Subgroups — C₂₃×#12917 #19160543

Order 5888.

384 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×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#350	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1764	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
9	C₂₃×#504	—	2	2	no	—
10	C₂₃×#504	—	2	2	no	—
11	C₂×C₂₃×M₃₂	—	2	1	no	—
12	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₂₃	—	2	1	no	—
15	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#104	C₄	1	1	yes	characteristic
18	C₂₃×#416	C₄	1	1	yes	characteristic
19	C₂₃×#405	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#364	—	2	2	no	—
22	C₂₃×C₄⋊C₁₆	—	2	1	no	—
23	C₂₃×C₄⋊C₁₆	—	2	1	no	—
24	C₂₃×#350	—	2	2	no	—
25	C₂₃×#341	C₄	1	1	yes	characteristic
26	C₂₃×#339	C₄	1	1	yes	characteristic
27	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂₃×#131	—	2	1	no	—
29	C₂₃×#131	—	2	1	no	—
30	C₂₃×#131	D₄	1	2	yes	—
31	C₂₃×#131	Q₈	1	2	yes	—
32	C₂×M₁₆×C₂₃	—	2	1	no	—
33	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	—	2	1	no	—
36	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂₃×M₃₂	—	2	4	no	—
41	C₂₃×M₃₂	—	2	1	no	—
42	C₂₃×M₃₂	—	2	1	no	—
43	C₂₃×M₃₂	—	2	1	no	—
44	C₂₃×M₃₂	D₄	1	2	yes	—
45	C₂×C₁₆×C₂₃	—	2	4	no	—
46	C₂×C₁₆×C₂₃	—	2	1	no	—
47	C₂×C₁₆×C₂₃	—	2	1	no	—
48	C₂×C₁₆×C₂₃	—	2	1	no	—
49	C₂×C₁₆×C₂₃	—	2	1	no	—
50	C₂×C₁₆×C₂₃	—	2	1	no	—
51	C₂×C₁₆×C₂₃	—	2	1	no	—
52	C₂×C₁₆×C₂₃	—	2	1	no	—
53	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
55	C₂₃×#104	—	2	2	no	—
56	C₂₃×#103	C₈	1	1	yes	characteristic
57	C₂₃×#103	C₈	1	1	yes	characteristic
58	C₂₃×#102	C₈	1	1	yes	characteristic
59	C₂₃×#102	C₈	1	1	yes	characteristic
60	C₂₃×#97	—	2	1	no	—
61	C₄×C₈×C₂₃	—	2	1	no	—
62	C₄×C₈×C₂₃	—	2	1	no	—
63	C₄×C₈×C₂₃	—	2	1	no	—
64	#41×C₂₃	—	2	1	no	—
65	#41×C₂₃	—	2	1	no	—
66	#41×C₂₃	C₄⋊C₄	1	2	yes	—
67	#41×C₂₃	M₁₆	1	2	yes	—
68	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
69	C₂×D₄×C₂₃	—	2	1	no	—
70	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	—	2	1	no	—
72	C₂²×C₄×C₂₃	—	2	1	no	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	M₁₆×C₂₃	—	2	1	no	—
75	M₁₆×C₂₃	—	2	2	no	—
76	M₁₆×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	—	2	1	no	—
78	M₁₆×C₂₃	—	2	1	no	—
79	M₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	2	1	no	—
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	1	no	—
86	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
95	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
96	C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₄²×C₂₃	—	2	1	no	—
98	C₄²×C₂₃	—	2	1	no	—
99	C₄²×C₂₃	—	2	1	no	—
100	C₁₆×C₂₃	—	2	1	no	—
101	C₁₆×C₂₃	—	2	2	no	—
102	C₁₆×C₂₃	—	2	1	no	—
103	C₁₆×C₂₃	—	4	1	no	—
104	C₁₆×C₂₃	—	4	1	no	—
105	#12917	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₃	—	2	1	no	—
107	C₂³×C₂₃	#131	1	1	yes	characteristic
108	Q₈×C₂₃	—	2	1	no	—
109	Q₈×C₂₃	C₄⋊C₈	1	2	yes	—
110	D₄×C₂₃	—	2	2	no	—
111	D₄×C₂₃	—	2	2	no	—
112	D₄×C₂₃	—	2	1	no	—
113	D₄×C₂₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	4	1	no	—
121	C₂×C₄×C₂₃	—	4	1	no	—
122	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
128	C₈×C₂₃	—	2	2	no	—
129	C₈×C₂₃	—	2	1	no	—
130	C₈×C₂₃	—	2	1	no	—
131	C₈×C₂₃	—	4	1	no	—
132	C₈×C₂₃	#120	1	1	yes	characteristic
133	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₂₃	#123	1	1	yes	characteristic
136	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
137	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#2151	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×#350	C₂×C₂₃	1	1	yes	characteristic
140	#1764	C₂×C₂₃	1	1	yes	characteristic
141	#1379	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#1374	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	2	no	—
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	#424	1	1	yes	characteristic
149	C₂²×C₂₃	C₈×D₄	1	1	yes	characteristic
150	C₂²×C₂₃	#435	1	1	yes	characteristic
151	C₄×C₂₃	—	2	2	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	—	4	1	no	—
154	C₄×C₂₃	—	4	1	no	—
155	C₄×C₂₃	#385	1	1	yes	characteristic
156	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
157	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
158	C₄×C₂₃	#435	1	1	yes	characteristic
159	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
160	#504	—	2	2	no	—
161	#504	—	2	2	no	—
162	C₂×M₃₂	—	2	1	no	—
163	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
165	C₂²×C₁₆	—	2	1	no	—
166	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
167	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×#104	C₄×C₂₃	1	1	yes	characteristic
169	#416	C₄×C₂₃	1	1	yes	characteristic
170	#405	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
172	#364	—	2	2	no	—
173	C₄⋊C₁₆	—	2	1	no	—
174	C₄⋊C₁₆	—	2	1	no	—
175	#350	—	2	2	no	—
176	#341	C₄×C₂₃	1	1	yes	characteristic
177	#339	C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₂₃	—	2	1	no	—
179	C₂×C₂₃	—	2	1	no	—
180	C₂×C₂₃	—	4	2	no	—
181	C₂×C₂₃	#1850	1	1	yes	characteristic
182	C₂×C₂₃	#2170	1	1	yes	characteristic
183	C₂×C₂₃	#2171	1	1	yes	characteristic
184	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
185	#131	—	2	1	no	—
186	#131	—	2	1	no	—
187	#131	D₄×C₂₃	1	2	yes	—
188	#131	Q₈×C₂₃	1	2	yes	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
195	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
197	M₃₂	—	2	4	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	D₄×C₂₃	1	2	yes	—
202	C₂×C₁₆	—	2	4	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
212	#104	—	2	2	no	—
213	#103	C₈×C₂₃	1	1	yes	characteristic
214	#103	C₈×C₂₃	1	1	yes	characteristic
215	#102	C₈×C₂₃	1	1	yes	characteristic
216	#102	C₈×C₂₃	1	1	yes	characteristic
217	#97	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₂₃	#12917	1	1	yes	characteristic, direct product factor, Sylow
222	#41	—	2	1	no	—
223	#41	—	2	1	no	—
224	#41	C₄⋊C₄×C₂₃	1	2	yes	—
225	#41	M₁₆×C₂₃	1	2	yes	—
226	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	1	no	—
236	M₁₆	—	2	1	no	—
237	M₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
250	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
251	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
252	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	1	no	—
261	C₁₆	—	4	1	no	—
262	C₁₆	—	4	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	C₂₃×#131	1	1	yes	characteristic
265	Q₈	—	2	1	no	—
266	Q₈	C₂₃×C₄⋊C₈	1	2	yes	—
267	D₄	—	2	2	no	—
268	D₄	—	2	2	no	—
269	D₄	—	2	1	no	—
270	D₄	C₂₃×C₄⋊C₈	1	2	yes	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
284	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	4	1	no	—
289	C₈	C₂₃×#120	1	1	yes	characteristic
290	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
291	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
292	C₈	C₂₃×#123	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₂₃×#424	1	1	yes	characteristic
299	C₂²	C₈×D₄×C₂₃	1	1	yes	characteristic
300	C₂²	C₂₃×#435	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂₃×#385	1	1	yes	characteristic
306	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
307	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
308	C₄	C₂₃×#435	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	2	no	—
312	C₂	C₂₃×#1850	1	1	yes	characteristic
313	C₂	C₂₃×#2170	1	1	yes	characteristic
314	C₂	C₂₃×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₈×D₄
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#364	C₂	semidirect
3. C₂₃×#2151	C₂	semidirect
6. C₂₃×#1379	C₂	semidirect
7. C₂₃×#1374	C₂	semidirect
105. #12917	C₂₃	direct (N × H)
137. C₂×#364	C₂×C₂₃	semidirect
138. #2151	C₂×C₂₃	semidirect
141. #1379	C₂×C₂₃	semidirect
142. #1374	C₂×C₂₃	semidirect
221. C₂₃	#12917	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×#504	2
C₂×C₂₃×#364	2
C₂₃×#2151	2
C₂×C₂₃×#350	2
C₂₃×#1764	2
C₂₃×#1379	2
C₂₃×#1374	2
#12917	23

Subgroups — C23×#12917 #19160543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12917 #19160543