Subgroups — C₂₃×#17628 #19165254

Order 5888.

942 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3507	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2983	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2351	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1517	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1512	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1509	C₂	1	4	yes	maximal, complemented
8	C₂₃×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1502	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#559	C₄	1	2	yes	complemented
11	C₂₃×#557	C₄	1	2	yes	complemented
12	C₂₃×#556	—	2	2	no	—
13	C₂₃×#556	C₄	1	2	yes	complemented
14	C₂₃×#554	C₄	1	2	yes	complemented
15	C₂₃×#552	—	2	2	no	—
16	C₂₃×#539	—	2	2	no	—
17	C₂₃×#536	—	2	1	no	—
18	C₂₃×#533	C₂²	1	1	yes	characteristic
19	C₂₃×#532	C₂²	1	1	yes	characteristic
20	C₂₃×#529	C₂²	1	4	yes	—
21	C₂₃×#520	—	2	1	no	—
22	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#446	—	2	2	no	—
25	C₂₃×#446	—	2	2	no	—
26	C₂₃×#432	C₂²	1	4	yes	—
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂₃×#431	C₂²	1	1	yes	characteristic
29	C₂₃×#389	—	2	1	no	—
30	C₂₃×#389	—	2	2	no	—
31	C₂₃×#382	C₂²	1	2	yes	—
32	C₂₃×#382	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₂₃×#328	C₂²	1	2	yes	—
36	C₂₃×#328	C₂²	1	2	yes	—
37	C₂₃×#328	C₂²	1	4	yes	—
38	C₂₃×#327	C₂²	1	2	yes	—
39	C₂₃×#327	C₂²	1	2	yes	—
40	C₂₃×#327	C₂²	1	4	yes	—
41	C₂×#41×C₂₃	—	2	1	no	—
42	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
47	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
48	C₂₃×#126	—	2	4	no	—
49	C₂₃×#126	—	2	4	no	—
50	C₂₃×#125	—	2	4	no	—
51	C₂₃×#125	—	2	1	no	—
52	C₂₃×#125	C₂×C₄	1	4	yes	—
53	C₂₃×#124	—	2	2	no	—
54	C₂₃×#124	—	2	1	no	—
55	C₂₃×#124	C₂×C₄	1	4	yes	—
56	C₂₃×#123	—	2	4	no	—
57	C₂₃×#123	—	2	4	no	—
58	C₂₃×C₂²⋊Q₈	—	2	2	no	—
59	C₂₃×C₂²⋊Q₈	—	2	2	no	—
60	C₂₃×C₂²⋊Q₈	—	2	1	no	—
61	C₂₃×C₂²⋊Q₈	—	2	4	no	—
62	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
63	C₂₃×#121	—	2	1	no	—
64	C₂₃×#121	C₂³	1	2	yes	—
65	C₄×Q₈×C₂₃	—	2	2	no	—
66	C₄×Q₈×C₂₃	—	2	1	no	—
67	C₄×Q₈×C₂₃	—	4	4	no	—
68	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
69	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
70	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
72	C₄×Q₈×C₂₃	D₄	1	2	yes	—
73	C₄×Q₈×C₂₃	D₄	1	2	yes	—
74	C₄×D₄×C₂₃	—	2	4	no	—
75	C₄×D₄×C₂₃	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	1	no	—
77	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
79	C₂₃×#117	—	2	1	no	—
80	C₂₃×#117	—	2	2	no	—
81	C₂₃×#117	—	2	4	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
89	C₂×C₄²×C₂₃	—	2	2	no	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
93	C₂₃×C₄⋊C₈	—	2	1	no	—
94	C₂₃×C₄⋊C₈	—	2	1	no	—
95	C₂₃×C₄⋊C₈	—	2	4	no	—
96	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
97	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
98	C₂₃×#97	—	2	4	no	—
99	C₄×C₈×C₂₃	—	2	1	no	—
100	C₄×C₈×C₂₃	—	2	1	no	—
101	C₂₃×#95	—	2	2	no	—
102	C₂₃×#95	—	2	1	no	—
103	C₂₃×#95	C₂³	1	1	yes	characteristic
104	C₂₃×#95	C₂³	1	1	yes	characteristic
105	C₂₃×#95	C₂³	1	2	yes	—
106	#41×C₂₃	—	4	2	no	—
107	#41×C₂₃	—	4	2	no	—
108	C₂×Q₈×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	—	2	1	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×Q₈×C₂₃	—	2	2	no	—
112	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
113	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
114	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₂₃	—	2	2	no	—
116	C₂×D₄×C₂₃	—	2	1	no	—
117	C₂×D₄×C₂₃	—	2	1	no	—
118	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₂₃	—	2	1	no	—
120	C₂²×C₄×C₂₃	—	2	2	no	—
121	C₂²×C₄×C₂₃	—	2	4	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₈×C₂₃	—	2	2	no	—
130	C₂×C₈×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	4	no	—
133	C₄⋊C₄×C₂₃	—	2	1	no	—
134	C₄⋊C₄×C₂₃	—	2	4	no	—
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	2	4	no	—
139	C₄⋊C₄×C₂₃	—	2	2	no	—
140	C₄⋊C₄×C₂₃	—	4	4	no	—
141	C₄⋊C₄×C₂₃	—	4	2	no	—
142	C₄⋊C₄×C₂₃	—	4	4	no	—
143	C₄⋊C₄×C₂₃	—	4	2	no	—
144	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
145	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
146	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
147	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
148	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
149	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
150	C₂²⋊C₄×C₂₃	—	2	2	no	—
151	C₂²⋊C₄×C₂₃	—	2	4	no	—
152	C₂²⋊C₄×C₂₃	—	2	4	no	—
153	C₂²⋊C₄×C₂₃	—	2	2	no	—
154	C₂²⋊C₄×C₂₃	—	2	4	no	—
155	C₂²⋊C₄×C₂₃	—	2	2	no	—
156	C₄²×C₂₃	—	2	4	no	—
157	C₄²×C₂₃	—	2	2	no	—
158	C₄²×C₂₃	—	4	4	no	—
159	C₄²×C₂₃	—	4	2	no	—
160	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
162	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₂₃	C₂×D₄	1	4	yes	—
164	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
166	#17628	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₂₃	—	2	2	no	—
168	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₂₃	—	2	4	no	—
170	Q₈×C₂₃	—	2	4	no	—
171	Q₈×C₂₃	—	4	2	no	—
172	Q₈×C₂₃	—	4	2	no	—
173	Q₈×C₂₃	—	8	1	no	—
174	D₄×C₂₃	—	4	2	no	—
175	D₄×C₂₃	—	4	2	no	—
176	D₄×C₂₃	—	4	2	no	—
177	D₄×C₂₃	—	4	2	no	—
178	D₄×C₂₃	—	8	1	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	1	no	—
188	C₂×C₄×C₂₃	—	4	1	no	—
189	C₂×C₄×C₂₃	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	4	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	1	no	—
193	C₂×C₄×C₂₃	—	4	2	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	—	8	4	no	—
198	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
199	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
200	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₂₃	#137	1	2	yes	—
206	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	C₈×C₂₃	—	4	2	no	—
209	C₈×C₂₃	—	4	2	no	—
210	#3507	C₂×C₂₃	1	1	yes	characteristic
211	#2983	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#2351	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#1518	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1517	C₂×C₂₃	1	1	yes	characteristic
215	#1512	C₂×C₂₃	1	4	yes	complemented
216	#1509	C₂×C₂₃	1	4	yes	complemented
217	#1503	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#1502	C₂×C₂₃	1	1	yes	characteristic
219	C₂²×C₂₃	—	2	1	no	—
220	C₂²×C₂₃	—	2	1	no	—
221	C₂²×C₂₃	—	4	2	no	—
222	C₂²×C₂₃	—	4	2	no	—
223	C₂²×C₂₃	—	4	2	no	—
224	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
227	C₄×C₂₃	—	2	1	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	2	4	no	—
230	C₄×C₂₃	—	2	4	no	—
231	C₄×C₂₃	—	2	1	no	—
232	C₄×C₂₃	—	2	1	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	4	2	no	—
235	C₄×C₂₃	—	4	2	no	—
236	C₄×C₂₃	—	4	2	no	—
237	C₄×C₂₃	—	8	2	no	—
238	C₄×C₂₃	—	8	1	no	—
239	C₄×C₂₃	#417	1	2	yes	—
240	C₄×C₂₃	#417	1	2	yes	—
241	#559	C₄×C₂₃	1	2	yes	complemented
242	#557	C₄×C₂₃	1	2	yes	complemented
243	#556	—	2	2	no	—
244	#556	C₄×C₂₃	1	2	yes	complemented
245	#554	C₄×C₂₃	1	2	yes	complemented
246	#552	—	2	2	no	—
247	#539	—	2	2	no	—
248	#536	—	2	1	no	—
249	#533	C₂²×C₂₃	1	1	yes	characteristic
250	#532	C₂²×C₂₃	1	1	yes	characteristic
251	#529	C₂²×C₂₃	1	4	yes	—
252	#520	—	2	1	no	—
253	C₂×C₄×Q₈	—	2	1	no	—
254	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	#446	—	2	2	no	—
256	#446	—	2	2	no	—
257	#432	C₂²×C₂₃	1	4	yes	—
258	#431	C₂²×C₂₃	1	1	yes	characteristic
259	#431	C₂²×C₂₃	1	1	yes	characteristic
260	#389	—	2	1	no	—
261	#389	—	2	2	no	—
262	#382	C₂²×C₂₃	1	2	yes	—
263	#382	C₂²×C₂₃	1	1	yes	characteristic
264	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
265	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
266	#328	C₂²×C₂₃	1	2	yes	—
267	#328	C₂²×C₂₃	1	2	yes	—
268	#328	C₂²×C₂₃	1	4	yes	—
269	#327	C₂²×C₂₃	1	2	yes	—
270	#327	C₂²×C₂₃	1	2	yes	—
271	#327	C₂²×C₂₃	1	4	yes	—
272	C₂×C₂₃	—	2	1	no	—
273	C₂×C₂₃	—	2	1	no	—
274	C₂×C₂₃	—	8	2	no	—
275	C₂×C₂₃	#2884	1	1	yes	characteristic
276	C₂×C₂₃	C₂×#417	1	1	yes	characteristic
277	C₂×C₂₃	#2901	1	1	yes	characteristic
278	C₂×#41	—	2	1	no	—
279	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
280	C₄⋊Q₈	—	2	2	no	—
281	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
282	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
283	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
284	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
285	#126	—	2	4	no	—
286	#126	—	2	4	no	—
287	#125	—	2	4	no	—
288	#125	—	2	1	no	—
289	#125	C₂×C₄×C₂₃	1	4	yes	—
290	#124	—	2	2	no	—
291	#124	—	2	1	no	—
292	#124	C₂×C₄×C₂₃	1	4	yes	—
293	#123	—	2	4	no	—
294	#123	—	2	4	no	—
295	C₂²⋊Q₈	—	2	2	no	—
296	C₂²⋊Q₈	—	2	2	no	—
297	C₂²⋊Q₈	—	2	1	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
300	#121	—	2	1	no	—
301	#121	C₂³×C₂₃	1	2	yes	—
302	C₄×Q₈	—	2	2	no	—
303	C₄×Q₈	—	2	1	no	—
304	C₄×Q₈	—	4	4	no	—
305	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
306	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
307	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
308	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
309	C₄×Q₈	D₄×C₂₃	1	2	yes	—
310	C₄×Q₈	D₄×C₂₃	1	2	yes	—
311	C₄×D₄	—	2	4	no	—
312	C₄×D₄	—	2	2	no	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
315	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
316	#117	—	2	1	no	—
317	#117	—	2	2	no	—
318	#117	—	2	4	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
325	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
326	C₂×C₄²	—	2	2	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
329	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	4	no	—
333	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
334	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
335	#97	—	2	4	no	—
336	C₄×C₈	—	2	1	no	—
337	C₄×C₈	—	2	1	no	—
338	#95	—	2	2	no	—
339	#95	—	2	1	no	—
340	#95	C₂³×C₂₃	1	1	yes	characteristic
341	#95	C₂³×C₂₃	1	1	yes	characteristic
342	#95	C₂³×C₂₃	1	2	yes	—
343	C₂₃	#17628	1	1	yes	characteristic, direct product factor, Sylow
344	#41	—	4	2	no	—
345	#41	—	4	2	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
351	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
352	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
366	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	4	4	no	—
379	C₄⋊C₄	—	4	4	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
383	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
384	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
385	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
386	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
387	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	4	no	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	4	4	no	—
397	C₄²	—	4	2	no	—
398	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
399	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
400	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
401	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
403	C₄²	C₂×D₄×C₂₃	1	4	yes	—
404	C₂³	—	2	2	no	—
405	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
406	Q₈	—	2	4	no	—
407	Q₈	—	2	4	no	—
408	Q₈	—	4	2	no	—
409	Q₈	—	4	2	no	—
410	Q₈	—	8	1	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	8	1	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	8	4	no	—
435	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
436	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
437	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
438	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
439	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
440	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
441	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
442	C₂×C₄	C₂₃×#137	1	2	yes	—
443	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
445	C₈	—	4	2	no	—
446	C₈	—	4	2	no	—
447	C₂²	—	2	1	no	—
448	C₂²	—	2	1	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
453	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
454	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	4	no	—
458	C₄	—	2	4	no	—
459	C₄	—	2	1	no	—
460	C₄	—	2	1	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	8	2	no	—
466	C₄	—	8	1	no	—
467	C₄	C₂₃×#417	1	2	yes	—
468	C₄	C₂₃×#417	1	2	yes	—
469	C₂	—	2	1	no	—
470	C₂	—	2	1	no	—
471	C₂	—	8	2	no	—
472	C₂	C₂₃×#2884	1	1	yes	characteristic
473	C₂	C₂×C₂₃×#417	1	1	yes	characteristic
474	C₂	C₂₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2983	C₂	semidirect
3. C₂₃×#2351	C₂	semidirect
4. C₂₃×#1518	C₂	semidirect
6. C₂₃×#1512	C₂	semidirect
7. C₂₃×#1509	C₂	semidirect
8. C₂₃×#1503	C₂	semidirect
10. C₂₃×#559	C₄	semidirect
11. C₂₃×#557	C₄	semidirect
13. C₂₃×#556	C₄	semidirect
14. C₂₃×#554	C₄	semidirect
166. #17628	C₂₃	direct (N × H)
211. #2983	C₂×C₂₃	semidirect
212. #2351	C₂×C₂₃	semidirect
213. #1518	C₂×C₂₃	semidirect
215. #1512	C₂×C₂₃	semidirect
216. #1509	C₂×C₂₃	semidirect
217. #1503	C₂×C₂₃	semidirect
241. #559	C₄×C₂₃	semidirect
242. #557	C₄×C₂₃	semidirect
244. #556	C₄×C₂₃	semidirect
245. #554	C₄×C₂₃	semidirect
343. C₂₃	#17628	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×#3507	2
C₂₃×#2983	2
C₂₃×#2351	2
C₂₃×#1518	2
C₂₃×#1517	2
C₂₃×#1512	2
C₂₃×#1509	2
C₂₃×#1503	2
C₂₃×#1502	2
#17628	23

Subgroups — C23×#17628 #19165254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17628 #19165254