Subgroups — C₂₃×#18623 #19166249

Order 5888.

970 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3079	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1633	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1632	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₂₃	—	2	2	no	—
11	Q₈²×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#557	—	2	1	no	—
13	C₂₃×#554	—	2	2	no	—
14	C₂₃×#554	C₂²	1	2	yes	—
15	D₄×Q₈×C₂₃	—	2	2	no	—
16	D₄×Q₈×C₂₃	—	2	1	no	—
17	C₂₃×#544	—	2	2	no	—
18	C₂₃×#544	—	2	1	no	—
19	C₂₃×#533	—	2	1	no	—
20	C₂₃×#533	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂₃×#520	—	2	1	no	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#500	—	2	1	no	—
27	C₂₃×#494	—	2	2	no	—
28	C₂₃×#494	C₂²	1	4	yes	—
29	C₂₃×#487	—	2	1	no	—
30	C₂₃×#484	—	2	2	no	—
31	C₂₃×#477	C₂²	1	4	yes	—
32	C₂₃×#477	C₂²	1	2	yes	—
33	C₂₃×#476	C₂²	1	2	yes	—
34	C₂₃×#475	—	2	1	no	—
35	C₂₃×#474	—	2	1	no	—
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#428	—	2	1	no	—
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#417	—	2	1	no	—
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#333	C₂²	1	4	yes	—
42	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₂₃	—	2	2	no	—
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
49	C₂×Q₁₆×C₂₃	—	2	8	no	—
50	C₂×Q₁₆×C₂₃	—	4	8	no	—
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂²×C₈×C₂₃	—	2	1	no	—
53	C₂₃×C₄⋊Q₈	—	2	2	no	—
54	C₂₃×C₄⋊Q₈	—	2	2	no	—
55	C₂₃×C₄⋊Q₈	—	2	4	no	—
56	C₂₃×C₄⋊Q₈	—	2	2	no	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
65	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
67	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₂₃×#125	—	2	4	no	—
69	C₂₃×#125	—	2	2	no	—
70	C₂₃×#125	—	2	2	no	—
71	C₂₃×#125	—	2	1	no	—
72	C₂₃×C₂²⋊Q₈	—	2	2	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	—	2	4	no	—
76	C₂₃×C₂²⋊Q₈	—	2	2	no	—
77	C₂₃×C₂²⋊Q₈	—	2	1	no	—
78	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	2	1	no	—
83	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
84	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
85	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₂₃	—	2	4	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	2	1	no	—
89	C₄×D₄×C₂₃	—	2	1	no	—
90	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
91	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×#117	—	2	2	no	—
93	C₂₃×#117	—	2	1	no	—
94	C₂₃×#117	—	2	1	no	—
95	C₂₃×#117	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄²×C₂₃	—	2	1	no	—
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂₃×#107	—	2	2	no	—
105	C₂₃×#107	—	2	4	no	—
106	C₂₃×#106	—	2	2	no	—
107	C₂₃×C₄⋊C₈	—	2	1	no	—
108	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
110	C₂₃×#103	—	2	2	no	—
111	C₂₃×#103	—	2	4	no	—
112	C₂₃×#102	—	2	2	no	—
113	C₂₃×C₂²⋊C₈	—	2	1	no	—
114	C₄×C₈×C₂₃	—	2	1	no	—
115	C₄×C₈×C₂₃	—	2	1	no	—
116	C₄×C₈×C₂₃	C₂³	1	2	yes	—
117	#41×C₂₃	—	2	2	no	—
118	#41×C₂₃	Q₁₆	1	4	yes	complemented
119	C₂×Q₈×C₂₃	—	2	4	no	—
120	C₂×Q₈×C₂₃	—	2	2	no	—
121	C₂×Q₈×C₂₃	—	2	1	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×Q₈×C₂₃	—	4	4	no	—
125	C₂×Q₈×C₂₃	—	4	1	no	—
126	C₂×Q₈×C₂₃	—	8	4	no	—
127	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
128	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
130	C₂×D₄×C₂₃	—	2	1	no	—
131	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	4	1	no	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	Q₁₆×C₂₃	—	4	16	no	—
142	M₁₆×C₂₃	—	4	2	no	—
143	C₂×C₈×C₂₃	—	2	4	no	—
144	C₂×C₈×C₂₃	—	2	2	no	—
145	C₂×C₈×C₂₃	—	2	2	no	—
146	C₂×C₈×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	4	no	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	2	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	4	1	no	—
165	C₄⋊C₄×C₂₃	—	4	1	no	—
166	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₂₃	—	2	4	no	—
170	C₂²⋊C₄×C₂₃	—	2	2	no	—
171	C₂²⋊C₄×C₂₃	—	2	1	no	—
172	C₂²⋊C₄×C₂₃	—	2	1	no	—
173	C₂²⋊C₄×C₂₃	—	2	1	no	—
174	C₄²×C₂₃	—	2	4	no	—
175	C₄²×C₂₃	—	2	1	no	—
176	C₄²×C₂₃	—	2	1	no	—
177	C₄²×C₂₃	—	2	1	no	—
178	C₄²×C₂₃	—	2	2	no	—
179	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
183	#18623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₂₃	—	2	1	no	—
185	C₂³×C₂₃	#120	1	1	yes	characteristic
186	Q₈×C₂₃	—	2	1	no	—
187	Q₈×C₂₃	—	2	4	no	—
188	Q₈×C₂₃	—	4	4	no	—
189	Q₈×C₂₃	—	4	4	no	—
190	Q₈×C₂₃	—	4	4	no	—
191	Q₈×C₂₃	—	4	2	no	—
192	Q₈×C₂₃	—	8	4	no	—
193	Q₈×C₂₃	—	8	2	no	—
194	Q₈×C₂₃	C₂×Q₁₆	1	2	yes	complemented
195	D₄×C₂₃	—	2	4	no	—
196	D₄×C₂₃	—	2	1	no	—
197	D₄×C₂₃	C₂×Q₁₆	1	2	yes	complemented
198	C₂×C₄×C₂₃	—	2	1	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	1	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	2	no	—
209	C₂×C₄×C₂₃	—	4	2	no	—
210	C₂×C₄×C₂₃	—	8	2	no	—
211	C₂×C₄×C₂₃	—	8	4	no	—
212	C₂×C₄×C₂₃	—	8	2	no	—
213	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
214	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
215	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
217	C₂×C₄×C₂₃	#137	1	2	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₈×C₂₃	—	2	4	no	—
222	C₈×C₂₃	—	4	2	no	—
223	C₈×C₂₃	—	4	2	no	—
224	#3503	C₂×C₂₃	1	1	yes	characteristic
225	#3230	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#3079	C₂×C₂₃	1	1	yes	characteristic
228	#1633	C₂×C₂₃	1	4	yes	complemented
229	#1632	C₂×C₂₃	1	4	yes	complemented
230	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#1476	C₂×C₂₃	1	1	yes	characteristic
233	C₂²×C₂₃	—	2	1	no	—
234	C₂²×C₂₃	—	2	1	no	—
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	4	1	no	—
237	C₂²×C₂₃	—	4	1	no	—
238	C₂²×C₂₃	#451	1	2	yes	—
239	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
240	C₄×C₂₃	—	2	2	no	—
241	C₄×C₂₃	—	2	1	no	—
242	C₄×C₂₃	—	2	1	no	—
243	C₄×C₂₃	—	4	1	no	—
244	C₄×C₂₃	—	4	2	no	—
245	C₄×C₂₃	—	4	1	no	—
246	C₄×C₂₃	—	4	4	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	8	2	no	—
249	C₄×C₂₃	#451	1	2	yes	—
250	C₄×C₂₃	#453	1	2	yes	—
251	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
252	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
253	C₄×C₂₃	#578	1	2	yes	—
254	C₂²×Q₁₆	—	2	2	no	—
255	Q₈²	C₂²×C₂₃	1	2	yes	—
256	#557	—	2	1	no	—
257	#554	—	2	2	no	—
258	#554	C₂²×C₂₃	1	2	yes	—
259	D₄×Q₈	—	2	2	no	—
260	D₄×Q₈	—	2	1	no	—
261	#544	—	2	2	no	—
262	#544	—	2	1	no	—
263	#533	—	2	1	no	—
264	#533	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×C₄⋊Q₈	—	2	1	no	—
266	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
268	#520	—	2	1	no	—
269	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
270	#500	—	2	1	no	—
271	#494	—	2	2	no	—
272	#494	C₂²×C₂₃	1	4	yes	—
273	#487	—	2	1	no	—
274	#484	—	2	2	no	—
275	#477	C₂²×C₂₃	1	2	yes	—
276	#477	C₂²×C₂₃	1	4	yes	—
277	#476	C₂²×C₂₃	1	2	yes	—
278	#475	—	2	1	no	—
279	#474	—	2	1	no	—
280	#431	C₂²×C₂₃	1	1	yes	characteristic
281	#428	—	2	1	no	—
282	#423	C₂²×C₂₃	1	1	yes	characteristic
283	#417	—	2	1	no	—
284	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
285	#333	C₂²×C₂₃	1	4	yes	—
286	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
287	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
288	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
289	C₂×C₂₃	—	2	2	no	—
290	C₂×C₂₃	—	4	2	no	—
291	C₂×C₂₃	C₂×#451	1	1	yes	characteristic
292	C₂×C₂₃	#3011	1	1	yes	characteristic
293	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
294	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
295	C₂²×Q₈	—	2	2	no	—
296	C₂²×Q₈	—	2	1	no	—
297	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
298	C₂×Q₁₆	—	2	8	no	—
299	C₂×Q₁₆	—	4	8	no	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂²×C₈	—	2	1	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	4	no	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	—	2	2	no	—
307	C₄⋊Q₈	—	2	1	no	—
308	C₄⋊Q₈	—	2	1	no	—
309	C₄⋊Q₈	—	2	1	no	—
310	C₄⋊Q₈	—	2	1	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
313	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
314	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
315	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
316	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
317	#125	—	2	2	no	—
318	#125	—	2	4	no	—
319	#125	—	2	2	no	—
320	#125	—	2	1	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	4	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	2	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	—	2	1	no	—
327	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
333	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
340	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
341	#117	—	2	2	no	—
342	#117	—	2	1	no	—
343	#117	—	2	1	no	—
344	#117	D₄×C₂₃	1	1	yes	characteristic
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄²	—	2	1	no	—
352	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
353	#107	—	2	2	no	—
354	#107	—	2	4	no	—
355	#106	—	2	2	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
358	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
359	#103	—	2	2	no	—
360	#103	—	2	4	no	—
361	#102	—	2	2	no	—
362	C₂²⋊C₈	—	2	1	no	—
363	C₄×C₈	—	2	1	no	—
364	C₄×C₈	—	2	1	no	—
365	C₄×C₈	C₂³×C₂₃	1	2	yes	—
366	C₂₃	#18623	1	1	yes	characteristic, direct product factor, Sylow
367	#41	—	2	2	no	—
368	#41	Q₁₆×C₂₃	1	4	yes	complemented
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	4	4	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	—	4	1	no	—
376	C₂×Q₈	—	8	4	no	—
377	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
378	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
379	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	4	1	no	—
388	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	Q₁₆	—	4	16	no	—
392	M₁₆	—	4	2	no	—
393	C₂×C₈	—	2	4	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	4	1	no	—
415	C₄⋊C₄	—	4	1	no	—
416	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
417	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
418	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	2	no	—
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₄²	—	2	4	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	2	no	—
429	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
433	C₂³	—	2	1	no	—
434	C₂³	C₂₃×#120	1	1	yes	characteristic
435	Q₈	—	2	1	no	—
436	Q₈	—	2	4	no	—
437	Q₈	—	4	4	no	—
438	Q₈	—	4	4	no	—
439	Q₈	—	4	4	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	8	4	no	—
442	Q₈	—	8	2	no	—
443	Q₈	C₂×Q₁₆×C₂₃	1	2	yes	complemented
444	D₄	—	2	4	no	—
445	D₄	—	2	1	no	—
446	D₄	C₂×Q₁₆×C₂₃	1	2	yes	complemented
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	8	2	no	—
460	C₂×C₄	—	8	4	no	—
461	C₂×C₄	—	8	2	no	—
462	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
463	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
464	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
465	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
466	C₂×C₄	C₂₃×#137	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
470	C₈	—	2	4	no	—
471	C₈	—	4	2	no	—
472	C₈	—	4	2	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	1	no	—
478	C₂²	C₂₃×#451	1	2	yes	—
479	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	8	2	no	—
489	C₄	C₂₃×#451	1	2	yes	—
490	C₄	C₂₃×#453	1	2	yes	—
491	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
492	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
493	C₄	C₂₃×#578	1	2	yes	—
494	C₂	—	2	2	no	—
495	C₂	—	4	2	no	—
496	C₂	C₂×C₂₃×#451	1	1	yes	characteristic
497	C₂	C₂₃×#3011	1	1	yes	characteristic
498	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3230	C₂	semidirect
3. C₂×C₂₃×#494	C₂	semidirect
5. C₂₃×#1633	C₂	semidirect
6. C₂₃×#1632	C₂	semidirect
7. C₂₃×#1548	C₂	semidirect
8. C₂₃×#1493	C₂	semidirect
118. #41×C₂₃	Q₁₆	semidirect
183. #18623	C₂₃	direct (N × H)
194. Q₈×C₂₃	C₂×Q₁₆	semidirect
197. D₄×C₂₃	C₂×Q₁₆	semidirect
225. #3230	C₂×C₂₃	semidirect
226. C₂×#494	C₂×C₂₃	semidirect
228. #1633	C₂×C₂₃	semidirect
229. #1632	C₂×C₂₃	semidirect
230. #1548	C₂×C₂₃	semidirect
231. #1493	C₂×C₂₃	semidirect
366. C₂₃	#18623	direct (N × H)
368. #41	Q₁₆×C₂₃	semidirect
443. Q₈	C₂×Q₁₆×C₂₃	semidirect
446. D₄	C₂×Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3503	2
C₂₃×#3230	2
C₂×C₂₃×#494	2
C₂₃×#3079	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1548	2
C₂₃×#1493	2
C₂₃×#1476	2
#18623	23

Subgroups — C23×#18623 #19166249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18623 #19166249