Subgroups — C₂₃×#10945 #19158571

Order 5888.

994 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#104	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2896	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#410	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1733	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#104	—	2	2	no	—
16	C₂×C₂₃×#104	—	2	1	no	—
17	C₂×C₂₃×#104	—	2	1	no	—
18	C₂×C₂₃×#104	C₄	1	4	yes	complemented
19	C₂₃×#419	C₄	1	2	yes	complemented
20	C₂₃×#418	C₄	1	2	yes	complemented
21	C₂₃×#417	—	2	2	no	—
22	C₂₃×#416	—	2	2	no	—
23	C₂₃×#410	—	2	2	no	—
24	C₂₃×#410	C₄	1	4	yes	complemented
25	C₂×C₂₃×#99	—	2	2	no	—
26	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
28	C₂₃×#376	—	2	1	no	—
29	C₄³×C₂₃	—	2	1	no	—
30	C₂₃×#344	—	2	2	no	—
31	C₂₃×#339	—	2	2	no	—
32	C₂₃×#339	—	2	2	no	—
33	C₂×#41×C₂₃	—	2	1	no	—
34	C₂×#41×C₂₃	—	2	1	no	—
35	C₂×#41×C₂₃	—	2	1	no	—
36	C₂×#41×C₂₃	—	2	1	no	—
37	C₂×#41×C₂₃	—	2	1	no	—
38	C₂×#41×C₂₃	—	2	1	no	—
39	C₂×#41×C₂₃	—	2	1	no	—
40	C₂×#41×C₂₃	—	2	1	no	—
41	C₂×#41×C₂₃	—	2	1	no	—
42	C₂×#41×C₂₃	—	2	1	no	—
43	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
45	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₄×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	—	2	1	no	—
49	C₂³×C₄×C₂₃	—	2	1	no	—
50	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂×M₁₆×C₂₃	—	2	1	no	—
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	—	2	1	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	—	2	1	no	—
60	C₂×M₁₆×C₂₃	—	2	1	no	—
61	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
62	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₂₃	—	2	1	no	—
64	C₂²×C₈×C₂₃	—	2	1	no	—
65	C₂₃×#117	—	2	1	no	—
66	C₂₃×#117	—	2	1	no	—
67	C₂₃×#117	—	2	1	no	—
68	C₂₃×#117	—	2	1	no	—
69	C₂₃×#117	—	2	1	no	—
70	C₂₃×#117	—	2	1	no	—
71	C₂₃×#117	—	2	1	no	—
72	C₂₃×#117	C₂×C₄	1	2	yes	—
73	C₂₃×#117	C₂×C₄	1	2	yes	—
74	C₂₃×#117	C₂×C₄	1	2	yes	—
75	C₂₃×#117	D₄	1	2	yes	—
76	C₂₃×#117	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
83	C₂×C₄²×C₂₃	—	2	1	no	—
84	C₂×C₄²×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	—	2	1	no	—
86	C₂×C₄²×C₂₃	—	2	1	no	—
87	C₂×C₄²×C₂₃	—	2	1	no	—
88	C₂×C₄²×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	4	no	—
90	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
91	C₂×C₄²×C₂₃	D₄	1	2	yes	—
92	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
94	C₂₃×#104	—	2	4	no	—
95	C₂₃×#104	—	2	4	no	—
96	C₂₃×#103	—	2	4	no	—
97	C₂₃×#102	—	2	4	no	—
98	C₂₃×#99	—	2	8	no	—
99	C₂₃×#95	—	2	4	no	—
100	C₂⁴×C₂₃	—	2	1	no	—
101	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
102	#41×C₂₃	—	2	2	no	—
103	#41×C₂₃	—	2	2	no	—
104	#41×C₂₃	—	2	2	no	—
105	#41×C₂₃	—	2	2	no	—
106	#41×C₂₃	—	2	2	no	—
107	#41×C₂₃	—	2	2	no	—
108	#41×C₂₃	—	2	2	no	—
109	#41×C₂₃	—	2	2	no	—
110	#41×C₂₃	—	4	1	no	—
111	#41×C₂₃	—	4	1	no	—
112	#41×C₂₃	—	4	1	no	—
113	#41×C₂₃	—	4	1	no	—
114	#41×C₂₃	—	4	1	no	—
115	#41×C₂₃	—	4	1	no	—
116	#41×C₂₃	—	4	1	no	—
117	#41×C₂₃	—	4	1	no	—
118	C₂×Q₈×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	—	2	1	no	—
121	C₂×Q₈×C₂₃	—	2	1	no	—
122	C₂×Q₈×C₂₃	—	2	1	no	—
123	C₂×Q₈×C₂₃	C₄²	1	2	yes	complemented
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	—	2	1	no	—
131	C₂×D₄×C₂₃	—	4	1	no	—
132	C₂×D₄×C₂₃	—	4	1	no	—
133	C₂×D₄×C₂₃	—	4	1	no	—
134	C₂×D₄×C₂₃	—	4	1	no	—
135	C₂×D₄×C₂₃	C₄²	1	2	yes	complemented
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	2	2	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	2	no	—
143	C₂²×C₄×C₂₃	—	2	2	no	—
144	C₂²×C₄×C₂₃	—	2	2	no	—
145	C₂²×C₄×C₂₃	—	4	4	no	—
146	C₂²×C₄×C₂₃	—	4	1	no	—
147	C₂²×C₄×C₂₃	—	4	1	no	—
148	C₂²×C₄×C₂₃	—	4	1	no	—
149	C₂²×C₄×C₂₃	—	4	1	no	—
150	C₂²×C₄×C₂₃	—	4	1	no	—
151	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
165	M₁₆×C₂₃	—	2	2	no	—
166	M₁₆×C₂₃	—	2	2	no	—
167	M₁₆×C₂₃	—	2	2	no	—
168	M₁₆×C₂₃	—	2	2	no	—
169	M₁₆×C₂₃	—	4	1	no	—
170	M₁₆×C₂₃	—	4	1	no	—
171	M₁₆×C₂₃	—	4	1	no	—
172	M₁₆×C₂₃	—	4	1	no	—
173	C₂×C₈×C₂₃	—	2	2	no	—
174	C₂×C₈×C₂₃	—	2	2	no	—
175	C₂×C₈×C₂₃	—	4	1	no	—
176	C₂×C₈×C₂₃	—	4	1	no	—
177	C₂×C₈×C₂₃	—	4	1	no	—
178	C₂×C₈×C₂₃	—	4	1	no	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₄⋊C₄×C₂₃	—	2	2	no	—
183	C₄⋊C₄×C₂₃	—	4	1	no	—
184	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
185	C₂²⋊C₄×C₂₃	—	2	2	no	—
186	C₂²⋊C₄×C₂₃	—	2	2	no	—
187	C₂²⋊C₄×C₂₃	—	2	2	no	—
188	C₂²⋊C₄×C₂₃	—	2	2	no	—
189	C₂²⋊C₄×C₂₃	—	4	1	no	—
190	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
191	C₄²×C₂₃	—	2	2	no	—
192	C₄²×C₂₃	—	2	2	no	—
193	C₄²×C₂₃	—	2	2	no	—
194	C₄²×C₂₃	—	2	2	no	—
195	C₄²×C₂₃	—	2	2	no	—
196	C₄²×C₂₃	—	2	2	no	—
197	C₄²×C₂₃	—	2	1	no	—
198	C₄²×C₂₃	—	2	1	no	—
199	C₄²×C₂₃	—	2	2	no	—
200	C₄²×C₂₃	—	4	8	no	—
201	C₄²×C₂₃	—	4	1	no	—
202	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
203	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
204	#10945	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₂₃	—	2	2	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	—	4	1	no	—
208	C₂³×C₂₃	—	4	1	no	—
209	C₂³×C₂₃	—	8	2	no	—
210	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
212	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
213	C₂³×C₂₃	#120	1	1	yes	characteristic
214	C₂³×C₂₃	#121	1	1	yes	characteristic
215	C₂³×C₂₃	#121	1	1	yes	characteristic
216	C₂³×C₂₃	#123	1	1	yes	characteristic
217	Q₈×C₂₃	—	2	2	no	—
218	Q₈×C₂₃	—	2	2	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	D₄×C₂₃	—	2	2	no	—
222	D₄×C₂₃	—	2	2	no	—
223	D₄×C₂₃	—	4	2	no	—
224	D₄×C₂₃	—	4	2	no	—
225	D₄×C₂₃	—	4	2	no	—
226	D₄×C₂₃	—	4	2	no	—
227	D₄×C₂₃	—	4	1	no	—
228	D₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	2	4	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	4	no	—
234	C₂×C₄×C₂₃	—	2	4	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	4	no	—
237	C₂×C₄×C₂₃	—	4	4	no	—
238	C₂×C₄×C₂₃	—	4	4	no	—
239	C₂×C₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	2	no	—
243	C₂×C₄×C₂₃	—	4	2	no	—
244	C₂×C₄×C₂₃	—	4	2	no	—
245	C₂×C₄×C₂₃	—	4	2	no	—
246	C₂×C₄×C₂₃	—	4	2	no	—
247	C₂×C₄×C₂₃	—	8	2	no	—
248	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
252	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
253	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
255	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
256	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
260	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
264	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
265	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
266	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
267	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
268	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
269	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
270	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
271	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
272	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
273	C₂×C₄×C₂₃	#126	1	1	yes	characteristic
274	C₂×C₄×C₂₃	#126	1	1	yes	characteristic
275	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
276	C₈×C₂₃	—	4	2	no	—
277	C₈×C₂₃	—	4	2	no	—
278	C₂²×#104	C₂×C₂₃	1	1	yes	characteristic
279	#2896	C₂×C₂₃	1	1	yes	characteristic
280	C₂×#410	C₂×C₂₃	1	1	yes	characteristic
281	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
282	C₂×#344	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1733	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂×#339	C₂×C₂₃	1	1	yes	characteristic, complemented
285	C₂²×C₂₃	—	2	2	no	—
286	C₂²×C₂₃	—	2	2	no	—
287	C₂²×C₂₃	—	2	2	no	—
288	C₂²×C₂₃	—	4	1	no	—
289	C₂²×C₂₃	—	4	1	no	—
290	C₂²×C₂₃	—	4	1	no	—
291	C₂²×C₂₃	—	4	1	no	—
292	C₂²×C₂₃	—	4	2	no	—
293	C₂²×C₂₃	—	8	2	no	—
294	C₂²×C₂₃	—	8	2	no	—
295	C₂²×C₂₃	—	8	2	no	—
296	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
297	C₂²×C₂₃	#386	1	1	yes	characteristic
298	C₂²×C₂₃	#386	1	1	yes	characteristic
299	C₂²×C₂₃	#386	1	1	yes	characteristic
300	C₂²×C₂₃	#388	1	1	yes	characteristic
301	C₂²×C₂₃	#388	1	1	yes	characteristic
302	C₂²×C₂₃	#390	1	1	yes	characteristic
303	C₄×C₂₃	—	2	4	no	—
304	C₄×C₂₃	—	4	4	no	—
305	C₄×C₂₃	—	4	2	no	—
306	C₄×C₂₃	—	4	4	no	—
307	C₄×C₂₃	—	4	2	no	—
308	C₄×C₂₃	—	8	2	no	—
309	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
310	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
311	C₄×C₂₃	#388	1	1	yes	characteristic
312	C₄×C₂₃	#388	1	1	yes	characteristic
313	C₄×C₂₃	#388	1	1	yes	characteristic
314	C₄×C₂₃	#388	1	1	yes	characteristic
315	C₄×C₂₃	#390	1	1	yes	characteristic
316	C₄×C₂₃	#390	1	1	yes	characteristic
317	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
318	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
319	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
323	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#104	—	2	2	no	—
325	C₂×#104	—	2	1	no	—
326	C₂×#104	—	2	1	no	—
327	C₂×#104	C₄×C₂₃	1	4	yes	complemented
328	#419	C₄×C₂₃	1	2	yes	complemented
329	#418	C₄×C₂₃	1	2	yes	complemented
330	#417	—	2	2	no	—
331	#416	—	2	2	no	—
332	#410	—	2	2	no	—
333	#410	C₄×C₂₃	1	4	yes	complemented
334	C₂×#99	—	2	2	no	—
335	C₄×C₄⋊C₄	—	2	2	no	—
336	C₄×C₂²⋊C₄	—	2	2	no	—
337	#376	—	2	1	no	—
338	C₄³	—	2	1	no	—
339	#344	—	2	2	no	—
340	#339	—	2	2	no	—
341	#339	—	2	2	no	—
342	C₂×C₂₃	—	2	1	no	—
343	C₂×C₂₃	—	2	1	no	—
344	C₂×C₂₃	—	2	1	no	—
345	C₂×C₂₃	—	2	1	no	—
346	C₂×C₂₃	—	2	1	no	—
347	C₂×C₂₃	—	2	1	no	—
348	C₂×C₂₃	—	8	2	no	—
349	C₂×C₂₃	#1437	1	1	yes	characteristic
350	C₂×C₂₃	#1906	1	2	yes	—
351	C₂×#41	—	2	1	no	—
352	C₂×#41	—	2	1	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	—	2	1	no	—
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×#41	—	2	1	no	—
360	C₂×#41	—	2	1	no	—
361	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
362	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
363	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
369	C₂×M₁₆	—	2	1	no	—
370	C₂×M₁₆	—	2	1	no	—
371	C₂×M₁₆	—	2	1	no	—
372	C₂×M₁₆	—	2	1	no	—
373	C₂×M₁₆	—	2	1	no	—
374	C₂×M₁₆	—	2	1	no	—
375	C₂×M₁₆	—	2	1	no	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
380	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
381	C₂²×C₈	—	2	1	no	—
382	C₂²×C₈	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	—	2	1	no	—
385	#117	—	2	1	no	—
386	#117	—	2	1	no	—
387	#117	—	2	1	no	—
388	#117	—	2	1	no	—
389	#117	—	2	1	no	—
390	#117	C₂×C₄×C₂₃	1	2	yes	—
391	#117	C₂×C₄×C₂₃	1	2	yes	—
392	#117	C₂×C₄×C₂₃	1	2	yes	—
393	#117	D₄×C₂₃	1	2	yes	—
394	#117	D₄×C₂₃	1	2	yes	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	—	2	1	no	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	—	2	1	no	—
406	C₂×C₄²	—	2	1	no	—
407	C₂×C₄²	—	2	4	no	—
408	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
409	C₂×C₄²	D₄×C₂₃	1	2	yes	—
410	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
411	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
412	#104	—	2	4	no	—
413	#104	—	2	4	no	—
414	#103	—	2	4	no	—
415	#102	—	2	4	no	—
416	#99	—	2	8	no	—
417	#95	—	2	4	no	—
418	C₂₃	#10945	1	1	yes	characteristic, direct product factor, Sylow
419	C₂⁴	—	2	1	no	—
420	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
421	#41	—	2	2	no	—
422	#41	—	2	2	no	—
423	#41	—	2	2	no	—
424	#41	—	2	2	no	—
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	2	no	—
429	#41	—	4	1	no	—
430	#41	—	4	1	no	—
431	#41	—	4	1	no	—
432	#41	—	4	1	no	—
433	#41	—	4	1	no	—
434	#41	—	4	1	no	—
435	#41	—	4	1	no	—
436	#41	—	4	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	C₄²×C₂₃	1	2	yes	complemented
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	4	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	C₄²×C₂₃	1	2	yes	complemented
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	—	4	1	no	—
467	C₂²×C₄	—	4	1	no	—
468	C₂²×C₄	—	4	1	no	—
469	C₂²×C₄	—	4	1	no	—
470	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
484	M₁₆	—	2	2	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	4	1	no	—
489	M₁₆	—	4	1	no	—
490	M₁₆	—	4	1	no	—
491	M₁₆	—	4	1	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	4	1	no	—
495	C₂×C₈	—	4	1	no	—
496	C₂×C₈	—	4	1	no	—
497	C₂×C₈	—	4	1	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	4	1	no	—
503	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	2	1	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	4	8	no	—
520	C₄²	—	4	1	no	—
521	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
522	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
523	C₂³	—	2	2	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	—	8	2	no	—
528	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
529	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
530	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
531	C₂³	C₂₃×#120	1	1	yes	characteristic
532	C₂³	C₂₃×#121	1	1	yes	characteristic
533	C₂³	C₂₃×#121	1	1	yes	characteristic
534	C₂³	C₂₃×#123	1	1	yes	characteristic
535	Q₈	—	2	2	no	—
536	Q₈	—	2	2	no	—
537	Q₈	—	4	1	no	—
538	Q₈	—	4	1	no	—
539	D₄	—	2	2	no	—
540	D₄	—	2	2	no	—
541	D₄	—	4	2	no	—
542	D₄	—	4	2	no	—
543	D₄	—	4	2	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	1	no	—
546	D₄	—	4	1	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	2	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	2	no	—
565	C₂×C₄	—	8	2	no	—
566	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
567	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
569	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
570	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
571	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
572	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
573	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
574	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
575	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
576	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
577	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
578	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
579	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
580	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
581	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
582	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
583	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
584	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
585	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
586	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
587	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
588	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
589	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
590	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
591	C₂×C₄	C₂₃×#126	1	1	yes	characteristic
592	C₂×C₄	C₂₃×#126	1	1	yes	characteristic
593	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
594	C₈	—	4	2	no	—
595	C₈	—	4	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	4	1	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	8	2	no	—
605	C₂²	—	8	2	no	—
606	C₂²	—	8	2	no	—
607	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
608	C₂²	C₂₃×#386	1	1	yes	characteristic
609	C₂²	C₂₃×#386	1	1	yes	characteristic
610	C₂²	C₂₃×#386	1	1	yes	characteristic
611	C₂²	C₂₃×#388	1	1	yes	characteristic
612	C₂²	C₂₃×#388	1	1	yes	characteristic
613	C₂²	C₂₃×#390	1	1	yes	characteristic
614	C₄	—	2	4	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	4	no	—
617	C₄	—	4	4	no	—
618	C₄	—	4	2	no	—
619	C₄	—	8	2	no	—
620	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
621	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
622	C₄	C₂₃×#388	1	1	yes	characteristic
623	C₄	C₂₃×#388	1	1	yes	characteristic
624	C₄	C₂₃×#388	1	1	yes	characteristic
625	C₄	C₂₃×#388	1	1	yes	characteristic
626	C₄	C₂₃×#390	1	1	yes	characteristic
627	C₄	C₂₃×#390	1	1	yes	characteristic
628	C₂	—	2	1	no	—
629	C₂	—	2	1	no	—
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	2	1	no	—
633	C₂	—	2	1	no	—
634	C₂	—	8	2	no	—
635	C₂	C₂₃×#1437	1	1	yes	characteristic
636	C₂	C₂₃×#1906	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#386
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₂₃×#117	C₂	semidirect
5. C₂×C₂₃×#344	C₂	semidirect
6. C₂₃×#1733	C₂	semidirect
7. C₂×C₂₃×#339	C₂	semidirect
18. C₂×C₂₃×#104	C₄	semidirect
19. C₂₃×#419	C₄	semidirect
20. C₂₃×#418	C₄	semidirect
24. C₂₃×#410	C₄	semidirect
123. C₂×Q₈×C₂₃	C₄²	semidirect
135. C₂×D₄×C₂₃	C₄²	semidirect
204. #10945	C₂₃	direct (N × H)
281. C₄×#117	C₂×C₂₃	semidirect
282. C₂×#344	C₂×C₂₃	semidirect
283. #1733	C₂×C₂₃	semidirect
284. C₂×#339	C₂×C₂₃	semidirect
327. C₂×#104	C₄×C₂₃	semidirect
328. #419	C₄×C₂₃	semidirect
329. #418	C₄×C₂₃	semidirect
333. #410	C₄×C₂₃	semidirect
418. C₂₃	#10945	direct (N × H)
442. C₂×Q₈	C₄²×C₂₃	semidirect
454. C₂×D₄	C₄²×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#104	2
C₂₃×#2896	2
C₂×C₂₃×#410	2
C₄×C₂₃×#117	2
C₂×C₂₃×#344	2
C₂₃×#1733	2
C₂×C₂₃×#339	2
#10945	23

Subgroups — C23×#10945 #19158571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10945 #19158571