Subgroups — C₂₃×#8042 #19155668

Order 5888.

654 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1296	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂×C₂₃×#117	—	2	1	no	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#432	—	2	2	no	—
9	C₂₃×#432	C₄	1	4	yes	complemented
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#423	C₄	1	4	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₂₃×#97	—	2	1	no	—
17	C₂₃×#384	C₂²	1	2	yes	—
18	C₂₃×#382	—	2	1	no	—
19	C₂₃×#380	—	2	1	no	—
20	C₂₃×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	—	4	4	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂₃×#117	—	4	4	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	—	2	4	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	—	2	2	no	—
47	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
48	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₂₃×C₂²⋊C₈	—	2	4	no	—
50	C₂₃×#97	—	2	4	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₂₃	—	4	8	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₂²⋊C₄×C₂₃	—	4	4	no	—
83	C₄²×C₂₃	—	4	4	no	—
84	C₄²×C₂₃	C₄²	1	2	yes	—
85	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
94	#8042	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	—	2	1	no	—
98	C₂³×C₂₃	—	2	1	no	—
99	C₂³×C₂₃	#95	1	2	yes	—
100	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	#95	1	2	yes	—
117	C₂×C₄×C₂₃	#95	1	2	yes	—
118	C₂×C₄×C₂₃	#95	1	2	yes	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₂₃	#137	1	2	yes	—
128	C₂×C₄×C₂₃	#137	1	2	yes	—
129	C₈×C₂₃	—	4	4	no	—
130	C₈×C₂₃	—	4	4	no	—
131	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
132	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
133	#2285	C₂×C₂₃	1	1	yes	characteristic
134	#1296	C₂×C₂₃	1	4	yes	complemented
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	2	2	no	—
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	—	4	2	no	—
139	C₂²×C₂₃	—	4	2	no	—
140	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
141	C₂²×C₂₃	#413	1	2	yes	—
142	C₂²×C₂₃	#417	1	2	yes	—
143	C₂²×C₂₃	#418	1	1	yes	characteristic
144	C₂²×C₂₃	#419	1	1	yes	characteristic
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	2	2	no	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	8	4	no	—
151	C₄×C₂₃	#417	1	2	yes	—
152	C₄×C₂₃	#418	1	1	yes	characteristic
153	C₄×C₂₃	#419	1	1	yes	characteristic
154	C₄×C₂₃	#428	1	2	yes	—
155	C₄×C₂₃	#428	1	2	yes	—
156	C₂²×M₁₆	—	2	1	no	—
157	C₂×#117	—	2	1	no	—
158	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
159	#432	—	2	2	no	—
160	#432	C₄×C₂₃	1	4	yes	complemented
161	#423	—	2	2	no	—
162	#423	C₄×C₂₃	1	4	yes	complemented
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
166	C₂×C₂²⋊C₈	—	2	1	no	—
167	C₂×#97	—	2	1	no	—
168	#384	C₂²×C₂₃	1	2	yes	—
169	#382	—	2	1	no	—
170	#380	—	2	1	no	—
171	#380	C₄×C₂₃	1	2	yes	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
174	C₂×C₂₃	—	4	2	no	—
175	C₂×C₂₃	#1557	1	2	yes	—
176	C₂×C₂₃	#1559	1	2	yes	—
177	C₂×C₂₃	#1736	1	1	yes	characteristic
178	C₂×C₂₃	#1736	1	1	yes	characteristic
179	C₂×C₂₃	#1742	1	1	yes	characteristic
180	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
182	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	4	4	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	#117	—	4	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
196	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₂₃	1	2	yes	—
199	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
205	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
206	C₂²⋊C₈	—	2	4	no	—
207	#97	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂₃	#8042	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
225	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
226	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
230	M₁₆	—	4	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	4	4	no	—
242	C₄²	C₄²×C₂₃	1	2	yes	—
243	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
244	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
245	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
246	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
247	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
248	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
249	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₂₃×#95	1	2	yes	—
257	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂₃×#95	1	2	yes	—
274	C₂×C₄	C₂₃×#95	1	2	yes	—
275	C₂×C₄	C₂₃×#95	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
283	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
284	C₂×C₄	C₂₃×#137	1	2	yes	—
285	C₂×C₄	C₂₃×#137	1	2	yes	—
286	C₈	—	4	4	no	—
287	C₈	—	4	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
294	C₂²	C₂₃×#413	1	2	yes	—
295	C₂²	C₂₃×#417	1	2	yes	—
296	C₂²	C₂₃×#418	1	1	yes	characteristic
297	C₂²	C₂₃×#419	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	8	4	no	—
304	C₄	C₂₃×#417	1	2	yes	—
305	C₄	C₂₃×#418	1	1	yes	characteristic
306	C₄	C₂₃×#419	1	1	yes	characteristic
307	C₄	C₂₃×#428	1	2	yes	—
308	C₄	C₂₃×#428	1	2	yes	—
309	C₂	—	4	2	no	—
310	C₂	C₂₃×#1557	1	2	yes	—
311	C₂	C₂₃×#1559	1	2	yes	—
312	C₂	C₂₃×#1736	1	1	yes	characteristic
313	C₂	C₂₃×#1736	1	1	yes	characteristic
314	C₂	C₂₃×#1742	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1296	C₂	semidirect
9. C₂₃×#432	C₄	semidirect
11. C₂₃×#423	C₄	semidirect
94. #8042	C₂₃	direct (N × H)
134. #1296	C₂×C₂₃	semidirect
160. #432	C₄×C₂₃	semidirect
162. #423	C₄×C₂₃	semidirect
210. C₂₃	#8042	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#432	2
C₂×C₂₃×#423	2
C₂₃×#2285	2
C₂₃×#1296	2
#8042	23

Subgroups — C23×#8042 #19155668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8042 #19155668