Subgroups — C₂₃×#13069 #19160695

Order 5888.

450 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×#3407	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#473	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1814	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#137	—	2	1	no	—
9	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
11	C₂₃×#504	—	2	2	no	—
12	C₂₃×#504	—	2	2	no	—
13	C₂×C₂₃×M₃₂	—	2	2	no	—
14	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂₃×#473	—	2	2	no	—
17	C₂₃×#473	C₄	1	4	yes	complemented
18	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
20	C₂₃×#427	C₂²	1	1	yes	characteristic
21	C₂₃×#421	—	2	1	no	—
22	C₂×C₂₃×#101	—	2	1	no	—
23	C₂₃×#362	—	2	2	no	—
24	C₂₃×#358	—	2	2	no	—
25	C₂₃×#344	—	2	1	no	—
26	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₂₃	—	2	1	no	—
28	C₂₃×#137	—	2	4	no	—
29	C₂₃×#137	—	4	2	no	—
30	C₂×Q₁₆×C₂₃	—	2	1	no	—
31	C₂×Q₁₆×C₂₃	—	2	1	no	—
32	C₂×Q₁₆×C₂₃	—	2	1	no	—
33	C₂×Q₁₆×C₂₃	—	2	1	no	—
34	C₂×Q₁₆×C₂₃	—	2	1	no	—
35	C₂×Q₁₆×C₂₃	—	4	1	no	—
36	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×SD₁₆×C₂₃	—	4	1	no	—
38	C₂₃×#131	—	2	1	no	—
39	C₂₃×#131	—	2	1	no	—
40	C₂₃×#131	C₂×C₄	1	2	yes	—
41	C₂₃×#131	D₄	1	2	yes	—
42	C₂×M₁₆×C₂₃	—	2	1	no	—
43	C₂×M₁₆×C₂₃	—	2	1	no	—
44	C₂×M₁₆×C₂₃	—	2	1	no	—
45	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₂₃	—	2	1	no	—
47	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×#117	—	2	1	no	—
49	C₂₃×M₃₂	—	2	4	no	—
50	C₂₃×M₃₂	—	2	1	no	—
51	C₂₃×M₃₂	—	2	1	no	—
52	C₂₃×M₃₂	—	2	1	no	—
53	C₂₃×M₃₂	D₄	1	2	yes	—
54	C₂×C₁₆×C₂₃	—	2	4	no	—
55	C₂×C₁₆×C₂₃	—	2	1	no	—
56	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
57	C₂₃×#108	—	2	1	no	—
58	C₂₃×#108	—	2	1	no	—
59	C₂₃×#108	—	2	1	no	—
60	C₂₃×#108	C₂×C₄	1	2	yes	—
61	C₂₃×#107	—	2	1	no	—
62	C₂₃×#106	—	2	1	no	—
63	C₂₃×#104	—	4	2	no	—
64	C₂₃×#101	—	2	4	no	—
65	#41×C₂₃	—	2	1	no	—
66	#41×C₂₃	—	2	1	no	—
67	#41×C₂₃	C₂²⋊C₄	1	2	yes	—
68	#41×C₂₃	D₈	1	2	yes	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	4	1	no	—
72	C₂×Q₈×C₂₃	—	4	1	no	—
73	C₂×Q₈×C₂₃	—	4	1	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	SD₁₆	1	1	yes	characteristic
76	C₂×D₄×C₂₃	—	2	1	no	—
77	C₂×D₄×C₂₃	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	—	2	1	no	—
79	C₂²×C₄×C₂₃	—	4	1	no	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₂₃	—	2	2	no	—
82	Q₁₆×C₂₃	—	2	2	no	—
83	Q₁₆×C₂₃	—	4	4	no	—
84	Q₁₆×C₂₃	—	4	1	no	—
85	Q₁₆×C₂₃	—	4	1	no	—
86	SD₁₆×C₂₃	—	4	4	no	—
87	M₁₆×C₂₃	—	2	1	no	—
88	M₁₆×C₂₃	—	2	2	no	—
89	M₁₆×C₂₃	—	2	2	no	—
90	M₁₆×C₂₃	—	2	2	no	—
91	M₁₆×C₂₃	—	4	1	no	—
92	M₁₆×C₂₃	—	4	1	no	—
93	M₁₆×C₂₃	—	4	1	no	—
94	M₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₂₃	—	2	1	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	4	1	no	—
99	C₂×C₈×C₂₃	—	4	1	no	—
100	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
106	C₄⋊C₄×C₂₃	—	4	1	no	—
107	C₂²⋊C₄×C₂₃	—	4	1	no	—
108	C₄²×C₂₃	—	4	1	no	—
109	C₁₆×C₂₃	—	2	2	no	—
110	C₁₆×C₂₃	—	2	2	no	—
111	#13069	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₂₃	—	2	1	no	—
113	C₂³×C₂₃	#135	1	1	yes	characteristic
114	Q₈×C₂₃	—	2	1	no	—
115	Q₈×C₂₃	—	4	2	no	—
116	Q₈×C₂₃	—	4	2	no	—
117	Q₈×C₂₃	—	8	1	no	—
118	Q₈×C₂₃	—	8	1	no	—
119	Q₈×C₂₃	#102	1	2	yes	—
120	D₄×C₂₃	—	2	2	no	—
121	D₄×C₂₃	—	2	2	no	—
122	D₄×C₂₃	—	2	1	no	—
123	D₄×C₂₃	#102	1	2	yes	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	8	1	no	—
129	C₂×C₄×C₂₃	—	8	1	no	—
130	C₂×C₄×C₂₃	—	8	1	no	—
131	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
137	C₈×C₂₃	—	2	2	no	—
138	C₈×C₂₃	—	4	2	no	—
139	C₈×C₂₃	—	8	1	no	—
140	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
142	C₈×C₂₃	#120	1	1	yes	characteristic
143	C₈×C₂₃	#123	1	1	yes	characteristic
144	#3407	C₂×C₂₃	1	1	yes	characteristic
145	C₂×#473	C₂×C₂₃	1	1	yes	characteristic
146	C₂×#362	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#2141	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#1814	C₂×C₂₃	1	1	yes	characteristic
149	#1389	C₂×C₂₃	1	1	yes	characteristic, complemented
150	#1385	C₂×C₂₃	1	1	yes	characteristic, complemented
151	C₂²×C₂₃	—	2	2	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	—	8	1	no	—
154	C₂²×C₂₃	#416	1	1	yes	characteristic
155	C₂²×C₂₃	#465	1	1	yes	characteristic
156	C₂²×C₂₃	#466	1	1	yes	characteristic
157	C₄×C₂₃	—	2	2	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	—	16	1	no	—
160	C₄×C₂₃	#386	1	1	yes	characteristic
161	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
162	C₄×C₂₃	#465	1	1	yes	characteristic
163	C₄×C₂₃	#466	1	1	yes	characteristic
164	C₂×#137	—	2	1	no	—
165	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
167	#504	—	2	2	no	—
168	#504	—	2	2	no	—
169	C₂×M₃₂	—	2	2	no	—
170	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
172	#473	—	2	2	no	—
173	#473	C₄×C₂₃	1	4	yes	complemented
174	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
176	#427	C₂²×C₂₃	1	1	yes	characteristic
177	#421	—	2	1	no	—
178	C₂×#101	—	2	1	no	—
179	#362	—	2	2	no	—
180	#358	—	2	2	no	—
181	#344	—	2	1	no	—
182	C₂×C₂₃	—	2	1	no	—
183	C₂×C₂₃	—	2	1	no	—
184	C₂×C₂₃	—	2	1	no	—
185	C₂×C₂₃	—	4	2	no	—
186	C₂×C₂₃	#1893	1	1	yes	characteristic
187	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
188	C₂²×Q₈	—	2	1	no	—
189	#137	—	2	4	no	—
190	#137	—	4	2	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	4	1	no	—
197	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
198	C₂×SD₁₆	—	4	1	no	—
199	#131	—	2	1	no	—
200	#131	—	2	1	no	—
201	#131	C₂×C₄×C₂₃	1	2	yes	—
202	#131	D₄×C₂₃	1	2	yes	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
209	#117	—	2	1	no	—
210	M₃₂	—	2	4	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	D₄×C₂₃	1	2	yes	—
215	C₂×C₁₆	—	2	4	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	C₂×C₄×C₂₃	1	2	yes	—
222	#107	—	2	1	no	—
223	#106	—	2	1	no	—
224	#104	—	4	2	no	—
225	#101	—	2	4	no	—
226	C₂₃	#13069	1	1	yes	characteristic, direct product factor, Sylow
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	C₂²⋊C₄×C₂₃	1	2	yes	—
230	#41	D₈×C₂₃	1	2	yes	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	SD₁₆×C₂₃	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	SD₁₆×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	4	4	no	—
246	Q₁₆	—	4	1	no	—
247	Q₁₆	—	4	1	no	—
248	SD₁₆	—	4	4	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
266	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
267	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
268	C₄⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	2	no	—
273	C₂³	—	2	1	no	—
274	C₂³	C₂₃×#135	1	1	yes	characteristic
275	Q₈	—	2	1	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	8	1	no	—
279	Q₈	—	8	1	no	—
280	Q₈	C₂₃×#102	1	2	yes	—
281	D₄	—	2	2	no	—
282	D₄	—	2	2	no	—
283	D₄	—	2	1	no	—
284	D₄	C₂₃×#102	1	2	yes	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
294	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
295	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
302	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
303	C₈	C₂₃×#120	1	1	yes	characteristic
304	C₈	C₂₃×#123	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	C₂₃×#416	1	1	yes	characteristic
309	C₂²	C₂₃×#465	1	1	yes	characteristic
310	C₂²	C₂₃×#466	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	16	1	no	—
314	C₄	C₂₃×#386	1	1	yes	characteristic
315	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
316	C₄	C₂₃×#465	1	1	yes	characteristic
317	C₄	C₂₃×#466	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	4	2	no	—
322	C₂	C₂₃×#1893	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1893
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#362	C₂	semidirect
4. C₂₃×#2141	C₂	semidirect
6. C₂₃×#1389	C₂	semidirect
7. C₂₃×#1385	C₂	semidirect
17. C₂₃×#473	C₄	semidirect
111. #13069	C₂₃	direct (N × H)
146. C₂×#362	C₂×C₂₃	semidirect
147. #2141	C₂×C₂₃	semidirect
149. #1389	C₂×C₂₃	semidirect
150. #1385	C₂×C₂₃	semidirect
173. #473	C₄×C₂₃	semidirect
226. C₂₃	#13069	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×#3407	2
C₂×C₂₃×#473	2
C₂×C₂₃×#362	2
C₂₃×#2141	2
C₂₃×#1814	2
C₂₃×#1389	2
C₂₃×#1385	2
#13069	23

Subgroups — C23×#13069 #19160695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13069 #19160695