Subgroups — C₂₃×#8707 #19156333

Order 5888.

680 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₂₃	—	2	1	no	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
11	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
14	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
15	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₂₃×#95	—	2	1	no	—
18	C₂₃×#336	—	2	1	no	—
19	C₂₃×#336	—	2	1	no	—
20	C₂³×C₄×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	—	2	1	no	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₂₃	C₈	1	4	yes	complemented
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	—	2	4	no	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	—	4	4	no	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	4	no	—
43	C₂₃×C₂²⋊C₈	—	2	4	no	—
44	C₂₃×C₂²⋊C₈	—	4	2	no	—
45	C₄×C₈×C₂₃	—	2	4	no	—
46	C₂₃×#95	—	2	2	no	—
47	C₂₃×#95	—	2	2	no	—
48	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	4	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	2	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
69	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
85	C₄⋊C₄×C₂₃	M₁₆	1	4	yes	—
86	C₄²×C₂₃	—	2	2	no	—
87	C₄²×C₂₃	—	2	2	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	—	4	4	no	—
90	C₄²×C₂₃	—	4	4	no	—
91	C₄²×C₂₃	#41	1	4	yes	—
92	#8707	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	—	2	1	no	—
97	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
98	C₂³×C₂₃	#103	1	2	yes	—
99	C₂³×C₂₃	#104	1	2	yes	—
100	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	4	8	no	—
118	C₂×C₄×C₂₃	—	4	8	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₂₃	#103	1	2	yes	—
125	C₂×C₄×C₂₃	#117	1	2	yes	—
126	C₂×C₄×C₂₃	#123	1	2	yes	—
127	C₂×C₄×C₂₃	#123	1	2	yes	—
128	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
133	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₈×C₂₃	—	4	4	no	—
135	C₈×C₂₃	—	8	2	no	—
136	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
137	#1797	C₂×C₂₃	1	1	yes	characteristic
138	C₄×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic
139	#1279	C₂×C₂₃	1	4	yes	complemented
140	C₂²×C₂₃	—	2	2	no	—
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	2	2	no	—
144	C₂²×C₂₃	—	2	2	no	—
145	C₂²×C₂₃	—	2	2	no	—
146	C₂²×C₂₃	Q₈⋊C₈	1	4	yes	—
147	C₂²×C₂₃	#381	1	1	yes	characteristic
148	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₂₃	#407	1	1	yes	characteristic
150	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
151	C₂²×C₂₃	#418	1	1	yes	characteristic
152	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
153	C₂²×C₂₃	#421	1	1	yes	characteristic
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	2	2	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	4	1	no	—
158	C₄×C₂₃	—	4	4	no	—
159	C₄×C₂₃	—	4	4	no	—
160	C₄×C₂₃	#431	1	2	yes	—
161	C₄×C₂₃	#432	1	2	yes	—
162	C₄×C₂₃	#481	1	2	yes	—
163	C₄×C₂₃	#484	1	2	yes	—
164	C₂²×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄²	—	2	1	no	—
167	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
176	C₂×#95	—	2	1	no	—
177	#336	—	2	1	no	—
178	#336	—	2	1	no	—
179	C₂×C₂₃	—	2	4	no	—
180	C₂×C₂₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
181	C₂×C₂₃	#1478	1	1	yes	characteristic
182	C₂×C₂₃	#1487	1	1	yes	characteristic
183	C₂×C₂₃	#1490	1	1	yes	characteristic
184	C₂×C₂₃	#1817	1	1	yes	characteristic
185	C₂×C₂₃	#1824	1	1	yes	characteristic
186	C₂×C₂₃	#1826	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	C₈×C₂₃	1	4	yes	complemented
199	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	4	4	no	—
205	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂²⋊C₈	—	4	2	no	—
212	C₄×C₈	—	2	4	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	C₂₃	#8707	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
236	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
237	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
238	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
253	C₄⋊C₄	M₁₆×C₂₃	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	#41×C₂₃	1	4	yes	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
265	C₂³	C₂₃×#103	1	2	yes	—
266	C₂³	C₂₃×#104	1	2	yes	—
267	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
291	C₂×C₄	C₂₃×#103	1	2	yes	—
292	C₂×C₄	C₂₃×#117	1	2	yes	—
293	C₂×C₄	C₂₃×#123	1	2	yes	—
294	C₂×C₄	C₂₃×#123	1	2	yes	—
295	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
297	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
298	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
300	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
301	C₈	—	4	4	no	—
302	C₈	—	8	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂₃×Q₈⋊C₈	1	4	yes	—
310	C₂²	C₂₃×#381	1	1	yes	characteristic
311	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
312	C₂²	C₂₃×#407	1	1	yes	characteristic
313	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
314	C₂²	C₂₃×#418	1	1	yes	characteristic
315	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
316	C₂²	C₂₃×#421	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	1	no	—
323	C₄	C₂₃×#431	1	2	yes	—
324	C₄	C₂₃×#432	1	2	yes	—
325	C₄	C₂₃×#481	1	2	yes	—
326	C₄	C₂₃×#484	1	2	yes	—
327	C₂	—	2	4	no	—
328	C₂	C₂×C₂₃×Q₈⋊C₈	1	1	yes	characteristic
329	C₂	C₂₃×#1478	1	1	yes	characteristic
330	C₂	C₂₃×#1487	1	1	yes	characteristic
331	C₂	C₂₃×#1490	1	1	yes	characteristic
332	C₂	C₂₃×#1817	1	1	yes	characteristic
333	C₂	C₂₃×#1824	1	1	yes	characteristic
334	C₂	C₂₃×#1826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1279	C₂	semidirect
31. C₂×C₄⋊C₄×C₂₃	C₈	semidirect
92. #8707	C₂₃	direct (N × H)
139. #1279	C₂×C₂₃	semidirect
198. C₂×C₄⋊C₄	C₈×C₂₃	semidirect
215. C₂₃	#8707	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×C₄×C₄⋊C₄×C₂₃	2
C₂₃×#1797	2
C₄×C₂₃×C₂²⋊C₈	2
C₂₃×#1279	2
#8707	23

Subgroups — C23×#8707 #19156333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8707 #19156333