Subgroups — C₂₃×#18622 #19166248

Order 5888.

1066 conjugacy classes of subgroups.

The subgroup lattice has 536 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, complemented
2	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3038	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1623	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1622	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, complemented
10	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
11	C₂²×SD₁₆×C₂₃	—	2	2	no	—
12	Q₈²×C₂₃	C₂²	1	2	yes	complemented
13	C₂₃×#557	—	2	1	no	—
14	C₂₃×#554	—	2	2	no	—
15	C₂₃×#554	C₂²	1	2	yes	complemented
16	D₄×Q₈×C₂₃	—	2	2	no	—
17	D₄×Q₈×C₂₃	—	2	1	no	—
18	C₂₃×#544	—	2	2	no	—
19	C₂₃×#544	—	2	1	no	—
20	C₂₃×#533	—	2	1	no	—
21	C₂₃×#533	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
25	C₂₃×#520	—	2	1	no	—
26	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#492	—	2	1	no	—
28	C₂₃×#492	—	2	2	no	—
29	C₂₃×#492	C₂²	1	4	yes	—
30	C₂₃×#486	—	2	1	no	—
31	C₂₃×#460	—	2	1	no	—
32	C₂₃×#460	C₂²	1	4	yes	—
33	C₂₃×#460	C₂²	1	2	yes	—
34	C₂₃×#459	—	2	1	no	—
35	C₂₃×#459	C₂²	1	2	yes	—
36	C₂₃×#450	—	2	2	no	—
37	C₂₃×#431	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#418	—	2	1	no	—
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#333	C₂²	1	4	yes	complemented
42	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
43	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
44	C₂₃×D₄⋊C₈	C₂²	1	2	yes	complemented
45	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₂₃	—	2	2	no	—
47	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	2	1	no	—
51	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
52	C₂×SD₁₆×C₂₃	—	2	8	no	—
53	C₂×SD₁₆×C₂₃	—	2	2	no	—
54	C₂×SD₁₆×C₂₃	—	2	4	no	—
55	C₂×SD₁₆×C₂₃	—	4	8	no	—
56	C₂×D₈×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	2	no	—
60	C₂₃×C₄⋊Q₈	—	2	2	no	—
61	C₂₃×C₄⋊Q₈	—	2	4	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	—	2	1	no	—
65	C₂₃×C₄⋊Q₈	—	2	1	no	—
66	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
67	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
70	C₂₃×#127	—	2	1	no	—
71	C₂₃×#127	—	2	1	no	—
72	C₂₃×#127	—	2	1	no	—
73	C₂₃×#127	C₂³	1	2	yes	—
74	C₂₃×#125	—	2	4	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	2	no	—
77	C₂₃×#125	—	2	1	no	—
78	C₂₃×C₂²⋊Q₈	—	2	2	no	—
79	C₂₃×C₂²⋊Q₈	—	2	4	no	—
80	C₂₃×C₂²⋊Q₈	—	2	4	no	—
81	C₂₃×C₂²⋊Q₈	—	2	4	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
85	C₄×Q₈×C₂₃	—	2	2	no	—
86	C₄×Q₈×C₂₃	—	2	2	no	—
87	C₄×Q₈×C₂₃	—	2	2	no	—
88	C₄×Q₈×C₂₃	—	2	1	no	—
89	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
91	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₄×D₄×C₂₃	—	2	4	no	—
93	C₄×D₄×C₂₃	—	2	2	no	—
94	C₄×D₄×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
97	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂₃×#117	—	2	2	no	—
99	C₂₃×#117	—	2	1	no	—
100	C₂₃×#117	—	2	1	no	—
101	C₂₃×#117	D₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂₃×C₄⋊C₈	—	2	1	no	—
109	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
110	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
111	C₂₃×#102	—	2	2	no	—
112	C₂₃×#102	—	2	2	no	—
113	C₂₃×#102	—	2	4	no	—
114	C₂₃×C₂²⋊C₈	—	2	1	no	—
115	C₄×C₈×C₂₃	—	2	1	no	—
116	C₄×C₈×C₂₃	—	2	1	no	—
117	C₄×C₈×C₂₃	C₂³	1	2	yes	—
118	C₂⁴×C₂₃	—	4	1	no	—
119	#41×C₂₃	—	2	2	no	—
120	#41×C₂₃	SD₁₆	1	4	yes	complemented
121	C₂×Q₈×C₂₃	—	2	4	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	2	1	no	—
124	C₂×Q₈×C₂₃	—	4	4	no	—
125	C₂×Q₈×C₂₃	—	8	4	no	—
126	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	—	2	2	no	—
131	C₂×D₄×C₂₃	—	2	2	no	—
132	C₂×D₄×C₂₃	—	2	2	no	—
133	C₂×D₄×C₂₃	—	4	4	no	—
134	C₂×D₄×C₂₃	—	4	1	no	—
135	C₂×D₄×C₂₃	—	4	1	no	—
136	C₂×D₄×C₂₃	—	4	1	no	—
137	C₂×D₄×C₂₃	—	4	2	no	—
138	C₂×D₄×C₂₃	—	4	2	no	—
139	C₂×D₄×C₂₃	—	8	2	no	—
140	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	2	2	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	SD₁₆×C₂₃	—	4	16	no	—
150	SD₁₆×C₂₃	—	4	4	no	—
151	SD₁₆×C₂₃	—	8	4	no	—
152	D₈×C₂₃	—	4	4	no	—
153	M₁₆×C₂₃	—	4	2	no	—
154	C₂×C₈×C₂₃	—	2	4	no	—
155	C₂×C₈×C₂₃	—	2	2	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	4	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	2	no	—
169	C₄⋊C₄×C₂₃	—	2	2	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²⋊C₄×C₂₃	—	2	4	no	—
177	C₂²⋊C₄×C₂₃	—	2	2	no	—
178	C₂²⋊C₄×C₂₃	—	2	1	no	—
179	C₂²⋊C₄×C₂₃	—	2	1	no	—
180	C₂²⋊C₄×C₂₃	—	2	1	no	—
181	C₄²×C₂₃	—	2	4	no	—
182	C₄²×C₂₃	—	2	1	no	—
183	C₄²×C₂₃	—	2	1	no	—
184	C₄²×C₂₃	—	2	1	no	—
185	C₄²×C₂₃	—	2	2	no	—
186	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
190	#18622	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₂₃	—	2	1	no	—
192	C₂³×C₂₃	—	4	2	no	—
193	C₂³×C₂₃	—	8	2	no	—
194	C₂³×C₂₃	—	8	2	no	—
195	C₂³×C₂₃	—	8	2	no	—
196	C₂³×C₂₃	#120	1	1	yes	characteristic
197	Q₈×C₂₃	—	2	1	no	—
198	Q₈×C₂₃	—	2	4	no	—
199	Q₈×C₂₃	—	4	4	no	—
200	Q₈×C₂₃	—	4	4	no	—
201	Q₈×C₂₃	—	4	2	no	—
202	Q₈×C₂₃	—	8	4	no	—
203	Q₈×C₂₃	C₂×SD₁₆	1	2	yes	complemented
204	D₄×C₂₃	—	2	4	no	—
205	D₄×C₂₃	—	2	1	no	—
206	D₄×C₂₃	—	4	4	no	—
207	D₄×C₂₃	—	4	2	no	—
208	D₄×C₂₃	—	4	2	no	—
209	D₄×C₂₃	—	4	4	no	—
210	D₄×C₂₃	—	8	2	no	—
211	D₄×C₂₃	—	8	2	no	—
212	D₄×C₂₃	—	8	2	no	—
213	D₄×C₂₃	C₂×SD₁₆	1	2	yes	complemented
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	4	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	4	2	no	—
224	C₂×C₄×C₂₃	—	4	2	no	—
225	C₂×C₄×C₂₃	—	8	2	no	—
226	C₂×C₄×C₂₃	—	8	4	no	—
227	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
228	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
229	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
231	C₂×C₄×C₂₃	#137	1	2	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	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	—	2	4	no	—
236	C₈×C₂₃	—	4	2	no	—
237	C₈×C₂₃	—	4	2	no	—
238	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#3208	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#3038	C₂×C₂₃	1	1	yes	characteristic
242	#1623	C₂×C₂₃	1	4	yes	complemented
243	#1622	C₂×C₂₃	1	4	yes	complemented
244	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
247	C₂²×C₂₃	—	2	1	no	—
248	C₂²×C₂₃	—	2	1	no	—
249	C₂²×C₂₃	—	2	2	no	—
250	C₂²×C₂₃	—	4	1	no	—
251	C₂²×C₂₃	—	4	1	no	—
252	C₂²×C₂₃	—	8	2	no	—
253	C₂²×C₂₃	—	8	2	no	—
254	C₂²×C₂₃	—	8	2	no	—
255	C₂²×C₂₃	—	16	4	no	—
256	C₂²×C₂₃	#448	1	2	yes	—
257	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
258	C₄×C₂₃	—	2	2	no	—
259	C₄×C₂₃	—	2	1	no	—
260	C₄×C₂₃	—	2	1	no	—
261	C₄×C₂₃	—	4	1	no	—
262	C₄×C₂₃	—	4	2	no	—
263	C₄×C₂₃	—	4	1	no	—
264	C₄×C₂₃	—	4	4	no	—
265	C₄×C₂₃	—	4	2	no	—
266	C₄×C₂₃	#448	1	2	yes	—
267	C₄×C₂₃	#453	1	2	yes	—
268	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
269	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
270	C₄×C₂₃	#577	1	2	yes	—
271	C₂×AGL(1,ℤ/8)	—	2	1	no	—
272	C₂²×SD₁₆	—	2	2	no	—
273	Q₈²	C₂²×C₂₃	1	2	yes	complemented
274	#557	—	2	1	no	—
275	#554	—	2	2	no	—
276	#554	C₂²×C₂₃	1	2	yes	complemented
277	D₄×Q₈	—	2	2	no	—
278	D₄×Q₈	—	2	1	no	—
279	#544	—	2	2	no	—
280	#544	—	2	1	no	—
281	#533	—	2	1	no	—
282	#533	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×C₄⋊Q₈	—	2	1	no	—
284	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
285	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
286	#520	—	2	1	no	—
287	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
288	#492	—	2	2	no	—
289	#492	—	2	1	no	—
290	#492	C₂²×C₂₃	1	4	yes	—
291	#486	—	2	1	no	—
292	#460	—	2	1	no	—
293	#460	C₂²×C₂₃	1	4	yes	—
294	#460	C₂²×C₂₃	1	2	yes	—
295	#459	—	2	1	no	—
296	#459	C₂²×C₂₃	1	2	yes	—
297	#450	—	2	2	no	—
298	#431	C₂²×C₂₃	1	1	yes	characteristic
299	#423	C₂²×C₂₃	1	1	yes	characteristic
300	#418	—	2	1	no	—
301	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
302	#333	C₂²×C₂₃	1	4	yes	complemented
303	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
304	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
305	D₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
306	C₂×C₂₃	—	2	2	no	—
307	C₂×C₂₃	—	4	2	no	—
308	C₂×C₂₃	—	16	2	no	—
309	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
310	C₂×C₂₃	#3010	1	1	yes	characteristic
311	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
312	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
313	C₂²×Q₈	—	2	2	no	—
314	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	—	2	1	no	—
318	AGL(1,ℤ/8)	—	4	4	no	—
319	C₂×SD₁₆	—	2	8	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	4	no	—
322	C₂×SD₁₆	—	4	8	no	—
323	C₂×D₈	—	2	2	no	—
324	C₂×M₁₆	—	2	1	no	—
325	C₂²×C₈	—	2	1	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	—	2	4	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	—	2	1	no	—
332	C₄⋊Q₈	—	2	1	no	—
333	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
334	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
335	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
337	#127	—	2	1	no	—
338	#127	—	2	1	no	—
339	#127	—	2	1	no	—
340	#127	C₂³×C₂₃	1	2	yes	—
341	#125	—	2	2	no	—
342	#125	—	2	4	no	—
343	#125	—	2	2	no	—
344	#125	—	2	1	no	—
345	C₂²⋊Q₈	—	2	2	no	—
346	C₂²⋊Q₈	—	2	4	no	—
347	C₂²⋊Q₈	—	2	4	no	—
348	C₂²⋊Q₈	—	2	2	no	—
349	C₂²⋊Q₈	—	2	4	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
352	C₄×Q₈	—	2	2	no	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
357	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
359	C₄×D₄	—	2	4	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	—	2	1	no	—
363	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
364	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
365	#117	—	2	2	no	—
366	#117	—	2	1	no	—
367	#117	—	2	1	no	—
368	#117	D₄×C₂₃	1	1	yes	characteristic
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	2	no	—
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
377	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
378	#102	—	2	2	no	—
379	#102	—	2	2	no	—
380	#102	—	2	4	no	—
381	C₂²⋊C₈	—	2	1	no	—
382	C₄×C₈	—	2	1	no	—
383	C₄×C₈	—	2	1	no	—
384	C₄×C₈	C₂³×C₂₃	1	2	yes	—
385	C₂₃	#18622	1	1	yes	characteristic, direct product factor, Sylow
386	C₂⁴	—	4	1	no	—
387	#41	—	2	2	no	—
388	#41	SD₁₆×C₂₃	1	4	yes	complemented
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	4	4	no	—
393	C₂×Q₈	—	8	4	no	—
394	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
395	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	4	4	no	—
402	C₂×D₄	—	4	1	no	—
403	C₂×D₄	—	4	1	no	—
404	C₂×D₄	—	4	1	no	—
405	C₂×D₄	—	4	2	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	—	8	2	no	—
408	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
415	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
417	SD₁₆	—	4	16	no	—
418	SD₁₆	—	4	4	no	—
419	SD₁₆	—	8	4	no	—
420	D₈	—	4	4	no	—
421	M₁₆	—	4	2	no	—
422	C₂×C₈	—	2	4	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
442	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	1	no	—
447	C₂²⋊C₄	—	2	1	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₄²	—	2	4	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	2	no	—
454	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
458	C₂³	—	2	1	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	8	2	no	—
461	C₂³	—	8	2	no	—
462	C₂³	—	8	2	no	—
463	C₂³	C₂₃×#120	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	4	no	—
466	Q₈	—	4	4	no	—
467	Q₈	—	4	4	no	—
468	Q₈	—	4	2	no	—
469	Q₈	—	8	4	no	—
470	Q₈	C₂×SD₁₆×C₂₃	1	2	yes	complemented
471	D₄	—	2	4	no	—
472	D₄	—	2	1	no	—
473	D₄	—	4	4	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	4	no	—
477	D₄	—	8	2	no	—
478	D₄	—	8	2	no	—
479	D₄	—	8	2	no	—
480	D₄	C₂×SD₁₆×C₂₃	1	2	yes	complemented
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	8	2	no	—
493	C₂×C₄	—	8	4	no	—
494	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
495	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
496	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
497	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
498	C₂×C₄	C₂₃×#137	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
500	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
502	C₈	—	2	4	no	—
503	C₈	—	4	2	no	—
504	C₈	—	4	2	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	1	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	4	1	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	8	2	no	—
511	C₂²	—	8	2	no	—
512	C₂²	—	8	2	no	—
513	C₂²	—	16	4	no	—
514	C₂²	C₂₃×#448	1	2	yes	—
515	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
516	C₄	—	2	2	no	—
517	C₄	—	2	1	no	—
518	C₄	—	2	1	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	4	no	—
523	C₄	—	4	2	no	—
524	C₄	C₂₃×#448	1	2	yes	—
525	C₄	C₂₃×#453	1	2	yes	—
526	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
527	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
528	C₄	C₂₃×#577	1	2	yes	—
529	C₂	—	2	2	no	—
530	C₂	—	4	2	no	—
531	C₂	—	16	2	no	—
532	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
533	C₂	C₂₃×#3010	1	1	yes	characteristic
534	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	C₂₃×#3038	C₂
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
1. C₂₃×#3503	C₂	semidirect
2. C₂₃×#3208	C₂	semidirect
3. C₂×C₂₃×#492	C₂	semidirect
5. C₂₃×#1623	C₂	semidirect
6. C₂₃×#1622	C₂	semidirect
7. C₂₃×#1548	C₂	semidirect
8. C₂₃×#1493	C₂	semidirect
9. C₂₃×#1476	C₂	semidirect
12. Q₈²×C₂₃	C₂²	semidirect
15. C₂₃×#554	C₂²	semidirect
41. C₂₃×#333	C₂²	semidirect
42. C₂₃×Q₈⋊C₈	C₂²	semidirect
43. C₂₃×Q₈⋊C₈	C₂²	semidirect
44. C₂₃×D₄⋊C₈	C₂²	semidirect
120. #41×C₂₃	SD₁₆	semidirect
190. #18622	C₂₃	direct (N × H)
203. Q₈×C₂₃	C₂×SD₁₆	semidirect
213. D₄×C₂₃	C₂×SD₁₆	semidirect
238. #3503	C₂×C₂₃	semidirect
239. #3208	C₂×C₂₃	semidirect
240. C₂×#492	C₂×C₂₃	semidirect
242. #1623	C₂×C₂₃	semidirect
243. #1622	C₂×C₂₃	semidirect
244. #1548	C₂×C₂₃	semidirect
245. #1493	C₂×C₂₃	semidirect
246. #1476	C₂×C₂₃	semidirect
273. Q₈²	C₂²×C₂₃	semidirect
276. #554	C₂²×C₂₃	semidirect
302. #333	C₂²×C₂₃	semidirect
303. Q₈⋊C₈	C₂²×C₂₃	semidirect
304. Q₈⋊C₈	C₂²×C₂₃	semidirect
305. D₄⋊C₈	C₂²×C₂₃	semidirect
385. C₂₃	#18622	direct (N × H)
388. #41	SD₁₆×C₂₃	semidirect
470. Q₈	C₂×SD₁₆×C₂₃	semidirect
480. D₄	C₂×SD₁₆×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×#3208	2
C₂×C₂₃×#492	2
C₂₃×#3038	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1548	2
C₂₃×#1493	2
C₂₃×#1476	2
#18622	23

Subgroups — C23×#18622 #19166248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18622 #19166248