Subgroups — C₂₃×#17893 #19165519

Order 5888.

684 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1594	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1592	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1590	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1589	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic
13	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#446	C₂²	1	2	yes	complemented
16	C₂₃×#442	C₄	1	2	yes	—
17	C₂₃×#440	C₄	1	2	yes	—
18	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
19	C₄×D₈×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#436	—	2	1	no	—
21	C₂₃×#436	—	2	1	no	—
22	C₂₃×#436	C₂²	1	2	yes	—
23	C₂₃×#436	C₂²	1	2	yes	—
24	C₂₃×#436	C₂²	1	2	yes	complemented
25	C₂₃×#435	—	2	2	no	—
26	C₂₃×#432	—	2	1	no	—
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂₃×#431	C₂²	1	1	yes	characteristic
29	C₂₃×#428	C₄	1	1	yes	characteristic
30	C₂₃×#423	—	2	1	no	—
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₃×#418	C₄	1	1	yes	characteristic
33	C₂₃×#417	C₄	1	1	yes	characteristic
34	C₂₃×#407	—	2	2	no	—
35	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#335	C₂²	1	2	yes	complemented
38	C₂₃×#334	C₂²	1	2	yes	complemented
39	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
42	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
43	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₂₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₂₃	C₂²	1	2	yes	complemented
47	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₂₃	—	2	1	no	—
50	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
51	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₂₃	—	2	2	no	—
54	C₂×M₁₆×C₂₃	—	2	2	no	—
55	C₂×M₁₆×C₂₃	—	2	2	no	—
56	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	1	no	—
60	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₄×D₄×C₂₃	—	2	1	no	—
63	C₄×D₄×C₂₃	—	2	1	no	—
64	C₄×D₄×C₂₃	—	2	1	no	—
65	C₄×D₄×C₂₃	—	2	1	no	—
66	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
67	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₄×D₄×C₂₃	C₂³	1	2	yes	—
69	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#117	—	2	1	no	—
71	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×#107	C₂×C₄	1	2	yes	—
77	C₂₃×#106	C₂×C₄	1	2	yes	—
78	C₂₃×C₄⋊C₈	—	2	2	no	—
79	C₂₃×C₄⋊C₈	—	2	1	no	—
80	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
81	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
83	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₂₃×#103	C₂×C₄	1	2	yes	—
86	C₂₃×#102	C₂×C₄	1	2	yes	—
87	C₂₃×#102	C₂×C₄	1	2	yes	—
88	C₂₃×#102	C₂×C₄	1	2	yes	—
89	C₂₃×C₂²⋊C₈	—	2	1	no	—
90	C₂₃×C₂²⋊C₈	—	2	2	no	—
91	C₂₃×C₂²⋊C₈	—	2	2	no	—
92	C₂₃×C₂²⋊C₈	—	2	2	no	—
93	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
94	C₂₃×#97	—	2	2	no	—
95	C₂₃×#97	C₂×C₄	1	2	yes	—
96	C₄×C₈×C₂₃	—	2	1	no	—
97	C₄×C₈×C₂₃	—	2	1	no	—
98	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₂₃	C₂³	1	2	yes	—
101	C₄×C₈×C₂₃	C₂³	1	2	yes	—
102	C₂⁴×C₂₃	—	2	1	no	—
103	#41×C₂₃	—	2	2	no	—
104	#41×C₂₃	—	2	2	no	—
105	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₂₃	—	2	1	no	—
107	C₂×D₄×C₂₃	—	2	1	no	—
108	C₂×D₄×C₂₃	—	2	2	no	—
109	C₂×D₄×C₂₃	—	2	2	no	—
110	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₂₃	—	2	1	no	—
113	C₂²×C₄×C₂₃	—	2	1	no	—
114	C₂²×C₄×C₂₃	—	2	1	no	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	8	1	no	—
119	C₂²×C₄×C₂₃	—	8	1	no	—
120	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
123	SD₁₆×C₂₃	—	2	4	no	—
124	D₈×C₂₃	—	2	4	no	—
125	M₁₆×C₂₃	—	2	2	no	—
126	M₁₆×C₂₃	—	8	2	no	—
127	M₁₆×C₂₃	—	8	2	no	—
128	M₁₆×C₂₃	—	8	2	no	—
129	M₁₆×C₂₃	M₁₆	1	4	yes	complemented
130	C₂×C₈×C₂₃	—	2	1	no	—
131	C₂×C₈×C₂₃	—	2	2	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	—	2	1	no	—
135	C₂×C₈×C₂₃	—	4	2	no	—
136	C₂×C₈×C₂₃	—	8	1	no	—
137	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₂₃	—	2	1	no	—
148	C₂²⋊C₄×C₂₃	—	2	2	no	—
149	C₂²⋊C₄×C₂₃	—	2	1	no	—
150	C₂²⋊C₄×C₂₃	—	2	2	no	—
151	C₂²⋊C₄×C₂₃	—	2	1	no	—
152	C₄²×C₂₃	—	2	1	no	—
153	C₄²×C₂₃	—	2	1	no	—
154	C₄²×C₂₃	—	2	1	no	—
155	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₂₃	#41	1	1	yes	characteristic
158	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
159	#17893	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₂₃	—	2	1	no	—
161	C₂³×C₂₃	—	2	2	no	—
162	C₂³×C₂₃	—	4	2	no	—
163	C₂³×C₂₃	—	8	1	no	—
164	C₂³×C₂₃	—	8	1	no	—
165	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
166	Q₈×C₂₃	—	2	1	no	—
167	Q₈×C₂₃	—	2	1	no	—
168	D₄×C₂₃	—	2	1	no	—
169	D₄×C₂₃	—	2	2	no	—
170	D₄×C₂₃	—	2	2	no	—
171	D₄×C₂₃	—	2	1	no	—
172	D₄×C₂₃	—	4	2	no	—
173	D₄×C₂₃	—	4	2	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	1	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	4	1	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	8	1	no	—
187	C₂×C₄×C₂₃	—	8	2	no	—
188	C₂×C₄×C₂₃	—	8	1	no	—
189	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₂₃	#131	1	2	yes	—
194	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
197	C₈×C₂₃	—	2	2	no	—
198	C₈×C₂₃	—	2	2	no	—
199	C₈×C₂₃	—	2	2	no	—
200	C₈×C₂₃	—	2	2	no	—
201	C₈×C₂₃	—	4	1	no	—
202	C₈×C₂₃	—	4	2	no	—
203	C₈×C₂₃	—	4	1	no	—
204	C₈×C₂₃	—	4	1	no	—
205	C₈×C₂₃	C₂×M₁₆	1	4	yes	—
206	#2989	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
208	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
209	#1594	C₂×C₂₃	1	2	yes	complemented
210	#1592	C₂×C₂₃	1	2	yes	complemented
211	#1590	C₂×C₂₃	1	2	yes	complemented
212	#1589	C₂×C₂₃	1	2	yes	complemented
213	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂²×C₂₃	—	2	1	no	—
218	C₂²×C₂₃	—	2	1	no	—
219	C₂²×C₂₃	—	4	1	no	—
220	C₂²×C₂₃	—	4	2	no	—
221	C₂²×C₂₃	—	4	2	no	—
222	C₂²×C₂₃	—	4	1	no	—
223	C₂²×C₂₃	—	4	1	no	—
224	C₂²×C₂₃	—	4	2	no	—
225	C₂²×C₂₃	—	8	4	no	—
226	C₂²×C₂₃	#436	1	2	yes	—
227	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	2	1	no	—
230	C₄×C₂₃	—	2	1	no	—
231	C₄×C₂₃	—	4	1	no	—
232	C₄×C₂₃	—	4	1	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	4	1	no	—
235	C₄×C₂₃	#436	1	2	yes	—
236	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
237	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
238	C₄×C₂₃	#576	1	1	yes	characteristic
239	C₄×C₂₃	#577	1	1	yes	characteristic
240	C₂×AGL(1,ℤ/8)	C₄×C₂₃	1	1	yes	characteristic
241	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
243	#446	C₂²×C₂₃	1	2	yes	complemented
244	#442	C₄×C₂₃	1	2	yes	—
245	#440	C₄×C₂₃	1	2	yes	—
246	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
247	C₄×D₈	C₂²×C₂₃	1	2	yes	—
248	#436	—	2	1	no	—
249	#436	—	2	1	no	—
250	#436	C₂²×C₂₃	1	2	yes	—
251	#436	C₂²×C₂₃	1	2	yes	—
252	#436	C₂²×C₂₃	1	2	yes	complemented
253	#435	—	2	2	no	—
254	#432	—	2	1	no	—
255	#431	C₂²×C₂₃	1	1	yes	characteristic
256	#431	C₂²×C₂₃	1	1	yes	characteristic
257	#428	C₄×C₂₃	1	1	yes	characteristic
258	#423	—	2	1	no	—
259	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
260	#418	C₄×C₂₃	1	1	yes	characteristic
261	#417	C₄×C₂₃	1	1	yes	characteristic
262	#407	—	2	2	no	—
263	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
265	#335	C₂²×C₂₃	1	2	yes	complemented
266	#334	C₂²×C₂₃	1	2	yes	complemented
267	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
269	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
270	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
271	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
272	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
273	#322	C₂²×C₂₃	1	2	yes	complemented
274	C₈²	C₂²×C₂₃	1	2	yes	complemented
275	C₂×C₂₃	—	2	2	no	—
276	C₂×C₂₃	—	8	1	no	—
277	C₂×C₂₃	—	8	2	no	—
278	C₂×C₂₃	C₂×#436	1	1	yes	characteristic
279	C₂×C₂₃	#2946	1	1	yes	characteristic
280	C₂×C₂₃	#2956	1	1	yes	characteristic
281	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
282	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	AGL(1,ℤ/8)	—	2	4	no	—
285	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
286	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	2	2	no	—
290	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
291	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
295	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
296	C₄×D₄	—	2	1	no	—
297	C₄×D₄	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
301	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
302	C₄×D₄	C₂³×C₂₃	1	2	yes	—
303	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
304	#117	—	2	1	no	—
305	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
310	#107	C₂×C₄×C₂₃	1	2	yes	—
311	#106	C₂×C₄×C₂₃	1	2	yes	—
312	C₄⋊C₈	—	2	2	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
315	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
317	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
319	#103	C₂×C₄×C₂₃	1	2	yes	—
320	#102	C₂×C₄×C₂₃	1	2	yes	—
321	#102	C₂×C₄×C₂₃	1	2	yes	—
322	#102	C₂×C₄×C₂₃	1	2	yes	—
323	C₂²⋊C₈	—	2	1	no	—
324	C₂²⋊C₈	—	2	2	no	—
325	C₂²⋊C₈	—	2	2	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
328	#97	—	2	2	no	—
329	#97	C₂×C₄×C₂₃	1	2	yes	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	—	2	1	no	—
332	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
333	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₄×C₈	C₂³×C₂₃	1	2	yes	—
335	C₄×C₈	C₂³×C₂₃	1	2	yes	—
336	C₂₃	#17893	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁴	—	2	1	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
346	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	8	1	no	—
354	C₂²×C₄	—	8	1	no	—
355	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
357	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
358	SD₁₆	—	2	4	no	—
359	D₈	—	2	4	no	—
360	M₁₆	—	2	2	no	—
361	M₁₆	—	8	2	no	—
362	M₁₆	—	8	2	no	—
363	M₁₆	—	8	2	no	—
364	M₁₆	M₁₆×C₂₃	1	4	yes	complemented
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	4	2	no	—
371	C₂×C₈	—	8	1	no	—
372	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
373	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
380	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
381	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
391	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
392	C₄²	#41×C₂₃	1	1	yes	characteristic
393	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₂³	—	2	1	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	8	1	no	—
398	C₂³	—	8	1	no	—
399	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	1	no	—
402	D₄	—	2	1	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	2	no	—
405	D₄	—	2	1	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	—	8	1	no	—
422	C₂×C₄	—	8	1	no	—
423	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
424	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
427	C₂×C₄	C₂₃×#131	1	2	yes	—
428	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	2	2	no	—
434	C₈	—	2	2	no	—
435	C₈	—	4	1	no	—
436	C₈	—	4	2	no	—
437	C₈	—	4	1	no	—
438	C₈	—	4	1	no	—
439	C₈	C₂×M₁₆×C₂₃	1	4	yes	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	8	4	no	—
449	C₂²	C₂₃×#436	1	2	yes	—
450	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	1	no	—
458	C₄	C₂₃×#436	1	2	yes	—
459	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
460	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
461	C₄	C₂₃×#576	1	1	yes	characteristic
462	C₄	C₂₃×#577	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	8	2	no	—
465	C₂	—	8	1	no	—
466	C₂	C₂×C₂₃×#436	1	1	yes	characteristic
467	C₂	C₂₃×#2946	1	1	yes	characteristic
468	C₂	C₂₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2989	C₂	semidirect
2. C₂₃×#2975	C₂	semidirect
4. C₂₃×#1594	C₂	semidirect
5. C₂₃×#1592	C₂	semidirect
6. C₂₃×#1590	C₂	semidirect
7. C₂₃×#1589	C₂	semidirect
8. C₂₃×#1565	C₂	semidirect
9. C₂₃×#1476	C₂	semidirect
10. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
15. C₂₃×#446	C₂²	semidirect
24. C₂₃×#436	C₂²	semidirect
37. C₂₃×#335	C₂²	semidirect
38. C₂₃×#334	C₂²	semidirect
39. C₂₃×Q₈⋊C₈	C₂²	semidirect
40. C₂₃×Q₈⋊C₈	C₂²	semidirect
43. C₂₃×D₄⋊C₈	C₂²	semidirect
44. C₂₃×D₄⋊C₈	C₂²	semidirect
45. C₂₃×#322	C₂²	semidirect
46. C₈²×C₂₃	C₂²	semidirect
129. M₁₆×C₂₃	M₁₆	semidirect
159. #17893	C₂₃	direct (N × H)
206. #2989	C₂×C₂₃	semidirect
207. #2975	C₂×C₂₃	semidirect
209. #1594	C₂×C₂₃	semidirect
210. #1592	C₂×C₂₃	semidirect
211. #1590	C₂×C₂₃	semidirect
212. #1589	C₂×C₂₃	semidirect
213. #1565	C₂×C₂₃	semidirect
214. #1476	C₂×C₂₃	semidirect
215. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
216. C₈×M₁₆	C₂×C₂₃	semidirect
243. #446	C₂²×C₂₃	semidirect
252. #436	C₂²×C₂₃	semidirect
265. #335	C₂²×C₂₃	semidirect
266. #334	C₂²×C₂₃	semidirect
267. Q₈⋊C₈	C₂²×C₂₃	semidirect
268. Q₈⋊C₈	C₂²×C₂₃	semidirect
271. D₄⋊C₈	C₂²×C₂₃	semidirect
272. D₄⋊C₈	C₂²×C₂₃	semidirect
273. #322	C₂²×C₂₃	semidirect
274. C₈²	C₂²×C₂₃	semidirect
336. C₂₃	#17893	direct (N × H)
364. M₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2989	2
C₂₃×#2975	2
C₄×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#1594	2
C₂₃×#1592	2
C₂₃×#1590	2
C₂₃×#1589	2
C₂₃×#1565	2
C₂₃×#1476	2
C₂×C₂₃×D₄⋊C₈	2
C₈×M₁₆×C₂₃	2
#17893	23

Subgroups — C23×#17893 #19165519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17893 #19165519