Subgroups — C₂₃×#19864 #19167490

Order 5888.

578 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3360	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3357	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3341	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3340	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1723	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1691	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1676	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1675	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1646	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1638	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1634	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#556	C₂²	1	1	yes	characteristic
17	C₂₃×#555	C₂²	1	1	yes	characteristic
18	C₂₃×#555	C₂²	1	1	yes	characteristic
19	C₂₃×#554	C₂²	1	1	yes	characteristic
20	C₂₃×#500	C₂²	1	1	yes	characteristic
21	C₂₃×#500	C₂²	1	1	yes	characteristic
22	C₂₃×#498	C₂²	1	1	yes	characteristic
23	C₂₃×#494	C₂²	1	1	yes	characteristic
24	C₂₃×#488	—	2	1	no	—
25	C₂₃×#487	C₂²	1	1	yes	characteristic
26	C₂₃×#487	C₂²	1	1	yes	characteristic
27	C₂₃×#485	—	2	1	no	—
28	C₂₃×#485	—	2	1	no	—
29	C₂₃×#482	—	2	1	no	—
30	C₂₃×#482	—	2	1	no	—
31	C₂₃×#482	—	2	1	no	—
32	C₂₃×#480	—	2	1	no	—
33	C₂₃×#479	—	2	1	no	—
34	C₂₃×#477	C₂²	1	1	yes	characteristic
35	C₂₃×#477	C₂²	1	1	yes	characteristic
36	C₂₃×#476	C₂²	1	1	yes	characteristic
37	C₂₃×#476	C₂²	1	1	yes	characteristic
38	C₂₃×#476	C₂²	1	1	yes	characteristic
39	C₂₃×#476	C₂²	1	1	yes	characteristic
40	C₂₃×#474	C₂²	1	1	yes	characteristic
41	C₂₃×#468	—	2	1	no	—
42	C₂₃×#466	—	2	1	no	—
43	C₂₃×#465	—	2	1	no	—
44	C₂₃×#463	—	2	1	no	—
45	C₂₃×#461	C₂²	1	1	yes	characteristic
46	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×SD₁₆×C₂₃	—	2	1	no	—
48	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
50	C₂₃×#436	C₂²	1	1	yes	characteristic
51	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#428	—	2	1	no	—
54	C₂₃×#427	—	2	1	no	—
55	C₂₃×#427	—	2	1	no	—
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#335	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×D₄⋊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×Q₁₆×C₂₃	—	4	1	no	—
68	C₂×SD₁₆×C₂₃	—	2	1	no	—
69	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
70	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
71	C₂×M₁₆×C₂₃	—	2	1	no	—
72	C₂²×C₈×C₂₃	—	2	1	no	—
73	C₂²×C₈×C₂₃	—	2	1	no	—
74	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₂₃×#126	—	4	1	no	—
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×#124	—	2	1	no	—
81	C₂₃×#124	—	4	1	no	—
82	C₂₃×#124	—	4	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	4	1	no	—
86	C₂₃×#121	—	2	1	no	—
87	C₂₃×#121	—	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	C₂³	1	1	yes	characteristic
92	C₄×D₄×C₂₃	—	2	1	no	—
93	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂₃×#117	—	2	1	no	—
97	C₂₃×#117	—	2	1	no	—
98	C₂₃×#117	—	2	1	no	—
99	C₂₃×#117	—	2	1	no	—
100	C₂₃×#117	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂₃×#107	—	2	1	no	—
103	C₂₃×#107	—	2	1	no	—
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#107	—	2	1	no	—
108	C₂₃×#107	—	2	1	no	—
109	C₂₃×#107	D₄	1	1	yes	characteristic
110	C₂₃×#106	—	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	—	2	1	no	—
116	C₂₃×#106	D₄	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	—	2	1	no	—
118	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#103	—	2	1	no	—
127	C₂₃×#103	—	4	1	no	—
128	C₂₃×#103	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
135	C₂₃×#102	D₄	1	1	yes	characteristic
136	C₂₃×#102	D₄	1	1	yes	characteristic
137	C₂₃×C₂²⋊C₈	—	2	1	no	—
138	C₂₃×C₂²⋊C₈	—	2	1	no	—
139	C₂₃×C₂²⋊C₈	—	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	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂×D₄×C₂₃	—	4	1	no	—
150	C₂×D₄×C₂₃	—	4	1	no	—
151	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
158	Q₁₆×C₂₃	—	8	2	no	—
159	SD₁₆×C₂₃	—	2	2	no	—
160	SD₁₆×C₂₃	—	4	2	no	—
161	D₈×C₂₃	—	4	1	no	—
162	M₁₆×C₂₃	—	4	2	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	—	2	1	no	—
168	C₂×C₈×C₂₃	—	4	2	no	—
169	C₂×C₈×C₂₃	—	4	2	no	—
170	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	4	1	no	—
184	C₄⋊C₄×C₂₃	—	4	1	no	—
185	C₄⋊C₄×C₂₃	—	4	1	no	—
186	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²⋊C₄×C₂₃	—	2	1	no	—
191	C₂²⋊C₄×C₂₃	—	2	1	no	—
192	C₂²⋊C₄×C₂₃	—	2	1	no	—
193	C₂²⋊C₄×C₂₃	—	4	1	no	—
194	C₂²⋊C₄×C₂₃	—	4	1	no	—
195	C₂²⋊C₄×C₂₃	—	4	1	no	—
196	C₂²⋊C₄×C₂₃	—	4	1	no	—
197	C₂²⋊C₄×C₂₃	—	4	1	no	—
198	C₂²⋊C₄×C₂₃	—	4	1	no	—
199	C₂²⋊C₄×C₂₃	—	4	1	no	—
200	C₄²×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	4	1	no	—
202	C₄²×C₂₃	—	4	1	no	—
203	C₄²×C₂₃	—	4	1	no	—
204	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
205	#19864	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₂₃	—	2	1	no	—
207	C₂³×C₂₃	—	2	1	no	—
208	C₂³×C₂₃	—	2	1	no	—
209	Q₈×C₂₃	—	2	1	no	—
210	Q₈×C₂₃	—	2	1	no	—
211	Q₈×C₂₃	—	4	1	no	—
212	Q₈×C₂₃	—	4	1	no	—
213	Q₈×C₂₃	—	8	1	no	—
214	Q₈×C₂₃	—	8	1	no	—
215	Q₈×C₂₃	—	8	1	no	—
216	D₄×C₂₃	—	2	1	no	—
217	D₄×C₂₃	—	2	1	no	—
218	D₄×C₂₃	—	2	1	no	—
219	D₄×C₂₃	—	2	1	no	—
220	D₄×C₂₃	—	2	1	no	—
221	D₄×C₂₃	—	2	1	no	—
222	D₄×C₂₃	—	16	1	no	—
223	D₄×C₂₃	—	16	1	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	16	1	no	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
243	C₈×C₂₃	—	2	1	no	—
244	C₈×C₂₃	—	2	1	no	—
245	C₈×C₂₃	—	2	1	no	—
246	C₈×C₂₃	—	4	1	no	—
247	C₈×C₂₃	—	4	1	no	—
248	C₈×C₂₃	—	4	1	no	—
249	C₈×C₂₃	—	4	1	no	—
250	C₈×C₂₃	—	4	1	no	—
251	C₈×C₂₃	#135	1	2	yes	—
252	#3360	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#3357	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#3341	C₂×C₂₃	1	1	yes	characteristic
255	#3340	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#1723	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#1691	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#1683	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#1676	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1675	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1646	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1638	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1634	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
266	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
267	C₂²×C₂₃	—	4	1	no	—
268	C₂²×C₂₃	—	4	1	no	—
269	C₂²×C₂₃	—	4	1	no	—
270	C₂²×C₂₃	—	4	1	no	—
271	C₂²×C₂₃	—	4	1	no	—
272	C₂²×C₂₃	—	4	1	no	—
273	C₂²×C₂₃	—	4	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	D₄²	1	1	yes	characteristic
277	C₄×C₂₃	—	2	1	no	—
278	C₄×C₂₃	—	4	1	no	—
279	C₄×C₂₃	—	4	1	no	—
280	C₄×C₂₃	—	4	1	no	—
281	C₄×C₂₃	—	4	1	no	—
282	C₄×C₂₃	—	4	1	no	—
283	C₄×C₂₃	—	8	1	no	—
284	C₄×C₂₃	—	8	1	no	—
285	C₄×C₂₃	—	8	1	no	—
286	C₄×C₂₃	—	8	1	no	—
287	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
288	C₄×C₂₃	#575	1	1	yes	characteristic
289	C₄×C₂₃	#578	1	1	yes	characteristic
290	C₄×C₂₃	#578	1	1	yes	characteristic
291	#556	C₂²×C₂₃	1	1	yes	characteristic
292	#555	C₂²×C₂₃	1	1	yes	characteristic
293	#555	C₂²×C₂₃	1	1	yes	characteristic
294	#554	C₂²×C₂₃	1	1	yes	characteristic
295	#500	C₂²×C₂₃	1	1	yes	characteristic
296	#500	C₂²×C₂₃	1	1	yes	characteristic
297	#498	C₂²×C₂₃	1	1	yes	characteristic
298	#494	C₂²×C₂₃	1	1	yes	characteristic
299	#488	—	2	1	no	—
300	#487	C₂²×C₂₃	1	1	yes	characteristic
301	#487	C₂²×C₂₃	1	1	yes	characteristic
302	#485	—	2	1	no	—
303	#485	—	2	1	no	—
304	#482	—	2	1	no	—
305	#482	—	2	1	no	—
306	#482	—	2	1	no	—
307	#480	—	2	1	no	—
308	#479	—	2	1	no	—
309	#477	C₂²×C₂₃	1	1	yes	characteristic
310	#477	C₂²×C₂₃	1	1	yes	characteristic
311	#476	C₂²×C₂₃	1	1	yes	characteristic
312	#476	C₂²×C₂₃	1	1	yes	characteristic
313	#476	C₂²×C₂₃	1	1	yes	characteristic
314	#476	C₂²×C₂₃	1	1	yes	characteristic
315	#474	C₂²×C₂₃	1	1	yes	characteristic
316	#468	—	2	1	no	—
317	#466	—	2	1	no	—
318	#465	—	2	1	no	—
319	#463	—	2	1	no	—
320	#461	C₂²×C₂₃	1	1	yes	characteristic
321	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
322	C₄×SD₁₆	—	2	1	no	—
323	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
324	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
325	#436	C₂²×C₂₃	1	1	yes	characteristic
326	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
327	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
328	#428	—	2	1	no	—
329	#427	—	2	1	no	—
330	#427	—	2	1	no	—
331	#335	C₂²×C₂₃	1	1	yes	characteristic
332	#335	C₂²×C₂₃	1	1	yes	characteristic
333	#333	C₂²×C₂₃	1	1	yes	characteristic
334	#333	C₂²×C₂₃	1	1	yes	characteristic
335	#333	C₂²×C₂₃	1	1	yes	characteristic
336	#332	C₂²×C₂₃	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
338	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
339	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
340	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
341	#322	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₂₃	—	8	1	no	—
343	C₂×C₂₃	—	8	1	no	—
344	C₂×C₂₃	—	8	1	no	—
345	C₂×C₂₃	#3269	1	1	yes	characteristic
346	C₂×C₂₃	#3283	1	1	yes	characteristic
347	C₂×C₂₃	#3288	1	1	yes	characteristic
348	C₂×Q₁₆	—	4	1	no	—
349	C₂×SD₁₆	—	2	1	no	—
350	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
351	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
352	C₂×M₁₆	—	2	1	no	—
353	C₂²×C₈	—	2	1	no	—
354	C₂²×C₈	—	2	1	no	—
355	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
359	#126	—	4	1	no	—
360	#125	—	2	1	no	—
361	#124	—	2	1	no	—
362	#124	—	4	1	no	—
363	#124	—	4	1	no	—
364	C₂²⋊Q₈	—	2	1	no	—
365	C₂²⋊Q₈	—	2	1	no	—
366	C₂²⋊Q₈	—	4	1	no	—
367	#121	—	2	1	no	—
368	#121	—	2	1	no	—
369	#121	—	2	1	no	—
370	#121	—	2	1	no	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
375	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
376	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
377	#117	—	2	1	no	—
378	#117	—	2	1	no	—
379	#117	—	2	1	no	—
380	#117	—	2	1	no	—
381	#117	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	#107	—	2	1	no	—
384	#107	—	2	1	no	—
385	#107	—	2	1	no	—
386	#107	—	2	1	no	—
387	#107	—	2	1	no	—
388	#107	—	2	1	no	—
389	#107	—	2	1	no	—
390	#107	D₄×C₂₃	1	1	yes	characteristic
391	#106	—	2	1	no	—
392	#106	—	2	1	no	—
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	#106	—	2	1	no	—
396	#106	—	2	1	no	—
397	#106	D₄×C₂₃	1	1	yes	characteristic
398	C₄⋊C₈	—	2	1	no	—
399	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
403	#103	—	2	1	no	—
404	#103	—	2	1	no	—
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	—	2	1	no	—
408	#103	—	4	1	no	—
409	#103	D₄×C₂₃	1	1	yes	characteristic
410	#102	—	2	1	no	—
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	—	2	1	no	—
414	#102	—	2	1	no	—
415	#102	D₄×C₂₃	1	1	yes	characteristic
416	#102	D₄×C₂₃	1	1	yes	characteristic
417	#102	D₄×C₂₃	1	1	yes	characteristic
418	C₂²⋊C₈	—	2	1	no	—
419	C₂²⋊C₈	—	2	1	no	—
420	C₂²⋊C₈	—	2	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
425	C₂₃	#19864	1	1	yes	characteristic, direct product factor, Sylow
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₂×D₄	—	4	1	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	Q₁₆	—	8	2	no	—
441	SD₁₆	—	2	2	no	—
442	SD₁₆	—	4	2	no	—
443	D₈	—	4	1	no	—
444	M₁₆	—	4	2	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	—	4	2	no	—
451	C₂×C₈	—	4	2	no	—
452	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	4	1	no	—
466	C₄⋊C₄	—	4	1	no	—
467	C₄⋊C₄	—	4	1	no	—
468	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	4	1	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₂²⋊C₄	—	4	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	4	1	no	—
484	C₄²	—	4	1	no	—
485	C₄²	—	4	1	no	—
486	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
487	C₂³	—	2	1	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	1	no	—
490	Q₈	—	2	1	no	—
491	Q₈	—	2	1	no	—
492	Q₈	—	4	1	no	—
493	Q₈	—	4	1	no	—
494	Q₈	—	8	1	no	—
495	Q₈	—	8	1	no	—
496	Q₈	—	8	1	no	—
497	D₄	—	2	1	no	—
498	D₄	—	2	1	no	—
499	D₄	—	2	1	no	—
500	D₄	—	2	1	no	—
501	D₄	—	2	1	no	—
502	D₄	—	2	1	no	—
503	D₄	—	16	1	no	—
504	D₄	—	16	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	16	1	no	—
521	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
523	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
524	C₈	—	2	1	no	—
525	C₈	—	2	1	no	—
526	C₈	—	2	1	no	—
527	C₈	—	4	1	no	—
528	C₈	—	4	1	no	—
529	C₈	—	4	1	no	—
530	C₈	—	4	1	no	—
531	C₈	—	4	1	no	—
532	C₈	C₂₃×#135	1	2	yes	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	1	no	—
538	C₂²	—	4	1	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	D₄²×C₂₃	1	1	yes	characteristic
543	C₄	—	2	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	8	1	no	—
550	C₄	—	8	1	no	—
551	C₄	—	8	1	no	—
552	C₄	—	8	1	no	—
553	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
554	C₄	C₂₃×#575	1	1	yes	characteristic
555	C₄	C₂₃×#578	1	1	yes	characteristic
556	C₄	C₂₃×#578	1	1	yes	characteristic
557	C₂	—	8	1	no	—
558	C₂	—	8	1	no	—
559	C₂	—	8	1	no	—
560	C₂	C₂₃×#3269	1	1	yes	characteristic
561	C₂	C₂₃×#3283	1	1	yes	characteristic
562	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3341	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×#3360	C₂	semidirect
2. C₂₃×#3357	C₂	semidirect
4. C₂₃×#3340	C₂	semidirect
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1691	C₂	semidirect
7. C₂₃×#1683	C₂	semidirect
8. C₂₃×#1676	C₂	semidirect
9. C₂₃×#1675	C₂	semidirect
10. C₂₃×#1646	C₂	semidirect
11. C₂₃×#1638	C₂	semidirect
12. C₂₃×#1634	C₂	semidirect
13. C₂₃×#1632	C₂	semidirect
14. C₂₃×#1581	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
205. #19864	C₂₃	direct (N × H)
252. #3360	C₂×C₂₃	semidirect
253. #3357	C₂×C₂₃	semidirect
255. #3340	C₂×C₂₃	semidirect
256. #1723	C₂×C₂₃	semidirect
257. #1691	C₂×C₂₃	semidirect
258. #1683	C₂×C₂₃	semidirect
259. #1676	C₂×C₂₃	semidirect
260. #1675	C₂×C₂₃	semidirect
261. #1646	C₂×C₂₃	semidirect
262. #1638	C₂×C₂₃	semidirect
263. #1634	C₂×C₂₃	semidirect
264. #1632	C₂×C₂₃	semidirect
265. #1581	C₂×C₂₃	semidirect
266. SD₁₆⋊C₈	C₂×C₂₃	semidirect
425. C₂₃	#19864	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×#3360	2
C₂₃×#3357	2
C₂₃×#3341	2
C₂₃×#3340	2
C₂₃×#1723	2
C₂₃×#1691	2
C₂₃×#1683	2
C₂₃×#1676	2
C₂₃×#1675	2
C₂₃×#1646	2
C₂₃×#1638	2
C₂₃×#1634	2
C₂₃×#1632	2
C₂₃×#1581	2
C₂₃×SD₁₆⋊C₈	2
#19864	23

Subgroups — C23×#19864 #19167490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19864 #19167490