Subgroups — C₂₃×#19826 #19167452

Order 5888.

600 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3361	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3361	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3331	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1720	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1697	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1684	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1677	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1625	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1583	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×#554	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	C₂₃×#547	C₂²	1	1	yes	characteristic
20	C₂₃×#498	C₂²	1	1	yes	characteristic
21	C₂₃×#494	C₂²	1	1	yes	characteristic
22	C₂₃×#488	—	2	1	no	—
23	C₂₃×#488	—	2	1	no	—
24	C₂₃×#487	C₂²	1	1	yes	characteristic
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×#483	—	2	1	no	—
29	C₂₃×#483	—	2	1	no	—
30	C₂₃×#481	—	2	1	no	—
31	C₂₃×#478	—	2	1	no	—
32	C₂₃×#478	—	2	1	no	—
33	C₂₃×#477	C₂²	1	1	yes	characteristic
34	C₂₃×#477	C₂²	1	1	yes	characteristic
35	C₂₃×#476	C₂²	1	1	yes	characteristic
36	C₂₃×#476	C₂²	1	1	yes	characteristic
37	C₂₃×#475	C₂²	1	1	yes	characteristic
38	C₂₃×#470	—	2	1	no	—
39	C₂₃×#465	—	2	1	no	—
40	C₂₃×#463	—	2	1	no	—
41	C₂₃×#463	—	2	1	no	—
42	C₂₃×#462	C₂²	1	1	yes	characteristic
43	C₂₃×#459	C₂²	1	1	yes	characteristic
44	C₂₃×#459	C₂²	1	1	yes	characteristic
45	C₂₃×#452	—	2	1	no	—
46	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
50	C₄×D₈×C₂₃	—	2	1	no	—
51	C₄×D₈×C₂₃	—	2	1	no	—
52	C₂₃×#436	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#106	—	2	1	no	—
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×#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×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×Q₁₆×C₂₃	—	4	1	no	—
70	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
72	C₂×D₈×C₂₃	—	2	1	no	—
73	C₂×D₈×C₂₃	—	2	1	no	—
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₂₃×#127	C₂³	1	1	yes	characteristic
80	C₂₃×#125	—	2	1	no	—
81	C₂₃×#125	—	2	1	no	—
82	C₂₃×#124	—	2	1	no	—
83	C₂₃×#124	—	2	1	no	—
84	C₂₃×#123	—	4	1	no	—
85	C₂₃×#123	—	4	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	—	2	1	no	—
88	C₂₃×C₂²⋊Q₈	—	2	1	no	—
89	C₂₃×C₂²⋊Q₈	—	4	1	no	—
90	C₂₃×C₂²⋊Q₈	—	4	1	no	—
91	C₂₃×#121	—	2	1	no	—
92	C₄×Q₈×C₂₃	C₂³	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	C₂³	1	1	yes	characteristic
99	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂₃×#107	—	2	1	no	—
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#106	—	2	1	no	—
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	D₄	1	1	yes	characteristic
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	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×#103	—	2	1	no	—
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	D₄	1	1	yes	characteristic
125	C₂₃×#103	D₄	1	1	yes	characteristic
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	—	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	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	—	2	1	no	—
149	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
155	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	4	1	no	—
160	C₂²×C₄×C₂₃	—	4	1	no	—
161	C₂²×C₄×C₂₃	—	4	1	no	—
162	C₂²×C₄×C₂₃	—	4	1	no	—
163	Q₁₆×C₂₃	—	8	1	no	—
164	Q₁₆×C₂₃	—	8	2	no	—
165	SD₁₆×C₂₃	—	2	2	no	—
166	SD₁₆×C₂₃	—	4	1	no	—
167	D₈×C₂₃	—	4	2	no	—
168	D₈×C₂₃	—	4	2	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	—	4	2	no	—
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	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
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	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
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	C₂²⋊C₄×C₂₃	—	4	1	no	—
205	C₄²×C₂₃	—	2	1	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
208	#19826	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₂₃	—	2	1	no	—
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	—	4	1	no	—
217	Q₈×C₂₃	—	4	1	no	—
218	Q₈×C₂₃	—	8	1	no	—
219	Q₈×C₂₃	—	8	1	no	—
220	D₄×C₂₃	—	2	1	no	—
221	D₄×C₂₃	—	2	1	no	—
222	D₄×C₂₃	—	2	1	no	—
223	D₄×C₂₃	—	4	1	no	—
224	D₄×C₂₃	—	4	1	no	—
225	D₄×C₂₃	—	4	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	8	1	no	—
228	D₄×C₂₃	—	8	2	no	—
229	D₄×C₂₃	#135	1	2	yes	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	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	—	8	2	no	—
244	C₂×C₄×C₂₃	—	16	1	no	—
245	C₂×C₄×C₂₃	—	16	1	no	—
246	C₂×C₄×C₂₃	—	16	1	no	—
247	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
250	C₈×C₂₃	—	2	1	no	—
251	C₈×C₂₃	—	2	1	no	—
252	C₈×C₂₃	—	2	1	no	—
253	C₈×C₂₃	—	2	1	no	—
254	C₈×C₂₃	—	4	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	#3361	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#3361	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#3335	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#3331	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1720	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1697	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1684	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1677	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1672	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1625	C₂×C₂₃	1	1	yes	characteristic
272	#1583	C₂×C₂₃	1	1	yes	characteristic, complemented
273	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
274	C₂²×C₂₃	—	2	2	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	4	1	no	—
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	D₄²	1	1	yes	characteristic
284	C₄×C₂₃	—	2	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	4	1	no	—
288	C₄×C₂₃	—	4	1	no	—
289	C₄×C₂₃	—	4	1	no	—
290	C₄×C₂₃	—	4	1	no	—
291	C₄×C₂₃	—	8	1	no	—
292	C₄×C₂₃	—	8	1	no	—
293	C₄×C₂₃	—	8	1	no	—
294	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
295	C₄×C₂₃	#575	1	1	yes	characteristic
296	C₄×C₂₃	#577	1	1	yes	characteristic
297	C₄×C₂₃	#578	1	1	yes	characteristic
298	#559	C₂²×C₂₃	1	1	yes	characteristic
299	#554	C₂²×C₂₃	1	1	yes	characteristic
300	#554	C₂²×C₂₃	1	1	yes	characteristic
301	#547	C₂²×C₂₃	1	1	yes	characteristic
302	#498	C₂²×C₂₃	1	1	yes	characteristic
303	#494	C₂²×C₂₃	1	1	yes	characteristic
304	#488	—	2	1	no	—
305	#488	—	2	1	no	—
306	#487	C₂²×C₂₃	1	1	yes	characteristic
307	#486	C₂²×C₂₃	1	1	yes	characteristic
308	#486	C₂²×C₂₃	1	1	yes	characteristic
309	#486	C₂²×C₂₃	1	1	yes	characteristic
310	#483	—	2	1	no	—
311	#483	—	2	1	no	—
312	#481	—	2	1	no	—
313	#478	—	2	1	no	—
314	#478	—	2	1	no	—
315	#477	C₂²×C₂₃	1	1	yes	characteristic
316	#477	C₂²×C₂₃	1	1	yes	characteristic
317	#476	C₂²×C₂₃	1	1	yes	characteristic
318	#476	C₂²×C₂₃	1	1	yes	characteristic
319	#475	C₂²×C₂₃	1	1	yes	characteristic
320	#470	—	2	1	no	—
321	#465	—	2	1	no	—
322	#463	—	2	1	no	—
323	#463	—	2	1	no	—
324	#462	C₂²×C₂₃	1	1	yes	characteristic
325	#459	C₂²×C₂₃	1	1	yes	characteristic
326	#459	C₂²×C₂₃	1	1	yes	characteristic
327	#452	—	2	1	no	—
328	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
329	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
331	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
332	C₄×D₈	—	2	1	no	—
333	C₄×D₈	—	2	1	no	—
334	#436	C₂²×C₂₃	1	1	yes	characteristic
335	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#106	—	2	1	no	—
337	#417	—	2	1	no	—
338	#335	C₂²×C₂₃	1	1	yes	characteristic
339	#334	C₂²×C₂₃	1	1	yes	characteristic
340	#333	C₂²×C₂₃	1	1	yes	characteristic
341	#333	C₂²×C₂₃	1	1	yes	characteristic
342	#332	C₂²×C₂₃	1	1	yes	characteristic
343	#331	C₂²×C₂₃	1	1	yes	characteristic
344	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
346	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
347	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	#322	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×C₂₃	—	4	2	no	—
350	C₂×C₂₃	—	8	1	no	—
351	C₂×C₂₃	—	16	1	no	—
352	C₂×C₂₃	#3269	1	1	yes	characteristic
353	C₂×C₂₃	#3282	1	1	yes	characteristic
354	C₂×C₂₃	#3288	1	1	yes	characteristic
355	C₂×#41	—	2	1	no	—
356	C₂×Q₁₆	—	2	1	no	—
357	C₂×Q₁₆	—	4	1	no	—
358	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
359	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
360	C₂×D₈	—	2	1	no	—
361	C₂×D₈	—	2	1	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂²×C₈	—	2	1	no	—
364	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
367	#127	C₂³×C₂₃	1	1	yes	characteristic
368	#125	—	2	1	no	—
369	#125	—	2	1	no	—
370	#124	—	2	1	no	—
371	#124	—	2	1	no	—
372	#123	—	4	1	no	—
373	#123	—	4	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	4	1	no	—
378	C₂²⋊Q₈	—	4	1	no	—
379	#121	—	2	1	no	—
380	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₄×D₄	—	2	1	no	—
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	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	#107	—	2	1	no	—
392	#107	—	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	—	2	1	no	—
398	#106	D₄×C₂₃	1	1	yes	characteristic
399	#106	D₄×C₂₃	1	1	yes	characteristic
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
406	#103	—	2	1	no	—
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	—	2	1	no	—
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	—	4	1	no	—
423	#102	D₄×C₂₃	1	1	yes	characteristic
424	#102	D₄×C₂₃	1	1	yes	characteristic
425	C₂²⋊C₈	—	2	1	no	—
426	C₂²⋊C₈	—	2	1	no	—
427	C₄×C₈	—	2	1	no	—
428	C₄×C₈	—	2	1	no	—
429	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₂₃	#19826	1	1	yes	characteristic, direct product factor, Sylow
433	#41	—	4	2	no	—
434	#41	—	8	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
445	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	4	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	—	4	1	no	—
452	Q₁₆	—	8	1	no	—
453	Q₁₆	—	8	2	no	—
454	SD₁₆	—	2	2	no	—
455	SD₁₆	—	4	1	no	—
456	D₈	—	4	2	no	—
457	D₈	—	4	2	no	—
458	M₁₆	—	8	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	4	2	no	—
465	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
483	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
485	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	4	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	2	1	no	—
502	Q₈	—	2	1	no	—
503	Q₈	—	2	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	8	1	no	—
507	Q₈	—	8	1	no	—
508	D₄	—	2	1	no	—
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	D₄	—	8	2	no	—
517	D₄	C₂₃×#135	1	2	yes	—
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	—	2	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	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	8	2	no	—
532	C₂×C₄	—	16	1	no	—
533	C₂×C₄	—	16	1	no	—
534	C₂×C₄	—	16	1	no	—
535	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
537	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
538	C₈	—	2	1	no	—
539	C₈	—	2	1	no	—
540	C₈	—	2	1	no	—
541	C₈	—	2	1	no	—
542	C₈	—	4	1	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	1	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	D₄²×C₂₃	1	1	yes	characteristic
557	C₄	—	2	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	4	1	no	—
562	C₄	—	4	1	no	—
563	C₄	—	4	1	no	—
564	C₄	—	8	1	no	—
565	C₄	—	8	1	no	—
566	C₄	—	8	1	no	—
567	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
568	C₄	C₂₃×#575	1	1	yes	characteristic
569	C₄	C₂₃×#577	1	1	yes	characteristic
570	C₄	C₂₃×#578	1	1	yes	characteristic
571	C₂	—	4	2	no	—
572	C₂	—	8	1	no	—
573	C₂	—	16	1	no	—
574	C₂	C₂₃×#3269	1	1	yes	characteristic
575	C₂	C₂₃×#3282	1	1	yes	characteristic
576	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#1625	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×#3361	C₂	semidirect
2. C₂₃×#3361	C₂	semidirect
3. C₂₃×#3335	C₂	semidirect
4. C₂₃×#3331	C₂	semidirect
5. C₂₃×#1720	C₂	semidirect
6. C₂₃×#1697	C₂	semidirect
7. C₂₃×#1684	C₂	semidirect
8. C₂₃×#1677	C₂	semidirect
9. C₂₃×#1672	C₂	semidirect
10. C₂₃×#1640	C₂	semidirect
11. C₂₃×#1632	C₂	semidirect
12. C₂₃×#1632	C₂	semidirect
14. C₂₃×#1583	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
208. #19826	C₂₃	direct (N × H)
259. #3361	C₂×C₂₃	semidirect
260. #3361	C₂×C₂₃	semidirect
261. #3335	C₂×C₂₃	semidirect
262. #3331	C₂×C₂₃	semidirect
263. #1720	C₂×C₂₃	semidirect
264. #1697	C₂×C₂₃	semidirect
265. #1684	C₂×C₂₃	semidirect
266. #1677	C₂×C₂₃	semidirect
267. #1672	C₂×C₂₃	semidirect
268. #1640	C₂×C₂₃	semidirect
269. #1632	C₂×C₂₃	semidirect
270. #1632	C₂×C₂₃	semidirect
272. #1583	C₂×C₂₃	semidirect
273. SD₁₆⋊C₈	C₂×C₂₃	semidirect
432. C₂₃	#19826	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×#3361	2
C₂₃×#3335	2
C₂₃×#3331	2
C₂₃×#1720	2
C₂₃×#1697	2
C₂₃×#1684	2
C₂₃×#1677	2
C₂₃×#1672	2
C₂₃×#1640	2
C₂₃×#1632	2
C₂₃×#1625	2
C₂₃×#1583	2
C₂₃×SD₁₆⋊C₈	2
#19826	23

Subgroups — C23×#19826 #19167452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19826 #19167452