Subgroups — C₂₃×#3290 #15670909

Order 2944.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#577	C₂	1	2	yes	maximal, complemented
3	C₂₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#471	C₂	1	2	yes	maximal, complemented
8	C₂₃×#455	C₂	1	2	yes	maximal, complemented
9	C₂₃×#454	C₂	1	2	yes	maximal, complemented
10	C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	complemented
13	C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₂₃×#137	—	2	2	no	—
15	C₂₃×#137	—	2	2	no	—
16	C₂₃×#137	C₂²	1	2	yes	—
17	C₂₃×#137	C₂²	1	2	yes	complemented
18	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
19	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
20	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
21	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
22	C₂₃×#135	—	2	1	no	—
23	C₂₃×#135	—	2	2	no	—
24	C₂₃×#135	—	2	1	no	—
25	C₂₃×#135	C₂²	1	1	yes	characteristic
26	C₂₃×#135	C₂²	1	1	yes	characteristic
27	C₂×Q₁₆×C₂₃	—	2	1	no	—
28	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×SD₁₆×C₂₃	—	2	2	no	—
30	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
31	C₂×D₈×C₂₃	—	2	1	no	—
32	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#131	C₂²	1	1	yes	characteristic
34	C₂₃×#131	C₂²	1	2	yes	complemented
35	C₂₃×#131	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#124	C₂²	1	2	yes	complemented
37	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
39	C₂₃×#104	C₂²	1	1	yes	characteristic
40	C₂₃×#104	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#104	C₂²	1	2	yes	complemented
42	C₂₃×#101	C₂²	1	2	yes	complemented
43	C₂₃×#100	C₂²	1	2	yes	complemented
44	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
45	#41×C₂₃	—	2	2	no	—
46	#41×C₂₃	—	2	2	no	—
47	#41×C₂₃	—	2	2	no	—
48	#41×C₂₃	—	2	2	no	—
49	#41×C₂₃	—	4	2	no	—
50	#41×C₂₃	C₂³	1	1	yes	characteristic
51	#41×C₂₃	C₂³	1	2	yes	—
52	#41×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₂₃	—	2	2	no	—
54	C₂×Q₈×C₂₃	—	2	2	no	—
55	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₂×D₄×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	4	2	no	—
59	C₂×D₄×C₂₃	C₂³	1	2	yes	—
60	Q₁₆×C₂₃	—	2	2	no	—
61	Q₁₆×C₂₃	—	2	2	no	—
62	Q₁₆×C₂₃	—	2	1	no	—
63	Q₁₆×C₂₃	—	2	1	no	—
64	Q₁₆×C₂₃	D₄	1	1	yes	characteristic
65	Q₁₆×C₂₃	D₄	1	1	yes	characteristic
66	SD₁₆×C₂₃	—	2	2	no	—
67	SD₁₆×C₂₃	—	2	2	no	—
68	SD₁₆×C₂₃	—	2	2	no	—
69	SD₁₆×C₂₃	—	2	2	no	—
70	SD₁₆×C₂₃	—	2	2	no	—
71	SD₁₆×C₂₃	—	2	2	no	—
72	SD₁₆×C₂₃	D₄	1	1	yes	characteristic
73	SD₁₆×C₂₃	D₄	1	1	yes	characteristic
74	SD₁₆×C₂₃	D₄	1	2	yes	—
75	D₈×C₂₃	—	2	2	no	—
76	D₈×C₂₃	—	2	2	no	—
77	D₈×C₂₃	—	2	1	no	—
78	D₈×C₂₃	—	2	1	no	—
79	D₈×C₂₃	D₄	1	1	yes	characteristic
80	D₈×C₂₃	D₄	1	1	yes	characteristic
81	M₁₆×C₂₃	—	2	1	no	—
82	M₁₆×C₂₃	—	2	2	no	—
83	M₁₆×C₂₃	—	2	1	no	—
84	M₁₆×C₂₃	C₂³	1	1	yes	characteristic
85	M₁₆×C₂₃	C₂³	1	2	yes	—
86	M₁₆×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	1	no	—
90	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₂₃	—	4	2	no	—
93	C₄²×C₂₃	C₂³	1	1	yes	characteristic
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	4	2	no	—
96	Q₈×C₂₃	—	2	2	no	—
97	Q₈×C₂₃	—	2	2	no	—
98	Q₈×C₂₃	—	4	2	no	—
99	Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₂₃	C₂×D₄	1	2	yes	—
101	Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	D₄×C₂₃	—	2	1	no	—
103	D₄×C₂₃	—	2	2	no	—
104	D₄×C₂₃	—	2	1	no	—
105	D₄×C₂₃	—	2	2	no	—
106	D₄×C₂₃	—	2	2	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	D₄×C₂₃	C₂×D₄	1	2	yes	—
112	D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	2	2	no	—
124	C₈×C₂₃	—	2	1	no	—
125	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
129	#3290	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	2	no	—
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	4	2	no	—
134	C₂²×C₂₃	—	8	2	no	—
135	C₂²×C₂₃	—	16	2	no	—
136	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	4	1	no	—
141	C₄×C₂₃	—	4	2	no	—
142	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	#578	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#577	C₂×C₂₃	1	2	yes	complemented
146	#576	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#495	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#473	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#472	C₂×C₂₃	1	1	yes	characteristic, complemented
150	#471	C₂×C₂₃	1	2	yes	complemented
151	#455	C₂×C₂₃	1	2	yes	complemented
152	#454	C₂×C₂₃	1	2	yes	complemented
153	#443	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#443	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	—	4	1	no	—
157	C₂×C₂₃	—	4	2	no	—
158	C₂×C₂₃	—	4	1	no	—
159	C₂×C₂₃	—	8	2	no	—
160	C₂×C₂₃	D₄²	1	1	yes	characteristic
161	2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	complemented
162	H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	—
163	#137	—	2	2	no	—
164	#137	—	2	2	no	—
165	#137	C₂²×C₂₃	1	2	yes	—
166	#137	C₂²×C₂₃	1	2	yes	complemented
167	AGL(1,ℤ/8)	—	2	2	no	—
168	AGL(1,ℤ/8)	—	2	2	no	—
169	AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
170	AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	complemented
171	#135	—	2	1	no	—
172	#135	—	2	2	no	—
173	#135	—	2	1	no	—
174	#135	C₂²×C₂₃	1	1	yes	characteristic
175	#135	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×Q₁₆	—	2	1	no	—
177	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
182	#131	C₂²×C₂₃	1	1	yes	characteristic
183	#131	C₂²×C₂₃	1	2	yes	complemented
184	#131	C₂²×C₂₃	1	1	yes	characteristic, complemented
185	#124	C₂²×C₂₃	1	2	yes	complemented
186	#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
187	#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
188	#104	C₂²×C₂₃	1	1	yes	characteristic
189	#104	C₂²×C₂₃	1	1	yes	characteristic, complemented
190	#104	C₂²×C₂₃	1	2	yes	complemented
191	#101	C₂²×C₂₃	1	2	yes	complemented
192	#100	C₂²×C₂₃	1	2	yes	complemented
193	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
194	C₂₃	#3290	1	1	yes	characteristic, direct product factor, Sylow
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	4	2	no	—
200	#41	C₂³×C₂₃	1	1	yes	characteristic
201	#41	C₂³×C₂₃	1	2	yes	—
202	#41	C₂³×C₂₃	1	1	yes	characteristic
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	C₂³×C₂₃	1	2	yes	—
210	Q₁₆	—	2	2	no	—
211	Q₁₆	—	2	2	no	—
212	Q₁₆	—	2	1	no	—
213	Q₁₆	—	2	1	no	—
214	Q₁₆	D₄×C₂₃	1	1	yes	characteristic
215	Q₁₆	D₄×C₂₃	1	1	yes	characteristic
216	SD₁₆	—	2	2	no	—
217	SD₁₆	—	2	2	no	—
218	SD₁₆	—	2	2	no	—
219	SD₁₆	—	2	2	no	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	D₄×C₂₃	1	1	yes	characteristic
223	SD₁₆	D₄×C₂₃	1	1	yes	characteristic
224	SD₁₆	D₄×C₂₃	1	2	yes	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	D₈	—	2	1	no	—
228	D₈	—	2	1	no	—
229	D₈	D₄×C₂₃	1	1	yes	characteristic
230	D₈	D₄×C₂₃	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	C₂³×C₂₃	1	1	yes	characteristic
235	M₁₆	C₂³×C₂₃	1	2	yes	—
236	M₁₆	C₂³×C₂₃	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	C₂³×C₂₃	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	4	2	no	—
246	Q₈	—	2	2	no	—
247	Q₈	—	2	2	no	—
248	Q₈	—	4	2	no	—
249	Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
250	Q₈	C₂×D₄×C₂₃	1	2	yes	—
251	Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
261	D₄	C₂×D₄×C₂₃	1	2	yes	—
262	D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	1	no	—
275	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
277	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	16	2	no	—
285	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	2	no	—
291	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
292	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	8	2	no	—
298	C₂	D₄²×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#578	C₂	semidirect
2. C₂₃×#577	C₂	semidirect
3. C₂₃×#576	C₂	semidirect
4. C₂₃×#495	C₂	semidirect
5. C₂₃×#473	C₂	semidirect
6. C₂₃×#472	C₂	semidirect
7. C₂₃×#471	C₂	semidirect
8. C₂₃×#455	C₂	semidirect
9. C₂₃×#454	C₂	semidirect
10. C₂₃×#443	C₂	semidirect
11. C₂₃×#443	C₂	semidirect
12. C₂₃×2₋¹⁺⁴	C₂²	semidirect
17. C₂₃×#137	C₂²	semidirect
21. C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
28. C₂×Q₁₆×C₂₃	C₂²	semidirect
30. C₂×SD₁₆×C₂₃	C₂²	semidirect
34. C₂₃×#131	C₂²	semidirect
35. C₂₃×#131	C₂²	semidirect
36. C₂₃×#124	C₂²	semidirect
37. C₂₃×#108	C₂²	semidirect
38. C₂₃×#108	C₂²	semidirect
40. C₂₃×#104	C₂²	semidirect
41. C₂₃×#104	C₂²	semidirect
42. C₂₃×#101	C₂²	semidirect
43. C₂₃×#100	C₂²	semidirect
44. C₄×C₈×C₂₃	C₂²	semidirect
129. #3290	C₂₃	direct (N × H)
144. #578	C₂×C₂₃	semidirect
145. #577	C₂×C₂₃	semidirect
146. #576	C₂×C₂₃	semidirect
147. #495	C₂×C₂₃	semidirect
148. #473	C₂×C₂₃	semidirect
149. #472	C₂×C₂₃	semidirect
150. #471	C₂×C₂₃	semidirect
151. #455	C₂×C₂₃	semidirect
152. #454	C₂×C₂₃	semidirect
153. #443	C₂×C₂₃	semidirect
154. #443	C₂×C₂₃	semidirect
161. 2₋¹⁺⁴	C₂²×C₂₃	semidirect
166. #137	C₂²×C₂₃	semidirect
170. AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
177. C₂×Q₁₆	C₂²×C₂₃	semidirect
179. C₂×SD₁₆	C₂²×C₂₃	semidirect
183. #131	C₂²×C₂₃	semidirect
184. #131	C₂²×C₂₃	semidirect
185. #124	C₂²×C₂₃	semidirect
186. #108	C₂²×C₂₃	semidirect
187. #108	C₂²×C₂₃	semidirect
189. #104	C₂²×C₂₃	semidirect
190. #104	C₂²×C₂₃	semidirect
191. #101	C₂²×C₂₃	semidirect
192. #100	C₂²×C₂₃	semidirect
193. C₄×C₈	C₂²×C₂₃	semidirect
194. C₂₃	#3290	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×#578	2
C₂₃×#577	2
C₂₃×#576	2
C₂₃×#495	2
C₂₃×#473	2
C₂₃×#472	2
C₂₃×#471	2
C₂₃×#455	2
C₂₃×#454	2
C₂₃×#443	2
#3290	23

Subgroups — C23×#3290 #15670909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3290 #15670909