Subgroups — C₂₃×#33668 #19181294

Order 5888.

552 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2235	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2232	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2229	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2211	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2188	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2185	C₂	1	1	yes	maximal, characteristic
9	C₄×C₂₃×Q₃₂	C₂	1	2	yes	maximal, complemented
10	C₄×C₂₃×SD₃₂	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#529	—	2	1	no	—
14	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
15	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
16	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
17	C₂₃×#499	C₂²	1	1	yes	characteristic
18	C₂₃×#495	C₂²	1	1	yes	characteristic
19	C₂₃×#488	—	2	1	no	—
20	C₂₃×#467	C₂²	1	1	yes	characteristic
21	C₂₃×#466	C₂²	1	1	yes	characteristic
22	C₂₃×#465	—	2	1	no	—
23	C₄×Q₁₆×C₂₃	—	2	1	no	—
24	C₄×Q₁₆×C₂₃	—	2	1	no	—
25	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
28	C₄×SD₁₆×C₂₃	—	2	1	no	—
29	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	—	2	1	no	—
32	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#367	C₂²	1	2	yes	—
34	C₂₃×#366	C₂²	1	2	yes	—
35	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
36	C₂₃×#358	C₂²	1	2	yes	—
37	C₂₃×#358	C₂²	1	2	yes	—
38	C₂₃×#358	C₂²	1	2	yes	—
39	C₂₃×#357	C₂²	1	2	yes	—
40	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂×Q₁₆×C₂₃	—	2	4	no	—
45	C₂×Q₁₆×C₂₃	—	2	1	no	—
46	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
48	C₂×SD₁₆×C₂₃	—	2	1	no	—
49	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#126	—	4	1	no	—
53	C₂₃×#125	—	2	1	no	—
54	C₂₃×#124	—	2	1	no	—
55	C₂₃×#123	—	4	1	no	—
56	C₂₃×C₂²⋊Q₈	—	2	1	no	—
57	C₂₃×#121	—	2	1	no	—
58	C₄×Q₈×C₂₃	—	2	1	no	—
59	C₄×Q₈×C₂₃	—	2	2	no	—
60	C₄×Q₈×C₂₃	—	2	2	no	—
61	C₄×Q₈×C₂₃	—	4	1	no	—
62	C₄×Q₈×C₂₃	—	4	1	no	—
63	C₄×D₄×C₂₃	—	2	1	no	—
64	C₄×D₄×C₂₃	—	4	1	no	—
65	C₂₃×#117	—	4	1	no	—
66	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
68	C₂×C₄²×C₂₃	—	4	1	no	—
69	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
70	C₂₃×Q₃₂	—	2	4	no	—
71	C₂₃×SD₃₂	—	2	4	no	—
72	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
73	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
74	C₂₃×#107	—	2	1	no	—
75	C₂₃×#107	C₂³	1	2	yes	—
76	C₂₃×#107	C₂³	1	1	yes	characteristic
77	C₂₃×#107	C₂³	1	1	yes	characteristic
78	C₂₃×#106	—	2	1	no	—
79	C₂₃×#103	—	2	2	no	—
80	C₂₃×#103	—	2	2	no	—
81	C₂₃×#103	—	2	1	no	—
82	C₂₃×#103	—	2	1	no	—
83	C₂₃×#102	—	2	1	no	—
84	C₂₃×#102	—	2	1	no	—
85	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₂₃	—	2	4	no	—
90	C₂×Q₈×C₂₃	—	2	2	no	—
91	C₂×Q₈×C₂₃	—	2	1	no	—
92	C₂×Q₈×C₂₃	—	4	2	no	—
93	C₂×Q₈×C₂₃	—	4	1	no	—
94	C₂×D₄×C₂₃	—	2	1	no	—
95	C₂×D₄×C₂₃	—	4	1	no	—
96	C₂²×C₄×C₂₃	—	4	1	no	—
97	C₂²×C₄×C₂₃	—	4	1	no	—
98	C₂²×C₄×C₂₃	—	4	1	no	—
99	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	Q₁₆×C₂₃	—	2	4	no	—
103	Q₁₆×C₂₃	—	2	2	no	—
104	Q₁₆×C₂₃	—	2	2	no	—
105	Q₁₆×C₂₃	#41	1	4	yes	—
106	SD₁₆×C₂₃	—	4	2	no	—
107	D₈×C₂₃	—	2	2	no	—
108	C₂×C₈×C₂₃	—	2	2	no	—
109	C₂×C₈×C₂₃	—	4	1	no	—
110	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₂₃	—	2	1	no	—
115	C₄⋊C₄×C₂₃	—	2	1	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	4	1	no	—
118	C₄⋊C₄×C₂₃	—	4	2	no	—
119	C₄⋊C₄×C₂₃	—	4	2	no	—
120	C₄⋊C₄×C₂₃	—	4	1	no	—
121	C₄⋊C₄×C₂₃	—	4	1	no	—
122	C₄⋊C₄×C₂₃	—	4	1	no	—
123	C₄⋊C₄×C₂₃	—	4	1	no	—
124	C₂²⋊C₄×C₂₃	—	4	1	no	—
125	C₂²⋊C₄×C₂₃	—	4	1	no	—
126	C₂²⋊C₄×C₂₃	—	4	1	no	—
127	C₂²⋊C₄×C₂₃	—	4	1	no	—
128	C₂²⋊C₄×C₂₃	—	4	1	no	—
129	C₄²×C₂₃	—	4	2	no	—
130	C₄²×C₂₃	—	4	2	no	—
131	C₄²×C₂₃	—	4	1	no	—
132	C₄²×C₂₃	D₈	1	1	yes	characteristic
133	C₄²×C₂₃	D₈	1	1	yes	characteristic
134	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₂₃	—	2	2	no	—
137	C₁₆×C₂₃	—	2	2	no	—
138	#33668	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₂₃	—	4	1	no	—
140	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
141	Q₈×C₂₃	—	2	4	no	—
142	Q₈×C₂₃	—	4	4	no	—
143	Q₈×C₂₃	—	4	2	no	—
144	Q₈×C₂₃	—	4	1	no	—
145	Q₈×C₂₃	—	4	1	no	—
146	D₄×C₂₃	—	4	1	no	—
147	D₄×C₂₃	—	4	1	no	—
148	D₄×C₂₃	—	8	2	no	—
149	C₂×C₄×C₂₃	—	2	2	no	—
150	C₂×C₄×C₂₃	—	2	1	no	—
151	C₂×C₄×C₂₃	—	2	1	no	—
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	4	4	no	—
154	C₂×C₄×C₂₃	—	4	2	no	—
155	C₂×C₄×C₂₃	—	4	2	no	—
156	C₂×C₄×C₂₃	—	4	1	no	—
157	C₂×C₄×C₂₃	—	4	1	no	—
158	C₂×C₄×C₂₃	—	4	1	no	—
159	C₂×C₄×C₂₃	—	8	2	no	—
160	C₂×C₄×C₂₃	—	8	2	no	—
161	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₂₃	—	2	1	no	—
168	C₈×C₂₃	—	2	1	no	—
169	C₈×C₂₃	—	2	1	no	—
170	C₈×C₂₃	C₂×#41	1	2	yes	—
171	#3168	C₂×C₂₃	1	1	yes	characteristic
172	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
173	#2235	C₂×C₂₃	1	1	yes	characteristic, complemented
174	#2232	C₂×C₂₃	1	1	yes	characteristic
175	#2229	C₂×C₂₃	1	2	yes	complemented
176	#2211	C₂×C₂₃	1	2	yes	complemented
177	#2188	C₂×C₂₃	1	1	yes	characteristic, complemented
178	#2185	C₂×C₂₃	1	1	yes	characteristic
179	C₄×Q₃₂	C₂×C₂₃	1	2	yes	complemented
180	C₄×SD₃₂	C₂×C₂₃	1	2	yes	complemented
181	#2162	C₂×C₂₃	1	1	yes	characteristic, complemented
182	C₂²×C₂₃	—	2	1	no	—
183	C₂²×C₂₃	—	2	1	no	—
184	C₂²×C₂₃	—	8	1	no	—
185	C₂²×C₂₃	—	8	1	no	—
186	C₂²×C₂₃	—	8	1	no	—
187	C₂²×C₂₃	#508	1	2	yes	—
188	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
189	C₄×C₂₃	—	2	1	no	—
190	C₄×C₂₃	—	2	1	no	—
191	C₄×C₂₃	—	2	1	no	—
192	C₄×C₂₃	—	2	1	no	—
193	C₄×C₂₃	—	4	4	no	—
194	C₄×C₂₃	—	8	1	no	—
195	C₄×C₂₃	—	8	2	no	—
196	C₄×C₂₃	—	8	1	no	—
197	C₄×C₂₃	—	8	1	no	—
198	C₄×C₂₃	#510	1	2	yes	—
199	C₄×C₂₃	#525	1	1	yes	characteristic
200	C₄×C₂₃	#575	1	1	yes	characteristic
201	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
202	#529	—	2	1	no	—
203	C₂×C₄×Q₈	—	2	1	no	—
204	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
205	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
206	#499	C₂²×C₂₃	1	1	yes	characteristic
207	#495	C₂²×C₂₃	1	1	yes	characteristic
208	#488	—	2	1	no	—
209	#467	C₂²×C₂₃	1	1	yes	characteristic
210	#466	C₂²×C₂₃	1	1	yes	characteristic
211	#465	—	2	1	no	—
212	C₄×Q₁₆	—	2	1	no	—
213	C₄×Q₁₆	—	2	1	no	—
214	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
215	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
216	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
217	C₄×SD₁₆	—	2	1	no	—
218	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
219	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×#103	—	2	1	no	—
221	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
222	#367	C₂²×C₂₃	1	2	yes	—
223	#366	C₂²×C₂₃	1	2	yes	—
224	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
225	#358	C₂²×C₂₃	1	2	yes	—
226	#358	C₂²×C₂₃	1	2	yes	—
227	#358	C₂²×C₂₃	1	2	yes	—
228	#357	C₂²×C₂₃	1	2	yes	—
229	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
230	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
231	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
232	C₂×C₂₃	—	2	2	no	—
233	C₂×C₂₃	—	16	1	no	—
234	C₂×C₂₃	#3100	1	1	yes	characteristic
235	C₂×C₂₃	C₂×#508	1	1	yes	characteristic
236	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
237	C₂²×Q₈	—	2	1	no	—
238	C₂×Q₁₆	—	2	4	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
241	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
244	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
246	#126	—	4	1	no	—
247	#125	—	2	1	no	—
248	#124	—	2	1	no	—
249	#123	—	4	1	no	—
250	C₂²⋊Q₈	—	2	1	no	—
251	#121	—	2	1	no	—
252	C₄×Q₈	—	2	2	no	—
253	C₄×Q₈	—	2	2	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	4	1	no	—
256	C₄×Q₈	—	4	1	no	—
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	—	4	1	no	—
259	#117	—	4	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	4	1	no	—
262	C₂×C₄²	—	4	1	no	—
263	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
264	Q₃₂	—	2	4	no	—
265	SD₃₂	—	2	4	no	—
266	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
267	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
268	#107	—	2	1	no	—
269	#107	C₂³×C₂₃	1	2	yes	—
270	#107	C₂³×C₂₃	1	1	yes	characteristic
271	#107	C₂³×C₂₃	1	1	yes	characteristic
272	#106	—	2	1	no	—
273	#103	—	2	2	no	—
274	#103	—	2	2	no	—
275	#103	—	2	1	no	—
276	#103	—	2	1	no	—
277	#102	—	2	1	no	—
278	#102	—	2	1	no	—
279	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
280	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
281	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
282	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
283	C₂₃	#33668	1	1	yes	characteristic, direct product factor, Sylow
284	C₂×Q₈	—	2	4	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	4	2	no	—
288	C₂×Q₈	—	4	1	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
295	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
297	Q₁₆	—	2	4	no	—
298	Q₁₆	—	2	2	no	—
299	Q₁₆	—	2	2	no	—
300	Q₁₆	#41×C₂₃	1	4	yes	—
301	SD₁₆	—	4	2	no	—
302	D₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
308	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	4	1	no	—
321	C₂²⋊C₄	—	4	1	no	—
322	C₂²⋊C₄	—	4	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₄²	—	4	2	no	—
325	C₄²	—	4	2	no	—
326	C₄²	—	4	1	no	—
327	C₄²	D₈×C₂₃	1	1	yes	characteristic
328	C₄²	D₈×C₂₃	1	1	yes	characteristic
329	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
330	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	2	no	—
333	C₂³	—	4	1	no	—
334	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
335	Q₈	—	2	4	no	—
336	Q₈	—	4	4	no	—
337	Q₈	—	4	2	no	—
338	Q₈	—	4	1	no	—
339	Q₈	—	4	1	no	—
340	D₄	—	4	1	no	—
341	D₄	—	4	1	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	8	2	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
356	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
357	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
358	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
359	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	C₂×#41×C₂₃	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	1	no	—
370	C₂²	C₂₃×#508	1	2	yes	—
371	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	4	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	2	no	—
379	C₄	—	8	1	no	—
380	C₄	—	8	1	no	—
381	C₄	C₂₃×#510	1	2	yes	—
382	C₄	C₂₃×#525	1	1	yes	characteristic
383	C₄	C₂₃×#575	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	16	1	no	—
386	C₂	C₂₃×#3100	1	1	yes	characteristic
387	C₂	C₂×C₂₃×#508	1	1	yes	characteristic
388	C₂	C₂×C₂₃×#510	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₈
prime radical	C₂₃×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₂₃	C₂	semidirect
3. C₂₃×#2235	C₂	semidirect
5. C₂₃×#2229	C₂	semidirect
6. C₂₃×#2211	C₂	semidirect
7. C₂₃×#2188	C₂	semidirect
9. C₄×C₂₃×Q₃₂	C₂	semidirect
10. C₄×C₂₃×SD₃₂	C₂	semidirect
11. C₂₃×#2162	C₂	semidirect
138. #33668	C₂₃	direct (N × H)
172. C₂×C₄×Q₁₆	C₂×C₂₃	semidirect
173. #2235	C₂×C₂₃	semidirect
175. #2229	C₂×C₂₃	semidirect
176. #2211	C₂×C₂₃	semidirect
177. #2188	C₂×C₂₃	semidirect
179. C₄×Q₃₂	C₂×C₂₃	semidirect
180. C₄×SD₃₂	C₂×C₂₃	semidirect
181. #2162	C₂×C₂₃	semidirect
283. C₂₃	#33668	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×#3168	2
C₂×C₄×Q₁₆×C₂₃	2
C₂₃×#2235	2
C₂₃×#2232	2
C₂₃×#2229	2
C₂₃×#2211	2
C₂₃×#2188	2
C₂₃×#2185	2
C₄×C₂₃×Q₃₂	2
C₄×C₂₃×SD₃₂	2
C₂₃×#2162	2
#33668	23

Subgroups — C23×#33668 #19181294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33668 #19181294