Subgroups — C₂₃×#29131 #19176757

Order 5888.

742 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2101	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2042	C₂	1	2	yes	maximal
5	C₂₃×#1936	C₂	1	2	yes	maximal
6	C₂₃×#1935	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#125	—	2	1	no	—
13	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#103	C₂²	1	2	yes	—
19	C₂×C₂₃×#103	C₂²	1	2	yes	—
20	C₂×C₂₃×#102	C₂²	1	2	yes	—
21	C₂×C₂₃×#102	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂₃×#391	C₂²	1	1	yes	characteristic
24	C₂₃×#390	C₂²	1	1	yes	characteristic
25	C₂₃×#389	—	2	1	no	—
26	C₂₃×#389	—	2	1	no	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#388	—	2	1	no	—
30	C₂₃×#388	—	2	1	no	—
31	C₂₃×#384	C₂²	1	1	yes	characteristic
32	C₂₃×#384	C₂²	1	1	yes	characteristic
33	C₂₃×#382	—	2	1	no	—
34	C₂₃×#382	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
37	C₄³×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂₃×#125	—	4	4	no	—
49	C₂₃×#124	—	2	2	no	—
50	C₂₃×#124	—	2	2	no	—
51	C₄×Q₈×C₂₃	—	2	2	no	—
52	C₄×Q₈×C₂₃	—	2	2	no	—
53	C₄×D₄×C₂₃	—	2	2	no	—
54	C₄×D₄×C₂₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄²×C₂₃	—	2	1	no	—
72	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×C₄⋊C₈	—	2	2	no	—
80	C₂₃×C₄⋊C₈	—	2	2	no	—
81	C₂₃×#103	—	2	4	no	—
82	C₂₃×#103	—	4	2	no	—
83	C₂₃×#102	—	2	4	no	—
84	C₂₃×#102	—	4	2	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	C₂×Q₈×C₂₃	—	2	1	no	—
94	C₂×Q₈×C₂₃	—	2	1	no	—
95	C₂×Q₈×C₂₃	—	2	1	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×Q₈×C₂₃	—	2	1	no	—
98	C₂×Q₈×C₂₃	#41	1	2	yes	—
99	C₂×D₄×C₂₃	—	2	1	no	—
100	C₂×D₄×C₂₃	—	2	1	no	—
101	C₂×D₄×C₂₃	—	2	1	no	—
102	C₂×D₄×C₂₃	—	2	1	no	—
103	C₂×D₄×C₂₃	—	2	1	no	—
104	C₂×D₄×C₂₃	#41	1	2	yes	—
105	C₂²×C₄×C₂₃	—	2	1	no	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	1	no	—
108	C₂²×C₄×C₂₃	—	2	2	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	4	2	no	—
111	C₂²×C₄×C₂₃	—	4	2	no	—
112	C₂²×C₄×C₂₃	—	4	2	no	—
113	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₄⋊C₄×C₂₃	—	2	2	no	—
127	C₄⋊C₄×C₂₃	—	2	2	no	—
128	C₄⋊C₄×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	1	no	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	4	2	no	—
135	C₄⋊C₄×C₂₃	—	4	2	no	—
136	C₄⋊C₄×C₂₃	—	4	4	no	—
137	C₄⋊C₄×C₂₃	—	4	4	no	—
138	C₄⋊C₄×C₂₃	—	4	2	no	—
139	C₄⋊C₄×C₂₃	—	4	2	no	—
140	C₄⋊C₄×C₂₃	—	4	2	no	—
141	C₂²⋊C₄×C₂₃	—	2	4	no	—
142	C₂²⋊C₄×C₂₃	—	2	4	no	—
143	C₂²⋊C₄×C₂₃	—	4	2	no	—
144	C₂²⋊C₄×C₂₃	—	4	2	no	—
145	C₂²⋊C₄×C₂₃	—	4	2	no	—
146	C₄²×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	—	2	1	no	—
149	C₄²×C₂₃	—	2	1	no	—
150	C₄²×C₂₃	—	2	1	no	—
151	C₄²×C₂₃	—	2	1	no	—
152	C₄²×C₂₃	—	4	1	no	—
153	C₄²×C₂₃	—	4	1	no	—
154	C₄²×C₂₃	—	4	2	no	—
155	C₄²×C₂₃	#41	1	2	yes	—
156	C₄²×C₂₃	#41	1	2	yes	—
157	#29131	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₂₃	—	2	2	no	—
159	C₂³×C₂₃	—	4	1	no	—
160	C₂³×C₂₃	—	4	1	no	—
161	C₂³×C₂₃	—	4	1	no	—
162	C₂³×C₂₃	—	4	1	no	—
163	C₂³×C₂₃	—	4	1	no	—
164	C₂³×C₂₃	—	4	1	no	—
165	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₂₃	—	2	2	no	—
167	Q₈×C₂₃	—	2	2	no	—
168	Q₈×C₂₃	—	4	1	no	—
169	Q₈×C₂₃	—	4	1	no	—
170	D₄×C₂₃	—	2	2	no	—
171	D₄×C₂₃	—	2	2	no	—
172	D₄×C₂₃	—	4	1	no	—
173	D₄×C₂₃	—	4	1	no	—
174	C₂×C₄×C₂₃	—	2	1	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	1	no	—
180	C₂×C₄×C₂₃	—	2	1	no	—
181	C₂×C₄×C₂₃	—	2	1	no	—
182	C₂×C₄×C₂₃	—	2	1	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	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	1	no	—
193	C₂×C₄×C₂₃	—	4	1	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	4	1	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	2	no	—
203	C₂×C₄×C₂₃	—	8	4	no	—
204	C₂×C₄×C₂₃	#124	1	2	yes	—
205	C₂×C₄×C₂₃	#124	1	2	yes	—
206	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄×C₂₃	#137	1	2	yes	—
208	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
214	C₈×C₂₃	—	4	2	no	—
215	C₈×C₂₃	—	4	2	no	—
216	#2859	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
218	#2101	C₂×C₂₃	1	2	yes	complemented
219	#2042	C₂×C₂₃	1	2	yes	—
220	#1936	C₂×C₂₃	1	2	yes	—
221	#1935	C₂×C₂₃	1	2	yes	complemented
222	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
223	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#1802	C₂×C₂₃	1	1	yes	characteristic
225	#1802	C₂×C₂₃	1	1	yes	characteristic
226	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₂²×C₂₃	—	4	2	no	—
228	C₂²×C₂₃	—	4	2	no	—
229	C₂²×C₂₃	—	4	2	no	—
230	C₂²×C₂₃	—	8	1	no	—
231	C₂²×C₂₃	—	8	1	no	—
232	C₂²×C₂₃	—	8	1	no	—
233	C₂²×C₂₃	—	8	1	no	—
234	C₂²×C₂₃	#525	1	1	yes	characteristic
235	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
236	C₂²×C₂₃	#537	1	1	yes	characteristic
237	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
239	C₂²×C₂₃	#575	1	1	yes	characteristic
240	C₂²×C₂₃	#575	1	1	yes	characteristic
241	C₄×C₂₃	—	2	2	no	—
242	C₄×C₂₃	—	4	2	no	—
243	C₄×C₂₃	—	4	1	no	—
244	C₄×C₂₃	—	4	1	no	—
245	C₄×C₂₃	—	4	1	no	—
246	C₄×C₂₃	—	4	1	no	—
247	C₄×C₂₃	—	8	2	no	—
248	C₄×C₂₃	—	8	1	no	—
249	C₄×C₂₃	—	8	1	no	—
250	C₄×C₂₃	—	8	2	no	—
251	C₄×C₂₃	#489	1	4	yes	—
252	C₄×C₂₃	#552	1	1	yes	characteristic
253	C₄×C₂₃	#552	1	1	yes	characteristic
254	C₄×C₂₃	#553	1	1	yes	characteristic
255	C₄×C₂₃	#553	1	1	yes	characteristic
256	C₂×#125	—	2	1	no	—
257	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×#103	C₂²×C₂₃	1	2	yes	—
263	C₂×#103	C₂²×C₂₃	1	2	yes	—
264	C₂×#102	C₂²×C₂₃	1	2	yes	—
265	C₂×#102	C₂²×C₂₃	1	2	yes	—
266	C₂×#97	C₂²×C₂₃	1	2	yes	—
267	#391	C₂²×C₂₃	1	1	yes	characteristic
268	#390	C₂²×C₂₃	1	1	yes	characteristic
269	#389	—	2	1	no	—
270	#389	—	2	1	no	—
271	#389	C₂²×C₂₃	1	2	yes	—
272	#389	C₂²×C₂₃	1	2	yes	—
273	#388	—	2	1	no	—
274	#388	—	2	1	no	—
275	#384	C₂²×C₂₃	1	1	yes	characteristic
276	#384	C₂²×C₂₃	1	1	yes	characteristic
277	#382	—	2	1	no	—
278	#382	C₂²×C₂₃	1	1	yes	characteristic
279	C₄×C₂²⋊C₄	—	2	1	no	—
280	C₄×C₂²⋊C₄	—	2	1	no	—
281	C₄³	C₂²×C₂₃	1	1	yes	characteristic
282	#340	C₂²×C₂₃	1	2	yes	—
283	#340	C₂²×C₂₃	1	2	yes	—
284	#340	C₂²×C₂₃	1	2	yes	—
285	#340	C₂²×C₂₃	1	2	yes	—
286	#336	C₂²×C₂₃	1	2	yes	—
287	C₂×C₂₃	—	8	2	no	—
288	C₂×C₂₃	#2562	1	1	yes	characteristic
289	C₂×C₂₃	#3115	1	1	yes	characteristic
290	C₂×C₂₃	#3116	1	1	yes	characteristic
291	C₂×C₂₃	C₂×#489	1	1	yes	characteristic
292	C₂×C₂₃	#3142	1	1	yes	characteristic
293	C₂×C₂₃	#3251	1	1	yes	characteristic
294	C₂×C₂₃	#3252	1	1	yes	characteristic
295	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
296	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
297	C₂³×C₄	—	2	1	no	—
298	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
299	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
300	#125	—	4	4	no	—
301	#124	—	2	2	no	—
302	#124	—	2	2	no	—
303	C₄×Q₈	—	2	2	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	1	no	—
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₂²⋊C₄	—	2	1	no	—
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₄²	—	2	1	no	—
324	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
325	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
326	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
327	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	2	no	—
333	#103	—	2	4	no	—
334	#103	—	4	2	no	—
335	#102	—	2	4	no	—
336	#102	—	4	2	no	—
337	#97	—	4	2	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	#95	—	2	1	no	—
341	#95	—	2	1	no	—
342	#95	—	2	2	no	—
343	#95	—	2	2	no	—
344	C₂₃	#29131	1	1	yes	characteristic, direct product factor, Sylow
345	C₂⁴	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	#41×C₂₃	1	2	yes	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	#41×C₂₃	1	2	yes	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	4	4	no	—
388	C₄⋊C₄	—	4	4	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	4	2	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	2	4	no	—
396	C₂²⋊C₄	—	4	2	no	—
397	C₂²⋊C₄	—	4	2	no	—
398	C₂²⋊C₄	—	4	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	4	1	no	—
406	C₄²	—	4	1	no	—
407	C₄²	—	4	2	no	—
408	C₄²	#41×C₂₃	1	2	yes	—
409	C₄²	#41×C₂₃	1	2	yes	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
418	Q₈	—	2	2	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	4	1	no	—
421	Q₈	—	4	1	no	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	1	no	—
425	D₄	—	4	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	8	4	no	—
456	C₂×C₄	C₂₃×#124	1	2	yes	—
457	C₂×C₄	C₂₃×#124	1	2	yes	—
458	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
459	C₂×C₄	C₂₃×#137	1	2	yes	—
460	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
464	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
465	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
466	C₈	—	4	2	no	—
467	C₈	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	8	1	no	—
472	C₂²	—	8	1	no	—
473	C₂²	—	8	1	no	—
474	C₂²	—	8	1	no	—
475	C₂²	C₂₃×#525	1	1	yes	characteristic
476	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
477	C₂²	C₂₃×#537	1	1	yes	characteristic
478	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
479	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
480	C₂²	C₂₃×#575	1	1	yes	characteristic
481	C₂²	C₂₃×#575	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	1	no	—
488	C₄	—	8	2	no	—
489	C₄	—	8	2	no	—
490	C₄	—	8	1	no	—
491	C₄	—	8	1	no	—
492	C₄	C₂₃×#489	1	4	yes	—
493	C₄	C₂₃×#552	1	1	yes	characteristic
494	C₄	C₂₃×#552	1	1	yes	characteristic
495	C₄	C₂₃×#553	1	1	yes	characteristic
496	C₄	C₂₃×#553	1	1	yes	characteristic
497	C₂	—	8	2	no	—
498	C₂	C₂₃×#2562	1	1	yes	characteristic
499	C₂	C₂₃×#3115	1	1	yes	characteristic
500	C₂	C₂₃×#3116	1	1	yes	characteristic
501	C₂	C₂×C₂₃×#489	1	1	yes	characteristic
502	C₂	C₂₃×#3142	1	1	yes	characteristic
503	C₂	C₂₃×#3251	1	1	yes	characteristic
504	C₂	C₂₃×#3252	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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2859	C₂	semidirect
3. C₂₃×#2101	C₂	semidirect
6. C₂₃×#1935	C₂	semidirect
7. C₂₃×#1803	C₂	semidirect
8. C₂₃×#1803	C₂	semidirect
11. C₂₃×#1767	C₂	semidirect
157. #29131	C₂₃	direct (N × H)
216. #2859	C₂×C₂₃	semidirect
218. #2101	C₂×C₂₃	semidirect
221. #1935	C₂×C₂₃	semidirect
222. #1803	C₂×C₂₃	semidirect
223. #1803	C₂×C₂₃	semidirect
226. #1767	C₂×C₂₃	semidirect
344. C₂₃	#29131	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×#2859	2
C₄×C₂₃×#124	2
C₂₃×#2101	2
C₂₃×#2042	2
C₂₃×#1936	2
C₂₃×#1935	2
C₂₃×#1803	2
C₂₃×#1802	2
C₂₃×#1767	2
#29131	23

Subgroups — C23×#29131 #19176757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29131 #19176757