Subgroups — C₂₃×#24068 #19171694

Order 5888.

1104 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#482	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2042	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2042	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1875	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1873	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#529	—	2	4	no	—
17	C₂₃×#529	—	2	4	no	—
18	C₂×C₂₃×#126	—	2	1	no	—
19	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#123	—	2	1	no	—
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	—	2	1	no	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#117	—	2	1	no	—
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#485	—	2	4	no	—
32	C₂₃×#482	—	2	4	no	—
33	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	C₂₃×#389	C₂²	1	1	yes	characteristic
48	C₂₃×#389	C₂²	1	1	yes	characteristic
49	C₂₃×#389	C₂²	1	1	yes	characteristic
50	C₂₃×#389	C₂²	1	1	yes	characteristic
51	C₂₃×#385	—	2	1	no	—
52	C₂₃×#385	—	2	1	no	—
53	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂₃×#336	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#126	—	2	8	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	2	no	—
77	C₂₃×#124	—	2	2	no	—
78	C₂₃×#124	—	2	2	no	—
79	C₂₃×#123	—	2	8	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	D₄	1	4	yes	—
82	C₂₃×#121	—	2	2	no	—
83	C₂₃×#121	D₄	1	4	yes	—
84	C₄×Q₈×C₂₃	—	2	2	no	—
85	C₄×Q₈×C₂₃	—	2	2	no	—
86	C₄×D₄×C₂₃	—	2	8	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₂₃×#117	—	2	8	no	—
90	C₂₃×#117	—	2	2	no	—
91	C₂₃×#117	—	2	2	no	—
92	C₂₃×#117	—	2	2	no	—
93	C₂₃×#117	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₄²×C₂₃	—	2	1	no	—
116	C₂×C₄²×C₂₃	—	2	2	no	—
117	C₂×C₄²×C₂₃	—	2	2	no	—
118	C₂×C₄²×C₂₃	—	2	1	no	—
119	C₂×C₄²×C₂₃	—	2	1	no	—
120	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
124	C₂₃×#107	—	2	2	no	—
125	C₂₃×#107	—	2	2	no	—
126	C₂₃×#106	—	2	2	no	—
127	C₂₃×#106	—	2	2	no	—
128	C₂₃×#103	—	2	2	no	—
129	C₂₃×#103	—	2	2	no	—
130	C₂₃×#103	—	2	2	no	—
131	C₂₃×#103	—	2	2	no	—
132	C₂₃×#102	—	2	2	no	—
133	C₂₃×#102	—	2	2	no	—
134	C₂₃×#102	—	2	2	no	—
135	C₂₃×#102	—	2	2	no	—
136	C₂₃×C₂²⋊C₈	—	2	2	no	—
137	C₂₃×C₂²⋊C₈	—	2	2	no	—
138	C₂₃×#95	—	2	1	no	—
139	C₂₃×#95	—	2	1	no	—
140	C₂₃×#95	—	2	1	no	—
141	C₂₃×#95	—	2	1	no	—
142	C₂⁴×C₂₃	—	2	1	no	—
143	C₂⁴×C₂₃	C₂×D₄	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	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₂₃	—	2	4	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	2	no	—
157	C₂×D₄×C₂₃	—	4	2	no	—
158	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	2	2	no	—
165	C₂²×C₄×C₂₃	—	2	2	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	2	2	no	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	4	1	no	—
175	C₂²×C₄×C₂₃	—	4	1	no	—
176	C₂²×C₄×C₂₃	—	4	1	no	—
177	C₂²×C₄×C₂₃	—	4	2	no	—
178	C₂²×C₄×C₂₃	—	4	2	no	—
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	—	2	1	no	—
189	C₂×C₈×C₂₃	—	2	1	no	—
190	C₂×C₈×C₂₃	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	1	no	—
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	—	2	1	no	—
195	C₂×C₈×C₂₃	—	2	1	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	4	no	—
200	C₄⋊C₄×C₂₃	—	2	4	no	—
201	C₄⋊C₄×C₂₃	—	2	4	no	—
202	C₄⋊C₄×C₂₃	—	2	4	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	2	1	no	—
207	C₄⋊C₄×C₂₃	—	2	1	no	—
208	C₄⋊C₄×C₂₃	—	2	1	no	—
209	C₄⋊C₄×C₂₃	—	2	1	no	—
210	C₄⋊C₄×C₂₃	—	2	1	no	—
211	C₄⋊C₄×C₂₃	—	4	2	no	—
212	C₄⋊C₄×C₂₃	—	4	2	no	—
213	C₄⋊C₄×C₂₃	—	4	2	no	—
214	C₄⋊C₄×C₂₃	—	4	2	no	—
215	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
217	C₄⋊C₄×C₂₃	#41	1	2	yes	—
218	C₄⋊C₄×C₂₃	#41	1	2	yes	—
219	C₂²⋊C₄×C₂₃	—	2	4	no	—
220	C₂²⋊C₄×C₂₃	—	2	4	no	—
221	C₂²⋊C₄×C₂₃	—	2	4	no	—
222	C₂²⋊C₄×C₂₃	—	2	4	no	—
223	C₂²⋊C₄×C₂₃	—	2	4	no	—
224	C₂²⋊C₄×C₂₃	—	2	4	no	—
225	C₂²⋊C₄×C₂₃	—	4	2	no	—
226	C₄²×C₂₃	—	2	4	no	—
227	C₄²×C₂₃	—	2	4	no	—
228	C₄²×C₂₃	—	2	2	no	—
229	C₄²×C₂₃	—	2	2	no	—
230	C₄²×C₂₃	—	2	1	no	—
231	C₄²×C₂₃	—	2	1	no	—
232	C₄²×C₂₃	—	2	1	no	—
233	C₄²×C₂₃	—	2	1	no	—
234	C₄²×C₂₃	—	4	2	no	—
235	#24068	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂³×C₂₃	—	2	1	no	—
237	C₂³×C₂₃	—	2	1	no	—
238	C₂³×C₂₃	—	2	1	no	—
239	C₂³×C₂₃	—	2	1	no	—
240	C₂³×C₂₃	—	2	1	no	—
241	C₂³×C₂₃	—	2	1	no	—
242	C₂³×C₂₃	—	2	2	no	—
243	C₂³×C₂₃	—	4	1	no	—
244	C₂³×C₂₃	—	4	1	no	—
245	C₂³×C₂₃	—	4	1	no	—
246	C₂³×C₂₃	—	4	1	no	—
247	C₂³×C₂₃	—	4	1	no	—
248	C₂³×C₂₃	—	4	1	no	—
249	C₂³×C₂₃	#120	1	2	yes	—
250	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
251	Q₈×C₂₃	—	2	2	no	—
252	Q₈×C₂₃	—	2	2	no	—
253	Q₈×C₂₃	—	4	1	no	—
254	Q₈×C₂₃	—	4	1	no	—
255	D₄×C₂₃	—	2	2	no	—
256	D₄×C₂₃	—	2	2	no	—
257	D₄×C₂₃	—	4	8	no	—
258	D₄×C₂₃	—	4	1	no	—
259	D₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	2	no	—
267	C₂×C₄×C₂₃	—	2	2	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	1	no	—
270	C₂×C₄×C₂₃	—	2	1	no	—
271	C₂×C₄×C₂₃	—	2	2	no	—
272	C₂×C₄×C₂₃	—	2	2	no	—
273	C₂×C₄×C₂₃	—	2	2	no	—
274	C₂×C₄×C₂₃	—	2	2	no	—
275	C₂×C₄×C₂₃	—	2	2	no	—
276	C₂×C₄×C₂₃	—	2	1	no	—
277	C₂×C₄×C₂₃	—	2	1	no	—
278	C₂×C₄×C₂₃	—	2	1	no	—
279	C₂×C₄×C₂₃	—	2	1	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	4	4	no	—
283	C₂×C₄×C₂₃	—	4	8	no	—
284	C₂×C₄×C₂₃	—	4	4	no	—
285	C₂×C₄×C₂₃	—	4	1	no	—
286	C₂×C₄×C₂₃	—	4	1	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	#120	1	2	yes	—
302	C₂×C₄×C₂₃	#135	1	2	yes	—
303	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
307	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
308	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
309	C₈×C₂₃	—	4	1	no	—
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	—	4	1	no	—
312	C₈×C₂₃	—	4	1	no	—
313	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
314	C₂×#485	C₂×C₂₃	1	1	yes	characteristic, complemented
315	C₂×#482	C₂×C₂₃	1	1	yes	characteristic
316	#2296	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
318	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#2042	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#2042	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1875	C₂×C₂₃	1	1	yes	characteristic
322	#1873	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1817	C₂×C₂₃	1	1	yes	characteristic, complemented
328	C₂²×C₂₃	—	2	2	no	—
329	C₂²×C₂₃	—	2	2	no	—
330	C₂²×C₂₃	—	2	2	no	—
331	C₂²×C₂₃	—	4	1	no	—
332	C₂²×C₂₃	—	4	1	no	—
333	C₂²×C₂₃	—	4	1	no	—
334	C₂²×C₂₃	—	4	1	no	—
335	C₂²×C₂₃	—	4	2	no	—
336	C₂²×C₂₃	—	4	2	no	—
337	C₂²×C₂₃	—	4	2	no	—
338	C₂²×C₂₃	—	8	1	no	—
339	C₂²×C₂₃	—	8	1	no	—
340	C₂²×C₂₃	—	8	1	no	—
341	C₂²×C₂₃	—	8	1	no	—
342	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
343	C₂²×C₂₃	#525	1	1	yes	characteristic
344	C₂²×C₂₃	#536	1	1	yes	characteristic
345	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
346	C₂²×C₂₃	#575	1	1	yes	characteristic
347	C₂²×C₂₃	#575	1	1	yes	characteristic
348	C₂²×C₂₃	#575	1	1	yes	characteristic
349	C₄×C₂₃	—	2	2	no	—
350	C₄×C₂₃	—	2	2	no	—
351	C₄×C₂₃	—	4	2	no	—
352	C₄×C₂₃	—	4	2	no	—
353	C₄×C₂₃	—	4	2	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	4	2	no	—
356	C₄×C₂₃	—	4	1	no	—
357	C₄×C₂₃	—	4	1	no	—
358	C₄×C₂₃	—	4	2	no	—
359	C₄×C₂₃	—	8	1	no	—
360	C₄×C₂₃	—	8	1	no	—
361	C₄×C₂₃	#547	1	1	yes	characteristic
362	C₄×C₂₃	#547	1	1	yes	characteristic
363	C₄×C₂₃	#547	1	1	yes	characteristic
364	C₄×C₂₃	#547	1	1	yes	characteristic
365	#529	—	2	4	no	—
366	#529	—	2	4	no	—
367	C₂×#126	—	2	1	no	—
368	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#123	—	2	1	no	—
371	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₄×D₄	—	2	1	no	—
375	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×#117	—	2	1	no	—
377	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
379	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
380	#485	—	2	4	no	—
381	#482	—	2	4	no	—
382	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
389	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
392	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
394	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
396	#389	C₂²×C₂₃	1	1	yes	characteristic
397	#389	C₂²×C₂₃	1	1	yes	characteristic
398	#389	C₂²×C₂₃	1	1	yes	characteristic
399	#389	C₂²×C₂₃	1	1	yes	characteristic
400	#385	—	2	1	no	—
401	#385	—	2	1	no	—
402	C₄×C₄⋊C₄	—	2	1	no	—
403	C₄×C₄⋊C₄	—	2	1	no	—
404	#340	C₂²×C₂₃	1	1	yes	characteristic
405	#340	C₂²×C₂₃	1	1	yes	characteristic
406	#340	C₂²×C₂₃	1	1	yes	characteristic
407	#340	C₂²×C₂₃	1	1	yes	characteristic
408	#336	C₂²×C₂₃	1	1	yes	characteristic
409	#336	C₂²×C₂₃	1	1	yes	characteristic
410	#336	C₂²×C₂₃	1	1	yes	characteristic
411	#336	C₂²×C₂₃	1	1	yes	characteristic
412	C₂×C₂₃	—	4	2	no	—
413	C₂×C₂₃	—	8	2	no	—
414	C₂×C₂₃	#2406	1	1	yes	characteristic
415	C₂×C₂₃	#3002	1	1	yes	characteristic
416	C₂×C₂₃	#3005	1	1	yes	characteristic
417	C₂×C₂₃	#3102	1	1	yes	characteristic
418	C₂×C₂₃	#3111	1	1	yes	characteristic
419	C₂×C₂₃	#3233	1	1	yes	characteristic
420	C₂×C₂₃	#3240	1	1	yes	characteristic
421	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₂²×D₄	—	2	1	no	—
423	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
424	C₂³×C₄	—	2	1	no	—
425	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
426	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
428	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	#126	—	2	8	no	—
433	#125	—	2	2	no	—
434	#125	—	2	2	no	—
435	#124	—	2	2	no	—
436	#124	—	2	2	no	—
437	#123	—	2	8	no	—
438	C₂²⋊Q₈	—	2	2	no	—
439	C₂²⋊Q₈	D₄×C₂₃	1	4	yes	—
440	#121	—	2	2	no	—
441	#121	D₄×C₂₃	1	4	yes	—
442	C₄×Q₈	—	2	2	no	—
443	C₄×Q₈	—	2	2	no	—
444	C₄×D₄	—	2	8	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	#117	—	2	8	no	—
448	#117	—	2	2	no	—
449	#117	—	2	2	no	—
450	#117	—	2	2	no	—
451	#117	—	2	2	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	1	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	—	2	1	no	—
458	C₂×C₄⋊C₄	—	2	1	no	—
459	C₂×C₄⋊C₄	—	2	1	no	—
460	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
462	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
463	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
464	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
465	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
466	C₂×C₂²⋊C₄	—	2	1	no	—
467	C₂×C₂²⋊C₄	—	2	1	no	—
468	C₂×C₂²⋊C₄	—	2	1	no	—
469	C₂×C₂²⋊C₄	—	2	1	no	—
470	C₂×C₂²⋊C₄	—	2	1	no	—
471	C₂×C₂²⋊C₄	—	2	1	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₄²	—	2	1	no	—
474	C₂×C₄²	—	2	1	no	—
475	C₂×C₄²	—	2	2	no	—
476	C₂×C₄²	—	2	2	no	—
477	C₂×C₄²	—	2	1	no	—
478	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
481	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
482	#107	—	2	2	no	—
483	#107	—	2	2	no	—
484	#106	—	2	2	no	—
485	#106	—	2	2	no	—
486	#103	—	2	2	no	—
487	#103	—	2	2	no	—
488	#103	—	2	2	no	—
489	#103	—	2	2	no	—
490	#102	—	2	2	no	—
491	#102	—	2	2	no	—
492	#102	—	2	2	no	—
493	#102	—	2	2	no	—
494	C₂²⋊C₈	—	2	2	no	—
495	C₂²⋊C₈	—	2	2	no	—
496	#95	—	2	1	no	—
497	#95	—	2	1	no	—
498	#95	—	2	1	no	—
499	#95	—	2	1	no	—
500	C₂₃	#24068	1	1	yes	characteristic, direct product factor, Sylow
501	C₂⁴	—	2	1	no	—
502	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	4	2	no	—
517	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	—	4	1	no	—
534	C₂²×C₄	—	4	1	no	—
535	C₂²×C₄	—	4	1	no	—
536	C₂²×C₄	—	4	2	no	—
537	C₂²×C₄	—	4	2	no	—
538	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
539	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
540	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
541	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
542	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
543	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
544	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	4	2	no	—
571	C₄⋊C₄	—	4	2	no	—
572	C₄⋊C₄	—	4	2	no	—
573	C₄⋊C₄	—	4	2	no	—
574	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
575	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
576	C₄⋊C₄	#41×C₂₃	1	2	yes	—
577	C₄⋊C₄	#41×C₂₃	1	2	yes	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	—	2	4	no	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	4	2	no	—
585	C₄²	—	2	4	no	—
586	C₄²	—	2	4	no	—
587	C₄²	—	2	2	no	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	1	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	4	2	no	—
594	C₂³	—	2	1	no	—
595	C₂³	—	2	1	no	—
596	C₂³	—	2	1	no	—
597	C₂³	—	2	1	no	—
598	C₂³	—	2	1	no	—
599	C₂³	—	2	1	no	—
600	C₂³	—	2	2	no	—
601	C₂³	—	4	1	no	—
602	C₂³	—	4	1	no	—
603	C₂³	—	4	1	no	—
604	C₂³	—	4	1	no	—
605	C₂³	—	4	1	no	—
606	C₂³	—	4	1	no	—
607	C₂³	C₂₃×#120	1	2	yes	—
608	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
609	Q₈	—	2	2	no	—
610	Q₈	—	2	2	no	—
611	Q₈	—	4	1	no	—
612	Q₈	—	4	1	no	—
613	D₄	—	2	2	no	—
614	D₄	—	2	2	no	—
615	D₄	—	4	8	no	—
616	D₄	—	4	1	no	—
617	D₄	—	4	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	4	4	no	—
641	C₂×C₄	—	4	8	no	—
642	C₂×C₄	—	4	4	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	C₂₃×#120	1	2	yes	—
660	C₂×C₄	C₂₃×#135	1	2	yes	—
661	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
662	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
663	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
664	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
665	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
666	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
667	C₈	—	4	1	no	—
668	C₈	—	4	1	no	—
669	C₈	—	4	1	no	—
670	C₈	—	4	1	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	2	no	—
673	C₂²	—	2	2	no	—
674	C₂²	—	4	1	no	—
675	C₂²	—	4	1	no	—
676	C₂²	—	4	1	no	—
677	C₂²	—	4	1	no	—
678	C₂²	—	4	2	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	4	2	no	—
681	C₂²	—	8	1	no	—
682	C₂²	—	8	1	no	—
683	C₂²	—	8	1	no	—
684	C₂²	—	8	1	no	—
685	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
686	C₂²	C₂₃×#525	1	1	yes	characteristic
687	C₂²	C₂₃×#536	1	1	yes	characteristic
688	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
689	C₂²	C₂₃×#575	1	1	yes	characteristic
690	C₂²	C₂₃×#575	1	1	yes	characteristic
691	C₂²	C₂₃×#575	1	1	yes	characteristic
692	C₄	—	2	2	no	—
693	C₄	—	2	2	no	—
694	C₄	—	4	2	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	2	no	—
702	C₄	—	8	1	no	—
703	C₄	—	8	1	no	—
704	C₄	C₂₃×#547	1	1	yes	characteristic
705	C₄	C₂₃×#547	1	1	yes	characteristic
706	C₄	C₂₃×#547	1	1	yes	characteristic
707	C₄	C₂₃×#547	1	1	yes	characteristic
708	C₂	—	4	2	no	—
709	C₂	—	8	2	no	—
710	C₂	C₂₃×#2406	1	1	yes	characteristic
711	C₂	C₂₃×#3002	1	1	yes	characteristic
712	C₂	C₂₃×#3005	1	1	yes	characteristic
713	C₂	C₂₃×#3102	1	1	yes	characteristic
714	C₂	C₂₃×#3111	1	1	yes	characteristic
715	C₂	C₂₃×#3233	1	1	yes	characteristic
716	C₂	C₂₃×#3240	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
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
2. C₂×C₂₃×#485	C₂	semidirect
4. C₂₃×#2296	C₂	semidirect
5. C₂₃×#2059	C₂	semidirect
6. C₂₃×#2059	C₂	semidirect
7. C₂₃×#2042	C₂	semidirect
8. C₂₃×#2042	C₂	semidirect
10. C₂₃×#1873	C₂	semidirect
11. C₂₃×#1865	C₂	semidirect
12. C₂₃×#1864	C₂	semidirect
13. C₂₃×#1863	C₂	semidirect
14. C₂₃×#1862	C₂	semidirect
15. C₂₃×#1817	C₂	semidirect
235. #24068	C₂₃	direct (N × H)
314. C₂×#485	C₂×C₂₃	semidirect
316. #2296	C₂×C₂₃	semidirect
317. #2059	C₂×C₂₃	semidirect
318. #2059	C₂×C₂₃	semidirect
319. #2042	C₂×C₂₃	semidirect
320. #2042	C₂×C₂₃	semidirect
322. #1873	C₂×C₂₃	semidirect
323. #1865	C₂×C₂₃	semidirect
324. #1864	C₂×C₂₃	semidirect
325. #1863	C₂×C₂₃	semidirect
326. #1862	C₂×C₂₃	semidirect
327. #1817	C₂×C₂₃	semidirect
500. C₂₃	#24068	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#529	2
C₂×C₂₃×#485	2
C₂×C₂₃×#482	2
C₂₃×#2296	2
C₂₃×#2059	2
C₂₃×#2042	2
C₂₃×#1875	2
C₂₃×#1873	2
C₂₃×#1865	2
C₂₃×#1864	2
C₂₃×#1863	2
C₂₃×#1862	2
C₂₃×#1817	2
#24068	23

Subgroups — C23×#24068 #19171694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24068 #19171694