Subgroups — C₂₃×#21703 #19169329

Order 5888.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#497	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#440	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2326	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1973	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1969	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1968	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1958	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#497	C₄	1	8	yes	complemented
27	C₂₃×#441	—	2	4	no	—
28	C₂₃×#440	—	2	4	no	—
29	C₄×Q₁₆×C₂₃	—	2	4	no	—
30	C₄×SD₁₆×C₂₃	—	2	4	no	—
31	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#103	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×#97	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#391	C₂²	1	1	yes	characteristic
50	C₂₃×#391	C₂²	1	1	yes	characteristic
51	C₂₃×#391	C₂²	1	1	yes	characteristic
52	C₂₃×#390	C₂²	1	1	yes	characteristic
53	C₂₃×#388	—	2	2	no	—
54	C₂₃×#387	—	2	2	no	—
55	C₂₃×#387	—	2	2	no	—
56	C₂₃×#385	—	2	2	no	—
57	C₂₃×#384	C₂²	1	1	yes	characteristic
58	C₂₃×#382	C₂²	1	1	yes	characteristic
59	C₂₃×#376	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂×Q₁₆×C₂₃	—	2	2	no	—
66	C₂×Q₁₆×C₂₃	—	2	2	no	—
67	C₂×Q₁₆×C₂₃	—	2	2	no	—
68	C₂×Q₁₆×C₂₃	—	2	2	no	—
69	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
70	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
71	C₂×SD₁₆×C₂₃	—	2	2	no	—
72	C₂×SD₁₆×C₂₃	—	2	2	no	—
73	C₂×SD₁₆×C₂₃	—	2	2	no	—
74	C₂×SD₁₆×C₂₃	—	2	2	no	—
75	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
76	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	—	2	2	no	—
82	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
83	C₂₃×#124	—	2	2	no	—
84	C₂₃×#124	C₂×C₄	1	4	yes	—
85	C₄×Q₈×C₂₃	—	2	2	no	—
86	C₄×Q₈×C₂₃	—	2	2	no	—
87	C₄×Q₈×C₂₃	—	2	2	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×D₄×C₂₃	—	2	2	no	—
92	C₄×D₄×C₂₃	—	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	2	no	—
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
105	C₂×C₄²×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂₃×#107	—	2	2	no	—
112	C₂₃×#107	—	2	2	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×#106	—	2	2	no	—
115	C₂₃×#103	—	2	2	no	—
116	C₂₃×#103	—	2	2	no	—
117	C₂₃×#103	—	2	2	no	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#97	—	2	2	no	—
124	C₂₃×#97	C₂×C₄	1	4	yes	complemented
125	C₂₃×#97	D₄	1	4	yes	—
126	C₄×C₈×C₂₃	—	2	2	no	—
127	C₄×C₈×C₂₃	—	2	2	no	—
128	C₄×C₈×C₂₃	D₄	1	4	yes	—
129	C₂₃×#95	—	2	2	no	—
130	C₂₃×#95	—	2	2	no	—
131	C₂₃×#95	—	2	2	no	—
132	C₂₃×#95	—	2	2	no	—
133	C₂₃×#95	D₄	1	2	yes	—
134	C₂₃×#95	D₄	1	2	yes	—
135	C₂⁴×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
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	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
152	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
153	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
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	—	4	2	no	—
168	C₂²×C₄×C₂₃	—	8	2	no	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₂₃	—	2	8	no	—
175	Q₁₆×C₂₃	—	2	8	no	—
176	SD₁₆×C₂₃	—	2	8	no	—
177	SD₁₆×C₂₃	—	2	8	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
183	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
184	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
186	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
187	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
188	C₂×C₈×C₂₃	#41	1	2	yes	—
189	C₂×C₈×C₂₃	#41	1	2	yes	—
190	C₄⋊C₄×C₂₃	—	2	8	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	4	2	no	—
201	C₄⋊C₄×C₂₃	—	4	2	no	—
202	C₄⋊C₄×C₂₃	—	8	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	8	no	—
204	C₂²⋊C₄×C₂₃	—	4	2	no	—
205	C₂²⋊C₄×C₂₃	—	8	2	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	4	2	no	—
209	C₄²×C₂₃	—	4	2	no	—
210	C₄²×C₂₃	—	4	2	no	—
211	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
212	C₄²×C₂₃	C₂×D₄	1	2	yes	—
213	C₄²×C₂₃	C₂×D₄	1	2	yes	—
214	C₄²×C₂₃	#41	1	2	yes	—
215	#21703	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₂₃	—	2	2	no	—
217	C₂³×C₂₃	—	4	1	no	—
218	C₂³×C₂₃	—	4	1	no	—
219	C₂³×C₂₃	—	4	2	no	—
220	C₂³×C₂₃	—	4	2	no	—
221	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	Q₈×C₂₃	—	2	2	no	—
223	Q₈×C₂₃	—	2	2	no	—
224	Q₈×C₂₃	—	2	2	no	—
225	Q₈×C₂₃	—	2	2	no	—
226	Q₈×C₂₃	—	2	2	no	—
227	Q₈×C₂₃	—	2	2	no	—
228	Q₈×C₂₃	—	4	1	no	—
229	Q₈×C₂₃	—	4	1	no	—
230	Q₈×C₂₃	—	4	1	no	—
231	Q₈×C₂₃	—	4	1	no	—
232	Q₈×C₂₃	—	4	1	no	—
233	Q₈×C₂₃	—	4	1	no	—
234	D₄×C₂₃	—	2	2	no	—
235	D₄×C₂₃	—	2	2	no	—
236	D₄×C₂₃	—	4	1	no	—
237	D₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	2	2	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	2	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	2	no	—
262	C₂×C₄×C₂₃	—	8	4	no	—
263	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
264	C₂×C₄×C₂₃	#127	1	2	yes	—
265	C₂×C₄×C₂₃	#127	1	2	yes	—
266	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
267	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
272	C₈×C₂₃	—	2	2	no	—
273	C₈×C₂₃	—	2	2	no	—
274	C₈×C₂₃	C₄×D₄	1	4	yes	—
275	C₈×C₂₃	C₄×D₄	1	4	yes	—
276	C₂×#497	C₂×C₂₃	1	1	yes	characteristic
277	C₂×#441	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂×#440	C₂×C₂₃	1	1	yes	characteristic
279	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
281	#2327	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#2326	C₂×C₂₃	1	1	yes	characteristic
283	#1973	C₂×C₂₃	1	1	yes	characteristic
284	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#1969	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1968	C₂×C₂₃	1	1	yes	characteristic
287	#1958	C₂×C₂₃	1	1	yes	characteristic
288	#1956	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#1847	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1726	C₂×C₂₃	1	1	yes	characteristic, complemented
291	C₂²×C₂₃	—	4	2	no	—
292	C₂²×C₂₃	—	4	4	no	—
293	C₂²×C₂₃	—	8	1	no	—
294	C₂²×C₂₃	—	8	1	no	—
295	C₂²×C₂₃	—	8	2	no	—
296	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
297	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
298	C₂²×C₂₃	#532	1	1	yes	characteristic
299	C₂²×C₂₃	#577	1	2	yes	—
300	C₂²×C₂₃	#578	1	2	yes	—
301	C₄×C₂₃	—	2	2	no	—
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	2	4	no	—
304	C₄×C₂₃	—	4	2	no	—
305	C₄×C₂₃	—	4	2	no	—
306	C₄×C₂₃	—	4	2	no	—
307	C₄×C₂₃	—	4	2	no	—
308	C₄×C₂₃	—	8	1	no	—
309	C₄×C₂₃	—	8	1	no	—
310	C₄×C₂₃	—	8	1	no	—
311	C₄×C₂₃	—	8	1	no	—
312	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
313	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
314	C₄×C₂₃	#532	1	1	yes	characteristic
315	C₄×C₂₃	#532	1	1	yes	characteristic
316	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
317	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
318	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
319	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
326	#497	C₄×C₂₃	1	8	yes	complemented
327	#441	—	2	4	no	—
328	#440	—	2	4	no	—
329	C₄×Q₁₆	—	2	4	no	—
330	C₄×SD₁₆	—	2	4	no	—
331	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
349	#391	C₂²×C₂₃	1	1	yes	characteristic
350	#391	C₂²×C₂₃	1	1	yes	characteristic
351	#391	C₂²×C₂₃	1	1	yes	characteristic
352	#390	C₂²×C₂₃	1	1	yes	characteristic
353	#388	—	2	2	no	—
354	#387	—	2	2	no	—
355	#387	—	2	2	no	—
356	#385	—	2	2	no	—
357	#384	C₂²×C₂₃	1	1	yes	characteristic
358	#382	C₂²×C₂₃	1	1	yes	characteristic
359	#376	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×C₂₃	—	8	2	no	—
361	C₂×C₂₃	C₄×#127	1	1	yes	characteristic
362	C₂×C₂₃	#2947	1	2	yes	—
363	C₂×C₂₃	#3154	1	1	yes	characteristic
364	C₂×C₂₃	#3156	1	1	yes	characteristic
365	C₂×C₂₃	#3178	1	2	yes	—
366	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
368	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
370	C₂³×C₄	—	2	1	no	—
371	C₂×Q₁₆	—	2	2	no	—
372	C₂×Q₁₆	—	2	2	no	—
373	C₂×Q₁₆	—	2	2	no	—
374	C₂×Q₁₆	—	2	2	no	—
375	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
376	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
377	C₂×SD₁₆	—	2	2	no	—
378	C₂×SD₁₆	—	2	2	no	—
379	C₂×SD₁₆	—	2	2	no	—
380	C₂×SD₁₆	—	2	2	no	—
381	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
382	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
383	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
385	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₄⋊Q₈	—	2	2	no	—
388	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
389	#124	—	2	2	no	—
390	#124	C₂×C₄×C₂₃	1	4	yes	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	2	no	—
393	C₄×Q₈	—	2	2	no	—
394	C₄×Q₈	—	2	2	no	—
395	C₄×Q₈	—	2	2	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₂×C₄⋊C₄	—	2	2	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	—	2	1	no	—
413	C₂×C₄²	—	2	1	no	—
414	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
417	#107	—	2	2	no	—
418	#107	—	2	2	no	—
419	#106	—	2	2	no	—
420	#106	—	2	2	no	—
421	#103	—	2	2	no	—
422	#103	—	2	2	no	—
423	#103	—	2	2	no	—
424	#103	—	2	2	no	—
425	#103	—	2	2	no	—
426	#103	—	2	2	no	—
427	#102	—	2	2	no	—
428	#102	—	2	2	no	—
429	#97	—	2	2	no	—
430	#97	C₂×C₄×C₂₃	1	4	yes	complemented
431	#97	D₄×C₂₃	1	4	yes	—
432	C₄×C₈	—	2	2	no	—
433	C₄×C₈	—	2	2	no	—
434	C₄×C₈	D₄×C₂₃	1	4	yes	—
435	#95	—	2	2	no	—
436	#95	—	2	2	no	—
437	#95	—	2	2	no	—
438	#95	—	2	2	no	—
439	#95	D₄×C₂₃	1	2	yes	—
440	#95	D₄×C₂₃	1	2	yes	—
441	C₂₃	#21703	1	1	yes	characteristic, direct product factor, Sylow
442	C₂⁴	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
459	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
460	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	8	2	no	—
476	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
477	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
481	Q₁₆	—	2	8	no	—
482	Q₁₆	—	2	8	no	—
483	SD₁₆	—	2	8	no	—
484	SD₁₆	—	2	8	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
490	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
491	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
492	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
493	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
494	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
495	C₂×C₈	#41×C₂₃	1	2	yes	—
496	C₂×C₈	#41×C₂₃	1	2	yes	—
497	C₄⋊C₄	—	2	8	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	4	2	no	—
507	C₄⋊C₄	—	4	2	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	8	2	no	—
510	C₂²⋊C₄	—	2	8	no	—
511	C₂²⋊C₄	—	4	2	no	—
512	C₂²⋊C₄	—	8	2	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	4	2	no	—
516	C₄²	—	4	2	no	—
517	C₄²	—	4	2	no	—
518	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
519	C₄²	C₂×D₄×C₂₃	1	2	yes	—
520	C₄²	C₂×D₄×C₂₃	1	2	yes	—
521	C₄²	#41×C₂₃	1	2	yes	—
522	C₂³	—	2	2	no	—
523	C₂³	—	4	2	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
528	Q₈	—	2	2	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	2	2	no	—
531	Q₈	—	2	2	no	—
532	Q₈	—	2	2	no	—
533	Q₈	—	2	2	no	—
534	Q₈	—	4	1	no	—
535	Q₈	—	4	1	no	—
536	Q₈	—	4	1	no	—
537	Q₈	—	4	1	no	—
538	Q₈	—	4	1	no	—
539	Q₈	—	4	1	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	2	no	—
542	D₄	—	4	1	no	—
543	D₄	—	4	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	8	4	no	—
569	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
570	C₂×C₄	C₂₃×#127	1	2	yes	—
571	C₂×C₄	C₂₃×#127	1	2	yes	—
572	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
573	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
574	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
575	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
576	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
577	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
578	C₈	—	2	2	no	—
579	C₈	—	2	2	no	—
580	C₈	C₄×D₄×C₂₃	1	4	yes	—
581	C₈	C₄×D₄×C₂₃	1	4	yes	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	8	2	no	—
585	C₂²	—	8	1	no	—
586	C₂²	—	8	1	no	—
587	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
588	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
589	C₂²	C₂₃×#532	1	1	yes	characteristic
590	C₂²	C₂₃×#577	1	2	yes	—
591	C₂²	C₂₃×#578	1	2	yes	—
592	C₄	—	2	4	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	8	1	no	—
600	C₄	—	8	1	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
604	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
605	C₄	C₂₃×#532	1	1	yes	characteristic
606	C₄	C₂₃×#532	1	1	yes	characteristic
607	C₂	—	8	2	no	—
608	C₂	C₄×C₂₃×#127	1	1	yes	characteristic
609	C₂	C₂₃×#2947	1	2	yes	—
610	C₂	C₂₃×#3154	1	1	yes	characteristic
611	C₂	C₂₃×#3156	1	1	yes	characteristic
612	C₂	C₂₃×#3178	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#441	C₂	semidirect
4. C₂×C₄×Q₁₆×C₂₃	C₂	semidirect
6. C₂₃×#2327	C₂	semidirect
9. C₂₃×#1971	C₂	semidirect
10. C₂₃×#1969	C₂	semidirect
13. C₂₃×#1956	C₂	semidirect
14. C₂₃×#1847	C₂	semidirect
15. C₂₃×#1726	C₂	semidirect
26. C₂₃×#497	C₄	semidirect
69. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
70. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
82. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
124. C₂₃×#97	C₂×C₄	semidirect
215. #21703	C₂₃	direct (N × H)
277. C₂×#441	C₂×C₂₃	semidirect
279. C₂×C₄×Q₁₆	C₂×C₂₃	semidirect
281. #2327	C₂×C₂₃	semidirect
284. #1971	C₂×C₂₃	semidirect
285. #1969	C₂×C₂₃	semidirect
288. #1956	C₂×C₂₃	semidirect
289. #1847	C₂×C₂₃	semidirect
290. #1726	C₂×C₂₃	semidirect
326. #497	C₄×C₂₃	semidirect
375. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
376. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
388. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
430. #97	C₂×C₄×C₂₃	semidirect
441. C₂₃	#21703	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×#497	2
C₂×C₂₃×#441	2
C₂×C₂₃×#440	2
C₂×C₄×Q₁₆×C₂₃	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#2327	2
C₂₃×#2326	2
C₂₃×#1973	2
C₂₃×#1971	2
C₂₃×#1969	2
C₂₃×#1968	2
C₂₃×#1958	2
C₂₃×#1956	2
C₂₃×#1847	2
C₂₃×#1726	2
#21703	23

Subgroups — C23×#21703 #19169329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21703 #19169329