Subgroups — C₂₃×#29113 #19176739

Order 5888.

910 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	C₂₃×#2857	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2297	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2072	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2071	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2024	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2023	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1936	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1935	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1825	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1788	C₂	1	2	yes	maximal
11	C₂₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#103	C₂²	1	2	yes	—
18	C₂×C₂₃×#103	C₂²	1	2	yes	—
19	C₂×C₂₃×#102	C₂²	1	2	yes	—
20	C₂×C₂₃×#102	C₂²	1	2	yes	—
21	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂₃×#391	—	2	1	no	—
24	C₂₃×#391	C₂²	1	1	yes	characteristic
25	C₂₃×#391	C₂²	1	1	yes	characteristic
26	C₂₃×#390	C₂²	1	1	yes	characteristic
27	C₂₃×#390	C₂²	1	1	yes	characteristic
28	C₂₃×#389	—	2	1	no	—
29	C₂₃×#389	C₂²	1	2	yes	—
30	C₂₃×#386	—	2	1	no	—
31	C₂₃×#386	—	2	1	no	—
32	C₂₃×#384	C₂²	1	1	yes	characteristic
33	C₂₃×#380	—	2	1	no	—
34	C₂₃×#380	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	1	yes	characteristic
43	C₂×#41×C₂₃	—	2	2	no	—
44	C₂×#41×C₂₃	—	2	2	no	—
45	C₂×#41×C₂₃	—	2	4	no	—
46	C₂×#41×C₂₃	—	4	2	no	—
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	—	2	1	no	—
52	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂₃×C₄⋊Q₈	—	4	2	no	—
58	C₂₃×C₂²⋊Q₈	—	4	4	no	—
59	C₂₃×#117	—	2	4	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	—	4	2	no	—
76	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×#103	—	2	4	no	—
81	C₂₃×#103	—	4	2	no	—
82	C₂₃×#102	—	2	4	no	—
83	C₂₃×#102	—	4	2	no	—
84	C₂₃×C₂²⋊C₈	—	2	4	no	—
85	C₂₃×#97	—	4	2	no	—
86	C₂₃×#95	—	2	1	no	—
87	C₂₃×#95	—	2	1	no	—
88	C₂₃×#95	—	2	1	no	—
89	C₂₃×#95	—	2	1	no	—
90	C₂₃×#95	—	2	2	no	—
91	C₂₃×#95	—	2	2	no	—
92	C₂⁴×C₂₃	—	2	1	no	—
93	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
94	#41×C₂₃	—	4	8	no	—
95	#41×C₂₃	—	4	8	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×Q₈×C₂₃	—	2	2	no	—
98	C₂×Q₈×C₂₃	—	2	2	no	—
99	C₂×Q₈×C₂₃	—	4	2	no	—
100	C₂×Q₈×C₂₃	—	4	2	no	—
101	C₂×Q₈×C₂₃	—	4	2	no	—
102	C₂×Q₈×C₂₃	#41	1	2	yes	—
103	C₂×D₄×C₂₃	—	2	4	no	—
104	C₂×D₄×C₂₃	—	2	2	no	—
105	C₂×D₄×C₂₃	—	2	1	no	—
106	C₂×D₄×C₂₃	—	2	2	no	—
107	C₂×D₄×C₂₃	—	4	2	no	—
108	C₂×D₄×C₂₃	—	4	2	no	—
109	C₂×D₄×C₂₃	#41	1	2	yes	—
110	C₂²×C₄×C₂₃	—	2	1	no	—
111	C₂²×C₄×C₂₃	—	2	1	no	—
112	C₂²×C₄×C₂₃	—	2	2	no	—
113	C₂²×C₄×C₂₃	—	2	4	no	—
114	C₂²×C₄×C₂₃	—	2	1	no	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	2	no	—
117	C₂²×C₄×C₂₃	—	2	1	no	—
118	C₂²×C₄×C₂₃	—	2	1	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	2	2	no	—
121	C₂²×C₄×C₂₃	—	4	1	no	—
122	C₂²×C₄×C₂₃	—	4	1	no	—
123	C₂²×C₄×C₂₃	—	4	1	no	—
124	C₂²×C₄×C₂₃	—	4	1	no	—
125	C₂²×C₄×C₂₃	—	4	2	no	—
126	C₂²×C₄×C₂₃	—	4	2	no	—
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×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	#41	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₄⋊C₄×C₂₃	—	2	2	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	4	2	no	—
147	C₄⋊C₄×C₂₃	—	4	4	no	—
148	C₄⋊C₄×C₂₃	—	4	2	no	—
149	C₄⋊C₄×C₂₃	—	4	2	no	—
150	C₂²⋊C₄×C₂₃	—	4	2	no	—
151	C₂²⋊C₄×C₂₃	—	4	2	no	—
152	C₂²⋊C₄×C₂₃	—	4	4	no	—
153	C₂²⋊C₄×C₂₃	—	4	2	no	—
154	C₂²⋊C₄×C₂₃	—	4	2	no	—
155	C₂²⋊C₄×C₂₃	—	4	2	no	—
156	C₄²×C₂₃	—	2	2	no	—
157	C₄²×C₂₃	—	2	2	no	—
158	C₄²×C₂₃	—	4	1	no	—
159	C₄²×C₂₃	—	4	1	no	—
160	C₄²×C₂₃	—	4	2	no	—
161	#29113	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₂₃	—	2	1	no	—
163	C₂³×C₂₃	—	2	1	no	—
164	C₂³×C₂₃	—	2	1	no	—
165	C₂³×C₂₃	—	2	1	no	—
166	C₂³×C₂₃	—	2	1	no	—
167	C₂³×C₂₃	—	2	1	no	—
168	C₂³×C₂₃	—	2	2	no	—
169	C₂³×C₂₃	—	4	1	no	—
170	C₂³×C₂₃	—	4	1	no	—
171	C₂³×C₂₃	—	4	1	no	—
172	C₂³×C₂₃	—	4	1	no	—
173	C₂³×C₂₃	—	4	1	no	—
174	C₂³×C₂₃	—	4	1	no	—
175	C₂³×C₂₃	#124	1	2	yes	—
176	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₂₃	—	2	4	no	—
178	Q₈×C₂₃	—	4	2	no	—
179	Q₈×C₂₃	—	8	4	no	—
180	D₄×C₂₃	—	2	4	no	—
181	D₄×C₂₃	—	4	8	no	—
182	D₄×C₂₃	—	4	2	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	1	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	1	no	—
193	C₂×C₄×C₂₃	—	2	1	no	—
194	C₂×C₄×C₂₃	—	2	1	no	—
195	C₂×C₄×C₂₃	—	2	1	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	4	no	—
200	C₂×C₄×C₂₃	—	4	8	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	1	no	—
205	C₂×C₄×C₂₃	—	4	1	no	—
206	C₂×C₄×C₂₃	—	4	2	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	1	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	2	no	—
215	C₂×C₄×C₂₃	#124	1	2	yes	—
216	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
217	C₂×C₄×C₂₃	#137	1	2	yes	—
218	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
221	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
222	C₈×C₂₃	—	4	2	no	—
223	C₈×C₂₃	—	4	2	no	—
224	#2857	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#2297	C₂×C₂₃	1	1	yes	characteristic
226	#2072	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#2071	C₂×C₂₃	1	1	yes	characteristic, complemented
228	#2024	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#2023	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#1936	C₂×C₂₃	1	2	yes	complemented
231	#1935	C₂×C₂₃	1	2	yes	complemented
232	#1825	C₂×C₂₃	1	2	yes	complemented
233	#1788	C₂×C₂₃	1	2	yes	—
234	#1749	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	2	2	no	—
237	C₂²×C₂₃	—	2	2	no	—
238	C₂²×C₂₃	—	4	1	no	—
239	C₂²×C₂₃	—	4	1	no	—
240	C₂²×C₂₃	—	4	1	no	—
241	C₂²×C₂₃	—	4	1	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	—	4	2	no	—
244	C₂²×C₂₃	—	4	2	no	—
245	C₂²×C₂₃	—	8	1	no	—
246	C₂²×C₂₃	—	8	1	no	—
247	C₂²×C₂₃	—	8	1	no	—
248	C₂²×C₂₃	—	8	1	no	—
249	C₂²×C₂₃	#525	1	1	yes	characteristic
250	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
251	C₂²×C₂₃	#534	1	1	yes	characteristic
252	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
254	C₂²×C₂₃	#575	1	1	yes	characteristic
255	C₂²×C₂₃	#575	1	1	yes	characteristic
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	2	2	no	—
258	C₄×C₂₃	—	2	2	no	—
259	C₄×C₂₃	—	4	2	no	—
260	C₄×C₂₃	—	4	1	no	—
261	C₄×C₂₃	—	4	1	no	—
262	C₄×C₂₃	—	8	2	no	—
263	C₄×C₂₃	—	8	1	no	—
264	C₄×C₂₃	—	8	1	no	—
265	C₄×C₂₃	—	8	2	no	—
266	C₄×C₂₃	#551	1	2	yes	—
267	C₄×C₂₃	#551	1	2	yes	—
268	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
269	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₂²⋊Q₈	—	2	1	no	—
271	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
272	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×#103	C₂²×C₂₃	1	2	yes	—
274	C₂×#103	C₂²×C₂₃	1	2	yes	—
275	C₂×#102	C₂²×C₂₃	1	2	yes	—
276	C₂×#102	C₂²×C₂₃	1	2	yes	—
277	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
278	C₂×#97	C₂²×C₂₃	1	2	yes	—
279	#391	—	2	1	no	—
280	#391	C₂²×C₂₃	1	1	yes	characteristic
281	#391	C₂²×C₂₃	1	1	yes	characteristic
282	#390	C₂²×C₂₃	1	1	yes	characteristic
283	#390	C₂²×C₂₃	1	1	yes	characteristic
284	#389	—	2	1	no	—
285	#389	C₂²×C₂₃	1	2	yes	—
286	#386	—	2	1	no	—
287	#386	—	2	1	no	—
288	#384	C₂²×C₂₃	1	1	yes	characteristic
289	#380	—	2	1	no	—
290	#380	C₂²×C₂₃	1	2	yes	—
291	C₄×C₂²⋊C₄	—	2	1	no	—
292	C₄×C₂²⋊C₄	—	2	1	no	—
293	#340	C₂²×C₂₃	1	2	yes	—
294	#340	C₂²×C₂₃	1	2	yes	—
295	#340	C₂²×C₂₃	1	2	yes	—
296	#340	C₂²×C₂₃	1	2	yes	—
297	#336	C₂²×C₂₃	1	1	yes	characteristic
298	#336	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₂₃	—	4	2	no	—
300	C₂×C₂₃	—	8	2	no	—
301	C₂×C₂₃	#2561	1	1	yes	characteristic
302	C₂×C₂₃	#3115	1	1	yes	characteristic
303	C₂×C₂₃	#3116	1	1	yes	characteristic
304	C₂×C₂₃	#3135	1	1	yes	characteristic
305	C₂×C₂₃	#3141	1	1	yes	characteristic
306	C₂×C₂₃	#3194	1	1	yes	characteristic
307	C₂×C₂₃	#3196	1	1	yes	characteristic
308	C₂×#41	—	2	2	no	—
309	C₂×#41	—	2	2	no	—
310	C₂×#41	—	2	4	no	—
311	C₂×#41	—	4	2	no	—
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
316	C₂³×C₄	—	2	1	no	—
317	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
318	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
321	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
322	C₄⋊Q₈	—	4	2	no	—
323	C₂²⋊Q₈	—	4	4	no	—
324	#117	—	2	4	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	—	4	2	no	—
341	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	#103	—	2	4	no	—
346	#103	—	4	2	no	—
347	#102	—	2	4	no	—
348	#102	—	4	2	no	—
349	C₂²⋊C₈	—	2	4	no	—
350	#97	—	4	2	no	—
351	#95	—	2	1	no	—
352	#95	—	2	1	no	—
353	#95	—	2	1	no	—
354	#95	—	2	1	no	—
355	#95	—	2	2	no	—
356	#95	—	2	2	no	—
357	C₂₃	#29113	1	1	yes	characteristic, direct product factor, Sylow
358	C₂⁴	—	2	1	no	—
359	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
360	#41	—	4	8	no	—
361	#41	—	4	8	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	4	2	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	—	4	2	no	—
368	C₂×Q₈	#41×C₂₃	1	2	yes	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	2	no	—
375	C₂×D₄	#41×C₂₃	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	4	1	no	—
388	C₂²×C₄	—	4	1	no	—
389	C₂²×C₄	—	4	1	no	—
390	C₂²×C₄	—	4	1	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	—	4	2	no	—
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×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	#41×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
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	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	4	4	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₂²⋊C₄	—	4	2	no	—
417	C₂²⋊C₄	—	4	2	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	2	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₂²⋊C₄	—	4	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	4	1	no	—
425	C₄²	—	4	1	no	—
426	C₄²	—	4	2	no	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	1	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	C₂₃×#124	1	2	yes	—
441	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
442	Q₈	—	2	4	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	8	4	no	—
445	D₄	—	2	4	no	—
446	D₄	—	4	8	no	—
447	D₄	—	4	2	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	1	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	4	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	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	C₂₃×#124	1	2	yes	—
481	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
482	C₂×C₄	C₂₃×#137	1	2	yes	—
483	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
484	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
486	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	4	1	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	8	1	no	—
502	C₂²	—	8	1	no	—
503	C₂²	C₂₃×#525	1	1	yes	characteristic
504	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
505	C₂²	C₂₃×#534	1	1	yes	characteristic
506	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
507	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
508	C₂²	C₂₃×#575	1	1	yes	characteristic
509	C₂²	C₂₃×#575	1	1	yes	characteristic
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	2	2	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	4	2	no	—
516	C₄	—	8	2	no	—
517	C₄	—	8	2	no	—
518	C₄	—	8	1	no	—
519	C₄	—	8	1	no	—
520	C₄	C₂₃×#551	1	2	yes	—
521	C₄	C₂₃×#551	1	2	yes	—
522	C₂	—	4	2	no	—
523	C₂	—	8	2	no	—
524	C₂	C₂₃×#2561	1	1	yes	characteristic
525	C₂	C₂₃×#3115	1	1	yes	characteristic
526	C₂	C₂₃×#3116	1	1	yes	characteristic
527	C₂	C₂₃×#3135	1	1	yes	characteristic
528	C₂	C₂₃×#3141	1	1	yes	characteristic
529	C₂	C₂₃×#3194	1	1	yes	characteristic
530	C₂	C₂₃×#3196	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	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
1. C₂₃×#2857	C₂	semidirect
3. C₂₃×#2072	C₂	semidirect
4. C₂₃×#2071	C₂	semidirect
5. C₂₃×#2024	C₂	semidirect
6. C₂₃×#2023	C₂	semidirect
7. C₂₃×#1936	C₂	semidirect
8. C₂₃×#1935	C₂	semidirect
9. C₂₃×#1825	C₂	semidirect
11. C₂₃×#1749	C₂	semidirect
161. #29113	C₂₃	direct (N × H)
224. #2857	C₂×C₂₃	semidirect
226. #2072	C₂×C₂₃	semidirect
227. #2071	C₂×C₂₃	semidirect
228. #2024	C₂×C₂₃	semidirect
229. #2023	C₂×C₂₃	semidirect
230. #1936	C₂×C₂₃	semidirect
231. #1935	C₂×C₂₃	semidirect
232. #1825	C₂×C₂₃	semidirect
234. #1749	C₂×C₂₃	semidirect
357. C₂₃	#29113	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2857	2
C₂₃×#2297	2
C₂₃×#2072	2
C₂₃×#2071	2
C₂₃×#2024	2
C₂₃×#2023	2
C₂₃×#1936	2
C₂₃×#1935	2
C₂₃×#1825	2
C₂₃×#1788	2
C₂₃×#1749	2
#29113	23

Subgroups — C23×#29113 #19176739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29113 #19176739