Subgroups — C₂₃×#20787 #19168413

Order 5888.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1841	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1833	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1827	C₂	1	2	yes	maximal
5	C₂₃×#1813	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1812	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1772	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1757	C₂	1	2	yes	maximal
9	C₂₃×#1747	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
12	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#108	C₂²	1	2	yes	—
14	C₂₃×#427	C₂²	1	2	yes	—
15	C₂₃×#413	C₂²	1	2	yes	—
16	C₂₃×#410	C₂²	1	2	yes	—
17	C₂₃×#405	C₄	1	2	yes	complemented
18	C₂₃×#405	C₄	1	2	yes	complemented
19	C₂₃×#405	C₂²	1	4	yes	—
20	C₂₃×#405	C₂²	1	1	yes	characteristic
21	C₂₃×#405	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₂₃	C₄	1	2	yes	complemented
23	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	complemented
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#344	C₂²	1	4	yes	—
27	C₂₃×#339	C₂²	1	4	yes	—
28	C₂₃×#338	C₂²	1	2	yes	—
29	C₂₃×H(3,ℤ/4)	C₂²	1	2	yes	—
30	C₂₃×H(3,ℤ/4)	C₂²	1	4	yes	—
31	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×#131	—	2	2	no	—
33	C₂₃×#131	—	2	2	no	—
34	C₂₃×#131	—	2	2	no	—
35	C₂₃×#131	—	2	2	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	4	yes	—
39	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₂₃	—	2	1	no	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	—	2	1	no	—
48	C₂₃×#117	C₂×C₄	1	2	yes	—
49	C₂₃×#117	C₂³	1	4	yes	—
50	C₂₃×#117	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×#108	—	2	2	no	—
54	C₂₃×#108	—	2	2	no	—
55	C₂₃×#107	—	2	2	no	—
56	C₂₃×#106	—	2	2	no	—
57	C₂₃×#101	—	2	2	no	—
58	C₂₃×#100	—	2	2	no	—
59	C₂₃×#99	—	2	4	no	—
60	C₂₃×#97	—	2	4	no	—
61	C₂₃×#97	C₂×C₄	1	2	yes	—
62	C₂₃×#97	C₂×C₄	1	2	yes	—
63	C₂₃×#97	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₂₃	—	2	4	no	—
65	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
72	#41×C₂₃	—	2	2	no	—
73	#41×C₂₃	—	2	2	no	—
74	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
75	C₂×D₄×C₂₃	#41	1	2	yes	—
76	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	—	2	2	no	—
78	C₂²×C₄×C₂₃	—	2	1	no	—
79	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₂₃	—	2	2	no	—
82	M₁₆×C₂₃	—	2	2	no	—
83	M₁₆×C₂₃	—	2	4	no	—
84	M₁₆×C₂₃	—	2	4	no	—
85	M₁₆×C₂₃	—	2	4	no	—
86	M₁₆×C₂₃	—	2	4	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	4	no	—
92	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	#41	1	2	yes	—
99	C₂×C₈×C₂₃	#41	1	2	yes	—
100	C₂×C₈×C₂₃	#41	1	2	yes	—
101	C₂×C₈×C₂₃	#41	1	2	yes	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₄⋊C₄×C₂₃	—	2	4	no	—
105	C₂²⋊C₄×C₂₃	—	2	2	no	—
106	C₂²⋊C₄×C₂₃	—	2	1	no	—
107	C₂²⋊C₄×C₂₃	—	2	4	no	—
108	C₄²×C₂₃	—	2	2	no	—
109	C₄²×C₂₃	—	2	1	no	—
110	C₄²×C₂₃	—	2	4	no	—
111	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
114	#20787	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₂₃	#117	1	2	yes	—
116	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
117	Q₈×C₂₃	—	2	2	no	—
118	D₄×C₂₃	—	2	4	no	—
119	D₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	4	4	no	—
127	C₂×C₄×C₂₃	—	4	4	no	—
128	C₂×C₄×C₂₃	#117	1	2	yes	—
129	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
133	C₈×C₂₃	—	2	2	no	—
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	4	4	no	—
136	C₈×C₂₃	#117	1	1	yes	characteristic
137	C₈×C₂₃	#117	1	1	yes	characteristic
138	C₈×C₂₃	#117	1	1	yes	characteristic
139	C₈×C₂₃	#117	1	2	yes	—
140	C₈×C₂₃	#117	1	2	yes	—
141	C₈×C₂₃	#117	1	1	yes	characteristic
142	C₄×#131	C₂×C₂₃	1	1	yes	characteristic
143	#1841	C₂×C₂₃	1	2	yes	complemented
144	#1833	C₂×C₂₃	1	2	yes	complemented
145	#1827	C₂×C₂₃	1	2	yes	—
146	#1813	C₂×C₂₃	1	1	yes	characteristic
147	#1812	C₂×C₂₃	1	1	yes	characteristic
148	#1772	C₂×C₂₃	1	2	yes	complemented
149	#1757	C₂×C₂₃	1	2	yes	—
150	#1747	C₂×C₂₃	1	2	yes	complemented
151	C₂²×C₂₃	—	2	2	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	—	4	2	no	—
154	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
155	C₂²×C₂₃	#529	1	2	yes	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	2	2	no	—
158	C₄×C₂₃	—	4	2	no	—
159	C₄×C₂₃	—	4	1	no	—
160	C₄×C₂₃	—	8	4	no	—
161	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
162	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
163	C₄×C₂₃	#529	1	2	yes	—
164	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
166	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×#108	C₂²×C₂₃	1	2	yes	—
168	#427	C₂²×C₂₃	1	2	yes	—
169	#413	C₂²×C₂₃	1	2	yes	—
170	#410	C₂²×C₂₃	1	2	yes	—
171	#405	C₄×C₂₃	1	2	yes	complemented
172	#405	C₄×C₂₃	1	2	yes	complemented
173	#405	C₂²×C₂₃	1	4	yes	—
174	#405	C₂²×C₂₃	1	1	yes	characteristic
175	#405	C₂²×C₂₃	1	1	yes	characteristic
176	C₄×M₁₆	C₄×C₂₃	1	2	yes	complemented
177	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	complemented
179	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
180	#344	C₂²×C₂₃	1	4	yes	—
181	#339	C₂²×C₂₃	1	4	yes	—
182	#338	C₂²×C₂₃	1	2	yes	—
183	H(3,ℤ/4)	C₂²×C₂₃	1	2	yes	—
184	H(3,ℤ/4)	C₂²×C₂₃	1	4	yes	—
185	C₂×C₂₃	—	2	1	no	—
186	C₂×C₂₃	—	2	2	no	—
187	C₂×C₂₃	—	4	2	no	—
188	C₂×C₂₃	#2283	1	1	yes	characteristic
189	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
190	#131	—	2	2	no	—
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	#131	—	2	2	no	—
194	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
195	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
196	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
197	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
198	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
200	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
201	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₄×Q₈	—	2	1	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	1	no	—
206	#117	C₂×C₄×C₂₃	1	2	yes	—
207	#117	C₂³×C₂₃	1	4	yes	—
208	#117	C₂³×C₂₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
210	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
211	#108	—	2	2	no	—
212	#108	—	2	2	no	—
213	#107	—	2	2	no	—
214	#106	—	2	2	no	—
215	#101	—	2	2	no	—
216	#100	—	2	2	no	—
217	#99	—	2	4	no	—
218	#97	—	2	4	no	—
219	#97	C₂×C₄×C₂₃	1	2	yes	—
220	#97	C₂×C₄×C₂₃	1	2	yes	—
221	#97	C₂×C₄×C₂₃	1	2	yes	—
222	C₄×C₈	—	2	4	no	—
223	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
224	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
225	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
226	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
227	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
228	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
229	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
230	C₂₃	#20787	1	1	yes	characteristic, direct product factor, Sylow
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
234	C₂×D₄	#41×C₂₃	1	2	yes	—
235	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
252	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
253	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
257	C₂×C₈	#41×C₂₃	1	2	yes	—
258	C₂×C₈	#41×C₂₃	1	2	yes	—
259	C₂×C₈	#41×C₂₃	1	2	yes	—
260	C₂×C₈	#41×C₂₃	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	4	no	—
270	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
271	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
272	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
273	C₂³	C₂₃×#117	1	2	yes	—
274	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₂₃×#117	1	2	yes	—
287	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
289	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
290	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	4	no	—
294	C₈	C₂₃×#117	1	1	yes	characteristic
295	C₈	C₂₃×#117	1	1	yes	characteristic
296	C₈	C₂₃×#117	1	1	yes	characteristic
297	C₈	C₂₃×#117	1	1	yes	characteristic
298	C₈	C₂₃×#117	1	2	yes	—
299	C₈	C₂₃×#117	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
304	C₂²	C₂₃×#529	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	4	no	—
310	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
311	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
312	C₄	C₂₃×#529	1	2	yes	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	C₂₃×#2283	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#117
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	#2283
prime radical	C₂×#41×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1841	C₂	semidirect
3. C₂₃×#1833	C₂	semidirect
7. C₂₃×#1772	C₂	semidirect
9. C₂₃×#1747	C₂	semidirect
17. C₂₃×#405	C₄	semidirect
18. C₂₃×#405	C₄	semidirect
22. C₄×M₁₆×C₂₃	C₄	semidirect
24. C₂×C₄×C₈×C₂₃	C₄	semidirect
114. #20787	C₂₃	direct (N × H)
143. #1841	C₂×C₂₃	semidirect
144. #1833	C₂×C₂₃	semidirect
148. #1772	C₂×C₂₃	semidirect
150. #1747	C₂×C₂₃	semidirect
171. #405	C₄×C₂₃	semidirect
172. #405	C₄×C₂₃	semidirect
176. C₄×M₁₆	C₄×C₂₃	semidirect
178. C₂×C₄×C₈	C₄×C₂₃	semidirect
230. C₂₃	#20787	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#131	2
C₂₃×#1841	2
C₂₃×#1833	2
C₂₃×#1827	2
C₂₃×#1813	2
C₂₃×#1812	2
C₂₃×#1772	2
C₂₃×#1757	2
C₂₃×#1747	2
#20787	23

Subgroups — C23×#20787 #19168413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20787 #19168413