Subgroups — C₂₃×#24106 #19171732

Order 5888.

1046 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#552	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2485	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2036	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2035	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1894	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1870	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1824	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#552	—	2	4	no	—
18	C₂₃×#552	—	2	4	no	—
19	C₂×C₂₃×#126	—	2	1	no	—
20	C₂×C₂₃×#125	—	2	1	no	—
21	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#123	—	2	1	no	—
23	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
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×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#485	—	2	4	no	—
32	C₂₃×#428	—	2	4	no	—
33	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#106	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×#103	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₂₃×#397	—	2	1	no	—
47	C₂₃×#394	—	2	1	no	—
48	C₂₃×#389	C₂²	1	1	yes	characteristic
49	C₂₃×#389	C₂²	1	1	yes	characteristic
50	C₂₃×#388	—	2	1	no	—
51	C₂₃×#387	—	2	1	no	—
52	C₂₃×#380	C₂²	1	1	yes	characteristic
53	C₂₃×#380	C₂²	1	1	yes	characteristic
54	C₂₃×#380	C₂²	1	1	yes	characteristic
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×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#340	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	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₂₃	—	2	2	no	—
69	C₂×M₁₆×C₂₃	—	2	2	no	—
70	C₂×M₁₆×C₂₃	—	2	2	no	—
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	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×#126	—	2	8	no	—
76	C₂₃×#125	—	2	8	no	—
77	C₂₃×#125	—	2	2	no	—
78	C₂₃×#125	—	2	2	no	—
79	C₂₃×#123	—	2	8	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	D₄	1	4	yes	—
84	C₄×Q₈×C₂₃	—	2	2	no	—
85	C₄×Q₈×C₂₃	—	2	2	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₂₃×#117	—	2	8	no	—
89	C₂₃×#117	—	2	2	no	—
90	C₂₃×#117	—	2	2	no	—
91	C₂₃×#117	—	2	2	no	—
92	C₂₃×#117	—	2	2	no	—
93	C₂×C₄⋊C₄×C₂₃	—	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	D₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
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×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	—	2	1	no	—
121	C₂×C₄²×C₂₃	—	2	1	no	—
122	C₂×C₄²×C₂₃	—	2	1	no	—
123	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
124	C₂₃×#107	—	2	2	no	—
125	C₂₃×#107	—	2	2	no	—
126	C₂₃×#107	—	2	2	no	—
127	C₂₃×#106	—	2	2	no	—
128	C₂₃×#106	—	2	2	no	—
129	C₂₃×#106	—	2	2	no	—
130	C₂₃×#103	—	2	2	no	—
131	C₂₃×#103	—	2	2	no	—
132	C₂₃×#103	—	2	2	no	—
133	C₂₃×#103	—	2	2	no	—
134	C₂₃×C₂²⋊C₈	—	2	2	no	—
135	C₂₃×C₂²⋊C₈	—	2	2	no	—
136	C₂₃×C₂²⋊C₈	—	2	2	no	—
137	C₂₃×#95	—	2	1	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	C₂×D₄	1	1	yes	characteristic
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	2	no	—
151	C₂×D₄×C₂₃	—	4	1	no	—
152	C₂×D₄×C₂₃	—	4	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
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	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	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	1	no	—
172	C₂²×C₄×C₂₃	—	4	2	no	—
173	C₂²×C₄×C₂₃	—	4	2	no	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	M₁₆×C₂₃	—	2	8	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
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	#41	1	2	yes	—
193	C₂×C₈×C₂₃	#41	1	2	yes	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	4	no	—
196	C₄⋊C₄×C₂₃	—	2	4	no	—
197	C₄⋊C₄×C₂₃	—	2	4	no	—
198	C₄⋊C₄×C₂₃	—	2	4	no	—
199	C₄⋊C₄×C₂₃	—	2	4	no	—
200	C₄⋊C₄×C₂₃	—	2	2	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	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	—	2	1	no	—
212	C₄⋊C₄×C₂₃	—	4	2	no	—
213	C₄⋊C₄×C₂₃	—	8	1	no	—
214	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
217	C₂²⋊C₄×C₂₃	—	2	4	no	—
218	C₂²⋊C₄×C₂₃	—	2	4	no	—
219	C₂²⋊C₄×C₂₃	—	2	4	no	—
220	C₂²⋊C₄×C₂₃	—	2	2	no	—
221	C₂²⋊C₄×C₂₃	—	2	2	no	—
222	C₂²⋊C₄×C₂₃	—	4	2	no	—
223	C₂²⋊C₄×C₂₃	—	4	2	no	—
224	C₂²⋊C₄×C₂₃	—	8	1	no	—
225	C₂²⋊C₄×C₂₃	—	8	1	no	—
226	C₄²×C₂₃	—	2	4	no	—
227	C₄²×C₂₃	—	2	4	no	—
228	C₄²×C₂₃	—	2	1	no	—
229	C₄²×C₂₃	—	2	1	no	—
230	C₄²×C₂₃	—	4	2	no	—
231	#24106	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	—	2	1	no	—
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	1	no	—
243	C₂³×C₂₃	—	2	1	no	—
244	C₂³×C₂₃	—	2	1	no	—
245	C₂³×C₂₃	#120	1	2	yes	—
246	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₂₃	—	2	2	no	—
248	Q₈×C₂₃	—	2	2	no	—
249	Q₈×C₂₃	—	4	1	no	—
250	Q₈×C₂₃	—	4	1	no	—
251	D₄×C₂₃	—	4	4	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	2	no	—
261	C₂×C₄×C₂₃	—	2	2	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	1	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	—	4	8	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	4	1	no	—
281	C₂×C₄×C₂₃	—	4	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	4	1	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	#120	1	2	yes	—
297	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
300	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
301	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
302	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
303	C₈×C₂₃	—	2	2	no	—
304	C₈×C₂₃	—	2	2	no	—
305	C₈×C₂₃	—	4	1	no	—
306	C₈×C₂₃	—	4	1	no	—
307	C₂×#552	C₂×C₂₃	1	1	yes	characteristic
308	C₂×#485	C₂×C₂₃	1	1	yes	characteristic, complemented
309	C₂×#428	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#2485	C₂×C₂₃	1	1	yes	characteristic
311	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#2036	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#2035	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#1894	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#1870	C₂×C₂₃	1	1	yes	characteristic
317	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
318	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1854	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1854	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1824	C₂×C₂₃	1	1	yes	characteristic, complemented
322	C₂²×C₂₃	—	2	2	no	—
323	C₂²×C₂₃	—	2	2	no	—
324	C₂²×C₂₃	—	2	2	no	—
325	C₂²×C₂₃	—	4	1	no	—
326	C₂²×C₂₃	—	4	1	no	—
327	C₂²×C₂₃	—	4	1	no	—
328	C₂²×C₂₃	—	4	1	no	—
329	C₂²×C₂₃	—	4	1	no	—
330	C₂²×C₂₃	—	4	1	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	1	no	—
336	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
337	C₂²×C₂₃	#547	1	1	yes	characteristic
338	C₂²×C₂₃	#547	1	1	yes	characteristic
339	C₂²×C₂₃	#575	1	1	yes	characteristic
340	C₂²×C₂₃	#575	1	1	yes	characteristic
341	C₂²×C₂₃	#577	1	1	yes	characteristic
342	C₂²×C₂₃	#578	1	1	yes	characteristic
343	C₄×C₂₃	—	2	2	no	—
344	C₄×C₂₃	—	4	1	no	—
345	C₄×C₂₃	—	4	2	no	—
346	C₄×C₂₃	—	4	1	no	—
347	C₄×C₂₃	—	4	2	no	—
348	C₄×C₂₃	—	4	2	no	—
349	C₄×C₂₃	—	4	1	no	—
350	C₄×C₂₃	—	4	2	no	—
351	C₄×C₂₃	—	8	1	no	—
352	C₄×C₂₃	—	8	1	no	—
353	C₄×C₂₃	—	8	1	no	—
354	C₄×C₂₃	—	8	1	no	—
355	C₄×C₂₃	#525	1	1	yes	characteristic
356	C₄×C₂₃	#536	1	1	yes	characteristic
357	C₄×C₂₃	#547	1	1	yes	characteristic
358	C₄×C₂₃	#547	1	1	yes	characteristic
359	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
360	#552	—	2	4	no	—
361	#552	—	2	4	no	—
362	C₂×#126	—	2	1	no	—
363	C₂×#125	—	2	1	no	—
364	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#123	—	2	1	no	—
366	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#117	—	2	1	no	—
371	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
373	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
374	#485	—	2	4	no	—
375	#428	—	2	4	no	—
376	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
389	#397	—	2	1	no	—
390	#394	—	2	1	no	—
391	#389	C₂²×C₂₃	1	1	yes	characteristic
392	#389	C₂²×C₂₃	1	1	yes	characteristic
393	#388	—	2	1	no	—
394	#387	—	2	1	no	—
395	#380	C₂²×C₂₃	1	1	yes	characteristic
396	#380	C₂²×C₂₃	1	1	yes	characteristic
397	#380	C₂²×C₂₃	1	1	yes	characteristic
398	#340	C₂²×C₂₃	1	1	yes	characteristic
399	#340	C₂²×C₂₃	1	1	yes	characteristic
400	#340	C₂²×C₂₃	1	1	yes	characteristic
401	#340	C₂²×C₂₃	1	1	yes	characteristic
402	#340	C₂²×C₂₃	1	1	yes	characteristic
403	#340	C₂²×C₂₃	1	1	yes	characteristic
404	#336	C₂²×C₂₃	1	1	yes	characteristic
405	#336	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×C₂₃	—	4	2	no	—
407	C₂×C₂₃	—	8	1	no	—
408	C₂×C₂₃	#2406	1	1	yes	characteristic
409	C₂×C₂₃	#3005	1	1	yes	characteristic
410	C₂×C₂₃	#3013	1	1	yes	characteristic
411	C₂×C₂₃	#3348	1	1	yes	characteristic
412	C₂×C₂₃	#3349	1	1	yes	characteristic
413	C₂×C₂₃	#3351	1	1	yes	characteristic
414	C₂×C₂₃	#3352	1	1	yes	characteristic
415	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
416	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
417	C₂³×C₄	—	2	1	no	—
418	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
419	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×M₁₆	—	2	2	no	—
421	C₂×M₁₆	—	2	2	no	—
422	C₂×M₁₆	—	2	2	no	—
423	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
424	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
426	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
427	#126	—	2	8	no	—
428	#125	—	2	8	no	—
429	#125	—	2	2	no	—
430	#125	—	2	2	no	—
431	#123	—	2	8	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	—	2	2	no	—
435	C₂²⋊Q₈	D₄×C₂₃	1	4	yes	—
436	C₄×Q₈	—	2	2	no	—
437	C₄×Q₈	—	2	2	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	#117	—	2	8	no	—
441	#117	—	2	2	no	—
442	#117	—	2	2	no	—
443	#117	—	2	2	no	—
444	#117	—	2	2	no	—
445	C₂×C₄⋊C₄	—	2	1	no	—
446	C₂×C₄⋊C₄	—	2	2	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	—	2	1	no	—
451	C₂×C₄⋊C₄	—	2	1	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
455	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
456	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
458	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
459	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
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×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₂²⋊C₄	—	2	1	no	—
465	C₂×C₂²⋊C₄	—	2	1	no	—
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	D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
472	C₂×C₄²	—	2	1	no	—
473	C₂×C₄²	—	2	1	no	—
474	C₂×C₄²	—	2	1	no	—
475	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
476	#107	—	2	2	no	—
477	#107	—	2	2	no	—
478	#107	—	2	2	no	—
479	#106	—	2	2	no	—
480	#106	—	2	2	no	—
481	#106	—	2	2	no	—
482	#103	—	2	2	no	—
483	#103	—	2	2	no	—
484	#103	—	2	2	no	—
485	#103	—	2	2	no	—
486	C₂²⋊C₈	—	2	2	no	—
487	C₂²⋊C₈	—	2	2	no	—
488	C₂²⋊C₈	—	2	2	no	—
489	#95	—	2	1	no	—
490	#95	—	2	1	no	—
491	#95	—	2	1	no	—
492	#95	—	2	1	no	—
493	#95	—	2	1	no	—
494	C₂₃	#24106	1	1	yes	characteristic, direct product factor, Sylow
495	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	1	no	—
502	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	4	1	no	—
505	C₂×D₄	—	4	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
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	1	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	—	4	2	no	—
527	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
529	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
532	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
534	M₁₆	—	2	8	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	#41×C₂₃	1	2	yes	—
546	C₂×C₈	#41×C₂₃	1	2	yes	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	4	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	4	no	—
552	C₄⋊C₄	—	2	4	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	4	2	no	—
566	C₄⋊C₄	—	8	1	no	—
567	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
568	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
569	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	2	4	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	4	2	no	—
576	C₂²⋊C₄	—	4	2	no	—
577	C₂²⋊C₄	—	8	1	no	—
578	C₂²⋊C₄	—	8	1	no	—
579	C₄²	—	2	4	no	—
580	C₄²	—	2	4	no	—
581	C₄²	—	2	1	no	—
582	C₄²	—	2	1	no	—
583	C₄²	—	4	2	no	—
584	C₂³	—	2	1	no	—
585	C₂³	—	2	1	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	2	1	no	—
588	C₂³	—	2	1	no	—
589	C₂³	—	2	1	no	—
590	C₂³	—	2	1	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	2	1	no	—
595	C₂³	—	2	1	no	—
596	C₂³	—	2	1	no	—
597	C₂³	C₂₃×#120	1	2	yes	—
598	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
599	Q₈	—	2	2	no	—
600	Q₈	—	2	2	no	—
601	Q₈	—	4	1	no	—
602	Q₈	—	4	1	no	—
603	D₄	—	4	4	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	4	8	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	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	C₂₃×#120	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
650	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
651	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
652	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
653	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
654	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
655	C₈	—	2	2	no	—
656	C₈	—	2	2	no	—
657	C₈	—	4	1	no	—
658	C₈	—	4	1	no	—
659	C₂²	—	2	2	no	—
660	C₂²	—	2	2	no	—
661	C₂²	—	2	2	no	—
662	C₂²	—	4	1	no	—
663	C₂²	—	4	1	no	—
664	C₂²	—	4	1	no	—
665	C₂²	—	4	1	no	—
666	C₂²	—	4	1	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	—	4	1	no	—
670	C₂²	—	4	1	no	—
671	C₂²	—	4	1	no	—
672	C₂²	—	4	1	no	—
673	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
674	C₂²	C₂₃×#547	1	1	yes	characteristic
675	C₂²	C₂₃×#547	1	1	yes	characteristic
676	C₂²	C₂₃×#575	1	1	yes	characteristic
677	C₂²	C₂₃×#575	1	1	yes	characteristic
678	C₂²	C₂₃×#577	1	1	yes	characteristic
679	C₂²	C₂₃×#578	1	1	yes	characteristic
680	C₄	—	2	2	no	—
681	C₄	—	4	1	no	—
682	C₄	—	4	2	no	—
683	C₄	—	4	1	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	2	no	—
686	C₄	—	4	1	no	—
687	C₄	—	4	2	no	—
688	C₄	—	8	1	no	—
689	C₄	—	8	1	no	—
690	C₄	—	8	1	no	—
691	C₄	—	8	1	no	—
692	C₄	C₂₃×#525	1	1	yes	characteristic
693	C₄	C₂₃×#536	1	1	yes	characteristic
694	C₄	C₂₃×#547	1	1	yes	characteristic
695	C₄	C₂₃×#547	1	1	yes	characteristic
696	C₂	—	4	2	no	—
697	C₂	—	8	1	no	—
698	C₂	C₂₃×#2406	1	1	yes	characteristic
699	C₂	C₂₃×#3005	1	1	yes	characteristic
700	C₂	C₂₃×#3013	1	1	yes	characteristic
701	C₂	C₂₃×#3348	1	1	yes	characteristic
702	C₂	C₂₃×#3349	1	1	yes	characteristic
703	C₂	C₂₃×#3351	1	1	yes	characteristic
704	C₂	C₂₃×#3352	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#485	C₂	semidirect
3. C₂×C₂₃×#428	C₂	semidirect
5. C₂₃×#2059	C₂	semidirect
6. C₂₃×#2059	C₂	semidirect
7. C₂₃×#2036	C₂	semidirect
8. C₂₃×#2035	C₂	semidirect
9. C₂₃×#1894	C₂	semidirect
11. C₂₃×#1865	C₂	semidirect
12. C₂₃×#1863	C₂	semidirect
13. C₂₃×#1854	C₂	semidirect
14. C₂₃×#1854	C₂	semidirect
15. C₂₃×#1824	C₂	semidirect
231. #24106	C₂₃	direct (N × H)
308. C₂×#485	C₂×C₂₃	semidirect
309. C₂×#428	C₂×C₂₃	semidirect
311. #2059	C₂×C₂₃	semidirect
312. #2059	C₂×C₂₃	semidirect
313. #2036	C₂×C₂₃	semidirect
314. #2035	C₂×C₂₃	semidirect
315. #1894	C₂×C₂₃	semidirect
317. #1865	C₂×C₂₃	semidirect
318. #1863	C₂×C₂₃	semidirect
319. #1854	C₂×C₂₃	semidirect
320. #1854	C₂×C₂₃	semidirect
321. #1824	C₂×C₂₃	semidirect
494. C₂₃	#24106	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×#552	2
C₂×C₂₃×#485	2
C₂×C₂₃×#428	2
C₂₃×#2485	2
C₂₃×#2059	2
C₂₃×#2036	2
C₂₃×#2035	2
C₂₃×#1894	2
C₂₃×#1870	2
C₂₃×#1865	2
C₂₃×#1863	2
C₂₃×#1854	2
C₂₃×#1824	2
#24106	23

Subgroups — C23×#24106 #19171732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24106 #19171732