Subgroups — C₂₃×#13246 #19160872

Order 5888.

422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1978	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×SD₃₂	—	2	1	no	—
11	C₂×C₂₃×SD₃₂	—	2	1	no	—
12	C₂×C₂₃×SD₃₂	—	2	2	no	—
13	C₂×C₂₃×SD₃₂	C₄	1	4	yes	complemented
14	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
18	C₂₃×#419	—	2	1	no	—
19	C₂₃×#418	—	2	1	no	—
20	C₂×C₂₃×#101	—	2	1	no	—
21	C₂×C₂₃×#100	—	2	1	no	—
22	C₂₃×#405	C₂²	1	1	yes	characteristic
23	C₂₃×#368	—	2	2	no	—
24	C₂₃×#362	—	2	2	no	—
25	C₂₃×#361	—	2	2	no	—
26	C₂₃×#346	—	2	1	no	—
27	C₄×C₁₆×C₂₃	—	2	1	no	—
28	C₂²×Q₈×C₂₃	—	2	1	no	—
29	C₂²×D₄×C₂₃	—	2	1	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	C₂×C₄	1	2	yes	—
36	C₂×D₈×C₂₃	—	2	1	no	—
37	C₂×D₈×C₂₃	—	2	1	no	—
38	C₂×D₈×C₂₃	—	2	1	no	—
39	C₂×D₈×C₂₃	—	2	1	no	—
40	C₂×D₈×C₂₃	—	2	1	no	—
41	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₂₃	—	2	1	no	—
43	C₂×M₁₆×C₂₃	—	2	1	no	—
44	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#117	D₄	1	1	yes	characteristic
47	C₂₃×SD₃₂	—	2	4	no	—
48	C₂₃×SD₃₂	—	2	4	no	—
49	C₂₃×M₃₂	—	2	1	no	—
50	C₂₃×M₃₂	D₄	1	2	yes	—
51	C₂×C₁₆×C₂₃	—	2	1	no	—
52	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
53	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
56	C₂₃×#108	—	2	1	no	—
57	C₂₃×#108	—	2	1	no	—
58	C₂₃×#108	—	2	1	no	—
59	C₂₃×#108	—	2	1	no	—
60	C₂₃×#103	—	4	2	no	—
61	C₂₃×#102	—	4	2	no	—
62	C₂₃×#101	—	4	2	no	—
63	C₂₃×#100	—	4	2	no	—
64	C₂₃×#97	—	2	1	no	—
65	C₄×C₈×C₂₃	—	2	1	no	—
66	C₂⁴×C₂₃	—	4	1	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	4	1	no	—
69	C₂×Q₈×C₂₃	—	4	1	no	—
70	C₂×Q₈×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	4	1	no	—
72	C₂×Q₈×C₂₃	—	4	1	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	4	1	no	—
75	C₂×D₄×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	4	1	no	—
77	C₂×D₄×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	4	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	1	no	—
84	Q₁₆×C₂₃	—	4	1	no	—
85	D₈×C₂₃	—	2	2	no	—
86	D₈×C₂₃	—	2	2	no	—
87	D₈×C₂₃	—	4	1	no	—
88	D₈×C₂₃	—	4	1	no	—
89	M₁₆×C₂₃	—	2	1	no	—
90	M₁₆×C₂₃	—	4	1	no	—
91	M₁₆×C₂₃	—	4	1	no	—
92	M₁₆×C₂₃	—	4	1	no	—
93	M₁₆×C₂₃	—	4	1	no	—
94	M₁₆×C₂₃	D₈	1	2	yes	—
95	C₂×C₈×C₂₃	—	2	1	no	—
96	C₂×C₈×C₂₃	—	4	1	no	—
97	C₂×C₈×C₂₃	—	4	1	no	—
98	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
104	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
105	C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₄²×C₂₃	—	2	1	no	—
107	C₁₆×C₂₃	—	2	2	no	—
108	C₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
109	C₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
110	#13246	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₂₃	—	4	2	no	—
112	C₂³×C₂₃	—	8	1	no	—
113	C₂³×C₂₃	—	8	1	no	—
114	C₂³×C₂₃	—	16	2	no	—
115	C₂³×C₂₃	#135	1	1	yes	characteristic
116	Q₈×C₂₃	—	4	2	no	—
117	Q₈×C₂₃	—	4	2	no	—
118	Q₈×C₂₃	—	8	1	no	—
119	Q₈×C₂₃	—	8	1	no	—
120	D₄×C₂₃	—	4	2	no	—
121	D₄×C₂₃	—	4	2	no	—
122	D₄×C₂₃	—	8	1	no	—
123	D₄×C₂₃	—	8	1	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	8	1	no	—
127	C₂×C₄×C₂₃	—	8	1	no	—
128	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	8	1	no	—
136	C₈×C₂₃	—	8	1	no	—
137	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₂₃	#121	1	1	yes	characteristic
140	C₈×C₂₃	#124	1	1	yes	characteristic
141	C₂²×SD₃₂	C₂×C₂₃	1	1	yes	characteristic
142	C₂×#368	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂×#362	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂×#361	C₂×C₂₃	1	1	yes	characteristic
145	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#1979	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#1978	C₂×C₂₃	1	1	yes	characteristic
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	—	8	2	no	—
150	C₂²×C₂₃	—	16	2	no	—
151	C₂²×C₂₃	—	16	2	no	—
152	C₂²×C₂₃	—	16	2	no	—
153	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
154	C₂²×C₂₃	#466	1	1	yes	characteristic
155	C₂²×C₂₃	#495	1	1	yes	characteristic
156	C₄×C₂₃	—	4	2	no	—
157	C₄×C₂₃	—	8	2	no	—
158	C₄×C₂₃	#390	1	1	yes	characteristic
159	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
160	C₄×C₂₃	#466	1	1	yes	characteristic
161	C₄×C₂₃	#493	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
163	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×SD₃₂	—	2	1	no	—
165	C₂×SD₃₂	—	2	1	no	—
166	C₂×SD₃₂	—	2	2	no	—
167	C₂×SD₃₂	C₄×C₂₃	1	4	yes	complemented
168	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
169	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
172	#419	—	2	1	no	—
173	#418	—	2	1	no	—
174	C₂×#101	—	2	1	no	—
175	C₂×#100	—	2	1	no	—
176	#405	C₂²×C₂₃	1	1	yes	characteristic
177	#368	—	2	2	no	—
178	#362	—	2	2	no	—
179	#361	—	2	2	no	—
180	#346	—	2	1	no	—
181	C₄×C₁₆	—	2	1	no	—
182	C₂×C₂₃	—	2	1	no	—
183	C₂×C₂₃	—	2	1	no	—
184	C₂×C₂₃	—	16	2	no	—
185	C₂×C₂₃	#1970	1	1	yes	characteristic
186	C₂×C₂₃	#2223	1	2	yes	—
187	C₂²×Q₈	—	2	1	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
205	#117	D₄×C₂₃	1	1	yes	characteristic
206	SD₃₂	—	2	4	no	—
207	SD₃₂	—	2	4	no	—
208	M₃₂	—	2	1	no	—
209	M₃₂	D₄×C₂₃	1	2	yes	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
212	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
215	#108	—	2	1	no	—
216	#108	—	2	1	no	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#103	—	4	2	no	—
220	#102	—	4	2	no	—
221	#101	—	4	2	no	—
222	#100	—	4	2	no	—
223	#97	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₂₃	#13246	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	—	4	1	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	4	1	no	—
244	Q₁₆	—	4	1	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	4	1	no	—
248	D₈	—	4	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	D₈×C₂₃	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
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	#41×C₂₃	1	1	yes	characteristic
263	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
264	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
269	C₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
270	C₂³	—	4	2	no	—
271	C₂³	—	8	1	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	16	2	no	—
274	C₂³	C₂₃×#135	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	1	no	—
278	Q₈	—	8	1	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	8	1	no	—
282	D₄	—	8	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
289	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
297	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
298	C₈	C₂₃×#121	1	1	yes	characteristic
299	C₈	C₂₃×#124	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	—	16	2	no	—
304	C₂²	—	16	2	no	—
305	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
306	C₂²	C₂₃×#466	1	1	yes	characteristic
307	C₂²	C₂₃×#495	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂₃×#390	1	1	yes	characteristic
311	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
312	C₄	C₂₃×#466	1	1	yes	characteristic
313	C₄	C₂₃×#493	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₂₃×#1970	1	1	yes	characteristic
318	C₂	C₂₃×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#466
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#368	C₂	semidirect
3. C₂×C₂₃×#362	C₂	semidirect
5. C₂₃×#2108	C₂	semidirect
6. C₂₃×#1979	C₂	semidirect
13. C₂×C₂₃×SD₃₂	C₄	semidirect
110. #13246	C₂₃	direct (N × H)
142. C₂×#368	C₂×C₂₃	semidirect
143. C₂×#362	C₂×C₂₃	semidirect
145. #2108	C₂×C₂₃	semidirect
146. #1979	C₂×C₂₃	semidirect
167. C₂×SD₃₂	C₄×C₂₃	semidirect
225. C₂₃	#13246	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×SD₃₂	2
C₂×C₂₃×#368	2
C₂×C₂₃×#362	2
C₂×C₂₃×#361	2
C₂₃×#2108	2
C₂₃×#1979	2
C₂₃×#1978	2
#13246	23

Subgroups — C23×#13246 #19160872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13246 #19160872