Subgroups — C₂₃×#19778 #19167404

Order 5888.

606 conjugacy classes of subgroups.

The subgroup lattice has 582 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×#3334	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3331	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1718	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1683	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1635	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×#554	C₂²	1	1	yes	characteristic
18	C₂₃×#547	C₂²	1	1	yes	characteristic
19	C₂₃×#547	C₂²	1	1	yes	characteristic
20	C₂₃×#500	C₂²	1	1	yes	characteristic
21	C₂₃×#498	C₂²	1	1	yes	characteristic
22	C₂₃×#488	—	2	1	no	—
23	C₂₃×#487	C₂²	1	1	yes	characteristic
24	C₂₃×#487	C₂²	1	1	yes	characteristic
25	C₂₃×#487	C₂²	1	1	yes	characteristic
26	C₂₃×#487	C₂²	1	1	yes	characteristic
27	C₂₃×#485	—	2	1	no	—
28	C₂₃×#483	—	2	1	no	—
29	C₂₃×#480	—	2	1	no	—
30	C₂₃×#480	—	2	1	no	—
31	C₂₃×#479	—	2	1	no	—
32	C₂₃×#476	C₂²	1	1	yes	characteristic
33	C₂₃×#476	C₂²	1	1	yes	characteristic
34	C₂₃×#474	C₂²	1	1	yes	characteristic
35	C₂₃×#474	C₂²	1	1	yes	characteristic
36	C₂₃×#474	C₂²	1	1	yes	characteristic
37	C₂₃×#470	—	2	1	no	—
38	C₂₃×#467	—	2	1	no	—
39	C₂₃×#466	—	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×#461	C₂²	1	1	yes	characteristic
44	C₂₃×#452	—	2	1	no	—
45	C₂₃×#452	—	2	1	no	—
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	C₂×C₂₃×#107	—	2	1	no	—
54	C₂₃×#417	—	2	1	no	—
55	C₂₃×#416	—	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×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	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×#41×C₂₃	—	2	1	no	—
68	C₂×#41×C₂₃	—	2	1	no	—
69	C₂×Q₁₆×C₂₃	—	2	1	no	—
70	C₂×Q₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
73	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×#126	—	4	1	no	—
82	C₂₃×#125	—	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	—	2	1	no	—
90	C₂₃×C₂²⋊Q₈	—	4	1	no	—
91	C₂₃×#121	—	2	1	no	—
92	C₂₃×#121	—	2	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#117	—	2	1	no	—
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×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#107	—	2	1	no	—
108	C₂₃×#107	D₄	1	1	yes	characteristic
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×#106	—	2	1	no	—
111	C₂₃×#106	—	4	1	no	—
112	C₂₃×#106	D₄	1	1	yes	characteristic
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	—	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	#41×C₂₃	—	4	2	no	—
145	#41×C₂₃	—	4	2	no	—
146	#41×C₂₃	—	8	1	no	—
147	#41×C₂₃	—	8	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	—	4	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	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	4	1	no	—
162	C₂²×C₄×C₂₃	—	4	1	no	—
163	C₂²×C₄×C₂₃	—	4	1	no	—
164	Q₁₆×C₂₃	—	8	1	no	—
165	Q₁₆×C₂₃	—	8	1	no	—
166	SD₁₆×C₂₃	—	2	2	no	—
167	SD₁₆×C₂₃	—	4	2	no	—
168	D₈×C₂₃	—	4	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	—	4	2	no	—
176	C₂×C₈×C₂₃	—	8	1	no	—
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	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	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	—	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	—	2	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	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	4	1	no	—
209	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
210	#19778	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂³×C₂₃	—	2	1	no	—
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×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	—	4	1	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	Q₈×C₂₃	—	4	1	no	—
222	Q₈×C₂₃	—	8	1	no	—
223	D₄×C₂₃	—	2	1	no	—
224	D₄×C₂₃	—	2	1	no	—
225	D₄×C₂₃	—	2	1	no	—
226	D₄×C₂₃	—	2	1	no	—
227	D₄×C₂₃	—	2	1	no	—
228	D₄×C₂₃	—	8	2	no	—
229	D₄×C₂₃	—	8	2	no	—
230	D₄×C₂₃	#135	1	2	yes	—
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	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	8	2	no	—
247	C₂×C₄×C₂₃	—	8	2	no	—
248	C₂×C₄×C₂₃	—	16	1	no	—
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
252	C₈×C₂₃	—	2	1	no	—
253	C₈×C₂₃	—	2	1	no	—
254	C₈×C₂₃	—	2	1	no	—
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	#3360	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#3334	C₂×C₂₃	1	1	yes	characteristic
263	#3331	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#3302	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1718	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1699	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1683	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1683	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1674	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1654	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1635	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
275	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₂²×C₂₃	—	2	2	no	—
277	C₂²×C₂₃	—	4	1	no	—
278	C₂²×C₂₃	—	4	1	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	4	1	no	—
281	C₂²×C₂₃	—	4	1	no	—
282	C₂²×C₂₃	—	4	1	no	—
283	C₂²×C₂₃	—	4	1	no	—
284	C₂²×C₂₃	—	4	1	no	—
285	C₂²×C₂₃	D₄²	1	1	yes	characteristic
286	C₄×C₂₃	—	2	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	—	4	1	no	—
292	C₄×C₂₃	—	8	1	no	—
293	C₄×C₂₃	—	8	1	no	—
294	C₄×C₂₃	—	8	1	no	—
295	C₄×C₂₃	—	8	1	no	—
296	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
297	C₄×C₂₃	#575	1	1	yes	characteristic
298	C₄×C₂₃	#576	1	1	yes	characteristic
299	C₄×C₂₃	#578	1	1	yes	characteristic
300	#556	C₂²×C₂₃	1	1	yes	characteristic
301	#554	C₂²×C₂₃	1	1	yes	characteristic
302	#547	C₂²×C₂₃	1	1	yes	characteristic
303	#547	C₂²×C₂₃	1	1	yes	characteristic
304	#500	C₂²×C₂₃	1	1	yes	characteristic
305	#498	C₂²×C₂₃	1	1	yes	characteristic
306	#488	—	2	1	no	—
307	#487	C₂²×C₂₃	1	1	yes	characteristic
308	#487	C₂²×C₂₃	1	1	yes	characteristic
309	#487	C₂²×C₂₃	1	1	yes	characteristic
310	#487	C₂²×C₂₃	1	1	yes	characteristic
311	#485	—	2	1	no	—
312	#483	—	2	1	no	—
313	#480	—	2	1	no	—
314	#480	—	2	1	no	—
315	#479	—	2	1	no	—
316	#476	C₂²×C₂₃	1	1	yes	characteristic
317	#476	C₂²×C₂₃	1	1	yes	characteristic
318	#474	C₂²×C₂₃	1	1	yes	characteristic
319	#474	C₂²×C₂₃	1	1	yes	characteristic
320	#474	C₂²×C₂₃	1	1	yes	characteristic
321	#470	—	2	1	no	—
322	#467	—	2	1	no	—
323	#466	—	2	1	no	—
324	#463	—	2	1	no	—
325	#462	C₂²×C₂₃	1	1	yes	characteristic
326	#462	C₂²×C₂₃	1	1	yes	characteristic
327	#461	C₂²×C₂₃	1	1	yes	characteristic
328	#452	—	2	1	no	—
329	#452	—	2	1	no	—
330	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
331	C₄×SD₁₆	—	2	1	no	—
332	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
333	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
334	#436	C₂²×C₂₃	1	1	yes	characteristic
335	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×#107	—	2	1	no	—
338	#417	—	2	1	no	—
339	#416	—	2	1	no	—
340	#335	C₂²×C₂₃	1	1	yes	characteristic
341	#335	C₂²×C₂₃	1	1	yes	characteristic
342	#333	C₂²×C₂₃	1	1	yes	characteristic
343	#332	C₂²×C₂₃	1	1	yes	characteristic
344	#332	C₂²×C₂₃	1	1	yes	characteristic
345	#332	C₂²×C₂₃	1	1	yes	characteristic
346	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
347	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
349	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	#322	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₂₃	—	4	2	no	—
352	C₂×C₂₃	—	8	1	no	—
353	C₂×C₂₃	—	8	1	no	—
354	C₂×C₂₃	#3269	1	1	yes	characteristic
355	C₂×C₂₃	#3280	1	1	yes	characteristic
356	C₂×C₂₃	#3289	1	1	yes	characteristic
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×Q₁₆	—	2	1	no	—
360	C₂×Q₁₆	—	2	1	no	—
361	C₂×SD₁₆	—	2	1	no	—
362	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
363	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
364	C₂×M₁₆	—	2	1	no	—
365	C₂²×C₈	—	2	1	no	—
366	C₂²×C₈	—	2	1	no	—
367	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
368	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
370	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	#126	—	4	1	no	—
372	#125	—	2	1	no	—
373	#124	—	2	1	no	—
374	#123	—	4	1	no	—
375	#123	—	4	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	4	1	no	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
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	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
389	#117	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	#107	—	2	1	no	—
394	#107	—	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	—	2	1	no	—
401	#106	—	4	1	no	—
402	#106	D₄×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	4	1	no	—
418	#103	D₄×C₂₃	1	1	yes	characteristic
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	D₄×C₂₃	1	1	yes	characteristic
425	#102	D₄×C₂₃	1	1	yes	characteristic
426	#102	D₄×C₂₃	1	1	yes	characteristic
427	C₂²⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	1	no	—
429	C₂²⋊C₈	—	2	1	no	—
430	C₄×C₈	—	2	1	no	—
431	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₂₃	#19778	1	1	yes	characteristic, direct product factor, Sylow
435	#41	—	4	2	no	—
436	#41	—	4	2	no	—
437	#41	—	8	1	no	—
438	#41	—	8	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	—	4	1	no	—
446	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
448	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	4	1	no	—
453	C₂²×C₄	—	4	1	no	—
454	C₂²×C₄	—	4	1	no	—
455	Q₁₆	—	8	1	no	—
456	Q₁₆	—	8	1	no	—
457	SD₁₆	—	2	2	no	—
458	SD₁₆	—	4	2	no	—
459	D₈	—	4	1	no	—
460	M₁₆	—	8	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	4	2	no	—
467	C₂×C₈	—	8	1	no	—
468	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	4	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	—	2	1	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	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	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	4	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	8	1	no	—
513	D₄	—	2	1	no	—
514	D₄	—	2	1	no	—
515	D₄	—	2	1	no	—
516	D₄	—	2	1	no	—
517	D₄	—	2	1	no	—
518	D₄	—	8	2	no	—
519	D₄	—	8	2	no	—
520	D₄	C₂₃×#135	1	2	yes	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	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	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	8	2	no	—
537	C₂×C₄	—	8	2	no	—
538	C₂×C₄	—	16	1	no	—
539	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
541	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
542	C₈	—	2	1	no	—
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	D₄²×C₂₃	1	1	yes	characteristic
561	C₄	—	2	1	no	—
562	C₄	—	4	1	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	—	8	1	no	—
571	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
572	C₄	C₂₃×#575	1	1	yes	characteristic
573	C₄	C₂₃×#576	1	1	yes	characteristic
574	C₄	C₂₃×#578	1	1	yes	characteristic
575	C₂	—	4	2	no	—
576	C₂	—	8	1	no	—
577	C₂	—	8	1	no	—
578	C₂	C₂₃×#3269	1	1	yes	characteristic
579	C₂	C₂₃×#3280	1	1	yes	characteristic
580	C₂	C₂₃×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3334	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
3. C₂₃×#3331	C₂	semidirect
4. C₂₃×#3302	C₂	semidirect
5. C₂₃×#1718	C₂	semidirect
6. C₂₃×#1699	C₂	semidirect
7. C₂₃×#1683	C₂	semidirect
8. C₂₃×#1683	C₂	semidirect
9. C₂₃×#1674	C₂	semidirect
10. C₂₃×#1654	C₂	semidirect
11. C₂₃×#1652	C₂	semidirect
12. C₂₃×#1644	C₂	semidirect
13. C₂₃×#1635	C₂	semidirect
14. C₂₃×#1581	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
210. #19778	C₂₃	direct (N × H)
261. #3360	C₂×C₂₃	semidirect
263. #3331	C₂×C₂₃	semidirect
264. #3302	C₂×C₂₃	semidirect
265. #1718	C₂×C₂₃	semidirect
266. #1699	C₂×C₂₃	semidirect
267. #1683	C₂×C₂₃	semidirect
268. #1683	C₂×C₂₃	semidirect
269. #1674	C₂×C₂₃	semidirect
270. #1654	C₂×C₂₃	semidirect
271. #1652	C₂×C₂₃	semidirect
272. #1644	C₂×C₂₃	semidirect
273. #1635	C₂×C₂₃	semidirect
274. #1581	C₂×C₂₃	semidirect
275. SD₁₆⋊C₈	C₂×C₂₃	semidirect
434. C₂₃	#19778	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×#3334	2
C₂₃×#3331	2
C₂₃×#3302	2
C₂₃×#1718	2
C₂₃×#1699	2
C₂₃×#1683	2
C₂₃×#1674	2
C₂₃×#1654	2
C₂₃×#1652	2
C₂₃×#1644	2
C₂₃×#1635	2
C₂₃×#1581	2
C₂₃×SD₁₆⋊C₈	2
#19778	23

Subgroups — C23×#19778 #19167404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19778 #19167404