Subgroups — C₂₃×#19751 #19167377

Order 5888.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3185	C₂	1	2	yes	maximal
2	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3165	C₂	1	2	yes	maximal
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×#1716	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1707	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1455	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×#501	—	2	4	no	—
14	C₂₃×#500	C₂²	1	2	yes	—
15	C₂₃×#500	C₂²	1	2	yes	—
16	C₂₃×#498	C₂²	1	2	yes	—
17	C₂₃×#498	C₂²	1	2	yes	—
18	C₂₃×#498	C₂²	1	2	yes	—
19	C₂₃×#498	C₂²	1	1	yes	characteristic
20	C₂₃×#494	C₂²	1	2	yes	—
21	C₂₃×#494	C₂²	1	1	yes	characteristic
22	C₂₃×#491	—	2	2	no	—
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×#487	C₂²	1	2	yes	—
28	C₂₃×#470	—	2	4	no	—
29	C₂₃×#470	—	2	2	no	—
30	C₂₃×#441	—	2	2	no	—
31	C₂₃×#441	—	2	4	no	—
32	C₂₃×#428	—	2	4	no	—
33	C₂₃×#419	—	2	2	no	—
34	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
35	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#322	C₂²	1	2	yes	—
37	C₂₃×#322	C₂²	1	1	yes	characteristic
38	C₈²×C₂₃	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₂₃	—	2	1	no	—
40	C₂×Q₁₆×C₂₃	—	2	2	no	—
41	C₂×Q₁₆×C₂₃	—	2	4	no	—
42	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
43	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
44	C₂₃×C₄⋊Q₈	—	2	2	no	—
45	C₂₃×C₄⋊Q₈	—	2	1	no	—
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₂₃×#125	—	2	1	no	—
51	C₂₃×#125	—	2	2	no	—
52	C₂₃×C₂²⋊Q₈	—	2	2	no	—
53	C₂₃×C₂²⋊Q₈	—	2	2	no	—
54	C₂₃×C₂²⋊Q₈	—	2	2	no	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₄×Q₈×C₂₃	—	2	2	no	—
57	C₄×Q₈×C₂₃	—	2	2	no	—
58	C₂₃×#117	—	2	1	no	—
59	C₂₃×#117	—	2	2	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×#107	—	2	4	no	—
63	C₂₃×#107	—	2	4	no	—
64	C₂₃×#106	—	2	4	no	—
65	C₂₃×#106	—	2	4	no	—
66	C₂₃×#106	—	2	2	no	—
67	C₂₃×#106	—	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×#97	D₄	1	2	yes	—
74	C₂₃×#97	D₄	1	4	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	Q₁₆×C₂₃	—	4	8	no	—
90	M₁₆×C₂₃	—	2	8	no	—
91	M₁₆×C₂₃	—	4	2	no	—
92	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
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	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₄⋊C₄×C₂₃	—	4	2	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₂²⋊C₄×C₂₃	—	4	1	no	—
109	C₂²⋊C₄×C₂₃	—	4	2	no	—
110	C₄²×C₂₃	—	4	2	no	—
111	C₄²×C₂₃	—	4	1	no	—
112	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₂₃	C₂×D₄	1	2	yes	—
114	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
115	#19751	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₃	#127	1	1	yes	characteristic
117	Q₈×C₂₃	—	4	4	no	—
118	Q₈×C₂₃	—	4	4	no	—
119	Q₈×C₂₃	—	4	4	no	—
120	Q₈×C₂₃	—	4	4	no	—
121	Q₈×C₂₃	—	8	2	no	—
122	C₂×C₄×C₂₃	—	2	4	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	4	2	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	8	2	no	—
129	C₂×C₄×C₂₃	#127	1	2	yes	—
130	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₂₃	—	2	4	no	—
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	#137	1	8	yes	—
137	#3185	C₂×C₂₃	1	2	yes	—
138	#3185	C₂×C₂₃	1	1	yes	characteristic
139	#3165	C₂×C₂₃	1	2	yes	—
140	#3139	C₂×C₂₃	1	1	yes	characteristic
141	#1723	C₂×C₂₃	1	2	yes	complemented
142	#1718	C₂×C₂₃	1	2	yes	complemented
143	#1716	C₂×C₂₃	1	2	yes	complemented
144	#1707	C₂×C₂₃	1	2	yes	complemented
145	#1455	C₂×C₂₃	1	1	yes	characteristic
146	C₂²×C₂₃	—	2	2	no	—
147	C₂²×C₂₃	—	2	1	no	—
148	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
149	C₄×C₂₃	—	2	2	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	8	2	no	—
152	C₄×C₂₃	—	8	2	no	—
153	C₄×C₂₃	—	8	2	no	—
154	C₄×C₂₃	—	8	2	no	—
155	C₄×C₂₃	C₂×#137	1	4	yes	—
156	C₄×C₂₃	#575	1	2	yes	—
157	#533	C₂²×C₂₃	1	1	yes	characteristic
158	#533	C₂²×C₂₃	1	2	yes	—
159	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
160	#501	—	2	4	no	—
161	#500	C₂²×C₂₃	1	2	yes	—
162	#500	C₂²×C₂₃	1	2	yes	—
163	#498	C₂²×C₂₃	1	2	yes	—
164	#498	C₂²×C₂₃	1	2	yes	—
165	#498	C₂²×C₂₃	1	2	yes	—
166	#498	C₂²×C₂₃	1	1	yes	characteristic
167	#494	C₂²×C₂₃	1	2	yes	—
168	#494	C₂²×C₂₃	1	1	yes	characteristic
169	#491	—	2	2	no	—
170	#487	C₂²×C₂₃	1	2	yes	—
171	#487	C₂²×C₂₃	1	2	yes	—
172	#487	C₂²×C₂₃	1	2	yes	—
173	#487	C₂²×C₂₃	1	2	yes	—
174	#487	C₂²×C₂₃	1	2	yes	—
175	#470	—	2	4	no	—
176	#470	—	2	2	no	—
177	#441	—	2	2	no	—
178	#441	—	2	4	no	—
179	#428	—	2	4	no	—
180	#419	—	2	2	no	—
181	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
182	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
183	#322	C₂²×C₂₃	1	2	yes	—
184	#322	C₂²×C₂₃	1	1	yes	characteristic
185	C₈²	C₂²×C₂₃	1	1	yes	characteristic
186	C₂×C₂₃	—	4	1	no	—
187	C₂×C₂₃	#3152	1	2	yes	—
188	C₂×C₂₃	#3153	1	1	yes	characteristic
189	C₂²×Q₈	—	2	1	no	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	4	no	—
192	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
193	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
197	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
198	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
200	#125	—	2	1	no	—
201	#125	—	2	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	#117	—	2	1	no	—
209	#117	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
212	#107	—	2	4	no	—
213	#107	—	2	4	no	—
214	#106	—	2	4	no	—
215	#106	—	2	4	no	—
216	#106	—	2	2	no	—
217	#106	—	2	4	no	—
218	#103	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	2	4	no	—
223	#97	D₄×C₂₃	1	2	yes	—
224	#97	D₄×C₂₃	1	4	yes	—
225	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₂₃	1	2	yes	—
227	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₂₃	1	2	yes	—
229	C₂₃	#19751	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	8	no	—
241	M₁₆	—	2	8	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
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	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₂₃	1	2	yes	—
265	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
266	C₂³	C₂₃×#127	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	8	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	C₂₃×#127	1	2	yes	—
280	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂₃×#137	1	8	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₂×C₂₃×#137	1	4	yes	—
297	C₄	C₂₃×#575	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	C₂₃×#3152	1	2	yes	—
300	C₂	C₂₃×#3153	1	1	yes	characteristic

	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×#1716	C₂	semidirect
8. C₂₃×#1707	C₂	semidirect
115. #19751	C₂₃	direct (N × H)
141. #1723	C₂×C₂₃	semidirect
142. #1718	C₂×C₂₃	semidirect
143. #1716	C₂×C₂₃	semidirect
144. #1707	C₂×C₂₃	semidirect
229. C₂₃	#19751	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×#3185	2
C₂₃×#3165	2
C₂₃×#3139	2
C₂₃×#1723	2
C₂₃×#1718	2
C₂₃×#1716	2
C₂₃×#1707	2
C₂₃×#1455	2
#19751	23

Subgroups — C23×#19751 #19167377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19751 #19167377