Subgroups — C₂₃×#12647 #19160273

Order 5888.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2140	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₂₃	—	2	1	no	—
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂₃×#442	—	2	2	no	—
13	C₂₃×#442	C₄	1	4	yes	complemented
14	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
15	C₂₃×#427	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#102	—	2	1	no	—
18	C₂₃×#405	—	2	1	no	—
19	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
20	C₂₃×#361	C₄	1	4	yes	complemented
21	C₂₃×#357	C₄	1	4	yes	complemented
22	C₂₃×#347	—	2	2	no	—
23	C₂₃×#339	—	2	1	no	—
24	C₂²×D₄×C₂₃	—	2	1	no	—
25	C₂³×C₄×C₂₃	—	4	1	no	—
26	C₂×D₈×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	—	2	1	no	—
29	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₂₃	—	2	1	no	—
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₂₃	—	2	4	no	—
36	C₄×D₄×C₂₃	—	4	2	no	—
37	C₂₃×#117	—	2	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂₃×M₃₂	—	2	1	no	—
42	C₂₃×M₃₂	—	2	1	no	—
43	C₂₃×M₃₂	—	2	1	no	—
44	C₂₃×M₃₂	C₂×C₄	1	2	yes	complemented
45	C₂×C₁₆×C₂₃	—	2	1	no	—
46	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
47	C₂₃×#108	—	2	1	no	—
48	C₂₃×#108	C₂×C₄	1	2	yes	complemented
49	C₂₃×#107	C₂×C₄	1	2	yes	complemented
50	C₂₃×#106	—	2	1	no	—
51	C₂₃×#106	—	2	1	no	—
52	C₂₃×#106	C₂×C₄	1	2	yes	—
53	C₂₃×#102	—	2	4	no	—
54	C₂₃×#97	—	4	1	no	—
55	C₂₃×#97	C₂×C₄	1	2	yes	—
56	C₂₃×#97	D₄	1	2	yes	—
57	C₄×C₈×C₂₃	—	4	1	no	—
58	C₂⁴×C₂₃	—	4	1	no	—
59	C₂×D₄×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	4	1	no	—
63	C₂×D₄×C₂₃	—	4	1	no	—
64	C₂×D₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	4	4	no	—
67	C₂²×C₄×C₂₃	—	4	1	no	—
68	C₂²×C₄×C₂₃	—	8	4	no	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	D₈×C₂₃	—	2	2	no	—
71	D₈×C₂₃	—	2	2	no	—
72	D₈×C₂₃	—	2	2	no	—
73	D₈×C₂₃	C₄²	1	4	yes	—
74	M₁₆×C₂₃	—	2	2	no	—
75	M₁₆×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	2	1	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	4	1	no	—
89	C₂²⋊C₄×C₂₃	—	4	4	no	—
90	C₂²⋊C₄×C₂₃	—	4	1	no	—
91	C₄²×C₂₃	—	2	1	no	—
92	C₄²×C₂₃	—	4	1	no	—
93	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
94	C₁₆×C₂₃	—	2	2	no	—
95	C₁₆×C₂₃	—	2	2	no	—
96	#12647	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	4	2	no	—
98	C₂³×C₂₃	—	4	2	no	—
99	C₂³×C₂₃	—	4	2	no	—
100	C₂³×C₂₃	—	8	2	no	—
101	C₂³×C₂₃	—	8	2	no	—
102	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	D₄×C₂₃	—	2	4	no	—
104	D₄×C₂₃	—	4	2	no	—
105	D₄×C₂₃	—	4	2	no	—
106	D₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	8	8	no	—
112	C₂×C₄×C₂₃	—	8	2	no	—
113	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
119	C₈×C₂₃	—	2	2	no	—
120	C₈×C₂₃	—	4	2	no	—
121	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
122	C₈×C₂₃	#117	1	1	yes	characteristic
123	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
125	C₂×#442	C₂×C₂₃	1	1	yes	characteristic
126	C₂×#361	C₂×C₂₃	1	1	yes	characteristic
127	#2140	C₂×C₂₃	1	1	yes	characteristic
128	C₂×#347	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#1773	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1386	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#1383	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	4	4	no	—
134	C₂²×C₂₃	—	8	2	no	—
135	C₂²×C₂₃	—	8	2	no	—
136	C₂²×C₂₃	—	8	4	no	—
137	C₂²×C₂₃	—	8	4	no	—
138	C₂²×C₂₃	#418	1	1	yes	characteristic
139	C₂²×C₂₃	#440	1	1	yes	characteristic
140	C₂²×C₂₃	#442	1	1	yes	characteristic
141	C₄×C₂₃	—	4	2	no	—
142	C₄×C₂₃	—	8	2	no	—
143	C₄×C₂₃	—	8	2	no	—
144	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₂₃	#417	1	1	yes	characteristic
146	C₄×C₂₃	#440	1	1	yes	characteristic
147	C₄×C₂₃	#442	1	1	yes	characteristic
148	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄×D₄	—	2	1	no	—
150	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
152	#442	—	2	2	no	—
153	#442	C₄×C₂₃	1	4	yes	complemented
154	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
155	#427	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#102	—	2	1	no	—
158	#405	—	2	1	no	—
159	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
160	#361	C₄×C₂₃	1	4	yes	complemented
161	#357	C₄×C₂₃	1	4	yes	complemented
162	#347	—	2	2	no	—
163	#339	—	2	1	no	—
164	C₂×C₂₃	—	2	1	no	—
165	C₂×C₂₃	—	2	1	no	—
166	C₂×C₂₃	—	8	4	no	—
167	C₂×C₂₃	#1762	1	1	yes	characteristic
168	C₂×C₂₃	AGL(1,ℤ/16)	1	2	yes	—
169	C₂²×D₄	—	2	1	no	—
170	C₂³×C₄	—	4	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
175	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
176	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₄×D₄	—	2	4	no	—
181	C₄×D₄	—	4	2	no	—
182	#117	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₂²⋊C₄	—	4	1	no	—
185	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	C₂×C₄×C₂₃	1	2	yes	complemented
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
192	#108	—	2	1	no	—
193	#108	C₂×C₄×C₂₃	1	2	yes	complemented
194	#107	C₂×C₄×C₂₃	1	2	yes	complemented
195	#106	—	2	1	no	—
196	#106	—	2	1	no	—
197	#106	C₂×C₄×C₂₃	1	2	yes	—
198	#102	—	2	4	no	—
199	#97	—	4	1	no	—
200	#97	C₂×C₄×C₂₃	1	2	yes	—
201	#97	D₄×C₂₃	1	2	yes	—
202	C₄×C₈	—	4	1	no	—
203	C₂₃	#12647	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	—	4	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	D₈	—	2	2	no	—
217	D₈	—	2	2	no	—
218	D₈	—	2	2	no	—
219	D₈	C₄²×C₂₃	1	4	yes	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
231	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
267	C₈	C₂₃×#117	1	1	yes	characteristic
268	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
269	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	C₂₃×#418	1	1	yes	characteristic
277	C₂²	C₂₃×#440	1	1	yes	characteristic
278	C₂²	C₂₃×#442	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
283	C₄	C₂₃×#417	1	1	yes	characteristic
284	C₄	C₂₃×#440	1	1	yes	characteristic
285	C₄	C₂₃×#442	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	4	no	—
289	C₂	C₂₃×#1762	1	1	yes	characteristic
290	C₂	C₂₃×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#442
commutator	C₈	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₃×#347	C₂	semidirect
5. C₂₃×#1773	C₂	semidirect
6. C₂₃×#1386	C₂	semidirect
7. C₂₃×#1383	C₂	semidirect
13. C₂₃×#442	C₄	semidirect
20. C₂₃×#361	C₄	semidirect
21. C₂₃×#357	C₄	semidirect
44. C₂₃×M₃₂	C₂×C₄	semidirect
46. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
48. C₂₃×#108	C₂×C₄	semidirect
49. C₂₃×#107	C₂×C₄	semidirect
96. #12647	C₂₃	direct (N × H)
128. C₂×#347	C₂×C₂₃	semidirect
129. #1773	C₂×C₂₃	semidirect
130. #1386	C₂×C₂₃	semidirect
131. #1383	C₂×C₂₃	semidirect
153. #442	C₄×C₂₃	semidirect
160. #361	C₄×C₂₃	semidirect
161. #357	C₄×C₂₃	semidirect
189. M₃₂	C₂×C₄×C₂₃	semidirect
191. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
193. #108	C₂×C₄×C₂₃	semidirect
194. #107	C₂×C₄×C₂₃	semidirect
203. C₂₃	#12647	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#442	2
C₂×C₂₃×#361	2
C₂₃×#2140	2
C₂×C₂₃×#347	2
C₂₃×#1773	2
C₂₃×#1386	2
C₂₃×#1383	2
#12647	23

Subgroups — C23×#12647 #19160273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12647 #19160273