Subgroups — C₂₃×#10672 #19158298

Order 5888.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1957	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1957	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1957	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#107	—	2	2	no	—
10	C₂×C₂₃×#106	—	2	2	no	—
11	C₂×C₂₃×#102	—	2	2	no	—
12	C₂×C₂₃×#102	—	2	2	no	—
13	C₂×C₂₃×#102	—	2	2	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#384	C₂²	1	1	yes	characteristic
18	C₂₃×#384	C₂²	1	1	yes	characteristic
19	C₂₃×#384	C₂²	1	1	yes	characteristic
20	C₂₃×#340	—	2	2	no	—
21	C₂²×D₄×C₂₃	—	2	1	no	—
22	C₂²×D₄×C₂₃	—	2	1	no	—
23	C₂²×D₄×C₂₃	—	2	1	no	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂²×C₈×C₂₃	D₄	1	2	yes	—
26	C₂²×C₈×C₂₃	D₄	1	2	yes	—
27	C₂₃×#127	—	2	1	no	—
28	C₂₃×#127	—	2	1	no	—
29	C₂₃×#127	—	2	1	no	—
30	C₂₃×#127	Q₈	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#107	—	4	4	no	—
42	C₂₃×#106	—	4	4	no	—
43	C₂₃×#102	—	4	4	no	—
44	C₂₃×#102	—	4	4	no	—
45	C₂₃×#102	—	4	4	no	—
46	C₄×C₈×C₂₃	—	2	1	no	—
47	C₄×C₈×C₂₃	—	2	1	no	—
48	C₄×C₈×C₂₃	—	2	1	no	—
49	C₄×C₈×C₂₃	—	2	1	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	—	2	1	no	—
52	C₂⁴×C₂₃	—	4	1	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	1	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂×D₄×C₂₃	—	4	2	no	—
60	C₂×D₄×C₂₃	—	4	2	no	—
61	C₂×D₄×C₂₃	—	4	2	no	—
62	C₂×D₄×C₂₃	—	4	2	no	—
63	C₂×D₄×C₂₃	—	4	2	no	—
64	C₂×D₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	—	4	1	no	—
66	C₂²×C₄×C₂₃	—	4	1	no	—
67	C₂²×C₄×C₂₃	—	4	1	no	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	D₈	1	4	yes	—
79	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
80	C₄⋊C₄×C₂₃	—	4	2	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₄⋊C₄×C₂₃	—	4	2	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₂₃	#41	1	1	yes	characteristic
91	C₄²×C₂₃	#41	1	1	yes	characteristic
92	C₄²×C₂₃	#41	1	1	yes	characteristic
93	#10672	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	4	2	no	—
95	C₂³×C₂₃	—	8	1	no	—
96	C₂³×C₂₃	—	8	1	no	—
97	C₂³×C₂₃	—	8	1	no	—
98	C₂³×C₂₃	—	8	1	no	—
99	C₂³×C₂₃	—	8	1	no	—
100	C₂³×C₂₃	—	8	1	no	—
101	C₂³×C₂₃	#127	1	1	yes	characteristic
102	D₄×C₂₃	—	4	4	no	—
103	D₄×C₂₃	—	4	4	no	—
104	D₄×C₂₃	—	4	4	no	—
105	D₄×C₂₃	—	8	2	no	—
106	D₄×C₂₃	—	8	2	no	—
107	D₄×C₂₃	—	8	2	no	—
108	C₂×C₄×C₂₃	—	8	1	no	—
109	C₂×C₄×C₂₃	—	8	1	no	—
110	C₂×C₄×C₂₃	—	8	1	no	—
111	C₂×C₄×C₂₃	—	8	1	no	—
112	C₂×C₄×C₂₃	—	8	1	no	—
113	C₂×C₄×C₂₃	—	8	1	no	—
114	C₂×C₄×C₂₃	—	8	1	no	—
115	C₂×C₄×C₂₃	—	8	1	no	—
116	C₂×C₄×C₂₃	—	8	1	no	—
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₂₃	#135	1	2	yes	—
126	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₂₃	#137	1	2	yes	—
129	C₈×C₂₃	—	2	4	no	—
130	C₈×C₂₃	—	2	4	no	—
131	C₈×C₂₃	—	4	2	no	—
132	#1957	C₂×C₂₃	1	1	yes	characteristic
133	#1957	C₂×C₂₃	1	1	yes	characteristic
134	#1957	C₂×C₂₃	1	1	yes	characteristic
135	#1845	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#1844	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#1834	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#1273	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂²×C₂₃	—	8	2	no	—
140	C₂²×C₂₃	—	8	2	no	—
141	C₂²×C₂₃	—	8	2	no	—
142	C₂²×C₂₃	—	16	1	no	—
143	C₂²×C₂₃	—	16	1	no	—
144	C₂²×C₂₃	—	16	1	no	—
145	C₂²×C₂₃	—	16	1	no	—
146	C₂²×C₂₃	#399	1	1	yes	characteristic
147	C₂²×C₂₃	#492	1	1	yes	characteristic
148	C₂²×C₂₃	#493	1	1	yes	characteristic
149	C₂²×C₂₃	#495	1	1	yes	characteristic
150	C₂²×C₂₃	#496	1	1	yes	characteristic
151	C₂²×C₂₃	#496	1	1	yes	characteristic
152	C₂²×C₂₃	#497	1	1	yes	characteristic
153	C₄×C₂₃	—	16	1	no	—
154	C₄×C₂₃	—	16	1	no	—
155	C₄×C₂₃	—	16	1	no	—
156	C₄×C₂₃	#474	1	2	yes	—
157	C₄×C₂₃	#476	1	2	yes	—
158	C₄×C₂₃	#478	1	2	yes	—
159	C₄×C₂₃	#480	1	2	yes	—
160	C₄×C₂₃	#481	1	2	yes	—
161	C₄×C₂₃	#482	1	2	yes	—
162	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×#107	—	2	2	no	—
164	C₂×#106	—	2	2	no	—
165	C₂×#102	—	2	2	no	—
166	C₂×#102	—	2	2	no	—
167	C₂×#102	—	2	2	no	—
168	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
171	#384	C₂²×C₂₃	1	1	yes	characteristic
172	#384	C₂²×C₂₃	1	1	yes	characteristic
173	#384	C₂²×C₂₃	1	1	yes	characteristic
174	#340	—	2	2	no	—
175	C₂×C₂₃	—	16	2	no	—
176	C₂×C₂₃	#1706	1	1	yes	characteristic
177	C₂×C₂₃	#1714	1	1	yes	characteristic
178	C₂×C₂₃	#1720	1	1	yes	characteristic
179	C₂×C₂₃	#1721	1	1	yes	characteristic
180	C₂×C₂₃	#2092	1	1	yes	characteristic
181	C₂×C₂₃	#2093	1	1	yes	characteristic
182	C₂×C₂₃	#2094	1	1	yes	characteristic
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂²×C₈	D₄×C₂₃	1	2	yes	—
187	C₂²×C₈	D₄×C₂₃	1	2	yes	—
188	C₂²×C₈	D₄×C₂₃	1	2	yes	—
189	#127	—	2	1	no	—
190	#127	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	Q₈×C₂₃	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
203	#107	—	4	4	no	—
204	#106	—	4	4	no	—
205	#102	—	4	4	no	—
206	#102	—	4	4	no	—
207	#102	—	4	4	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₂₃	#10672	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	—	4	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
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	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	D₈×C₂₃	1	4	yes	—
242	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
253	C₄²	#41×C₂₃	1	1	yes	characteristic
254	C₄²	#41×C₂₃	1	1	yes	characteristic
255	C₄²	#41×C₂₃	1	1	yes	characteristic
256	C₂³	—	4	2	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	8	1	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	—	8	1	no	—
262	C₂³	—	8	1	no	—
263	C₂³	C₂₃×#127	1	1	yes	characteristic
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	8	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
281	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
283	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
284	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
285	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
286	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
287	C₂×C₄	C₂₃×#135	1	2	yes	—
288	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
289	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
290	C₂×C₄	C₂₃×#137	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	16	1	no	—
298	C₂²	—	16	1	no	—
299	C₂²	—	16	1	no	—
300	C₂²	—	16	1	no	—
301	C₂²	C₂₃×#399	1	1	yes	characteristic
302	C₂²	C₂₃×#492	1	1	yes	characteristic
303	C₂²	C₂₃×#493	1	1	yes	characteristic
304	C₂²	C₂₃×#495	1	1	yes	characteristic
305	C₂²	C₂₃×#496	1	1	yes	characteristic
306	C₂²	C₂₃×#496	1	1	yes	characteristic
307	C₂²	C₂₃×#497	1	1	yes	characteristic
308	C₄	—	16	1	no	—
309	C₄	—	16	1	no	—
310	C₄	—	16	1	no	—
311	C₄	C₂₃×#474	1	2	yes	—
312	C₄	C₂₃×#476	1	2	yes	—
313	C₄	C₂₃×#478	1	2	yes	—
314	C₄	C₂₃×#480	1	2	yes	—
315	C₄	C₂₃×#481	1	2	yes	—
316	C₄	C₂₃×#482	1	2	yes	—
317	C₂	—	16	2	no	—
318	C₂	C₂₃×#1706	1	1	yes	characteristic
319	C₂	C₂₃×#1714	1	1	yes	characteristic
320	C₂	C₂₃×#1720	1	1	yes	characteristic
321	C₂	C₂₃×#1721	1	1	yes	characteristic
322	C₂	C₂₃×#2092	1	1	yes	characteristic
323	C₂	C₂₃×#2093	1	1	yes	characteristic
324	C₂	C₂₃×#2094	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1845	C₂	semidirect
5. C₂₃×#1844	C₂	semidirect
6. C₂₃×#1834	C₂	semidirect
7. C₂₃×#1273	C₂	semidirect
93. #10672	C₂₃	direct (N × H)
135. #1845	C₂×C₂₃	semidirect
136. #1844	C₂×C₂₃	semidirect
137. #1834	C₂×C₂₃	semidirect
138. #1273	C₂×C₂₃	semidirect
214. C₂₃	#10672	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×#1957	2
C₂₃×#1845	2
C₂₃×#1844	2
C₂₃×#1834	2
C₂₃×#1273	2
#10672	23

Subgroups — C23×#10672 #19158298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10672 #19158298