄	—	2	1	no	—
395	#107	—	2	1	no	—
396	#107	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	D₄×C₂₃	1	1	yes	characteristic
399	#106	—	2	1	no	—
400	#106	D₄×C₂₃	1	1	yes	characteristic
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	—	2	1	no	—
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	D₄×C₂₃	1	1	yes	characteristic
412	#103	D₄×C₂₃	1	1	yes	characteristic
413	#103	D₄×C₂₃	1	1	yes	characteristic
414	#102	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	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	#102	—	2	1	no	—
427	#102	D₄×C₂₃	1	1	yes	characteristic
428	C₂²⋊C₈	—	2	1	no	—
429	#97	—	2	1	no	—
430	C₄×C₈	—	2	1	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₂₃	#19925	1	1	yes	characteristic, direct product factor, Sylow
436	#41	—	4	2	no	—
437	#41	—	8	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	4	1	no	—
441	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	4	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	—	4	1	no	—
455	C₂²×C₄	—	4	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	Q₁₆	—	2	2	no	—
458	Q₁₆	—	8	2	no	—
459	SD₁₆	—	4	1	no	—
460	SD₁₆	—	4	2	no	—
461	SD₁₆	—	4	2	no	—
462	SD₁₆	—	8	1	no	—
463	D₈	—	4	2	no	—
464	D₈	—	8	2	no	—
465	M₁₆	—	8	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
472	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
473	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	4	1	no	—
500	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
501	C₂³	—	2	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	4	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	1	no	—
509	Q₈	—	8	1	no	—
510	Q₈	—	8	1	no	—
511	Q₈	—	16	1	no	—
512	Q₈	C₂₃×#135	1	2	yes	—
513	D₄	—	2	1	no	—
514	D₄	—	2	1	no	—
515	D₄	—	4	1	no	—
516	D₄	—	4	1	no	—
517	D₄	—	4	1	no	—
518	D₄	—	4	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	8	2	no	—
524	D₄	—	8	1	no	—
525	D₄	—	8	1	no	—
526	D₄	—	16	1	no	—
527	C₂×C₄	—	2	1	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	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	16	1	no	—
540	C₂×C₄	—	16	1	no	—
541	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
544	C₈	—	2	2	no	—
545	C₈	—	2	1	no	—
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	D₄²×C₂₃	1	1	yes	characteristic
563	C₄	—	2	2	no	—
564	C₄	—	2	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	8	1	no	—
572	C₄	—	8	1	no	—
573	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
574	C₄	C₂₃×#575	1	1	yes	characteristic
575	C₄	C₂₃×#577	1	1	yes	characteristic
576	C₄	C₂₃×#578	1	1	yes	characteristic
577	C₂	—	8	1	no	—
578	C₂	—	16	1	no	—
579	C₂	—	16	1	no	—
580	C₂	C₂₃×#3277	1	1	yes	characteristic
581	C₂	C₂₃×#3282	1	1	yes	characteristic
582	C₂	C₂₃×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#1621	C₂
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×#3375	C₂	semidirect
2. C₂₃×#3362	C₂	semidirect
3. C₂₃×#3361	C₂	semidirect
4. C₂₃×#3333	C₂	semidirect
5. C₂₃×#1709	C₂	semidirect
6. C₂₃×#1698	C₂	semidirect
7. C₂₃×#1697	C₂	semidirect
8. C₂₃×#1678	C₂	semidirect
9. C₂₃×#1678	C₂	semidirect
10. C₂₃×#1633	C₂	semidirect
11. C₂₃×#1632	C₂	semidirect
13. C₂₃×#1620	C₂	semidirect
14. C₂₃×#1590	C₂	semidirect
15. C₈×Q₁₆×C₂₃	C₂	semidirect
209. #19925	C₂₃	direct (N × H)
262. #3375	C₂×C₂₃	semidirect
263. #3362	C₂×C₂₃	semidirect
264. #3361	C₂×C₂₃	semidirect
265. #3333	C₂×C₂₃	semidirect
266. #1709	C₂×C₂₃	semidirect
267. #1698	C₂×C₂₃	semidirect
268. #1697	C₂×C₂₃	semidirect
269. #1678	C₂×C₂₃	semidirect
270. #1678	C₂×C₂₃	semidirect
271. #1633	C₂×C₂₃	semidirect
272. #1632	C₂×C₂₃	semidirect
274. #1620	C₂×C₂₃	semidirect
275. #1590	C₂×C₂₃	semidirect
276. C₈×Q₁₆	C₂×C₂₃	semidirect
435. C₂₃	#19925	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×#3375	2
C₂₃×#3362	2
C₂₃×#3361	2
C₂₃×#3333	2
C₂₃×#1709	2
C₂₃×#1698	2
C₂₃×#1697	2
C₂₃×#1678	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1621	2
C₂₃×#1620	2
C₂₃×#1590	2
C₈×Q₁₆×C₂₃	2
#19925	23

Subgroups — C23×#19925 #19167551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19925 #19167551