Subgroups — C₂₃×#19199 #19166825

Order 5888.

650 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3227	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3226	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1703	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1683	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#131	—	2	1	no	—
11	C₂₃×#532	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂₃×#500	—	2	1	no	—
15	C₂₃×#498	—	2	1	no	—
16	C₂₃×#487	—	2	1	no	—
17	C₂₃×#487	—	2	1	no	—
18	C₂₃×#487	C₂²	1	2	yes	—
19	C₂₃×#487	C₂²	1	2	yes	—
20	C₂₃×#486	—	2	1	no	—
21	C₂₃×#486	—	2	1	no	—
22	C₂₃×#481	—	2	2	no	—
23	C₂₃×#480	—	2	2	no	—
24	C₂₃×#476	C₂²	1	4	yes	—
25	C₂₃×#474	C₂²	1	4	yes	—
26	C₂₃×#436	—	2	1	no	—
27	C₂₃×#436	C₂²	1	2	yes	—
28	C₈×D₄×C₂₃	—	2	1	no	—
29	C₈×D₄×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#431	C₂²	1	1	yes	characteristic
31	C₂₃×#431	C₂²	1	1	yes	characteristic
32	C₂₃×#423	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	—	2	1	no	—
34	C₂×C₂₃×#103	—	2	1	no	—
35	C₂₃×#408	—	2	1	no	—
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#335	C₂²	1	2	yes	—
38	C₂₃×#334	C₂²	1	2	yes	—
39	C₂₃×#333	C₂²	1	4	yes	—
40	C₂₃×#332	C₂²	1	4	yes	—
41	C₂×#41×C₂₃	—	2	1	no	—
42	C₂²×Q₈×C₂₃	—	2	1	no	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂₃×#131	—	4	4	no	—
45	C₂×M₁₆×C₂₃	—	2	1	no	—
46	C₂×M₁₆×C₂₃	—	2	2	no	—
47	C₂²×C₈×C₂₃	—	2	2	no	—
48	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₂₃×C₄⋊Q₈	—	2	1	no	—
51	C₂₃×C₄⋊Q₈	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	—	2	1	no	—
53	C₂₃×C₄⋊Q₈	—	2	1	no	—
54	C₂₃×C₄⋊Q₈	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
57	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₂₃×#127	D₄	1	1	yes	characteristic
60	C₂₃×#124	—	2	1	no	—
61	C₂₃×#121	—	2	1	no	—
62	C₂₃×#121	D₄	1	2	yes	—
63	C₄×D₄×C₂₃	—	2	1	no	—
64	C₄×D₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#107	—	2	4	no	—
71	C₂₃×#106	—	2	4	no	—
72	C₂₃×C₄⋊C₈	—	2	1	no	—
73	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
76	C₂₃×#103	—	2	2	no	—
77	C₂₃×#103	—	2	2	no	—
78	C₂₃×#103	—	2	2	no	—
79	C₂₃×#103	—	2	2	no	—
80	C₂₃×#102	—	2	4	no	—
81	C₂₃×#102	—	2	4	no	—
82	C₂₃×C₂²⋊C₈	—	2	1	no	—
83	C₂₃×C₂²⋊C₈	—	2	1	no	—
84	C₂₃×C₂²⋊C₈	—	2	2	no	—
85	C₂₃×C₂²⋊C₈	—	2	2	no	—
86	C₄×C₈×C₂₃	—	2	1	no	—
87	C₄×C₈×C₂₃	—	2	1	no	—
88	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	#41×C₂₃	—	4	2	no	—
93	#41×C₂₃	—	4	2	no	—
94	C₂×Q₈×C₂₃	—	2	1	no	—
95	C₂×Q₈×C₂₃	—	2	2	no	—
96	C₂×Q₈×C₂₃	—	2	2	no	—
97	C₂×Q₈×C₂₃	—	4	1	no	—
98	C₂×Q₈×C₂₃	—	4	1	no	—
99	C₂×Q₈×C₂₃	—	4	2	no	—
100	C₂×Q₈×C₂₃	—	4	2	no	—
101	C₂×Q₈×C₂₃	—	8	2	no	—
102	C₂×D₄×C₂₃	—	2	2	no	—
103	C₂×D₄×C₂₃	—	2	1	no	—
104	C₂×D₄×C₂₃	—	2	1	no	—
105	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₂₃	—	2	2	no	—
107	C₂²×C₄×C₂₃	—	4	1	no	—
108	C₂²×C₄×C₂₃	—	4	1	no	—
109	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
112	M₁₆×C₂₃	—	4	4	no	—
113	M₁₆×C₂₃	—	4	2	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	—	4	4	no	—
120	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₂₃	—	2	1	no	—
125	C₄⋊C₄×C₂₃	—	2	2	no	—
126	C₄⋊C₄×C₂₃	—	2	2	no	—
127	C₄⋊C₄×C₂₃	—	2	2	no	—
128	C₄⋊C₄×C₂₃	—	2	2	no	—
129	C₄⋊C₄×C₂₃	—	4	1	no	—
130	C₄⋊C₄×C₂₃	—	4	1	no	—
131	C₄⋊C₄×C₂₃	—	4	1	no	—
132	C₄⋊C₄×C₂₃	—	4	1	no	—
133	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
134	C₂²⋊C₄×C₂₃	—	2	2	no	—
135	C₂²⋊C₄×C₂₃	—	2	2	no	—
136	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₂₃	#41	1	1	yes	characteristic
139	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
140	#19199	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₂₃	—	2	2	no	—
142	C₂³×C₂₃	#121	1	1	yes	characteristic
143	Q₈×C₂₃	—	4	2	no	—
144	Q₈×C₂₃	—	4	2	no	—
145	Q₈×C₂₃	—	4	2	no	—
146	Q₈×C₂₃	—	8	2	no	—
147	Q₈×C₂₃	—	8	2	no	—
148	Q₈×C₂₃	—	8	2	no	—
149	D₄×C₂₃	—	2	4	no	—
150	D₄×C₂₃	—	4	4	no	—
151	D₄×C₂₃	—	4	2	no	—
152	D₄×C₂₃	—	4	2	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	2	2	no	—
155	C₂×C₄×C₂₃	—	2	2	no	—
156	C₂×C₄×C₂₃	—	4	2	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	4	2	no	—
159	C₂×C₄×C₂₃	—	4	2	no	—
160	C₂×C₄×C₂₃	—	8	2	no	—
161	C₂×C₄×C₂₃	—	8	2	no	—
162	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₂₃	#135	1	2	yes	—
166	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
170	C₈×C₂₃	—	2	1	no	—
171	C₈×C₂₃	—	2	1	no	—
172	C₈×C₂₃	—	2	1	no	—
173	C₈×C₂₃	—	2	1	no	—
174	C₈×C₂₃	—	4	2	no	—
175	C₈×C₂₃	—	4	2	no	—
176	C₈×C₂₃	—	4	2	no	—
177	#3227	C₂×C₂₃	1	1	yes	characteristic
178	#3226	C₂×C₂₃	1	1	yes	characteristic
179	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
180	#2931	C₂×C₂₃	1	1	yes	characteristic
181	#1703	C₂×C₂₃	1	4	yes	complemented
182	#1683	C₂×C₂₃	1	4	yes	complemented
183	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
184	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
185	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
186	C₂²×C₂₃	—	2	1	no	—
187	C₂²×C₂₃	—	2	1	no	—
188	C₂²×C₂₃	—	4	2	no	—
189	C₂²×C₂₃	—	4	2	no	—
190	C₂²×C₂₃	—	4	2	no	—
191	C₂²×C₂₃	#463	1	2	yes	—
192	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
193	C₄×C₂₃	—	2	1	no	—
194	C₄×C₂₃	—	2	1	no	—
195	C₄×C₂₃	—	4	2	no	—
196	C₄×C₂₃	—	8	2	no	—
197	C₄×C₂₃	—	8	2	no	—
198	C₄×C₂₃	#463	1	2	yes	—
199	C₄×C₂₃	#471	1	2	yes	—
200	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
201	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₄×C₂₃	#578	1	2	yes	—
203	C₂×#131	—	2	1	no	—
204	#532	C₂²×C₂₃	1	1	yes	characteristic
205	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
206	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
207	#500	—	2	1	no	—
208	#498	—	2	1	no	—
209	#487	—	2	1	no	—
210	#487	—	2	1	no	—
211	#487	C₂²×C₂₃	1	2	yes	—
212	#487	C₂²×C₂₃	1	2	yes	—
213	#486	—	2	1	no	—
214	#486	—	2	1	no	—
215	#481	—	2	2	no	—
216	#480	—	2	2	no	—
217	#476	C₂²×C₂₃	1	4	yes	—
218	#474	C₂²×C₂₃	1	4	yes	—
219	#436	—	2	1	no	—
220	#436	C₂²×C₂₃	1	2	yes	—
221	C₈×D₄	—	2	1	no	—
222	C₈×D₄	C₂²×C₂₃	1	2	yes	—
223	#431	C₂²×C₂₃	1	1	yes	characteristic
224	#431	C₂²×C₂₃	1	1	yes	characteristic
225	#423	C₂²×C₂₃	1	1	yes	characteristic
226	C₂×#103	—	2	1	no	—
227	C₂×#103	—	2	1	no	—
228	#408	—	2	1	no	—
229	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
230	#335	C₂²×C₂₃	1	2	yes	—
231	#334	C₂²×C₂₃	1	2	yes	—
232	#333	C₂²×C₂₃	1	4	yes	—
233	#332	C₂²×C₂₃	1	4	yes	—
234	C₂×C₂₃	—	2	2	no	—
235	C₂×C₂₃	—	8	2	no	—
236	C₂×C₂₃	C₂×#463	1	1	yes	characteristic
237	C₂×C₂₃	#3059	1	1	yes	characteristic
238	C₂×C₂₃	C₂×#471	1	1	yes	characteristic
239	C₂×#41	—	2	1	no	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×Q₈	—	2	1	no	—
242	#131	—	4	4	no	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
247	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
255	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
256	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
257	#127	D₄×C₂₃	1	1	yes	characteristic
258	#124	—	2	1	no	—
259	#121	—	2	1	no	—
260	#121	D₄×C₂₃	1	2	yes	—
261	C₄×D₄	—	2	1	no	—
262	C₄×D₄	C₂³×C₂₃	1	2	yes	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
268	#107	—	2	4	no	—
269	#106	—	2	4	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
272	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
273	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
274	#103	—	2	2	no	—
275	#103	—	2	2	no	—
276	#103	—	2	2	no	—
277	#103	—	2	2	no	—
278	#102	—	2	4	no	—
279	#102	—	2	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
287	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
290	C₂₃	#19199	1	1	yes	characteristic, direct product factor, Sylow
291	#41	—	4	2	no	—
292	#41	—	4	2	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	—	8	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
310	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
311	M₁₆	—	4	4	no	—
312	M₁₆	—	4	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	4	4	no	—
319	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
321	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
322	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
336	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
337	C₄²	#41×C₂₃	1	1	yes	characteristic
338	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
339	C₂³	—	2	2	no	—
340	C₂³	C₂₃×#121	1	1	yes	characteristic
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	8	2	no	—
346	Q₈	—	8	2	no	—
347	D₄	—	2	4	no	—
348	D₄	—	4	4	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	8	2	no	—
360	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
361	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
362	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
363	C₂×C₄	C₂₃×#135	1	2	yes	—
364	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
365	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	2	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	C₂₃×#463	1	2	yes	—
381	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	8	2	no	—
387	C₄	C₂₃×#463	1	2	yes	—
388	C₄	C₂₃×#471	1	2	yes	—
389	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
390	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
391	C₄	C₂₃×#578	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	8	2	no	—
394	C₂	C₂×C₂₃×#463	1	1	yes	characteristic
395	C₂	C₂₃×#3059	1	1	yes	characteristic
396	C₂	C₂×C₂₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
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
3. C₂×C₂₃×#487	C₂	semidirect
5. C₂₃×#1703	C₂	semidirect
6. C₂₃×#1683	C₂	semidirect
7. C₂₃×#1564	C₂	semidirect
8. C₂₃×#1548	C₂	semidirect
9. C₂₃×#1546	C₂	semidirect
140. #19199	C₂₃	direct (N × H)
179. C₂×#487	C₂×C₂₃	semidirect
181. #1703	C₂×C₂₃	semidirect
182. #1683	C₂×C₂₃	semidirect
183. #1564	C₂×C₂₃	semidirect
184. #1548	C₂×C₂₃	semidirect
185. #1546	C₂×C₂₃	semidirect
290. C₂₃	#19199	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×#3227	2
C₂₃×#3226	2
C₂×C₂₃×#487	2
C₂₃×#2931	2
C₂₃×#1703	2
C₂₃×#1683	2
C₂₃×#1564	2
C₂₃×#1548	2
C₂₃×#1546	2
#19199	23

Subgroups — C23×#19199 #19166825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19199 #19166825