Subgroups — C₂₃×#21633 #19169259

Order 5888.

898 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1954	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1924	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1752	C₂	1	4	yes	maximal, complemented
9	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#124	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#446	—	2	4	no	—
15	C₈×D₄×C₂₃	—	2	4	no	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂×C₂₃×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#391	C₄	1	1	yes	characteristic
27	C₂₃×#390	C₄	1	1	yes	characteristic
28	C₂₃×#388	C₄	1	4	yes	—
29	C₂₃×#382	C₄	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
31	C₄³×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#336	C₂²	1	4	yes	—
33	C₂₃×#336	C₂²	1	4	yes	—
34	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	—	2	4	no	—
38	C₂²×C₈×C₂₃	—	2	8	no	—
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
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×#124	—	2	4	no	—
45	C₄×Q₈×C₂₃	—	2	2	no	—
46	C₄×Q₈×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	—	2	2	no	—
48	C₄×D₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊C₈	—	2	4	no	—
62	C₂₃×C₄⋊C₈	—	2	2	no	—
63	C₂₃×C₄⋊C₈	—	2	2	no	—
64	C₂₃×C₂²⋊C₈	—	2	4	no	—
65	C₂₃×C₂²⋊C₈	—	2	8	no	—
66	C₂₃×#97	—	2	4	no	—
67	C₂₃×#97	—	2	4	no	—
68	C₂₃×#97	—	2	4	no	—
69	C₄×C₈×C₂₃	—	2	2	no	—
70	C₄×C₈×C₂₃	D₄	1	4	yes	—
71	C₂₃×#95	C₂×C₄	1	4	yes	—
72	C₂₃×#95	C₂×C₄	1	4	yes	—
73	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂×Q₈×C₂₃	—	2	2	no	—
75	C₂×Q₈×C₂₃	—	2	2	no	—
76	C₂×Q₈×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	2	2	no	—
79	C₂×D₄×C₂₃	M₁₆	1	4	yes	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	—	2	4	no	—
83	C₂²×C₄×C₂₃	—	2	4	no	—
84	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	4	no	—
98	C₂×C₈×C₂₃	—	2	4	no	—
99	C₂×C₈×C₂₃	—	4	8	no	—
100	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₂₃	#41	1	4	yes	—
103	C₂×C₈×C₂₃	#41	1	4	yes	—
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₄⋊C₄×C₂₃	—	2	4	no	—
107	C₂²⋊C₄×C₂₃	—	2	8	no	—
108	C₂²⋊C₄×C₂₃	—	2	4	no	—
109	C₂²⋊C₄×C₂₃	—	4	4	no	—
110	C₄²×C₂₃	—	2	2	no	—
111	C₄²×C₂₃	—	2	2	no	—
112	C₄²×C₂₃	—	2	2	no	—
113	C₄²×C₂₃	—	2	2	no	—
114	C₄²×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	—	2	2	no	—
116	C₄²×C₂₃	—	4	2	no	—
117	C₄²×C₂₃	C₂×D₄	1	2	yes	—
118	C₄²×C₂₃	#41	1	2	yes	—
119	#21633	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	2	2	no	—
121	C₂³×C₂₃	—	2	4	no	—
122	C₂³×C₂₃	—	2	2	no	—
123	C₂³×C₂₃	—	2	4	no	—
124	C₂³×C₂₃	C₂×M₁₆	1	4	yes	—
125	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
126	Q₈×C₂₃	—	4	4	no	—
127	D₄×C₂₃	—	2	8	no	—
128	C₂×C₄×C₂₃	—	2	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	2	2	no	—
133	C₂×C₄×C₂₃	—	2	2	no	—
134	C₂×C₄×C₂₃	—	2	4	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	2	4	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	—	4	4	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₂₃	#131	1	2	yes	—
149	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
151	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
153	C₈×C₂₃	—	2	4	no	—
154	C₈×C₂₃	—	2	2	no	—
155	C₈×C₂₃	—	4	2	no	—
156	C₈×C₂₃	—	4	2	no	—
157	C₈×C₂₃	#124	1	4	yes	complemented
158	C₂×#446	C₂×C₂₃	1	1	yes	characteristic, complemented
159	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
160	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
161	#1984	C₂×C₂₃	1	1	yes	characteristic, complemented
162	#1954	C₂×C₂₃	1	1	yes	characteristic
163	#1924	C₂×C₂₃	1	4	yes	complemented
164	#1831	C₂×C₂₃	1	1	yes	characteristic, complemented
165	#1752	C₂×C₂₃	1	4	yes	complemented
166	C₄×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
167	C₂²×C₂₃	—	2	4	no	—
168	C₂²×C₂₃	—	2	8	no	—
169	C₂²×C₂₃	—	4	2	no	—
170	C₂²×C₂₃	—	4	4	no	—
171	C₂²×C₂₃	—	4	2	no	—
172	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₂₃	C₄×#41	1	2	yes	—
174	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
175	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
176	C₂²×C₂₃	#568	1	2	yes	—
177	C₄×C₂₃	—	2	2	no	—
178	C₄×C₂₃	—	4	2	no	—
179	C₄×C₂₃	—	4	2	no	—
180	C₄×C₂₃	—	4	2	no	—
181	C₄×C₂₃	—	4	2	no	—
182	C₄×C₂₃	—	4	2	no	—
183	C₄×C₂₃	#435	1	4	yes	—
184	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
185	C₄×C₂₃	#529	1	2	yes	—
186	C₄×C₂₃	#532	1	1	yes	characteristic
187	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
188	C₂×#124	C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
190	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
191	#446	—	2	4	no	—
192	C₈×D₄	—	2	4	no	—
193	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
195	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
196	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
198	C₂×#97	C₂²×C₂₃	1	2	yes	—
199	C₂×#97	C₂²×C₂₃	1	2	yes	—
200	C₂×#97	C₂²×C₂₃	1	2	yes	—
201	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
203	#391	C₄×C₂₃	1	1	yes	characteristic
204	#390	C₄×C₂₃	1	1	yes	characteristic
205	#388	C₄×C₂₃	1	4	yes	—
206	#382	C₄×C₂₃	1	1	yes	characteristic
207	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
208	C₄³	C₂²×C₂₃	1	1	yes	characteristic
209	#336	C₂²×C₂₃	1	4	yes	—
210	#336	C₂²×C₂₃	1	4	yes	—
211	C₂×C₂₃	—	4	4	no	—
212	C₂×C₂₃	C₄×#124	1	1	yes	characteristic
213	C₂×C₂₃	C₂×#435	1	1	yes	characteristic
214	C₂×C₂₃	#2932	1	1	yes	characteristic
215	C₂×C₂₃	#2969	1	2	yes	—
216	C₂×C₂₃	#2970	1	2	yes	—
217	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
218	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
219	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	2	8	no	—
222	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
223	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
224	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
226	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
227	#124	—	2	4	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	2	no	—
232	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
235	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
236	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
237	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
238	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
241	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
242	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
243	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	4	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	C₂²⋊C₈	—	2	4	no	—
248	C₂²⋊C₈	—	2	8	no	—
249	#97	—	2	4	no	—
250	#97	—	2	4	no	—
251	#97	—	2	4	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	D₄×C₂₃	1	4	yes	—
254	#95	C₂×C₄×C₂₃	1	4	yes	—
255	#95	C₂×C₄×C₂₃	1	4	yes	—
256	C₂₃	#21633	1	1	yes	characteristic, direct product factor, Sylow
257	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	M₁₆×C₂₃	1	4	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
269	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
270	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
273	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
275	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	4	8	no	—
284	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
286	C₂×C₈	#41×C₂₃	1	4	yes	—
287	C₂×C₈	#41×C₂₃	1	4	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₂²⋊C₄	—	2	8	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	4	4	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	C₂×D₄×C₂₃	1	2	yes	—
302	C₄²	#41×C₂₃	1	2	yes	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	4	no	—
305	C₂³	—	2	2	no	—
306	C₂³	—	2	4	no	—
307	C₂³	C₂×M₁₆×C₂₃	1	4	yes	—
308	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
309	Q₈	—	4	4	no	—
310	D₄	—	2	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
329	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
330	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
331	C₂×C₄	C₂₃×#131	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
334	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
336	C₈	—	2	4	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	C₂₃×#124	1	4	yes	complemented
341	C₂²	—	2	4	no	—
342	C₂²	—	2	8	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
347	C₂²	C₄×#41×C₂₃	1	2	yes	—
348	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
349	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
350	C₂²	C₂₃×#568	1	2	yes	—
351	C₄	—	2	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	C₂₃×#435	1	4	yes	—
358	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
359	C₄	C₂₃×#529	1	2	yes	—
360	C₄	C₂₃×#532	1	1	yes	characteristic
361	C₂	—	4	4	no	—
362	C₂	C₄×C₂₃×#124	1	1	yes	characteristic
363	C₂	C₂×C₂₃×#435	1	1	yes	characteristic
364	C₂	C₂₃×#2932	1	1	yes	characteristic
365	C₂	C₂₃×#2969	1	2	yes	—
366	C₂	C₂₃×#2970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#446	C₂	semidirect
4. C₂₃×#1984	C₂	semidirect
6. C₂₃×#1924	C₂	semidirect
7. C₂₃×#1831	C₂	semidirect
8. C₂₃×#1752	C₂	semidirect
9. C₄×C₂₃×#97	C₂	semidirect
119. #21633	C₂₃	direct (N × H)
157. C₈×C₂₃	#124	semidirect
158. C₂×#446	C₂×C₂₃	semidirect
161. #1984	C₂×C₂₃	semidirect
163. #1924	C₂×C₂₃	semidirect
164. #1831	C₂×C₂₃	semidirect
165. #1752	C₂×C₂₃	semidirect
166. C₄×#97	C₂×C₂₃	semidirect
256. C₂₃	#21633	direct (N × H)
340. C₈	C₂₃×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#446	2
C₂×C₈×D₄×C₂₃	2
C₄×C₂₃×#124	2
C₂₃×#1984	2
C₂₃×#1954	2
C₂₃×#1924	2
C₂₃×#1831	2
C₂₃×#1752	2
C₄×C₂₃×#97	2
#21633	23

Subgroups — C23×#21633 #19169259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21633 #19169259