Subgroups — C₂₃×#12604 #19160230

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#350	C₂	1	2	yes	maximal
3	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
8	C₄×#41×C₂₃	C₄	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#405	—	2	1	no	—
12	C₂₃×#405	—	2	1	no	—
13	C₂₃×#405	C₄	1	1	yes	characteristic
14	C₂₃×#405	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₂₃	—	2	1	no	—
16	C₄×M₁₆×C₂₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₃	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#350	C₄	1	8	yes	complemented
20	C₂₃×#346	—	2	1	no	—
21	C₂₃×#346	—	2	1	no	—
22	C₂₃×#346	—	2	1	no	—
23	C₄×C₁₆×C₂₃	—	2	1	no	—
24	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×#131	—	2	2	no	—
26	C₂₃×#131	—	2	2	no	—
27	C₂₃×#131	C₈	1	4	yes	—
28	C₂₃×#131	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₂₃	C₈	1	2	yes	—
38	C₄×D₄×C₂₃	—	2	2	no	—
39	C₄×D₄×C₂₃	C₈	1	2	yes	—
40	C₂₃×#117	—	2	1	no	—
41	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	—	2	1	no	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂₃×M₃₂	—	2	2	no	—
45	C₂₃×M₃₂	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₆×C₂₃	—	2	2	no	—
47	C₂×C₁₆×C₂₃	—	2	2	no	—
48	C₂×C₁₆×C₂₃	—	2	2	no	—
49	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
50	C₂₃×#97	—	2	1	no	—
51	C₂₃×#97	—	2	1	no	—
52	C₂₃×#97	—	2	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
60	#41×C₂₃	—	2	1	no	—
61	#41×C₂₃	—	2	1	no	—
62	#41×C₂₃	C₄²	1	2	yes	—
63	#41×C₂₃	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₂₃	—	2	1	no	—
66	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	—	2	2	no	—
69	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	M₁₆×C₂₃	—	2	1	no	—
72	M₁₆×C₂₃	—	2	1	no	—
73	M₁₆×C₂₃	—	2	4	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	C₄²	1	2	yes	—
76	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
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×C₈	1	2	yes	—
84	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	#41	1	2	yes	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	2	1	no	—
95	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×C₈	1	2	yes	—
98	C₁₆×C₂₃	—	2	4	no	—
99	C₁₆×C₂₃	—	2	4	no	—
100	#12604	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	2	1	no	—
102	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
103	Q₈×C₂₃	—	2	1	no	—
104	Q₈×C₂₃	C₄×C₈	1	2	yes	—
105	D₄×C₂₃	—	2	2	no	—
106	D₄×C₂₃	—	2	2	no	—
107	D₄×C₂₃	—	2	1	no	—
108	D₄×C₂₃	C₄×C₈	1	2	yes	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
117	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
123	C₈×C₂₃	—	2	2	no	—
124	C₈×C₂₃	—	2	2	no	—
125	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
126	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
127	C₈×C₂₃	C₄×D₄	1	2	yes	—
128	C₈×C₂₃	C₄×D₄	1	2	yes	—
129	C₄×#131	C₂×C₂₃	1	1	yes	characteristic
130	C₂×#350	C₂×C₂₃	1	2	yes	—
131	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#1379	C₂×C₂₃	1	2	yes	complemented
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	C₄×M₁₆	1	1	yes	characteristic
138	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₂₃	#431	1	1	yes	characteristic
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	4	2	no	—
144	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₄×C₂₃	#431	1	1	yes	characteristic
148	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
149	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
150	C₄×#41	C₄×C₂₃	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
152	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
153	#405	—	2	1	no	—
154	#405	—	2	1	no	—
155	#405	C₄×C₂₃	1	1	yes	characteristic
156	#405	C₂²×C₂₃	1	1	yes	characteristic
157	C₄×M₁₆	—	2	1	no	—
158	C₄×M₁₆	C₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄×C₈	—	2	1	no	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	#350	C₄×C₂₃	1	8	yes	complemented
162	#346	—	2	1	no	—
163	#346	—	2	1	no	—
164	#346	—	2	1	no	—
165	C₄×C₁₆	—	2	1	no	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	1	no	—
168	C₂×C₂₃	—	4	2	no	—
169	C₂×C₂₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
170	C₂×C₂₃	#2117	1	2	yes	—
171	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
172	#131	—	2	2	no	—
173	#131	—	2	2	no	—
174	#131	C₈×C₂₃	1	4	yes	—
175	#131	C₂×C₄×C₂₃	1	4	yes	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
184	C₄×Q₈	C₈×C₂₃	1	2	yes	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	C₈×C₂₃	1	2	yes	—
187	#117	—	2	1	no	—
188	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	C₂×C₄×C₂₃	1	4	yes	complemented
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
197	#97	—	2	1	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
204	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
206	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
207	C₂₃	#12604	1	1	yes	characteristic, direct product factor, Sylow
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₄²×C₂₃	1	2	yes	—
211	#41	C₂×C₈×C₂₃	1	2	yes	—
212	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₄²×C₂₃	1	2	yes	—
224	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
232	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₈	#41×C₂₃	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
244	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
245	C₄²	C₂×C₈×C₂₃	1	2	yes	—
246	C₁₆	—	2	4	no	—
247	C₁₆	—	2	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×C₈×C₂₃	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×C₈×C₂₃	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
263	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
264	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
273	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
274	C₈	C₄×D₄×C₂₃	1	2	yes	—
275	C₈	C₄×D₄×C₂₃	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×M₁₆×C₂₃	1	1	yes	characteristic
280	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
281	C₂²	C₂₃×#431	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
287	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
288	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
289	C₄	C₂₃×#431	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
294	C₂	C₂₃×#2117	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₃×M₃₂	C₂	semidirect
4. C₂×C₂₃×#346	C₂	semidirect
5. C₂₃×#1379	C₂	semidirect
19. C₂₃×#350	C₄	semidirect
45. C₂₃×M₃₂	C₂×C₄	semidirect
49. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
100. #12604	C₂₃	direct (N × H)
131. C₄×M₃₂	C₂×C₂₃	semidirect
132. C₂×#346	C₂×C₂₃	semidirect
133. #1379	C₂×C₂₃	semidirect
161. #350	C₄×C₂₃	semidirect
192. M₃₂	C₂×C₄×C₂₃	semidirect
196. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
207. C₂₃	#12604	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#131	2
C₂×C₂₃×#350	2
C₄×C₂₃×M₃₂	2
C₂×C₂₃×#346	2
C₂₃×#1379	2
#12604	23

Subgroups — C23×#12604 #19160230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12604 #19160230