Subgroups — C₂₃×#17776 #19165402

Order 5888.

988 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3508	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1557	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1553	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1547	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1540	C₂	1	2	yes	maximal, complemented
12	C₂₃×#559	—	2	1	no	—
13	C₂₃×#559	—	2	1	no	—
14	C₂₃×#550	—	2	1	no	—
15	C₂₃×#550	—	2	1	no	—
16	C₂₃×#546	—	2	2	no	—
17	C₂₃×#546	—	2	2	no	—
18	C₂₃×#540	C₄	1	4	yes	—
19	C₂₃×#535	C₄	1	2	yes	—
20	C₂₃×#533	C₂²	1	1	yes	characteristic
21	C₂₃×#532	C₂²	1	2	yes	—
22	C₂₃×#532	C₂²	1	2	yes	—
23	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
24	C₂₃×#518	C₄	1	2	yes	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#446	—	2	1	no	—
27	C₈×Q₈×C₂₃	—	2	1	no	—
28	C₂₃×#436	—	2	1	no	—
29	C₂₃×#436	—	2	2	no	—
30	C₂₃×#435	—	2	2	no	—
31	C₈×D₄×C₂₃	—	2	1	no	—
32	C₂₃×#432	C₂²	1	2	yes	complemented
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#431	C₂²	1	1	yes	characteristic
35	C₂₃×#431	C₂²	1	2	yes	complemented
36	C₂₃×#423	C₂²	1	1	yes	characteristic
37	C₂₃×#423	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	2	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂₃×#332	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#332	C₂²	1	2	yes	complemented
42	C₂₃×#332	C₂²	1	1	yes	characteristic, complemented
43	C₂₃×#332	C₂²	1	1	yes	characteristic, complemented
44	C₂₃×#332	C₂²	1	1	yes	characteristic, complemented
45	C₂₃×#332	C₂²	1	2	yes	complemented
46	C₂₃×#331	C₂²	1	2	yes	—
47	C₂₃×#331	C₂²	1	2	yes	—
48	C₂₃×#331	C₂²	1	2	yes	—
49	C₂₃×#331	C₂²	1	2	yes	—
50	C₂×#41×C₂₃	—	2	2	no	—
51	C₂×#41×C₂₃	—	2	2	no	—
52	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₂₃	—	2	1	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂×M₁₆×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×#127	—	2	2	no	—
63	C₂₃×#127	—	2	1	no	—
64	C₂₃×#127	—	2	1	no	—
65	C₂₃×#127	C₂×C₄	1	2	yes	—
66	C₂₃×#127	C₂³	1	2	yes	—
67	C₂₃×#125	C₂×C₄	1	2	yes	—
68	C₂₃×#124	—	2	2	no	—
69	C₂₃×#124	—	2	2	no	—
70	C₂₃×#124	—	2	2	no	—
71	C₂₃×#123	—	2	4	no	—
72	C₂₃×#123	—	2	4	no	—
73	C₂₃×C₂²⋊Q₈	—	2	1	no	—
74	C₂₃×C₂²⋊Q₈	—	2	1	no	—
75	C₂₃×#121	—	2	2	no	—
76	C₂₃×#121	—	2	2	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	—	2	4	no	—
79	C₂₃×#121	C₂×C₄	1	4	yes	—
80	C₂₃×#121	C₂×C₄	1	4	yes	—
81	C₂₃×#120	—	2	4	no	—
82	C₄×Q₈×C₂₃	—	2	1	no	—
83	C₄×Q₈×C₂₃	—	2	1	no	—
84	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₄×D₄×C₂₃	—	2	2	no	—
90	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
91	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
93	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
94	C₄×D₄×C₂₃	D₄	1	2	yes	—
95	C₄×D₄×C₂₃	D₄	1	2	yes	—
96	C₂₃×#117	C₂×C₄	1	2	yes	—
97	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
98	C₂₃×#117	C₂×C₄	1	2	yes	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
100	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×C₄⋊C₈	—	2	2	no	—
103	C₂₃×C₄⋊C₈	—	2	1	no	—
104	C₂₃×C₄⋊C₈	—	2	1	no	—
105	C₂₃×C₄⋊C₈	—	2	2	no	—
106	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
107	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₂₃×C₂²⋊C₈	—	2	1	no	—
113	C₂₃×C₂²⋊C₈	—	2	1	no	—
114	C₂₃×C₂²⋊C₈	—	2	2	no	—
115	C₂₃×C₂²⋊C₈	—	2	2	no	—
116	C₂₃×#97	—	2	2	no	—
117	C₄×C₈×C₂₃	—	2	1	no	—
118	C₄×C₈×C₂₃	—	2	1	no	—
119	C₄×C₈×C₂₃	—	2	2	no	—
120	C₂⁴×C₂₃	—	2	2	no	—
121	#41×C₂₃	—	4	2	no	—
122	#41×C₂₃	—	8	2	no	—
123	#41×C₂₃	—	8	2	no	—
124	#41×C₂₃	—	8	2	no	—
125	#41×C₂₃	—	8	2	no	—
126	C₂×Q₈×C₂₃	—	2	1	no	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×D₄×C₂₃	—	2	4	no	—
130	C₂×D₄×C₂₃	—	2	4	no	—
131	C₂×D₄×C₂₃	—	2	2	no	—
132	C₂×D₄×C₂₃	—	2	2	no	—
133	C₂×D₄×C₂₃	—	2	2	no	—
134	C₂×D₄×C₂₃	—	2	2	no	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	—	4	4	no	—
137	C₂×D₄×C₂₃	—	8	1	no	—
138	C₂×D₄×C₂₃	—	8	1	no	—
139	C₂×D₄×C₂₃	—	8	1	no	—
140	C₂×D₄×C₂₃	—	8	1	no	—
141	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
143	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	4	no	—
146	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
148	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	M₁₆×C₂₃	—	8	1	no	—
153	M₁₆×C₂₃	—	8	1	no	—
154	M₁₆×C₂₃	—	8	2	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	2	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	8	2	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
169	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
170	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
171	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
172	C₂²⋊C₄×C₂₃	—	2	2	no	—
173	C₂²⋊C₄×C₂₃	—	2	4	no	—
174	C₂²⋊C₄×C₂₃	—	2	4	no	—
175	C₂²⋊C₄×C₂₃	—	2	4	no	—
176	C₂²⋊C₄×C₂₃	—	2	4	no	—
177	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
178	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
179	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
180	C₄²×C₂₃	—	2	2	no	—
181	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
182	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₂₃	C₂×D₄	1	2	yes	—
184	C₄²×C₂₃	C₂×D₄	1	2	yes	—
185	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
188	#17776	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₂₃	—	2	4	no	—
190	C₂³×C₂₃	—	4	4	no	—
191	C₂³×C₂₃	—	8	2	no	—
192	C₂³×C₂₃	—	8	2	no	—
193	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
195	Q₈×C₂₃	—	2	2	no	—
196	Q₈×C₂₃	—	4	1	no	—
197	Q₈×C₂₃	—	4	1	no	—
198	Q₈×C₂₃	—	4	1	no	—
199	Q₈×C₂₃	—	4	1	no	—
200	D₄×C₂₃	—	2	2	no	—
201	D₄×C₂₃	—	4	2	no	—
202	D₄×C₂₃	—	4	4	no	—
203	D₄×C₂₃	—	4	2	no	—
204	D₄×C₂₃	—	4	2	no	—
205	D₄×C₂₃	—	4	2	no	—
206	D₄×C₂₃	—	4	2	no	—
207	D₄×C₂₃	—	4	4	no	—
208	D₄×C₂₃	—	8	2	no	—
209	D₄×C₂₃	—	8	2	no	—
210	D₄×C₂₃	—	8	4	no	—
211	D₄×C₂₃	—	8	4	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	4	2	no	—
224	C₂×C₄×C₂₃	—	8	4	no	—
225	C₂×C₄×C₂₃	—	8	4	no	—
226	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
235	C₈×C₂₃	—	4	1	no	—
236	C₈×C₂₃	—	4	2	no	—
237	C₈×C₂₃	—	4	2	no	—
238	C₈×C₂₃	—	4	1	no	—
239	C₈×C₂₃	—	4	1	no	—
240	C₈×C₂₃	—	4	1	no	—
241	#3508	C₂×C₂₃	1	1	yes	characteristic
242	#2991	C₂×C₂₃	1	1	yes	characteristic, complemented
243	#2990	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#1561	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#1557	C₂×C₂₃	1	2	yes	complemented
246	#1553	C₂×C₂₃	1	2	yes	complemented
247	#1547	C₂×C₂₃	1	2	yes	complemented
248	#1545	C₂×C₂₃	1	1	yes	characteristic, complemented
249	#1545	C₂×C₂₃	1	1	yes	characteristic, complemented
250	#1544	C₂×C₂₃	1	1	yes	characteristic, complemented
251	#1540	C₂×C₂₃	1	2	yes	complemented
252	C₂²×C₂₃	—	2	1	no	—
253	C₂²×C₂₃	—	2	2	no	—
254	C₂²×C₂₃	—	2	2	no	—
255	C₂²×C₂₃	—	2	2	no	—
256	C₂²×C₂₃	—	2	1	no	—
257	C₂²×C₂₃	—	2	1	no	—
258	C₂²×C₂₃	—	4	4	no	—
259	C₂²×C₂₃	—	4	4	no	—
260	C₂²×C₂₃	—	4	4	no	—
261	C₂²×C₂₃	—	16	4	no	—
262	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
263	C₄×C₂₃	—	2	1	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	2	2	no	—
266	C₄×C₂₃	—	2	2	no	—
267	C₄×C₂₃	—	2	1	no	—
268	C₄×C₂₃	—	2	1	no	—
269	C₄×C₂₃	—	4	2	no	—
270	C₄×C₂₃	—	4	2	no	—
271	C₄×C₂₃	#413	1	2	yes	—
272	C₄×C₂₃	#576	1	1	yes	characteristic
273	C₄×C₂₃	#576	1	1	yes	characteristic
274	C₄×C₂₃	#577	1	1	yes	characteristic
275	C₄×C₂₃	#577	1	1	yes	characteristic
276	#559	—	2	1	no	—
277	#559	—	2	1	no	—
278	#550	—	2	1	no	—
279	#550	—	2	1	no	—
280	#546	—	2	2	no	—
281	#546	—	2	2	no	—
282	#540	C₄×C₂₃	1	4	yes	—
283	#535	C₄×C₂₃	1	2	yes	—
284	#533	C₂²×C₂₃	1	1	yes	characteristic
285	#532	C₂²×C₂₃	1	2	yes	—
286	#532	C₂²×C₂₃	1	2	yes	—
287	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
288	#518	C₄×C₂₃	1	2	yes	—
289	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
290	#446	—	2	1	no	—
291	C₈×Q₈	—	2	1	no	—
292	#436	—	2	1	no	—
293	#436	—	2	2	no	—
294	#435	—	2	2	no	—
295	C₈×D₄	—	2	1	no	—
296	#432	C₂²×C₂₃	1	2	yes	complemented
297	#431	C₂²×C₂₃	1	1	yes	characteristic
298	#431	C₂²×C₂₃	1	1	yes	characteristic
299	#431	C₂²×C₂₃	1	2	yes	complemented
300	#423	C₂²×C₂₃	1	1	yes	characteristic
301	#423	C₂²×C₂₃	1	1	yes	characteristic
302	#423	C₂²×C₂₃	1	2	yes	complemented
303	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
304	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
305	#332	C₂²×C₂₃	1	2	yes	complemented
306	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
307	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
308	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
309	#332	C₂²×C₂₃	1	2	yes	complemented
310	#331	C₂²×C₂₃	1	2	yes	—
311	#331	C₂²×C₂₃	1	2	yes	—
312	#331	C₂²×C₂₃	1	2	yes	—
313	#331	C₂²×C₂₃	1	2	yes	—
314	C₂×C₂₃	—	4	2	no	—
315	C₂×C₂₃	—	4	1	no	—
316	C₂×C₂₃	—	8	4	no	—
317	C₂×C₂₃	C₂×#413	1	1	yes	characteristic
318	C₂×C₂₃	#2897	1	1	yes	characteristic
319	C₂×C₂₃	#2897	1	1	yes	characteristic
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
326	C₂×M₁₆	—	2	1	no	—
327	C₂×M₁₆	—	2	1	no	—
328	C₂×M₁₆	—	2	2	no	—
329	C₂²×C₈	—	2	2	no	—
330	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
331	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
332	#127	—	2	2	no	—
333	#127	—	2	1	no	—
334	#127	—	2	1	no	—
335	#127	C₂×C₄×C₂₃	1	2	yes	—
336	#127	C₂³×C₂₃	1	2	yes	—
337	#125	C₂×C₄×C₂₃	1	2	yes	—
338	#124	—	2	2	no	—
339	#124	—	2	2	no	—
340	#124	—	2	2	no	—
341	#123	—	2	4	no	—
342	#123	—	2	4	no	—
343	C₂²⋊Q₈	—	2	1	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	#121	—	2	2	no	—
346	#121	—	2	2	no	—
347	#121	—	2	4	no	—
348	#121	—	2	4	no	—
349	#121	C₂×C₄×C₂₃	1	4	yes	—
350	#121	C₂×C₄×C₂₃	1	4	yes	—
351	#120	—	2	4	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
355	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
361	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
362	C₄×D₄	D₄×C₂₃	1	2	yes	—
363	C₄×D₄	D₄×C₂₃	1	2	yes	—
364	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
365	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
366	#117	C₂×C₄×C₂₃	1	2	yes	—
367	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
368	#117	C₂×C₄×C₂₃	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	4	no	—
370	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
372	C₄⋊C₈	—	2	2	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	2	no	—
376	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
377	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
378	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
379	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₂²⋊C₈	—	2	1	no	—
383	C₂²⋊C₈	—	2	1	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	#97	—	2	2	no	—
387	C₄×C₈	—	2	1	no	—
388	C₄×C₈	—	2	1	no	—
389	C₄×C₈	—	2	2	no	—
390	C₂₃	#17776	1	1	yes	characteristic, direct product factor, Sylow
391	C₂⁴	—	2	2	no	—
392	#41	—	4	2	no	—
393	#41	—	8	2	no	—
394	#41	—	8	2	no	—
395	#41	—	8	2	no	—
396	#41	—	8	2	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	8	1	no	—
409	C₂×D₄	—	8	1	no	—
410	C₂×D₄	—	8	1	no	—
411	C₂×D₄	—	8	1	no	—
412	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
413	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
414	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
418	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
423	M₁₆	—	8	1	no	—
424	M₁₆	—	8	1	no	—
425	M₁₆	—	8	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	8	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
438	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
439	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
440	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
441	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
442	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
449	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
450	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
451	C₄²	—	2	2	no	—
452	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
453	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
454	C₄²	C₂×D₄×C₂₃	1	2	yes	—
455	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₄²	C₂×D₄×C₂₃	1	2	yes	—
458	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
459	C₂³	—	2	4	no	—
460	C₂³	—	4	4	no	—
461	C₂³	—	8	2	no	—
462	C₂³	—	8	2	no	—
463	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
464	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
465	Q₈	—	2	2	no	—
466	Q₈	—	4	1	no	—
467	Q₈	—	4	1	no	—
468	Q₈	—	4	1	no	—
469	Q₈	—	4	1	no	—
470	D₄	—	2	2	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	4	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	4	no	—
478	D₄	—	8	2	no	—
479	D₄	—	8	2	no	—
480	D₄	—	8	4	no	—
481	D₄	—	8	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	8	4	no	—
495	C₂×C₄	—	8	4	no	—
496	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
497	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
498	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
500	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
501	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
502	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
505	C₈	—	4	1	no	—
506	C₈	—	4	2	no	—
507	C₈	—	4	1	no	—
508	C₈	—	4	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	2	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	4	no	—
520	C₂²	—	16	4	no	—
521	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
522	C₄	—	2	1	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	C₂₃×#413	1	2	yes	—
531	C₄	C₂₃×#576	1	1	yes	characteristic
532	C₄	C₂₃×#576	1	1	yes	characteristic
533	C₄	C₂₃×#577	1	1	yes	characteristic
534	C₄	C₂₃×#577	1	1	yes	characteristic
535	C₂	—	4	1	no	—
536	C₂	—	4	2	no	—
537	C₂	—	8	4	no	—
538	C₂	C₂×C₂₃×#413	1	1	yes	characteristic
539	C₂	C₂₃×#2897	1	1	yes	characteristic
540	C₂	C₂₃×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2991	C₂	semidirect
3. C₂₃×#2990	C₂	semidirect
4. C₂₃×#1561	C₂	semidirect
5. C₂₃×#1557	C₂	semidirect
6. C₂₃×#1553	C₂	semidirect
7. C₂₃×#1547	C₂	semidirect
8. C₂₃×#1545	C₂	semidirect
9. C₂₃×#1545	C₂	semidirect
10. C₂₃×#1544	C₂	semidirect
11. C₂₃×#1540	C₂	semidirect
32. C₂₃×#432	C₂²	semidirect
35. C₂₃×#431	C₂²	semidirect
38. C₂₃×#423	C₂²	semidirect
39. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
40. C₂₃×#332	C₂²	semidirect
41. C₂₃×#332	C₂²	semidirect
42. C₂₃×#332	C₂²	semidirect
43. C₂₃×#332	C₂²	semidirect
44. C₂₃×#332	C₂²	semidirect
45. C₂₃×#332	C₂²	semidirect
188. #17776	C₂₃	direct (N × H)
242. #2991	C₂×C₂₃	semidirect
243. #2990	C₂×C₂₃	semidirect
244. #1561	C₂×C₂₃	semidirect
245. #1557	C₂×C₂₃	semidirect
246. #1553	C₂×C₂₃	semidirect
247. #1547	C₂×C₂₃	semidirect
248. #1545	C₂×C₂₃	semidirect
249. #1545	C₂×C₂₃	semidirect
250. #1544	C₂×C₂₃	semidirect
251. #1540	C₂×C₂₃	semidirect
296. #432	C₂²×C₂₃	semidirect
299. #431	C₂²×C₂₃	semidirect
302. #423	C₂²×C₂₃	semidirect
303. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
304. #332	C₂²×C₂₃	semidirect
305. #332	C₂²×C₂₃	semidirect
306. #332	C₂²×C₂₃	semidirect
307. #332	C₂²×C₂₃	semidirect
308. #332	C₂²×C₂₃	semidirect
309. #332	C₂²×C₂₃	semidirect
390. C₂₃	#17776	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×#3508	2
C₂₃×#2991	2
C₂₃×#2990	2
C₂₃×#1561	2
C₂₃×#1557	2
C₂₃×#1553	2
C₂₃×#1547	2
C₂₃×#1545	2
C₂₃×#1544	2
C₂₃×#1540	2
#17776	23

Subgroups — C23×#17776 #19165402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17776 #19165402