Subgroups — C₂₃×#21950 #19169576

Order 5888.

894 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	C₂₃×#2879	C₂	1	2	yes	maximal
2	C₂×C₂₃×#408	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1855	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1811	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1810	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#108	C₂²	1	4	yes	—
11	C₂×C₂₃×#108	C₂²	1	4	yes	—
12	C₂₃×#408	—	2	4	no	—
13	C₂₃×#408	—	2	4	no	—
14	C₂₃×#408	C₄	1	4	yes	—
15	C₂₃×#408	C₄	1	4	yes	—
16	C₂₃×#407	C₂²	1	4	yes	—
17	C₂₃×#407	C₂²	1	4	yes	—
18	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₂₃×#341	C₂²	1	4	yes	—
20	C₂₃×#341	C₂²	1	4	yes	—
21	C₂×#41×C₂₃	—	4	2	no	—
22	C₂×#41×C₂₃	—	4	2	no	—
23	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×#41×C₂₃	D₄	1	2	yes	—
26	C₂×#41×C₂₃	D₄	1	2	yes	—
27	C₂²×Q₈×C₂₃	Q₈	1	1	yes	characteristic
28	C₂²×D₄×C₂₃	Q₈	1	1	yes	characteristic
29	C₂²×D₄×C₂₃	Q₈	1	2	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂×M₁₆×C₂₃	—	2	4	no	—
33	C₂×M₁₆×C₂₃	—	4	2	no	—
34	C₂×M₁₆×C₂₃	—	4	4	no	—
35	C₂×M₁₆×C₂₃	—	4	4	no	—
36	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
39	C₂²×C₈×C₂₃	—	2	4	no	—
40	C₂²×C₈×C₂₃	—	4	2	no	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂₃×#108	—	4	4	no	—
46	C₂₃×#108	—	4	4	no	—
47	C₂₃×C₂²⋊C₈	—	2	8	no	—
48	C₂₃×C₂²⋊C₈	—	2	4	no	—
49	C₂₃×C₂²⋊C₈	—	2	4	no	—
50	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
51	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂⁴×C₂₃	C₂×Q₈	1	2	yes	—
53	#41×C₂₃	—	4	2	no	—
54	#41×C₂₃	—	4	2	no	—
55	#41×C₂₃	—	4	2	no	—
56	#41×C₂₃	—	4	2	no	—
57	#41×C₂₃	—	4	2	no	—
58	#41×C₂₃	—	4	4	no	—
59	#41×C₂₃	—	4	2	no	—
60	C₂×Q₈×C₂₃	—	4	2	no	—
61	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
63	C₂×D₄×C₂₃	—	4	4	no	—
64	C₂×D₄×C₂₃	—	4	2	no	—
65	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
68	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	4	no	—
72	C₂²×C₄×C₂₃	—	2	4	no	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₂₃	—	2	8	no	—
83	M₁₆×C₂₃	—	2	8	no	—
84	M₁₆×C₂₃	—	4	2	no	—
85	M₁₆×C₂₃	—	4	2	no	—
86	M₁₆×C₂₃	—	4	8	no	—
87	C₂×C₈×C₂₃	—	2	4	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	2	8	no	—
90	C₂×C₈×C₂₃	—	4	2	no	—
91	C₂×C₈×C₂₃	—	4	2	no	—
92	C₂×C₈×C₂₃	—	4	8	no	—
93	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
95	#21950	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	—	4	2	no	—
99	C₂³×C₂₃	—	4	4	no	—
100	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
104	Q₈×C₂₃	—	4	2	no	—
105	Q₈×C₂₃	—	4	2	no	—
106	D₄×C₂₃	—	4	2	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	4	no	—
109	D₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	4	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
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	4	yes	—
125	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
129	C₈×C₂₃	—	2	4	no	—
130	C₈×C₂₃	—	2	4	no	—
131	C₈×C₂₃	—	4	8	no	—
132	#2879	C₂×C₂₃	1	2	yes	—
133	C₂×#408	C₂×C₂₃	1	1	yes	characteristic
134	#1855	C₂×C₂₃	1	8	yes	complemented
135	#1811	C₂×C₂₃	1	2	yes	complemented
136	#1810	C₂×C₂₃	1	2	yes	complemented
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	4	no	—
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	2	8	no	—
144	C₂²×C₂₃	—	4	4	no	—
145	C₂²×C₂₃	—	8	4	no	—
146	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₂₃	#543	1	2	yes	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	4	no	—
151	C₄×C₂₃	#518	1	1	yes	characteristic
152	C₄×C₂₃	#518	1	1	yes	characteristic
153	C₄×C₂₃	#534	1	2	yes	—
154	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
156	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#108	C₂²×C₂₃	1	4	yes	—
159	C₂×#108	C₂²×C₂₃	1	4	yes	—
160	#408	—	2	4	no	—
161	#408	—	2	4	no	—
162	#408	C₄×C₂₃	1	4	yes	—
163	#408	C₄×C₂₃	1	4	yes	—
164	#407	C₂²×C₂₃	1	4	yes	—
165	#407	C₂²×C₂₃	1	4	yes	—
166	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
167	#341	C₂²×C₂₃	1	4	yes	—
168	#341	C₂²×C₂₃	1	4	yes	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	—	4	1	no	—
171	C₂×C₂₃	—	4	1	no	—
172	C₂×C₂₃	—	4	4	no	—
173	C₂×C₂₃	#2315	1	1	yes	characteristic
174	C₂×C₂₃	#2916	1	2	yes	—
175	C₂×#41	—	4	2	no	—
176	C₂×#41	—	4	2	no	—
177	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
178	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
179	C₂×#41	D₄×C₂₃	1	2	yes	—
180	C₂×#41	D₄×C₂₃	1	2	yes	—
181	C₂²×Q₈	Q₈×C₂₃	1	1	yes	characteristic
182	C₂²×D₄	Q₈×C₂₃	1	1	yes	characteristic
183	C₂²×D₄	Q₈×C₂₃	1	2	yes	—
184	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	4	2	no	—
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	—	4	4	no	—
190	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
192	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
196	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
199	#108	—	4	4	no	—
200	#108	—	4	4	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	C₂²⋊C₈	—	2	4	no	—
203	C₂²⋊C₈	—	2	8	no	—
204	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
205	C₂₃	#21950	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
207	C₂⁴	C₂×Q₈×C₂₃	1	2	yes	—
208	#41	—	4	2	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	#41	—	4	2	no	—
212	#41	—	4	2	no	—
213	#41	—	4	4	no	—
214	#41	—	4	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
217	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
221	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
222	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
234	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
236	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
237	M₁₆	—	2	8	no	—
238	M₁₆	—	2	8	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	8	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	8	no	—
248	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
249	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	4	no	—
254	C₂³	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
255	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
256	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
257	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
277	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
282	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	4	no	—
285	C₈	—	4	8	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
296	C₂²	C₂₃×#543	1	2	yes	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	4	no	—
300	C₄	C₂₃×#518	1	1	yes	characteristic
301	C₄	C₂₃×#518	1	1	yes	characteristic
302	C₄	C₂₃×#534	1	2	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	4	1	no	—
305	C₂	—	4	1	no	—
306	C₂	—	4	4	no	—
307	C₂	C₂₃×#2315	1	1	yes	characteristic
308	C₂	C₂₃×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1855	C₂	semidirect
4. C₂₃×#1811	C₂	semidirect
5. C₂₃×#1810	C₂	semidirect
95. #21950	C₂₃	direct (N × H)
134. #1855	C₂×C₂₃	semidirect
135. #1811	C₂×C₂₃	semidirect
136. #1810	C₂×C₂₃	semidirect
205. C₂₃	#21950	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2879	2
C₂×C₂₃×#408	2
C₂₃×#1855	2
C₂₃×#1811	2
C₂₃×#1810	2
#21950	23

Subgroups — C23×#21950 #19169576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21950 #19169576