Subgroups — C₂₃×#19855 #19167481

Order 5888.

622 conjugacy classes of subgroups.

The subgroup lattice has 604 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×#3394	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3305	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×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1641	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1630	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1628	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1624	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	C₂₃×#559	C₂²	1	1	yes	characteristic
17	C₂₃×#556	C₂²	1	1	yes	characteristic
18	C₂₃×#555	C₂²	1	1	yes	characteristic
19	C₂₃×#550	C₂²	1	1	yes	characteristic
20	C₂₃×#498	C₂²	1	1	yes	characteristic
21	C₂₃×#496	—	2	1	no	—
22	C₂₃×#495	—	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×#486	C₂²	1	1	yes	characteristic
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#485	—	2	1	no	—
29	C₂₃×#482	—	2	1	no	—
30	C₂₃×#482	—	2	1	no	—
31	C₂₃×#477	C₂²	1	1	yes	characteristic
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#474	C₂²	1	1	yes	characteristic
34	C₂₃×#466	—	2	1	no	—
35	C₂₃×#465	—	2	1	no	—
36	C₂₃×#464	—	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×#449	—	2	1	no	—
46	C₂₃×#446	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₂₃×#442	—	2	1	no	—
49	C₄×SD₁₆×C₂₃	—	2	1	no	—
50	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₂₃×#427	—	2	1	no	—
55	C₂₃×#416	—	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×Q₁₆×C₂₃	—	2	1	no	—
69	C₂×SD₁₆×C₂₃	—	2	1	no	—
70	C₂×SD₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	4	1	no	—
73	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
75	C₂×D₈×C₂₃	—	2	1	no	—
76	C₂×D₈×C₂₃	—	2	1	no	—
77	C₂×D₈×C₂₃	—	2	1	no	—
78	C₂×D₈×C₂₃	—	2	1	no	—
79	C₂×D₈×C₂₃	—	2	1	no	—
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂²×C₈×C₂₃	—	2	1	no	—
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×#127	—	2	1	no	—
85	C₂₃×#127	C₂³	1	1	yes	characteristic
86	C₂₃×#127	C₂³	1	1	yes	characteristic
87	C₂₃×#126	—	4	1	no	—
88	C₂₃×#124	—	2	1	no	—
89	C₂₃×#124	—	2	1	no	—
90	C₂₃×#124	—	2	1	no	—
91	C₂₃×#124	—	4	1	no	—
92	C₂₃×#123	—	4	1	no	—
93	C₂₃×C₂²⋊Q₈	—	2	1	no	—
94	C₂₃×#121	—	2	1	no	—
95	C₂₃×#121	—	2	1	no	—
96	C₂₃×#121	—	2	1	no	—
97	C₂₃×#121	—	4	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×#117	—	2	1	no	—
107	C₂₃×#117	—	2	1	no	—
108	C₂₃×#117	—	2	1	no	—
109	C₂₃×#107	—	2	1	no	—
110	C₂₃×#107	—	2	1	no	—
111	C₂₃×#107	—	2	1	no	—
112	C₂₃×#106	—	2	1	no	—
113	C₂₃×#106	—	2	1	no	—
114	C₂₃×#106	D₄	1	1	yes	characteristic
115	C₂₃×#106	D₄	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	—	2	1	no	—
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	D₄	1	1	yes	characteristic
126	C₂₃×#103	D₄	1	1	yes	characteristic
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	—	4	1	no	—
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×#97	—	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	C₂×D₄	1	1	yes	characteristic
149	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
158	C₂×D₄×C₂₃	—	4	1	no	—
159	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	4	1	no	—
164	C₂²×C₄×C₂₃	—	4	1	no	—
165	Q₁₆×C₂₃	—	8	1	no	—
166	SD₁₆×C₂₃	—	2	2	no	—
167	SD₁₆×C₂₃	—	4	1	no	—
168	SD₁₆×C₂₃	—	4	2	no	—
169	SD₁₆×C₂₃	—	8	1	no	—
170	SD₁₆×C₂₃	—	8	1	no	—
171	SD₁₆×C₂₃	—	8	2	no	—
172	D₈×C₂₃	—	4	2	no	—
173	D₈×C₂₃	—	8	1	no	—
174	D₈×C₂₃	—	8	1	no	—
175	D₈×C₂₃	—	8	1	no	—
176	D₈×C₂₃	—	8	1	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	—	4	2	no	—
183	C₂×C₈×C₂₃	—	8	1	no	—
184	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	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	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	C₂²⋊C₄×C₂₃	—	4	1	no	—
205	C₂²⋊C₄×C₂₃	—	4	1	no	—
206	C₂²⋊C₄×C₂₃	—	4	1	no	—
207	C₂²⋊C₄×C₂₃	—	4	1	no	—
208	C₂²⋊C₄×C₂₃	—	4	1	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	4	1	no	—
212	C₄²×C₂₃	—	4	1	no	—
213	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
214	#19855	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₂₃	—	2	1	no	—
216	C₂³×C₂₃	—	2	1	no	—
217	C₂³×C₂₃	—	4	1	no	—
218	C₂³×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	2	1	no	—
220	Q₈×C₂₃	—	2	1	no	—
221	Q₈×C₂₃	—	2	1	no	—
222	Q₈×C₂₃	—	2	1	no	—
223	Q₈×C₂₃	—	4	1	no	—
224	Q₈×C₂₃	—	4	1	no	—
225	Q₈×C₂₃	—	8	1	no	—
226	D₄×C₂₃	—	2	1	no	—
227	D₄×C₂₃	—	2	1	no	—
228	D₄×C₂₃	—	2	1	no	—
229	D₄×C₂₃	—	2	1	no	—
230	D₄×C₂₃	—	4	1	no	—
231	D₄×C₂₃	—	4	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	—	8	1	no	—
239	D₄×C₂₃	—	8	1	no	—
240	D₄×C₂₃	—	8	2	no	—
241	D₄×C₂₃	—	16	1	no	—
242	D₄×C₂₃	—	16	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	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	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	—	8	2	no	—
256	C₂×C₄×C₂₃	—	16	1	no	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
260	C₈×C₂₃	—	2	1	no	—
261	C₈×C₂₃	—	2	1	no	—
262	C₈×C₂₃	—	2	1	no	—
263	C₈×C₂₃	—	4	1	no	—
264	C₈×C₂₃	—	4	1	no	—
265	C₈×C₂₃	—	4	1	no	—
266	C₈×C₂₃	—	4	1	no	—
267	C₈×C₂₃	—	4	1	no	—
268	C₈×C₂₃	#135	1	2	yes	—
269	#3399	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#3394	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#3340	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#3305	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1715	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1692	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1667	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1665	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1641	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1630	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1628	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1624	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
283	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂²×C₂₃	—	4	1	no	—
285	C₂²×C₂₃	—	4	1	no	—
286	C₂²×C₂₃	—	4	1	no	—
287	C₂²×C₂₃	—	4	1	no	—
288	C₂²×C₂₃	—	4	1	no	—
289	C₂²×C₂₃	—	4	1	no	—
290	C₂²×C₂₃	—	8	1	no	—
291	C₂²×C₂₃	—	8	1	no	—
292	C₂²×C₂₃	—	8	1	no	—
293	C₂²×C₂₃	—	8	1	no	—
294	C₂²×C₂₃	—	8	1	no	—
295	C₂²×C₂₃	—	8	1	no	—
296	C₂²×C₂₃	D₄²	1	1	yes	characteristic
297	C₄×C₂₃	—	2	1	no	—
298	C₄×C₂₃	—	4	1	no	—
299	C₄×C₂₃	—	4	1	no	—
300	C₄×C₂₃	—	4	1	no	—
301	C₄×C₂₃	—	4	1	no	—
302	C₄×C₂₃	—	4	1	no	—
303	C₄×C₂₃	—	4	1	no	—
304	C₄×C₂₃	—	8	1	no	—
305	C₄×C₂₃	—	8	1	no	—
306	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
307	C₄×C₂₃	#575	1	1	yes	characteristic
308	C₄×C₂₃	#576	1	1	yes	characteristic
309	C₄×C₂₃	#577	1	1	yes	characteristic
310	#559	C₂²×C₂₃	1	1	yes	characteristic
311	#556	C₂²×C₂₃	1	1	yes	characteristic
312	#555	C₂²×C₂₃	1	1	yes	characteristic
313	#550	C₂²×C₂₃	1	1	yes	characteristic
314	#498	C₂²×C₂₃	1	1	yes	characteristic
315	#496	—	2	1	no	—
316	#495	—	2	1	no	—
317	#493	C₂²×C₂₃	1	1	yes	characteristic
318	#492	C₂²×C₂₃	1	1	yes	characteristic
319	#492	C₂²×C₂₃	1	1	yes	characteristic
320	#486	C₂²×C₂₃	1	1	yes	characteristic
321	#486	C₂²×C₂₃	1	1	yes	characteristic
322	#485	—	2	1	no	—
323	#482	—	2	1	no	—
324	#482	—	2	1	no	—
325	#477	C₂²×C₂₃	1	1	yes	characteristic
326	#477	C₂²×C₂₃	1	1	yes	characteristic
327	#474	C₂²×C₂₃	1	1	yes	characteristic
328	#466	—	2	1	no	—
329	#465	—	2	1	no	—
330	#464	—	2	1	no	—
331	#463	—	2	1	no	—
332	#463	—	2	1	no	—
333	#462	C₂²×C₂₃	1	1	yes	characteristic
334	#460	C₂²×C₂₃	1	1	yes	characteristic
335	#459	—	2	1	no	—
336	#459	C₂²×C₂₃	1	1	yes	characteristic
337	#459	C₂²×C₂₃	1	1	yes	characteristic
338	#459	C₂²×C₂₃	1	1	yes	characteristic
339	#449	—	2	1	no	—
340	#446	C₂²×C₂₃	1	1	yes	characteristic
341	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
342	#442	—	2	1	no	—
343	C₄×SD₁₆	—	2	1	no	—
344	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
345	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
346	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
347	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
348	#427	—	2	1	no	—
349	#416	—	2	1	no	—
350	#335	C₂²×C₂₃	1	1	yes	characteristic
351	#334	C₂²×C₂₃	1	1	yes	characteristic
352	#333	C₂²×C₂₃	1	1	yes	characteristic
353	#332	C₂²×C₂₃	1	1	yes	characteristic
354	#331	C₂²×C₂₃	1	1	yes	characteristic
355	#331	C₂²×C₂₃	1	1	yes	characteristic
356	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
357	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
358	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
359	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
360	#322	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₂₃	—	8	1	no	—
362	C₂×C₂₃	—	8	1	no	—
363	C₂×C₂₃	—	16	1	no	—
364	C₂×C₂₃	—	16	1	no	—
365	C₂×C₂₃	#3268	1	1	yes	characteristic
366	C₂×C₂₃	#3284	1	1	yes	characteristic
367	C₂×C₂₃	#3288	1	1	yes	characteristic
368	C₂×#41	—	2	1	no	—
369	C₂×Q₁₆	—	2	1	no	—
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	—	2	1	no	—
373	C₂×SD₁₆	—	4	1	no	—
374	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
375	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
376	C₂×D₈	—	2	1	no	—
377	C₂×D₈	—	2	1	no	—
378	C₂×D₈	—	2	1	no	—
379	C₂×D₈	—	2	1	no	—
380	C₂×D₈	—	2	1	no	—
381	C₂²×C₈	—	2	1	no	—
382	C₂²×C₈	—	2	1	no	—
383	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
385	#127	—	2	1	no	—
386	#127	C₂³×C₂₃	1	1	yes	characteristic
387	#127	C₂³×C₂₃	1	1	yes	characteristic
388	#126	—	4	1	no	—
389	#124	—	2	1	no	—
390	#124	—	2	1	no	—
391	#124	—	2	1	no	—
392	#124	—	4	1	no	—
393	#123	—	4	1	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	—	4	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
406	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
407	#117	—	2	1	no	—
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#107	—	2	1	no	—
411	#107	—	2	1	no	—
412	#107	—	2	1	no	—
413	#106	—	2	1	no	—
414	#106	—	2	1	no	—
415	#106	D₄×C₂₃	1	1	yes	characteristic
416	#106	D₄×C₂₃	1	1	yes	characteristic
417	C₄⋊C₈	—	2	1	no	—
418	C₄⋊C₈	—	2	1	no	—
419	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
423	#103	—	2	1	no	—
424	#103	—	2	1	no	—
425	#103	—	2	1	no	—
426	#103	D₄×C₂₃	1	1	yes	characteristic
427	#103	D₄×C₂₃	1	1	yes	characteristic
428	#102	—	2	1	no	—
429	#102	—	2	1	no	—
430	#102	—	2	1	no	—
431	#102	—	2	1	no	—
432	#102	—	2	1	no	—
433	#102	—	2	1	no	—
434	#102	—	2	1	no	—
435	#102	—	4	1	no	—
436	#102	D₄×C₂₃	1	1	yes	characteristic
437	#102	D₄×C₂₃	1	1	yes	characteristic
438	C₂²⋊C₈	—	2	1	no	—
439	C₂²⋊C₈	—	2	1	no	—
440	#97	—	2	1	no	—
441	C₄×C₈	—	2	1	no	—
442	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
443	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
444	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₂₃	#19855	1	1	yes	characteristic, direct product factor, Sylow
446	#41	—	4	2	no	—
447	#41	—	8	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
451	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	4	1	no	—
459	C₂×D₄	—	4	1	no	—
460	C₂×D₄	—	4	1	no	—
461	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	—	4	1	no	—
467	Q₁₆	—	8	1	no	—
468	SD₁₆	—	2	2	no	—
469	SD₁₆	—	4	1	no	—
470	SD₁₆	—	4	2	no	—
471	SD₁₆	—	8	1	no	—
472	SD₁₆	—	8	1	no	—
473	SD₁₆	—	8	2	no	—
474	D₈	—	4	2	no	—
475	D₈	—	8	1	no	—
476	D₈	—	8	1	no	—
477	D₈	—	8	1	no	—
478	D₈	—	8	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	4	2	no	—
485	C₂×C₈	—	8	1	no	—
486	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂²⋊C₄	—	2	1	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	4	1	no	—
504	C₂²⋊C₄	—	4	1	no	—
505	C₂²⋊C₄	—	4	1	no	—
506	C₂²⋊C₄	—	4	1	no	—
507	C₂²⋊C₄	—	4	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	4	1	no	—
514	C₄²	—	4	1	no	—
515	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	1	no	—
520	Q₈	—	2	1	no	—
521	Q₈	—	2	1	no	—
522	Q₈	—	2	1	no	—
523	Q₈	—	2	1	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	8	1	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	4	1	no	—
532	D₄	—	4	1	no	—
533	D₄	—	4	1	no	—
534	D₄	—	4	1	no	—
535	D₄	—	4	1	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	1	no	—
539	D₄	—	8	1	no	—
540	D₄	—	8	1	no	—
541	D₄	—	8	2	no	—
542	D₄	—	16	1	no	—
543	D₄	—	16	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	8	2	no	—
557	C₂×C₄	—	16	1	no	—
558	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
560	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
561	C₈	—	2	1	no	—
562	C₈	—	2	1	no	—
563	C₈	—	2	1	no	—
564	C₈	—	4	1	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	C₂₃×#135	1	2	yes	—
570	C₂²	—	4	1	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	—	8	1	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	8	1	no	—
581	C₂²	—	8	1	no	—
582	C₂²	D₄²×C₂₃	1	1	yes	characteristic
583	C₄	—	2	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	8	1	no	—
591	C₄	—	8	1	no	—
592	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
593	C₄	C₂₃×#575	1	1	yes	characteristic
594	C₄	C₂₃×#576	1	1	yes	characteristic
595	C₄	C₂₃×#577	1	1	yes	characteristic
596	C₂	—	8	1	no	—
597	C₂	—	8	1	no	—
598	C₂	—	16	1	no	—
599	C₂	—	16	1	no	—
600	C₂	C₂₃×#3268	1	1	yes	characteristic
601	C₂	C₂₃×#3284	1	1	yes	characteristic
602	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×#3394	C₂	semidirect
3. C₂₃×#3340	C₂	semidirect
4. C₂₃×#3305	C₂	semidirect
5. C₂₃×#1715	C₂	semidirect
6. C₂₃×#1692	C₂	semidirect
7. C₂₃×#1678	C₂	semidirect
8. C₂₃×#1667	C₂	semidirect
9. C₂₃×#1665	C₂	semidirect
10. C₂₃×#1641	C₂	semidirect
11. C₂₃×#1630	C₂	semidirect
12. C₂₃×#1628	C₂	semidirect
13. C₂₃×#1624	C₂	semidirect
14. C₂₃×#1582	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
214. #19855	C₂₃	direct (N × H)
269. #3399	C₂×C₂₃	semidirect
270. #3394	C₂×C₂₃	semidirect
271. #3340	C₂×C₂₃	semidirect
272. #3305	C₂×C₂₃	semidirect
273. #1715	C₂×C₂₃	semidirect
274. #1692	C₂×C₂₃	semidirect
275. #1678	C₂×C₂₃	semidirect
276. #1667	C₂×C₂₃	semidirect
277. #1665	C₂×C₂₃	semidirect
278. #1641	C₂×C₂₃	semidirect
279. #1630	C₂×C₂₃	semidirect
280. #1628	C₂×C₂₃	semidirect
281. #1624	C₂×C₂₃	semidirect
282. #1582	C₂×C₂₃	semidirect
283. SD₁₆⋊C₈	C₂×C₂₃	semidirect
445. C₂₃	#19855	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×#3399	2
C₂₃×#3394	2
C₂₃×#3340	2
C₂₃×#3305	2
C₂₃×#1715	2
C₂₃×#1692	2
C₂₃×#1678	2
C₂₃×#1667	2
C₂₃×#1665	2
C₂₃×#1641	2
C₂₃×#1630	2
C₂₃×#1628	2
C₂₃×#1624	2
C₂₃×#1582	2
C₂₃×SD₁₆⋊C₈	2
#19855	23

Subgroups — C23×#19855 #19167481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19855 #19167481