Subgroups — C₂₃×#11238 #19158864

Order 5888.

682 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#343	C₂	1	2	yes	maximal
4	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1736	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#106	—	2	2	no	—
12	C₂×C₂₃×#106	—	2	4	no	—
13	C₂₃×#424	—	2	2	no	—
14	C₂₃×#424	C₄	1	4	yes	complemented
15	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
16	C₂₃×#343	—	2	4	no	—
17	C₂₃×#341	—	2	2	no	—
18	C₂₃×#340	—	2	4	no	—
19	C₂³×C₄×C₂₃	—	2	2	no	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	—	2	1	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	—	2	2	no	—
33	C₂²×C₈×C₂₃	—	2	2	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₂₃	D₄	1	2	yes	—
37	C₂²×C₈×C₂₃	D₄	1	2	yes	—
38	C₂₃×#117	—	2	1	no	—
39	C₂₃×#117	—	2	1	no	—
40	C₂₃×#117	C₂×C₄	1	2	yes	—
41	C₂₃×#117	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
50	C₂×C₄²×C₂₃	—	2	1	no	—
51	C₂₃×#106	—	2	8	no	—
52	C₂₃×C₄⋊C₈	—	2	8	no	—
53	C₂⁴×C₂₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	—	8	4	no	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₂₃	#41	1	2	yes	—
67	C₂²×C₄×C₂₃	#41	1	2	yes	—
68	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	—	4	2	no	—
83	C₂×C₈×C₂₃	—	4	2	no	—
84	C₂×C₈×C₂₃	—	4	1	no	—
85	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	4	no	—
88	C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	—	4	2	no	—
91	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	#11238	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	#117	1	1	yes	characteristic
99	C₂³×C₂₃	#121	1	1	yes	characteristic
100	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂³×C₂₃	#123	1	2	yes	—
102	C₂³×C₂₃	#125	1	2	yes	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	—	8	4	no	—
108	C₂×C₄×C₂₃	—	8	4	no	—
109	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#121	1	2	yes	—
115	C₂×C₄×C₂₃	#121	1	2	yes	—
116	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₂₃	#125	1	2	yes	—
124	C₂×C₄×C₂₃	#126	1	2	yes	—
125	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	—	4	2	no	—
129	C₈×C₂₃	—	4	4	no	—
130	C₂²×#106	C₂×C₂₃	1	1	yes	characteristic
131	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
132	C₂×#343	C₂×C₂₃	1	2	yes	—
133	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
134	#1736	C₂×C₂₃	1	2	yes	—
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	2	2	no	—
137	C₂²×C₂₃	—	2	4	no	—
138	C₂²×C₂₃	—	2	2	no	—
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	#380	1	1	yes	characteristic
141	C₂²×C₂₃	#387	1	2	yes	—
142	C₂²×C₂₃	#394	1	1	yes	characteristic
143	C₂²×C₂₃	#398	1	2	yes	—
144	C₂²×C₂₃	#398	1	1	yes	characteristic
145	C₂²×C₂₃	#471	1	4	yes	—
146	C₄×C₂₃	—	4	4	no	—
147	C₄×C₂₃	—	8	4	no	—
148	C₄×C₂₃	#386	1	1	yes	characteristic
149	C₄×C₂₃	#389	1	2	yes	—
150	C₄×C₂₃	#391	1	1	yes	characteristic
151	C₄×C₂₃	#395	1	1	yes	characteristic
152	C₄×C₂₃	#396	1	2	yes	—
153	C₄×C₂₃	#400	1	1	yes	characteristic
154	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
156	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
159	C₂×#106	—	2	2	no	—
160	C₂×#106	—	2	4	no	—
161	#424	—	2	2	no	—
162	#424	C₄×C₂₃	1	4	yes	complemented
163	C₂×C₄⋊C₈	—	2	2	no	—
164	#343	—	2	4	no	—
165	#341	—	2	2	no	—
166	#340	—	2	4	no	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	#1441	1	1	yes	characteristic
170	C₂×C₂₃	#1815	1	2	yes	—
171	C₂×C₂₃	#2046	1	2	yes	—
172	C₂×C₂₃	#2090	1	2	yes	—
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
184	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
190	C₂²×C₈	D₄×C₂₃	1	2	yes	—
191	C₂²×C₈	D₄×C₂₃	1	2	yes	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	C₂×C₄×C₂₃	1	2	yes	—
195	#117	Q₈×C₂₃	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
202	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₄²	—	2	1	no	—
205	#106	—	2	8	no	—
206	C₄⋊C₈	—	2	8	no	—
207	C₂₃	#11238	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	#41×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
221	C₂²×C₄	#41×C₂₃	1	2	yes	—
222	C₂²×C₄	#41×C₂₃	1	2	yes	—
223	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	C₂₃×#117	1	1	yes	characteristic
253	C₂³	C₂₃×#121	1	1	yes	characteristic
254	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂³	C₂₃×#123	1	2	yes	—
256	C₂³	C₂₃×#125	1	2	yes	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
266	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
267	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
268	C₂×C₄	C₂₃×#121	1	2	yes	—
269	C₂×C₄	C₂₃×#121	1	2	yes	—
270	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
271	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
272	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
276	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
277	C₂×C₄	C₂₃×#125	1	2	yes	—
278	C₂×C₄	C₂₃×#126	1	2	yes	—
279	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
280	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	C₂₃×#380	1	1	yes	characteristic
290	C₂²	C₂₃×#387	1	2	yes	—
291	C₂²	C₂₃×#394	1	1	yes	characteristic
292	C₂²	C₂₃×#398	1	2	yes	—
293	C₂²	C₂₃×#398	1	1	yes	characteristic
294	C₂²	C₂₃×#471	1	4	yes	—
295	C₄	—	4	4	no	—
296	C₄	—	8	4	no	—
297	C₄	C₂₃×#386	1	1	yes	characteristic
298	C₄	C₂₃×#389	1	2	yes	—
299	C₄	C₂₃×#391	1	1	yes	characteristic
300	C₄	C₂₃×#395	1	1	yes	characteristic
301	C₄	C₂₃×#396	1	2	yes	—
302	C₄	C₂₃×#400	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	C₂₃×#1441	1	1	yes	characteristic
306	C₂	C₂₃×#1815	1	2	yes	—
307	C₂	C₂₃×#2046	1	2	yes	—
308	C₂	C₂₃×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂₃×#424	C₄	semidirect
96. #11238	C₂₃	direct (N × H)
162. #424	C₄×C₂₃	semidirect
207. C₂₃	#11238	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#106	2
C₂×C₂₃×#424	2
C₂×C₂₃×#343	2
C₂×C₂₃×#341	2
C₂₃×#1736	2
#11238	23

Subgroups — C23×#11238 #19158864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11238 #19158864