Subgroups — C₂₃×#19732 #19167358

Order 5888.

644 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×#3231	C₂	1	2	yes	maximal
2	C₂₃×#3230	C₂	1	2	yes	maximal
3	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1723	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1718	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1718	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1710	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1453	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#533	C₂²	1	2	yes	—
11	C₂₃×#533	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×#500	C₂²	1	2	yes	—
14	C₂₃×#500	C₂²	1	2	yes	—
15	C₂₃×#500	C₂²	1	2	yes	—
16	C₂₃×#498	C₂²	1	1	yes	characteristic
17	C₂₃×#498	C₂²	1	2	yes	—
18	C₂₃×#494	C₂²	1	1	yes	characteristic
19	C₂₃×#491	—	2	2	no	—
20	C₂₃×#487	C₂²	1	2	yes	—
21	C₂₃×#487	C₂²	1	2	yes	—
22	C₂₃×#487	C₂²	1	2	yes	—
23	C₂₃×#487	C₂²	1	2	yes	—
24	C₂₃×#487	C₂²	1	2	yes	—
25	C₂₃×#487	C₂²	1	2	yes	—
26	C₂₃×#487	C₂²	1	2	yes	—
27	C₂₃×#485	—	2	4	no	—
28	C₂₃×#484	—	2	4	no	—
29	C₂₃×#479	—	2	4	no	—
30	C₂₃×#477	—	2	4	no	—
31	C₂₃×#470	—	2	2	no	—
32	C₂₃×#441	—	2	2	no	—
33	C₂₃×#431	C₂²	1	2	yes	—
34	C₂₃×#419	—	2	2	no	—
35	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#322	C₂²	1	1	yes	characteristic
37	C₂₃×#322	C₂²	1	2	yes	—
38	C₈²×C₂₃	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₂₃	—	2	1	no	—
40	C₂×Q₁₆×C₂₃	—	2	2	no	—
41	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
42	C₂₃×C₄⋊Q₈	—	2	2	no	—
43	C₂₃×C₄⋊Q₈	—	2	1	no	—
44	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×#125	—	2	2	no	—
49	C₂₃×#125	—	2	1	no	—
50	C₂₃×C₂²⋊Q₈	—	2	2	no	—
51	C₂₃×C₂²⋊Q₈	—	2	2	no	—
52	C₂₃×C₂²⋊Q₈	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	2	2	no	—
54	C₄×Q₈×C₂₃	—	2	2	no	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₂₃×#117	—	2	1	no	—
57	C₂₃×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#107	—	2	4	no	—
61	C₂₃×#107	—	2	4	no	—
62	C₂₃×#107	—	2	4	no	—
63	C₂₃×#106	—	2	2	no	—
64	C₂₃×#106	—	2	4	no	—
65	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
66	C₂₃×#103	—	2	4	no	—
67	C₂₃×#103	—	2	4	no	—
68	C₂₃×#103	—	2	4	no	—
69	C₂₃×#103	—	2	4	no	—
70	C₂₃×#103	—	2	4	no	—
71	C₂₃×#103	—	2	4	no	—
72	C₂₃×#103	—	2	4	no	—
73	C₂₃×C₂²⋊C₈	D₄	1	4	yes	—
74	C₂₃×#97	D₄	1	2	yes	—
75	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₂₃	C₂³	1	2	yes	—
77	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₂₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₂₃	—	2	2	no	—
80	C₂×Q₈×C₂₃	—	2	2	no	—
81	C₂×Q₈×C₂₃	—	2	2	no	—
82	C₂×Q₈×C₂₃	—	2	2	no	—
83	C₂×Q₈×C₂₃	—	4	4	no	—
84	C₂×Q₈×C₂₃	—	4	1	no	—
85	C₂²×C₄×C₂₃	—	4	1	no	—
86	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₂₃	—	4	4	no	—
89	M₁₆×C₂₃	—	4	2	no	—
90	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	4	2	no	—
103	C₄⋊C₄×C₂₃	—	4	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₂²⋊C₄×C₂₃	—	4	1	no	—
107	C₂²⋊C₄×C₂₃	—	4	2	no	—
108	C₄²×C₂₃	—	4	2	no	—
109	C₄²×C₂₃	—	4	1	no	—
110	C₄²×C₂₃	C₂×D₄	1	2	yes	—
111	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
113	#19732	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₃	#127	1	1	yes	characteristic
115	Q₈×C₂₃	—	4	4	no	—
116	Q₈×C₂₃	—	4	4	no	—
117	Q₈×C₂₃	—	4	4	no	—
118	Q₈×C₂₃	—	4	4	no	—
119	Q₈×C₂₃	—	8	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	4	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	8	2	no	—
127	C₂×C₄×C₂₃	#127	1	2	yes	—
128	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
131	C₈×C₂₃	—	2	2	no	—
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	2	4	no	—
134	C₈×C₂₃	—	2	4	no	—
135	#3231	C₂×C₂₃	1	2	yes	—
136	#3230	C₂×C₂₃	1	2	yes	—
137	#3185	C₂×C₂₃	1	1	yes	characteristic
138	#3139	C₂×C₂₃	1	1	yes	characteristic
139	#1723	C₂×C₂₃	1	2	yes	complemented
140	#1718	C₂×C₂₃	1	2	yes	complemented
141	#1718	C₂×C₂₃	1	2	yes	complemented
142	#1710	C₂×C₂₃	1	2	yes	complemented
143	#1453	C₂×C₂₃	1	1	yes	characteristic
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	2	no	—
146	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	8	2	no	—
150	C₄×C₂₃	—	8	2	no	—
151	C₄×C₂₃	—	8	2	no	—
152	C₄×C₂₃	—	8	2	no	—
153	C₄×C₂₃	#575	1	2	yes	—
154	C₄×C₂₃	#578	1	4	yes	—
155	#533	C₂²×C₂₃	1	1	yes	characteristic
156	#533	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
158	#500	C₂²×C₂₃	1	2	yes	—
159	#500	C₂²×C₂₃	1	2	yes	—
160	#500	C₂²×C₂₃	1	2	yes	—
161	#498	C₂²×C₂₃	1	1	yes	characteristic
162	#498	C₂²×C₂₃	1	2	yes	—
163	#494	C₂²×C₂₃	1	1	yes	characteristic
164	#491	—	2	2	no	—
165	#487	C₂²×C₂₃	1	2	yes	—
166	#487	C₂²×C₂₃	1	2	yes	—
167	#487	C₂²×C₂₃	1	2	yes	—
168	#487	C₂²×C₂₃	1	2	yes	—
169	#487	C₂²×C₂₃	1	2	yes	—
170	#487	C₂²×C₂₃	1	2	yes	—
171	#487	C₂²×C₂₃	1	2	yes	—
172	#485	—	2	4	no	—
173	#484	—	2	4	no	—
174	#479	—	2	4	no	—
175	#477	—	2	4	no	—
176	#470	—	2	2	no	—
177	#441	—	2	2	no	—
178	#431	C₂²×C₂₃	1	2	yes	—
179	#419	—	2	2	no	—
180	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
181	#322	C₂²×C₂₃	1	1	yes	characteristic
182	#322	C₂²×C₂₃	1	2	yes	—
183	C₈²	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×C₂₃	—	4	1	no	—
185	C₂×C₂₃	#3153	1	1	yes	characteristic
186	C₂×C₂₃	#3156	1	2	yes	—
187	C₂²×Q₈	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
193	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
194	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
195	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
196	#125	—	2	2	no	—
197	#125	—	2	1	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₂²⋊Q₈	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	#117	—	2	1	no	—
205	#117	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	#107	—	2	4	no	—
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#106	—	2	2	no	—
212	#106	—	2	4	no	—
213	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
214	#103	—	2	4	no	—
215	#103	—	2	4	no	—
216	#103	—	2	4	no	—
217	#103	—	2	4	no	—
218	#103	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	C₂²⋊C₈	D₄×C₂₃	1	4	yes	—
222	#97	D₄×C₂₃	1	2	yes	—
223	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₂₃	1	2	yes	—
225	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₂₃	1	2	yes	—
227	C₂₃	#19732	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	Q₁₆	—	4	4	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₂₃	1	2	yes	—
261	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
262	C₂³	C₂₃×#127	1	1	yes	characteristic
263	Q₈	—	4	4	no	—
264	Q₈	—	4	4	no	—
265	Q₈	—	4	4	no	—
266	Q₈	—	4	4	no	—
267	Q₈	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	C₂₃×#127	1	2	yes	—
276	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₂₃×#575	1	2	yes	—
293	C₄	C₂₃×#578	1	4	yes	—
294	C₂	—	4	1	no	—
295	C₂	C₂₃×#3153	1	1	yes	characteristic
296	C₂	C₂₃×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1718	C₂	semidirect
7. C₂₃×#1718	C₂	semidirect
8. C₂₃×#1710	C₂	semidirect
113. #19732	C₂₃	direct (N × H)
139. #1723	C₂×C₂₃	semidirect
140. #1718	C₂×C₂₃	semidirect
141. #1718	C₂×C₂₃	semidirect
142. #1710	C₂×C₂₃	semidirect
227. C₂₃	#19732	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×#3231	2
C₂₃×#3230	2
C₂₃×#3185	2
C₂₃×#3139	2
C₂₃×#1723	2
C₂₃×#1718	2
C₂₃×#1710	2
C₂₃×#1453	2
#19732	23

Subgroups — C23×#19732 #19167358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19732 #19167358