Subgroups — C₂₃×#22606 #19170232

Order 5888.

704 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2924	C₂	1	2	yes	maximal
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1945	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1922	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1826	C₂	1	2	yes	maximal
6	C₂₃×#1810	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1772	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	2	yes	—
11	C₂₃×#433	—	2	4	no	—
12	C₂₃×#432	—	2	2	no	—
13	C₂₃×#431	—	2	2	no	—
14	C₂×C₂₃×#108	C₂²	1	2	yes	—
15	C₂₃×#427	C₄	1	1	yes	characteristic, complemented
16	C₂₃×#427	C₄	1	1	yes	characteristic, complemented
17	C₂₃×#427	C₄	1	2	yes	complemented
18	C₂₃×#427	C₂²	1	2	yes	—
19	C₂₃×#427	C₂²	1	1	yes	characteristic
20	C₂₃×#427	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#107	C₄	1	1	yes	characteristic, complemented
22	C₂×C₂₃×#107	C₄	1	1	yes	characteristic, complemented
23	C₂×C₂₃×#106	C₄	1	2	yes	complemented
24	C₂₃×#424	C₂²	1	4	yes	—
25	C₂₃×#407	C₂²	1	2	yes	—
26	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₃×#405	C₂²	1	4	yes	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#341	C₂²	1	2	yes	—
31	C₂₃×#339	C₂²	1	4	yes	—
32	C₂₃×#339	C₂²	1	4	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₂₃	—	2	4	no	—
37	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×#117	—	2	2	no	—
45	C₂₃×#117	—	2	2	no	—
46	C₂₃×#117	C₂×C₄	1	2	yes	—
47	C₂₃×#117	C₂×C₄	1	2	yes	—
48	C₂₃×#117	C₂³	1	2	yes	—
49	C₂₃×#117	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄²×C₂₃	—	2	2	no	—
55	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
56	C₂₃×#108	—	2	4	no	—
57	C₂₃×#107	C₂×C₄	1	2	yes	—
58	C₂₃×#107	C₂×C₄	1	2	yes	—
59	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
60	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
61	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
62	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
63	C₂₃×#106	C₂×C₄	1	2	yes	—
64	C₂₃×#106	C₂×C₄	1	2	yes	—
65	C₂₃×#106	C₂×C₄	1	2	yes	—
66	C₂₃×#106	C₂×C₄	1	2	yes	—
67	C₂₃×C₄⋊C₈	—	2	4	no	—
68	C₂₃×C₄⋊C₈	—	2	4	no	—
69	C₂₃×C₂²⋊C₈	—	2	2	no	—
70	C₂₃×C₂²⋊C₈	—	2	2	no	—
71	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
73	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
74	C₂₃×#97	—	2	4	no	—
75	C₄×C₈×C₂₃	—	2	4	no	—
76	C₄×C₈×C₂₃	—	2	4	no	—
77	C₂₃×#95	—	2	1	no	—
78	C₂⁴×C₂₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	—	2	2	no	—
83	C₂²×C₄×C₂₃	—	2	2	no	—
84	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
89	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₂₃	—	2	4	no	—
91	M₁₆×C₂₃	—	2	4	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	2	2	no	—
99	C₂×C₈×C₂₃	—	2	4	no	—
100	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₂₃	#41	1	4	yes	—
107	C₄⋊C₄×C₂₃	—	2	2	no	—
108	C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
110	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
113	C₂²⋊C₄×C₂₃	—	2	4	no	—
114	C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂²⋊C₄×C₂₃	—	2	2	no	—
116	C₂²⋊C₄×C₂₃	—	2	2	no	—
117	C₂²⋊C₄×C₂₃	—	2	2	no	—
118	C₄²×C₂₃	—	2	2	no	—
119	C₄²×C₂₃	—	2	2	no	—
120	C₄²×C₂₃	—	2	2	no	—
121	C₄²×C₂₃	—	2	2	no	—
122	C₄²×C₂₃	—	2	2	no	—
123	C₄²×C₂₃	—	2	2	no	—
124	#22606	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₂₃	—	2	2	no	—
126	C₂³×C₂₃	—	2	2	no	—
127	C₂³×C₂₃	—	4	1	no	—
128	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
129	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
130	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	2	1	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	1	no	—
135	C₂×C₄×C₂₃	—	2	1	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	4	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	2	2	no	—
142	C₂×C₄×C₂₃	—	2	2	no	—
143	C₂×C₄×C₂₃	—	4	2	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	—	4	4	no	—
146	C₂×C₄×C₂₃	—	8	2	no	—
147	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
152	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
153	C₈×C₂₃	—	2	1	no	—
154	C₈×C₂₃	—	2	1	no	—
155	C₈×C₂₃	—	2	2	no	—
156	C₈×C₂₃	—	2	2	no	—
157	C₈×C₂₃	—	4	4	no	—
158	C₈×C₂₃	#117	1	2	yes	—
159	C₈×C₂₃	#117	1	2	yes	—
160	#2924	C₂×C₂₃	1	2	yes	—
161	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
162	#1945	C₂×C₂₃	1	2	yes	complemented
163	#1922	C₂×C₂₃	1	4	yes	complemented
164	#1826	C₂×C₂₃	1	2	yes	—
165	#1810	C₂×C₂₃	1	1	yes	characteristic
166	#1772	C₂×C₂₃	1	2	yes	complemented
167	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
168	C₂²×C₂₃	—	2	1	no	—
169	C₂²×C₂₃	—	2	1	no	—
170	C₂²×C₂₃	—	2	1	no	—
171	C₂²×C₂₃	—	2	1	no	—
172	C₂²×C₂₃	—	2	1	no	—
173	C₂²×C₂₃	—	2	1	no	—
174	C₂²×C₂₃	—	2	1	no	—
175	C₂²×C₂₃	—	4	4	no	—
176	C₂²×C₂₃	#443	1	2	yes	—
177	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂²×C₂₃	#552	1	2	yes	—
179	C₄×C₂₃	—	2	1	no	—
180	C₄×C₂₃	—	2	1	no	—
181	C₄×C₂₃	—	4	4	no	—
182	C₄×C₂₃	—	4	2	no	—
183	C₄×C₂₃	—	4	2	no	—
184	C₄×C₂₃	—	4	2	no	—
185	C₄×C₂₃	—	4	2	no	—
186	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
187	C₄×C₂₃	#519	1	1	yes	characteristic
188	C₄×C₂₃	#547	1	2	yes	—
189	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
190	C₂×#117	C₂²×C₂₃	1	2	yes	—
191	#433	—	2	4	no	—
192	#432	—	2	2	no	—
193	#431	—	2	2	no	—
194	C₂×#108	C₂²×C₂₃	1	2	yes	—
195	#427	C₄×C₂₃	1	1	yes	characteristic, complemented
196	#427	C₄×C₂₃	1	1	yes	characteristic, complemented
197	#427	C₄×C₂₃	1	2	yes	complemented
198	#427	C₂²×C₂₃	1	2	yes	—
199	#427	C₂²×C₂₃	1	1	yes	characteristic
200	#427	C₂²×C₂₃	1	1	yes	characteristic
201	C₂×#107	C₄×C₂₃	1	1	yes	characteristic, complemented
202	C₂×#107	C₄×C₂₃	1	1	yes	characteristic, complemented
203	C₂×#106	C₄×C₂₃	1	2	yes	complemented
204	#424	C₂²×C₂₃	1	4	yes	—
205	#407	C₂²×C₂₃	1	2	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
207	#405	C₂²×C₂₃	1	4	yes	—
208	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
209	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
210	#341	C₂²×C₂₃	1	2	yes	—
211	#339	C₂²×C₂₃	1	4	yes	—
212	#339	C₂²×C₂₃	1	4	yes	—
213	#336	C₂²×C₂₃	1	2	yes	—
214	C₂×C₂₃	—	2	2	no	—
215	C₂×C₂₃	—	2	2	no	—
216	C₂×C₂₃	—	4	1	no	—
217	C₂×C₂₃	#2344	1	1	yes	characteristic
218	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
219	C₂×C₂₃	#2957	1	1	yes	characteristic
220	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
221	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
222	C₂²×C₈	—	2	4	no	—
223	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
226	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
229	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
230	#117	—	2	2	no	—
231	#117	—	2	2	no	—
232	#117	C₂×C₄×C₂₃	1	2	yes	—
233	#117	C₂×C₄×C₂₃	1	2	yes	—
234	#117	C₂³×C₂₃	1	2	yes	—
235	#117	C₂³×C₂₃	1	2	yes	—
236	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₂²⋊C₄	—	2	1	no	—
240	C₂×C₄²	—	2	2	no	—
241	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
242	#108	—	2	4	no	—
243	#107	C₂×C₄×C₂₃	1	2	yes	—
244	#107	C₂×C₄×C₂₃	1	2	yes	—
245	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
246	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
247	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
248	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
249	#106	C₂×C₄×C₂₃	1	2	yes	—
250	#106	C₂×C₄×C₂₃	1	2	yes	—
251	#106	C₂×C₄×C₂₃	1	2	yes	—
252	#106	C₂×C₄×C₂₃	1	2	yes	—
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	4	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
258	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
259	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
260	#97	—	2	4	no	—
261	C₄×C₈	—	2	4	no	—
262	C₄×C₈	—	2	4	no	—
263	#95	—	2	1	no	—
264	C₂₃	#22606	1	1	yes	characteristic, direct product factor, Sylow
265	C₂⁴	#41×C₂₃	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
276	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
288	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
289	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
293	C₂×C₈	#41×C₂₃	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
297	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
298	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
299	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
315	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
316	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
334	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
335	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
336	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
337	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
338	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	4	4	no	—
344	C₈	C₂₃×#117	1	2	yes	—
345	C₈	C₂₃×#117	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	4	no	—
354	C₂²	C₂₃×#443	1	2	yes	—
355	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
356	C₂²	C₂₃×#552	1	2	yes	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
365	C₄	C₂₃×#519	1	1	yes	characteristic
366	C₄	C₂₃×#547	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	C₂₃×#2344	1	1	yes	characteristic
371	C₂	C₂×C₂₃×#443	1	1	yes	characteristic
372	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1945	C₂	semidirect
4. C₂₃×#1922	C₂	semidirect
7. C₂₃×#1772	C₂	semidirect
15. C₂₃×#427	C₄	semidirect
16. C₂₃×#427	C₄	semidirect
17. C₂₃×#427	C₄	semidirect
21. C₂×C₂₃×#107	C₄	semidirect
22. C₂×C₂₃×#107	C₄	semidirect
23. C₂×C₂₃×#106	C₄	semidirect
124. #22606	C₂₃	direct (N × H)
162. #1945	C₂×C₂₃	semidirect
163. #1922	C₂×C₂₃	semidirect
166. #1772	C₂×C₂₃	semidirect
195. #427	C₄×C₂₃	semidirect
196. #427	C₄×C₂₃	semidirect
197. #427	C₄×C₂₃	semidirect
201. C₂×#107	C₄×C₂₃	semidirect
202. C₂×#107	C₄×C₂₃	semidirect
203. C₂×#106	C₄×C₂₃	semidirect
264. C₂₃	#22606	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×#2924	2
C₂×C₂₃×#427	2
C₂₃×#1945	2
C₂₃×#1922	2
C₂₃×#1826	2
C₂₃×#1810	2
C₂₃×#1772	2
C₈×C₂²⋊C₄×C₂₃	2
#22606	23

Subgroups — C23×#22606 #19170232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22606 #19170232