Subgroups — C₂₃×#33743 #19181369

Order 5888.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3176	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2244	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2217	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2175	C₂	1	2	yes	maximal, complemented
10	C₄×C₂₃×SD₃₂	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#541	—	2	1	no	—
13	C₂₃×#540	—	2	1	no	—
14	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#499	C₂²	1	1	yes	characteristic
19	C₂₃×#499	C₂²	1	1	yes	characteristic
20	C₂₃×#494	C₂²	1	1	yes	characteristic
21	C₂₃×#493	C₂²	1	1	yes	characteristic
22	C₂₃×#491	—	2	1	no	—
23	C₂₃×#490	—	2	1	no	—
24	C₂₃×#470	—	2	1	no	—
25	C₂₃×#469	—	2	1	no	—
26	C₂₃×#467	C₂²	1	2	yes	—
27	C₂₃×#466	C₂²	1	2	yes	—
28	C₂₃×#442	—	2	1	no	—
29	C₂₃×#441	—	2	1	no	—
30	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
31	C₄×D₈×C₂₃	C₂²	1	2	yes	—
32	C₂₃×#405	C₂²	1	1	yes	characteristic
33	C₂₃×#367	C₂²	1	2	yes	—
34	C₂₃×#366	C₂²	1	2	yes	—
35	C₂₃×#358	C₂²	1	2	yes	—
36	C₂₃×#358	C₂²	1	2	yes	—
37	C₂₃×#357	C₂²	1	2	yes	—
38	C₂₃×#357	C₂²	1	2	yes	—
39	C₂₃×#346	C₂²	1	1	yes	characteristic
40	C₂₃×#346	C₂²	1	1	yes	characteristic
41	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
42	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆×C₂₃	—	2	1	no	—
44	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
45	C₂×D₈×C₂₃	—	2	1	no	—
46	C₂×D₈×C₂₃	C₂³	1	2	yes	—
47	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊Q₈	—	2	1	no	—
50	C₂₃×#127	—	2	1	no	—
51	C₂₃×#126	—	4	2	no	—
52	C₂₃×#125	—	2	1	no	—
53	C₂₃×#125	—	2	1	no	—
54	C₂₃×#125	—	4	2	no	—
55	C₂₃×#123	—	4	2	no	—
56	C₂₃×C₂²⋊Q₈	—	2	2	no	—
57	C₂₃×#121	—	2	2	no	—
58	C₂₃×#121	—	4	2	no	—
59	C₄×Q₈×C₂₃	—	2	2	no	—
60	C₄×D₄×C₂₃	—	2	2	no	—
61	C₂₃×#117	D₄	1	1	yes	characteristic
62	C₂₃×SD₃₂	—	2	4	no	—
63	C₂₃×SD₃₂	—	2	4	no	—
64	C₂₃×M₃₂	—	2	2	no	—
65	C₂×C₁₆×C₂₃	—	2	2	no	—
66	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#107	C₂³	1	2	yes	—
71	C₂₃×#107	C₂³	1	2	yes	—
72	C₂₃×#106	—	2	1	no	—
73	C₂₃×#106	—	2	1	no	—
74	C₂₃×#103	—	2	2	no	—
75	C₂₃×#103	—	2	2	no	—
76	C₂₃×#102	—	2	2	no	—
77	C₂₃×#102	—	2	2	no	—
78	C₂₃×#97	D₄	1	1	yes	characteristic
79	C₂₃×#97	D₄	1	1	yes	characteristic
80	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₂₃	—	2	2	no	—
83	C₂×D₄×C₂₃	—	2	2	no	—
84	C₂×D₄×C₂₃	—	4	2	no	—
85	C₂×D₄×C₂₃	—	4	2	no	—
86	C₂²×C₄×C₂₃	—	4	2	no	—
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₂₃	—	2	2	no	—
89	Q₁₆×C₂₃	—	2	2	no	—
90	Q₁₆×C₂₃	—	4	2	no	—
91	D₈×C₂₃	—	2	2	no	—
92	D₈×C₂₃	—	2	2	no	—
93	D₈×C₂₃	—	4	2	no	—
94	M₁₆×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	2	1	no	—
96	C₂×C₈×C₂₃	—	2	1	no	—
97	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	4	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₄⋊C₄×C₂₃	—	4	2	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
109	C₂²⋊C₄×C₂₃	—	4	2	no	—
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₂²⋊C₄×C₂₃	—	4	2	no	—
112	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
113	C₄²×C₂₃	—	4	2	no	—
114	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₂₃	—	2	1	no	—
117	C₁₆×C₂₃	—	2	1	no	—
118	C₁₆×C₂₃	#41	1	2	yes	—
119	C₁₆×C₂₃	#41	1	2	yes	—
120	#33743	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₃	—	4	2	no	—
122	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
123	Q₈×C₂₃	—	4	2	no	—
124	Q₈×C₂₃	—	4	2	no	—
125	D₄×C₂₃	—	4	2	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	16	2	no	—
128	D₄×C₂₃	—	16	2	no	—
129	C₂×C₄×C₂₃	—	2	1	no	—
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	16	2	no	—
135	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	#135	1	2	yes	—
142	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
144	#3177	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#3176	C₂×C₂₃	1	1	yes	characteristic
146	#2253	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#2248	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#2245	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#2244	C₂×C₂₃	1	1	yes	characteristic, complemented
150	#2220	C₂×C₂₃	1	2	yes	complemented
151	#2217	C₂×C₂₃	1	2	yes	complemented
152	#2175	C₂×C₂₃	1	2	yes	complemented
153	C₄×SD₃₂	C₂×C₂₃	1	2	yes	complemented
154	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₂²×C₂₃	—	2	2	no	—
156	C₂²×C₂₃	—	2	1	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	—	8	2	no	—
159	C₂²×C₂₃	—	8	2	no	—
160	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₂₃	—	2	2	no	—
162	C₄×C₂₃	—	2	2	no	—
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	8	2	no	—
165	C₄×C₂₃	—	8	2	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	#532	1	1	yes	characteristic
168	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
169	#541	—	2	1	no	—
170	#540	—	2	1	no	—
171	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
172	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
173	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
174	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
175	#499	C₂²×C₂₃	1	1	yes	characteristic
176	#499	C₂²×C₂₃	1	1	yes	characteristic
177	#494	C₂²×C₂₃	1	1	yes	characteristic
178	#493	C₂²×C₂₃	1	1	yes	characteristic
179	#491	—	2	1	no	—
180	#490	—	2	1	no	—
181	#470	—	2	1	no	—
182	#469	—	2	1	no	—
183	#467	C₂²×C₂₃	1	2	yes	—
184	#466	C₂²×C₂₃	1	2	yes	—
185	#442	—	2	1	no	—
186	#441	—	2	1	no	—
187	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
188	C₄×D₈	C₂²×C₂₃	1	2	yes	—
189	#405	C₂²×C₂₃	1	1	yes	characteristic
190	#367	C₂²×C₂₃	1	2	yes	—
191	#366	C₂²×C₂₃	1	2	yes	—
192	#358	C₂²×C₂₃	1	2	yes	—
193	#358	C₂²×C₂₃	1	2	yes	—
194	#357	C₂²×C₂₃	1	2	yes	—
195	#357	C₂²×C₂₃	1	2	yes	—
196	#346	C₂²×C₂₃	1	1	yes	characteristic
197	#346	C₂²×C₂₃	1	1	yes	characteristic
198	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
199	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
200	C₂×C₂₃	—	4	1	no	—
201	C₂×C₂₃	—	16	2	no	—
202	C₂×C₂₃	#3169	1	1	yes	characteristic
203	C₂×C₂₃	#3416	1	2	yes	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	C₂³×C₂₃	1	2	yes	—
208	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	#127	—	2	1	no	—
212	#126	—	4	2	no	—
213	#125	—	2	1	no	—
214	#125	—	2	1	no	—
215	#125	—	4	2	no	—
216	#123	—	4	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	#117	D₄×C₂₃	1	1	yes	characteristic
223	SD₃₂	—	2	4	no	—
224	SD₃₂	—	2	4	no	—
225	M₃₂	—	2	2	no	—
226	C₂×C₁₆	—	2	2	no	—
227	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
228	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
231	#107	C₂³×C₂₃	1	2	yes	—
232	#107	C₂³×C₂₃	1	2	yes	—
233	#106	—	2	1	no	—
234	#106	—	2	1	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#102	—	2	2	no	—
238	#102	—	2	2	no	—
239	#97	D₄×C₂₃	1	1	yes	characteristic
240	#97	D₄×C₂₃	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
243	C₂₃	#33743	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	4	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	2	no	—
256	M₁₆	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
275	C₄²	—	4	2	no	—
276	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	#41×C₂₃	1	2	yes	—
281	C₁₆	#41×C₂₃	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
297	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
298	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	C₂₃×#135	1	2	yes	—
303	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
304	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₂₃×#532	1	1	yes	characteristic
318	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
319	C₂	—	4	1	no	—
320	C₂	—	16	2	no	—
321	C₂	C₂₃×#3169	1	1	yes	characteristic
322	C₂	C₂₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3177	C₂	semidirect
3. C₂₃×#2253	C₂	semidirect
4. C₂₃×#2248	C₂	semidirect
5. C₂₃×#2245	C₂	semidirect
6. C₂₃×#2244	C₂	semidirect
7. C₂₃×#2220	C₂	semidirect
8. C₂₃×#2217	C₂	semidirect
9. C₂₃×#2175	C₂	semidirect
10. C₄×C₂₃×SD₃₂	C₂	semidirect
11. C₂₃×#2108	C₂	semidirect
120. #33743	C₂₃	direct (N × H)
144. #3177	C₂×C₂₃	semidirect
146. #2253	C₂×C₂₃	semidirect
147. #2248	C₂×C₂₃	semidirect
148. #2245	C₂×C₂₃	semidirect
149. #2244	C₂×C₂₃	semidirect
150. #2220	C₂×C₂₃	semidirect
151. #2217	C₂×C₂₃	semidirect
152. #2175	C₂×C₂₃	semidirect
153. C₄×SD₃₂	C₂×C₂₃	semidirect
154. #2108	C₂×C₂₃	semidirect
243. C₂₃	#33743	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×#3177	2
C₂₃×#3176	2
C₂₃×#2253	2
C₂₃×#2248	2
C₂₃×#2245	2
C₂₃×#2244	2
C₂₃×#2220	2
C₂₃×#2217	2
C₂₃×#2175	2
C₄×C₂₃×SD₃₂	2
C₂₃×#2108	2
#33743	23

Subgroups — C23×#33743 #19181369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33743 #19181369