Subgroups — C₂₃×#19921 #19167547

Order 5888.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3394	C₂	1	1	yes	maximal, characteristic
2	Q₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3375	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1708	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1700	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1697	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1638	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1627	C₂	1	1	yes	maximal, characteristic
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×#556	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#498	—	2	1	no	—
21	C₂₃×#498	C₂²	1	1	yes	characteristic
22	C₂₃×#495	—	2	1	no	—
23	C₂₃×#492	C₂²	1	1	yes	characteristic
24	C₂₃×#489	—	2	1	no	—
25	C₂₃×#487	C₂²	1	1	yes	characteristic
26	C₂₃×#487	C₂²	1	1	yes	characteristic
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#486	C₂²	1	1	yes	characteristic
29	C₂₃×#485	—	2	1	no	—
30	C₂₃×#477	C₂²	1	1	yes	characteristic
31	C₂₃×#476	—	2	1	no	—
32	C₂₃×#476	C₂²	1	1	yes	characteristic
33	C₂₃×#475	—	2	1	no	—
34	C₂₃×#475	C₂²	1	1	yes	characteristic
35	C₂₃×#475	C₂²	1	1	yes	characteristic
36	C₂₃×#474	—	2	1	no	—
37	C₂₃×#470	—	2	1	no	—
38	C₂₃×#464	—	2	1	no	—
39	C₂₃×#464	—	2	1	no	—
40	C₂₃×#463	—	2	1	no	—
41	C₂₃×#462	C₂²	1	1	yes	characteristic
42	C₂₃×#462	C₂²	1	1	yes	characteristic
43	C₂₃×#460	C₂²	1	1	yes	characteristic
44	C₂₃×#459	C₂²	1	1	yes	characteristic
45	C₂₃×#451	—	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	—	2	1	no	—
53	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
54	C₂₃×#436	C₂²	1	1	yes	characteristic
55	C₂₃×#419	—	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×#332	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	C₂²×Q₈×C₂₃	—	2	1	no	—
68	C₂×Q₁₆×C₂₃	—	2	1	no	—
69	C₂×Q₁₆×C₂₃	—	2	1	no	—
70	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
75	C₂×D₈×C₂₃	—	2	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	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	4	1	no	—
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×#127	C₂³	1	1	yes	characteristic
85	C₂₃×#126	—	4	1	no	—
86	C₂₃×#124	—	2	1	no	—
87	C₂₃×#124	—	2	1	no	—
88	C₂₃×#124	—	2	1	no	—
89	C₂₃×#123	—	4	1	no	—
90	C₂₃×C₂²⋊Q₈	—	2	1	no	—
91	C₂₃×C₂²⋊Q₈	—	2	1	no	—
92	C₂₃×C₂²⋊Q₈	—	4	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
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	C₂₃×#117	—	2	1	no	—
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	D₄	1	1	yes	characteristic
106	C₂₃×#106	—	2	1	no	—
107	C₂₃×#106	—	2	1	no	—
108	C₂₃×#106	—	2	1	no	—
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×#106	—	2	1	no	—
111	C₂₃×#106	D₄	1	1	yes	characteristic
112	C₂₃×C₄⋊C₈	—	2	1	no	—
113	C₂₃×C₄⋊C₈	—	2	1	no	—
114	C₂₃×C₄⋊C₈	—	2	1	no	—
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×#103	—	2	1	no	—
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×#103	—	4	1	no	—
127	C₂₃×#103	D₄	1	1	yes	characteristic
128	C₂₃×#103	D₄	1	1	yes	characteristic
129	C₂₃×#103	D₄	1	1	yes	characteristic
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×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	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	4	1	no	—
148	C₂×Q₈×C₂₃	—	4	1	no	—
149	C₂×Q₈×C₂₃	—	4	2	no	—
150	C₂×Q₈×C₂₃	—	8	1	no	—
151	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	4	1	no	—
160	C₂²×C₄×C₂₃	—	4	1	no	—
161	Q₁₆×C₂₃	—	2	2	no	—
162	Q₁₆×C₂₃	—	8	1	no	—
163	Q₁₆×C₂₃	—	8	1	no	—
164	SD₁₆×C₂₃	—	2	2	no	—
165	SD₁₆×C₂₃	—	4	2	no	—
166	SD₁₆×C₂₃	—	4	2	no	—
167	SD₁₆×C₂₃	—	8	1	no	—
168	D₈×C₂₃	—	8	1	no	—
169	M₁₆×C₂₃	—	8	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	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	—	4	1	no	—
189	C₄⋊C₄×C₂₃	—	4	1	no	—
190	C₄⋊C₄×C₂₃	—	4	1	no	—
191	C₄⋊C₄×C₂₃	—	4	1	no	—
192	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₂₃	—	2	1	no	—
197	C₂²⋊C₄×C₂₃	—	4	1	no	—
198	C₂²⋊C₄×C₂₃	—	4	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	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	2	1	no	—
206	C₄²×C₂₃	—	4	1	no	—
207	C₄²×C₂₃	—	4	1	no	—
208	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
209	#19921	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₂₃	—	2	1	no	—
211	C₂³×C₂₃	—	4	1	no	—
212	Q₈×C₂₃	—	2	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	—	4	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	Q₈×C₂₃	—	8	2	no	—
222	Q₈×C₂₃	—	8	1	no	—
223	Q₈×C₂₃	—	16	1	no	—
224	Q₈×C₂₃	#135	1	2	yes	—
225	D₄×C₂₃	—	2	1	no	—
226	D₄×C₂₃	—	2	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	—	8	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	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	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	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	—	8	2	no	—
245	C₂×C₄×C₂₃	—	16	1	no	—
246	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
249	C₈×C₂₃	—	2	1	no	—
250	C₈×C₂₃	—	2	1	no	—
251	C₈×C₂₃	—	2	1	no	—
252	C₈×C₂₃	—	4	1	no	—
253	C₈×C₂₃	—	4	1	no	—
254	C₈×C₂₃	—	4	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	#137	1	2	yes	—
258	#3394	C₂×C₂₃	1	1	yes	characteristic
259	Q₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#3375	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#3328	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1708	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1700	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1697	C₂×C₂₃	1	1	yes	characteristic
265	#1672	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1666	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1638	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1627	C₂×C₂₃	1	1	yes	characteristic
271	#1590	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₈×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
273	C₂²×C₂₃	—	4	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	8	1	no	—
277	C₂²×C₂₃	—	8	1	no	—
278	C₂²×C₂₃	—	8	1	no	—
279	C₂²×C₂₃	D₄²	1	1	yes	characteristic
280	C₄×C₂₃	—	2	2	no	—
281	C₄×C₂₃	—	2	1	no	—
282	C₄×C₂₃	—	4	1	no	—
283	C₄×C₂₃	—	4	1	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	4	1	no	—
288	C₄×C₂₃	—	8	1	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	—	8	1	no	—
291	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
292	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
293	C₄×C₂₃	#577	1	1	yes	characteristic
294	C₄×C₂₃	#577	1	1	yes	characteristic
295	#559	C₂²×C₂₃	1	1	yes	characteristic
296	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
297	#556	C₂²×C₂₃	1	1	yes	characteristic
298	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
299	#498	—	2	1	no	—
300	#498	C₂²×C₂₃	1	1	yes	characteristic
301	#495	—	2	1	no	—
302	#492	C₂²×C₂₃	1	1	yes	characteristic
303	#489	—	2	1	no	—
304	#487	C₂²×C₂₃	1	1	yes	characteristic
305	#487	C₂²×C₂₃	1	1	yes	characteristic
306	#486	C₂²×C₂₃	1	1	yes	characteristic
307	#486	C₂²×C₂₃	1	1	yes	characteristic
308	#485	—	2	1	no	—
309	#477	C₂²×C₂₃	1	1	yes	characteristic
310	#476	—	2	1	no	—
311	#476	C₂²×C₂₃	1	1	yes	characteristic
312	#475	—	2	1	no	—
313	#475	C₂²×C₂₃	1	1	yes	characteristic
314	#475	C₂²×C₂₃	1	1	yes	characteristic
315	#474	—	2	1	no	—
316	#470	—	2	1	no	—
317	#464	—	2	1	no	—
318	#464	—	2	1	no	—
319	#463	—	2	1	no	—
320	#462	C₂²×C₂₃	1	1	yes	characteristic
321	#462	C₂²×C₂₃	1	1	yes	characteristic
322	#460	C₂²×C₂₃	1	1	yes	characteristic
323	#459	C₂²×C₂₃	1	1	yes	characteristic
324	#451	—	2	1	no	—
325	#446	C₂²×C₂₃	1	1	yes	characteristic
326	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
327	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
328	#440	—	2	1	no	—
329	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
330	C₄×SD₁₆	—	2	1	no	—
331	C₄×SD₁₆	—	2	1	no	—
332	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
333	#436	C₂²×C₂₃	1	1	yes	characteristic
334	#419	—	2	1	no	—
335	#335	C₂²×C₂₃	1	1	yes	characteristic
336	#334	C₂²×C₂₃	1	1	yes	characteristic
337	#333	C₂²×C₂₃	1	1	yes	characteristic
338	#332	C₂²×C₂₃	1	1	yes	characteristic
339	#332	C₂²×C₂₃	1	1	yes	characteristic
340	#331	C₂²×C₂₃	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	C₈²	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₂₃	—	8	1	no	—
347	C₂×C₂₃	—	16	1	no	—
348	C₂×C₂₃	#3278	1	1	yes	characteristic
349	C₂×C₂₃	#3282	1	1	yes	characteristic
350	C₂×C₂₃	#3290	1	1	yes	characteristic
351	C₂²×Q₈	—	2	1	no	—
352	C₂×Q₁₆	—	2	1	no	—
353	C₂×Q₁₆	—	2	1	no	—
354	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
355	C₂×SD₁₆	—	2	1	no	—
356	C₂×SD₁₆	—	2	1	no	—
357	C₂×SD₁₆	—	2	1	no	—
358	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
359	C₂×D₈	—	2	1	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₄⋊Q₈	—	2	1	no	—
362	C₄⋊Q₈	—	2	1	no	—
363	C₄⋊Q₈	—	2	1	no	—
364	C₄⋊Q₈	—	4	1	no	—
365	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
368	#127	C₂³×C₂₃	1	1	yes	characteristic
369	#126	—	4	1	no	—
370	#124	—	2	1	no	—
371	#124	—	2	1	no	—
372	#124	—	2	1	no	—
373	#123	—	4	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	4	1	no	—
377	C₄×Q₈	—	2	1	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
387	#117	—	2	1	no	—
388	#107	—	2	1	no	—
389	#107	D₄×C₂₃	1	1	yes	characteristic
390	#106	—	2	1	no	—
391	#106	—	2	1	no	—
392	#106	—	2	1	no	—
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	#106	D₄×C₂₃	1	1	yes	characteristic
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	—	2	1	no	—
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	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	4	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	—	4	1	no	—
420	#102	D₄×C₂₃	1	1	yes	characteristic
421	C₂²⋊C₈	—	2	1	no	—
422	#97	—	2	1	no	—
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	C₂₃	#19921	1	1	yes	characteristic, direct product factor, Sylow
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	4	1	no	—
433	C₂×Q₈	—	4	1	no	—
434	C₂×Q₈	—	4	2	no	—
435	C₂×Q₈	—	8	1	no	—
436	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	Q₁₆	—	2	2	no	—
447	Q₁₆	—	8	1	no	—
448	Q₁₆	—	8	1	no	—
449	SD₁₆	—	2	2	no	—
450	SD₁₆	—	4	2	no	—
451	SD₁₆	—	4	2	no	—
452	SD₁₆	—	8	1	no	—
453	D₈	—	8	1	no	—
454	M₁₆	—	8	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₂²⋊C₄	—	4	1	no	—
487	C₂²⋊C₄	—	4	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	4	1	no	—
492	C₄²	—	4	1	no	—
493	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
494	C₂³	—	2	1	no	—
495	C₂³	—	4	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	2	1	no	—
499	Q₈	—	2	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	8	2	no	—
506	Q₈	—	8	1	no	—
507	Q₈	—	16	1	no	—
508	Q₈	C₂₃×#135	1	2	yes	—
509	D₄	—	2	1	no	—
510	D₄	—	2	1	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	1	no	—
513	D₄	—	4	1	no	—
514	D₄	—	4	1	no	—
515	D₄	—	8	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	8	2	no	—
529	C₂×C₄	—	16	1	no	—
530	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
533	C₈	—	2	1	no	—
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	—	4	1	no	—
540	C₈	—	4	1	no	—
541	C₈	C₂₃×#137	1	2	yes	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	—	8	1	no	—
547	C₂²	—	8	1	no	—
548	C₂²	D₄²×C₂₃	1	1	yes	characteristic
549	C₄	—	2	2	no	—
550	C₄	—	2	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	8	1	no	—
558	C₄	—	8	1	no	—
559	C₄	—	8	1	no	—
560	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
561	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
562	C₄	C₂₃×#577	1	1	yes	characteristic
563	C₄	C₂₃×#577	1	1	yes	characteristic
564	C₂	—	8	1	no	—
565	C₂	—	16	1	no	—
566	C₂	C₂₃×#3278	1	1	yes	characteristic
567	C₂	C₂₃×#3282	1	1	yes	characteristic
568	C₂	C₂₃×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#459	C₂²
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
2. Q₈×SD₁₆×C₂₃	C₂	semidirect
3. C₂₃×#3375	C₂	semidirect
4. C₂₃×#3328	C₂	semidirect
5. C₂₃×#1708	C₂	semidirect
6. C₂₃×#1700	C₂	semidirect
8. C₂₃×#1672	C₂	semidirect
9. C₂₃×#1666	C₂	semidirect
10. C₂₃×#1645	C₂	semidirect
11. C₂₃×#1640	C₂	semidirect
12. C₂₃×#1638	C₂	semidirect
14. C₂₃×#1590	C₂	semidirect
15. C₈×Q₁₆×C₂₃	C₂	semidirect
209. #19921	C₂₃	direct (N × H)
259. Q₈×SD₁₆	C₂×C₂₃	semidirect
260. #3375	C₂×C₂₃	semidirect
261. #3328	C₂×C₂₃	semidirect
262. #1708	C₂×C₂₃	semidirect
263. #1700	C₂×C₂₃	semidirect
265. #1672	C₂×C₂₃	semidirect
266. #1666	C₂×C₂₃	semidirect
267. #1645	C₂×C₂₃	semidirect
268. #1640	C₂×C₂₃	semidirect
269. #1638	C₂×C₂₃	semidirect
271. #1590	C₂×C₂₃	semidirect
272. C₈×Q₁₆	C₂×C₂₃	semidirect
428. C₂₃	#19921	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×#3394	2
Q₈×SD₁₆×C₂₃	2
C₂₃×#3375	2
C₂₃×#3328	2
C₂₃×#1708	2
C₂₃×#1700	2
C₂₃×#1697	2
C₂₃×#1672	2
C₂₃×#1666	2
C₂₃×#1645	2
C₂₃×#1640	2
C₂₃×#1638	2
C₂₃×#1627	2
C₂₃×#1590	2
C₈×Q₁₆×C₂₃	2
#19921	23

Subgroups — C23×#19921 #19167547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19921 #19167547