Subgroups — C₂₃×#12623 #19160249

Order 5888.

470 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2120	C₂	1	2	yes	maximal
3	C₂₃×#1757	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#525	C₂²	1	1	yes	characteristic
9	C₂₃×#435	—	2	2	no	—
10	C₈×D₄×C₂₃	—	2	2	no	—
11	C₂₃×#424	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#413	C₂²	1	2	yes	—
13	C₂₃×#410	C₂²	1	2	yes	—
14	C₂₃×#408	C₂²	1	1	yes	characteristic
15	C₂₃×#407	C₄	1	1	yes	characteristic, complemented
16	C₂₃×#405	—	2	2	no	—
17	C₂₃×#405	C₄	1	1	yes	characteristic, complemented
18	C₂₃×#352	C₄	1	4	yes	complemented
19	C₂₃×AΣL(1,𝔽₁₆)	C₄	1	4	yes	complemented
20	C₂₃×#344	—	2	2	no	—
21	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×#131	—	4	1	no	—
24	C₂₃×#131	—	4	1	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	4	1	no	—
32	C₂²×C₈×C₂₃	—	4	1	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	D₄	1	2	yes	—
36	C₂₃×#123	C₂×C₄	1	2	yes	—
37	C₂₃×C₂²⋊Q₈	—	2	1	no	—
38	C₂₃×#121	—	2	1	no	—
39	C₂₃×#120	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₂₃	—	2	2	no	—
41	C₂₃×#117	—	2	2	no	—
42	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×#101	—	2	2	no	—
45	C₂₃×#100	C₂×C₄	1	4	yes	—
46	C₂₃×#99	—	2	2	no	—
47	C₂₃×#99	C₂×C₄	1	4	yes	—
48	C₂₃×C₂²⋊C₈	—	2	2	no	—
49	C₂₃×C₂²⋊C₈	—	2	2	no	—
50	C₂₃×#97	—	2	1	no	—
51	C₂₃×#97	—	4	2	no	—
52	C₄×C₈×C₂₃	—	2	1	no	—
53	C₄×C₈×C₂₃	—	4	2	no	—
54	C₂⁴×C₂₃	C₄²	1	1	yes	characteristic
55	#41×C₂₃	—	4	1	no	—
56	#41×C₂₃	—	4	1	no	—
57	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	4	1	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	M₁₆×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	4	1	no	—
70	M₁₆×C₂₃	—	4	2	no	—
71	M₁₆×C₂₃	—	4	2	no	—
72	M₁₆×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	1	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	—	4	1	no	—
83	C₂×C₈×C₂₃	—	4	1	no	—
84	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂²⋊C₄×C₂₃	—	2	4	no	—
93	C₂²⋊C₄×C₂₃	C₄²	1	2	yes	—
94	C₄²×C₂₃	—	2	1	no	—
95	C₄²×C₂₃	—	4	2	no	—
96	#12623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	1	no	—
98	C₂³×C₂₃	—	4	1	no	—
99	C₂³×C₂₃	—	4	2	no	—
100	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
101	C₂³×C₂₃	C₄×D₄	1	2	yes	—
102	Q₈×C₂₃	—	4	1	no	—
103	D₄×C₂₃	—	4	2	no	—
104	D₄×C₂₃	—	4	1	no	—
105	D₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
120	C₈×C₂₃	—	2	1	no	—
121	C₈×C₂₃	—	2	2	no	—
122	C₈×C₂₃	—	4	4	no	—
123	C₈×C₂₃	—	4	1	no	—
124	C₈×C₂₃	—	4	1	no	—
125	C₈×C₂₃	#99	1	1	yes	characteristic
126	C₈×C₂₃	#99	1	1	yes	characteristic
127	#2929	C₂×C₂₃	1	1	yes	characteristic
128	#2120	C₂×C₂₃	1	2	yes	—
129	#1757	C₂×C₂₃	1	2	yes	complemented
130	#1392	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#1391	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	4	2	no	—
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	—	4	2	no	—
139	C₂²×C₂₃	—	4	2	no	—
140	C₂²×C₂₃	—	4	1	no	—
141	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	2	no	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	—	8	4	no	—
147	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
148	C₄×C₂₃	#410	1	1	yes	characteristic
149	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
150	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic, complemented
151	#525	C₂²×C₂₃	1	1	yes	characteristic
152	#435	—	2	2	no	—
153	C₈×D₄	—	2	2	no	—
154	#424	C₄×C₂₃	1	1	yes	characteristic, complemented
155	#413	C₂²×C₂₃	1	2	yes	—
156	#410	C₂²×C₂₃	1	2	yes	—
157	#408	C₂²×C₂₃	1	1	yes	characteristic
158	#407	C₄×C₂₃	1	1	yes	characteristic, complemented
159	#405	—	2	2	no	—
160	#405	C₄×C₂₃	1	1	yes	characteristic, complemented
161	#352	C₄×C₂₃	1	4	yes	complemented
162	AΣL(1,𝔽₁₆)	C₄×C₂₃	1	4	yes	complemented
163	#344	—	2	2	no	—
164	C₂×C₂₃	—	2	1	no	—
165	C₂×C₂₃	—	4	2	no	—
166	C₂×C₂₃	—	4	2	no	—
167	C₂×C₂₃	—	4	1	no	—
168	C₂×C₂₃	C₄×#99	1	1	yes	characteristic
169	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
171	#131	—	4	1	no	—
172	#131	—	4	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	4	1	no	—
180	C₂²×C₈	—	4	1	no	—
181	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₂₃	1	2	yes	—
184	#123	C₂×C₄×C₂₃	1	2	yes	—
185	C₂²⋊Q₈	—	2	1	no	—
186	#121	—	2	1	no	—
187	#120	C₂×C₄×C₂₃	1	2	yes	—
188	C₄×D₄	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	#101	—	2	2	no	—
193	#100	C₂×C₄×C₂₃	1	4	yes	—
194	#99	—	2	2	no	—
195	#99	C₂×C₄×C₂₃	1	4	yes	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	1	no	—
199	#97	—	4	2	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	4	2	no	—
202	C₂₃	#12623	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	C₄²×C₂₃	1	1	yes	characteristic
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
209	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
235	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	C₄²×C₂₃	1	2	yes	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₂₃	1	2	yes	—
250	Q₈	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	1	no	—
253	D₄	—	4	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	C₂₃×#99	1	1	yes	characteristic
274	C₈	C₂₃×#99	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	4	no	—
290	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
291	C₄	C₂₃×#410	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	4	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	4	1	no	—
296	C₂	C₄×C₂₃×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#99
commutator	C₂³	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×#41	C₂³×C₂₃
socle	C₂×C₂₃	C₄×#99
prime radical	C₂₃×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1757	C₂	semidirect
4. C₂₃×#1392	C₂	semidirect
5. C₂₃×#1391	C₂	semidirect
7. C₂³×C₈×C₂₃	C₄	semidirect
11. C₂₃×#424	C₄	semidirect
15. C₂₃×#407	C₄	semidirect
17. C₂₃×#405	C₄	semidirect
18. C₂₃×#352	C₄	semidirect
19. C₂₃×AΣL(1,𝔽₁₆)	C₄	semidirect
96. #12623	C₂₃	direct (N × H)
129. #1757	C₂×C₂₃	semidirect
130. #1392	C₂×C₂₃	semidirect
131. #1391	C₂×C₂₃	semidirect
150. C₂³×C₈	C₄×C₂₃	semidirect
154. #424	C₄×C₂₃	semidirect
158. #407	C₄×C₂₃	semidirect
160. #405	C₄×C₂₃	semidirect
161. #352	C₄×C₂₃	semidirect
162. AΣL(1,𝔽₁₆)	C₄×C₂₃	semidirect
202. C₂₃	#12623	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×#2929	2
C₂₃×#2120	2
C₂₃×#1757	2
C₂₃×#1392	2
C₂₃×#1391	2
#12623	23

Subgroups — C23×#12623 #19160249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12623 #19160249