Subgroups — C₂₃×#24838 #19172464

Order 5888.

904 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2857	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2088	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2066	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1892	C₂	1	2	yes	maximal
10	C₂₃×#1807	C₂	1	2	yes	maximal
11	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#475	—	2	4	no	—
19	C₂×C₂₃×#106	C₂²	1	2	yes	—
20	C₂×C₂₃×#106	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#103	C₂²	1	2	yes	—
26	C₂×C₂₃×#103	C₂²	1	2	yes	—
27	C₂₃×#391	—	2	1	no	—
28	C₂₃×#391	C₂²	1	1	yes	characteristic
29	C₂₃×#389	—	2	1	no	—
30	C₂₃×#389	C₂²	1	1	yes	characteristic
31	C₂₃×#389	C₂²	1	2	yes	—
32	C₂₃×#387	—	2	1	no	—
33	C₂₃×#385	—	2	1	no	—
34	C₂₃×#384	C₂²	1	1	yes	characteristic
35	C₂₃×#380	—	2	1	no	—
36	C₂₃×#380	C₂²	1	2	yes	—
37	C₂₃×#380	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#340	C₂²	1	2	yes	—
43	C₂₃×#340	C₂²	1	2	yes	—
44	C₂₃×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₂₃	—	2	1	no	—
46	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₂₃	—	2	4	no	—
51	C₂²×C₈×C₂₃	—	2	4	no	—
52	C₂²×C₈×C₂₃	—	4	2	no	—
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂₃×C₄⋊Q₈	—	2	2	no	—
56	C₂₃×C₄⋊Q₈	—	2	2	no	—
57	C₂₃×C₂²⋊Q₈	—	2	2	no	—
58	C₂₃×C₂²⋊Q₈	—	4	4	no	—
59	C₂₃×C₂²⋊Q₈	Q₈	1	4	yes	complemented
60	C₄×Q₈×C₂₃	—	2	2	no	—
61	C₄×Q₈×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
69	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄²×C₂₃	—	2	2	no	—
78	C₂×C₄²×C₂₃	—	2	2	no	—
79	C₂×C₄²×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	—	2	1	no	—
81	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
85	C₂₃×#106	—	2	4	no	—
86	C₂₃×#106	—	4	2	no	—
87	C₂₃×C₄⋊C₈	—	2	4	no	—
88	C₂₃×C₄⋊C₈	—	2	2	no	—
89	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
90	C₂₃×#103	—	2	4	no	—
91	C₂₃×#103	—	4	2	no	—
92	C₂₃×#95	—	2	1	no	—
93	C₂₃×#95	—	2	1	no	—
94	C₂₃×#95	—	2	1	no	—
95	C₂₃×#95	—	2	1	no	—
96	C₂₃×#95	—	2	1	no	—
97	C₂₃×#95	—	2	2	no	—
98	C₂₃×#95	—	2	2	no	—
99	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×Q₈×C₂₃	—	2	1	no	—
101	C₂×Q₈×C₂₃	—	2	1	no	—
102	C₂×Q₈×C₂₃	—	2	1	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	4	2	no	—
106	C₂×Q₈×C₂₃	—	4	2	no	—
107	C₂×Q₈×C₂₃	—	4	2	no	—
108	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	2	2	no	—
111	C₂²×C₄×C₂₃	—	2	1	no	—
112	C₂²×C₄×C₂₃	—	2	1	no	—
113	C₂²×C₄×C₂₃	—	2	1	no	—
114	C₂²×C₄×C₂₃	—	2	1	no	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	1	no	—
118	C₂²×C₄×C₂₃	—	2	2	no	—
119	C₂²×C₄×C₂₃	—	2	1	no	—
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	2	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	—	4	1	no	—
124	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
131	C₂²×C₄×C₂₃	#41	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	2	no	—
136	C₂×C₈×C₂₃	—	2	2	no	—
137	C₂×C₈×C₂₃	—	2	2	no	—
138	C₂×C₈×C₂₃	—	4	8	no	—
139	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
140	C₄⋊C₄×C₂₃	—	2	4	no	—
141	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₄⋊C₄×C₂₃	—	4	2	no	—
151	C₄⋊C₄×C₂₃	—	4	2	no	—
152	C₄⋊C₄×C₂₃	—	4	2	no	—
153	C₄⋊C₄×C₂₃	—	4	2	no	—
154	C₄⋊C₄×C₂₃	—	4	2	no	—
155	C₄⋊C₄×C₂₃	—	4	4	no	—
156	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
157	C₂²⋊C₄×C₂₃	—	2	4	no	—
158	C₂²⋊C₄×C₂₃	—	4	4	no	—
159	C₂²⋊C₄×C₂₃	—	4	2	no	—
160	C₄²×C₂₃	—	2	2	no	—
161	C₄²×C₂₃	—	2	2	no	—
162	C₄²×C₂₃	—	2	1	no	—
163	C₄²×C₂₃	—	2	1	no	—
164	C₄²×C₂₃	—	4	2	no	—
165	C₄²×C₂₃	—	4	2	no	—
166	C₄²×C₂₃	C₂×D₄	1	2	yes	—
167	C₄²×C₂₃	#41	1	2	yes	—
168	#24838	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₂₃	—	2	1	no	—
170	C₂³×C₂₃	—	2	1	no	—
171	C₂³×C₂₃	—	2	1	no	—
172	C₂³×C₂₃	—	2	1	no	—
173	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
174	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
175	C₂³×C₂₃	#137	1	2	yes	—
176	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₂₃	—	2	2	no	—
178	Q₈×C₂₃	—	2	2	no	—
179	Q₈×C₂₃	—	4	1	no	—
180	Q₈×C₂₃	—	4	1	no	—
181	Q₈×C₂₃	—	8	4	no	—
182	C₂×C₄×C₂₃	—	2	1	no	—
183	C₂×C₄×C₂₃	—	2	1	no	—
184	C₂×C₄×C₂₃	—	2	1	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	8	no	—
193	C₂×C₄×C₂₃	—	2	1	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	1	no	—
198	C₂×C₄×C₂₃	—	4	4	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	1	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	1	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	2	no	—
209	C₂×C₄×C₂₃	—	4	1	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	4	1	no	—
212	C₂×C₄×C₂₃	—	4	1	no	—
213	C₂×C₄×C₂₃	—	4	2	no	—
214	C₂×C₄×C₂₃	—	4	1	no	—
215	C₂×C₄×C₂₃	#121	1	2	yes	—
216	C₂×C₄×C₂₃	#121	1	2	yes	—
217	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
218	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
219	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
220	C₂×C₄×C₂₃	#135	1	2	yes	—
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
227	C₈×C₂₃	—	2	4	no	—
228	C₈×C₂₃	—	4	2	no	—
229	C₂×#475	C₂×C₂₃	1	1	yes	characteristic, complemented
230	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
231	#2857	C₂×C₂₃	1	1	yes	characteristic
232	C₄×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
233	#2088	C₂×C₂₃	1	2	yes	complemented
234	#2066	C₂×C₂₃	1	2	yes	complemented
235	#1988	C₂×C₂₃	1	1	yes	characteristic, complemented
236	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
237	#1892	C₂×C₂₃	1	2	yes	—
238	#1807	C₂×C₂₃	1	2	yes	—
239	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂²×C₂₃	—	2	2	no	—
241	C₂²×C₂₃	—	2	2	no	—
242	C₂²×C₂₃	—	2	2	no	—
243	C₂²×C₂₃	—	2	2	no	—
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	—	2	2	no	—
246	C₂²×C₂₃	#475	1	4	yes	complemented
247	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
248	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
249	C₂²×C₂₃	#537	1	1	yes	characteristic
250	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
251	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
252	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
253	C₂²×C₂₃	#575	1	1	yes	characteristic
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	4	2	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	—	4	1	no	—
260	C₄×C₂₃	—	8	2	no	—
261	C₄×C₂₃	—	8	1	no	—
262	C₄×C₂₃	—	8	2	no	—
263	C₄×C₂₃	—	8	1	no	—
264	C₄×C₂₃	#465	1	4	yes	—
265	C₄×C₂₃	#546	1	1	yes	characteristic
266	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
267	C₄×C₂₃	#552	1	1	yes	characteristic
268	C₄×C₂₃	#554	1	1	yes	characteristic
269	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	C₂×C₂²⋊Q₈	—	2	1	no	—
272	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
274	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
275	#475	—	2	4	no	—
276	C₂×#106	C₂²×C₂₃	1	2	yes	—
277	C₂×#106	C₂²×C₂₃	1	2	yes	—
278	C₂×C₄⋊C₈	—	2	4	no	—
279	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
281	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×#103	C₂²×C₂₃	1	2	yes	—
283	C₂×#103	C₂²×C₂₃	1	2	yes	—
284	#391	—	2	1	no	—
285	#391	C₂²×C₂₃	1	1	yes	characteristic
286	#389	—	2	1	no	—
287	#389	C₂²×C₂₃	1	2	yes	—
288	#389	C₂²×C₂₃	1	1	yes	characteristic
289	#387	—	2	1	no	—
290	#385	—	2	1	no	—
291	#384	C₂²×C₂₃	1	1	yes	characteristic
292	#380	—	2	1	no	—
293	#380	C₂²×C₂₃	1	1	yes	characteristic
294	#380	C₂²×C₂₃	1	2	yes	—
295	C₄×C₄⋊C₄	—	2	1	no	—
296	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
297	C₄×C₂²⋊C₄	—	2	1	no	—
298	#340	C₂²×C₂₃	1	2	yes	—
299	#340	C₂²×C₂₃	1	2	yes	—
300	#340	C₂²×C₂₃	1	2	yes	—
301	#340	C₂²×C₂₃	1	2	yes	—
302	C₂×C₂₃	—	2	4	no	—
303	C₂×C₂₃	#2417	1	1	yes	characteristic
304	C₂×C₂₃	C₂×#465	1	1	yes	characteristic
305	C₂×C₂₃	#3056	1	1	yes	characteristic
306	C₂×C₂₃	C₂×#475	1	1	yes	characteristic
307	C₂×C₂₃	#3076	1	1	yes	characteristic
308	C₂×C₂₃	#3246	1	1	yes	characteristic
309	C₂×C₂₃	#3248	1	1	yes	characteristic
310	C₂²×Q₈	—	2	1	no	—
311	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
312	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
313	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×C₈	—	2	4	no	—
316	C₂²×C₈	—	2	4	no	—
317	C₂²×C₈	—	4	2	no	—
318	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
319	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	4	4	no	—
324	C₂²⋊Q₈	Q₈×C₂₃	1	4	yes	complemented
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₄²	—	2	2	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
347	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	#106	—	2	4	no	—
351	#106	—	4	2	no	—
352	C₄⋊C₈	—	2	4	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
355	#103	—	2	4	no	—
356	#103	—	4	2	no	—
357	#95	—	2	1	no	—
358	#95	—	2	1	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	#95	—	2	1	no	—
362	#95	—	2	2	no	—
363	#95	—	2	2	no	—
364	C₂₃	#24838	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	4	2	no	—
372	C₂×Q₈	—	4	2	no	—
373	C₂×Q₈	—	4	2	no	—
374	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	4	1	no	—
390	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
391	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
397	C₂²×C₄	#41×C₂₃	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	4	8	no	—
405	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	4	2	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	—	4	2	no	—
419	C₄⋊C₄	—	4	2	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	4	4	no	—
422	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	4	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	4	2	no	—
431	C₄²	—	4	2	no	—
432	C₄²	C₂×D₄×C₂₃	1	2	yes	—
433	C₄²	#41×C₂₃	1	2	yes	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
439	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
440	C₂³	C₂₃×#137	1	2	yes	—
441	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
442	Q₈	—	2	2	no	—
443	Q₈	—	2	2	no	—
444	Q₈	—	4	1	no	—
445	Q₈	—	4	1	no	—
446	Q₈	—	8	4	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	C₂₃×#121	1	2	yes	—
481	C₂×C₄	C₂₃×#121	1	2	yes	—
482	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
483	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
484	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
485	C₂×C₄	C₂₃×#135	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
487	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
490	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
491	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
492	C₈	—	2	4	no	—
493	C₈	—	4	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	C₂₃×#475	1	4	yes	complemented
501	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
502	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
503	C₂²	C₂₃×#537	1	1	yes	characteristic
504	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
505	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
506	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
507	C₂²	C₂₃×#575	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	2	no	—
514	C₄	—	8	2	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	2	no	—
517	C₄	—	8	1	no	—
518	C₄	C₂₃×#465	1	4	yes	—
519	C₄	C₂₃×#546	1	1	yes	characteristic
520	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
521	C₄	C₂₃×#552	1	1	yes	characteristic
522	C₄	C₂₃×#554	1	1	yes	characteristic
523	C₂	—	2	4	no	—
524	C₂	C₂₃×#2417	1	1	yes	characteristic
525	C₂	C₂×C₂₃×#465	1	1	yes	characteristic
526	C₂	C₂₃×#3056	1	1	yes	characteristic
527	C₂	C₂×C₂₃×#475	1	1	yes	characteristic
528	C₂	C₂₃×#3076	1	1	yes	characteristic
529	C₂	C₂₃×#3246	1	1	yes	characteristic
530	C₂	C₂₃×#3248	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#475	C₂	semidirect
5. C₂₃×#2088	C₂	semidirect
6. C₂₃×#2066	C₂	semidirect
7. C₂₃×#1988	C₂	semidirect
8. C₂₃×#1948	C₂	semidirect
11. C₂₃×#1803	C₂	semidirect
59. C₂₃×C₂²⋊Q₈	Q₈	semidirect
168. #24838	C₂₃	direct (N × H)
229. C₂×#475	C₂×C₂₃	semidirect
233. #2088	C₂×C₂₃	semidirect
234. #2066	C₂×C₂₃	semidirect
235. #1988	C₂×C₂₃	semidirect
236. #1948	C₂×C₂₃	semidirect
239. #1803	C₂×C₂₃	semidirect
246. C₂²×C₂₃	#475	semidirect
324. C₂²⋊Q₈	Q₈×C₂₃	semidirect
364. C₂₃	#24838	direct (N × H)
500. C₂²	C₂₃×#475	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#475	2
C₂²×C₂₃×C₄⋊C₈	2
C₂₃×#2857	2
C₄×C₂₃×C₂²⋊Q₈	2
C₂₃×#2088	2
C₂₃×#2066	2
C₂₃×#1988	2
C₂₃×#1948	2
C₂₃×#1892	2
C₂₃×#1807	2
C₂₃×#1803	2
#24838	23

Subgroups — C23×#24838 #19172464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24838 #19172464