Subgroups — C₂₃×#19630 #19167256

Order 5888.

858 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×#3226	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3152	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1722	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1713	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#137	—	2	4	no	—
10	C₂₃×#532	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂₃×#500	—	2	2	no	—
13	C₂₃×#500	C₂²	1	1	yes	characteristic
14	C₂₃×#500	C₂²	1	4	yes	—
15	C₂₃×#500	C₂²	1	2	yes	—
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#497	—	2	4	no	—
18	C₂₃×#494	C₂²	1	4	yes	—
19	C₂₃×#493	C₂²	1	1	yes	characteristic
20	C₂₃×#492	C₂²	1	4	yes	—
21	C₂₃×#486	C₂²	1	2	yes	—
22	C₂₃×#486	C₂²	1	4	yes	—
23	C₂₃×#480	—	2	4	no	—
24	C₂₃×#474	—	2	4	no	—
25	C₂₃×#466	—	2	2	no	—
26	C₄×D₈×C₂₃	—	2	2	no	—
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#107	—	2	2	no	—
29	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#322	C₂²	1	2	yes	—
32	C₈²×C₂₃	C₂²	1	2	yes	—
33	C₂×#41×C₂₃	—	2	2	no	—
34	C₂²×Q₈×C₂₃	—	2	2	no	—
35	C₂₃×#137	—	4	16	no	—
36	C₂×Q₁₆×C₂₃	—	2	8	no	—
37	C₂×SD₁₆×C₂₃	—	2	8	no	—
38	C₂×D₈×C₂₃	—	2	2	no	—
39	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
40	C₂²×C₈×C₂₃	D₄	1	2	yes	—
41	C₂₃×C₄⋊Q₈	—	2	2	no	—
42	C₂₃×C₄⋊Q₈	—	2	2	no	—
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
45	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₂₃×#127	C₂³	1	2	yes	—
47	C₂₃×#124	—	2	2	no	—
48	C₂₃×#121	—	2	2	no	—
49	C₂₃×#121	—	2	2	no	—
50	C₄×D₄×C₂₃	—	2	2	no	—
51	C₄×D₄×C₂₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×#107	—	2	4	no	—
56	C₂₃×#107	—	2	4	no	—
57	C₂₃×#107	—	2	2	no	—
58	C₂₃×#107	—	2	8	no	—
59	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
60	C₂₃×#102	—	2	4	no	—
61	C₂₃×#102	—	2	8	no	—
62	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
63	C₂₃×#97	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₂₃	C₂³	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	2	yes	—
68	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	#41×C₂₃	—	4	8	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	2	4	no	—
72	C₂×Q₈×C₂₃	—	4	2	no	—
73	C₂×Q₈×C₂₃	—	4	8	no	—
74	C₂×D₄×C₂₃	—	2	2	no	—
75	C₂×D₄×C₂₃	—	2	2	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	4	2	no	—
78	C₂²×C₄×C₂₃	—	4	2	no	—
79	C₂²×C₄×C₂₃	—	4	2	no	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₂₃	—	4	16	no	—
84	SD₁₆×C₂₃	—	4	16	no	—
85	D₈×C₂₃	—	4	4	no	—
86	M₁₆×C₂₃	D₈	1	8	yes	complemented
87	C₂×C₈×C₂₃	—	2	4	no	—
88	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₄⋊C₄×C₂₃	—	2	4	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	4	2	no	—
97	C₄⋊C₄×C₂₃	—	4	2	no	—
98	C₂²⋊C₄×C₂₃	—	4	2	no	—
99	C₂²⋊C₄×C₂₃	—	4	2	no	—
100	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
104	#19630	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	4	2	no	—
106	C₂³×C₂₃	#127	1	1	yes	characteristic
107	Q₈×C₂₃	—	4	8	no	—
108	Q₈×C₂₃	—	4	4	no	—
109	Q₈×C₂₃	—	8	4	no	—
110	D₄×C₂₃	—	4	4	no	—
111	D₄×C₂₃	—	4	4	no	—
112	D₄×C₂₃	—	4	4	no	—
113	D₄×C₂₃	—	8	4	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	8	4	no	—
119	C₂×C₄×C₂₃	—	8	4	no	—
120	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₂₃	—	2	2	no	—
128	C₈×C₂₃	—	2	4	no	—
129	C₈×C₂₃	C₂×D₈	1	8	yes	—
130	C₈×C₂₃	#137	1	4	yes	complemented
131	#3226	C₂×C₂₃	1	2	yes	complemented
132	#3169	C₂×C₂₃	1	1	yes	characteristic
133	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#3152	C₂×C₂₃	1	2	yes	complemented
135	#1722	C₂×C₂₃	1	4	yes	complemented
136	#1713	C₂×C₂₃	1	2	yes	complemented
137	#1706	C₂×C₂₃	1	2	yes	complemented
138	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	8	2	no	—
142	C₂²×C₂₃	—	8	2	no	—
143	C₂²×C₂₃	—	8	2	no	—
144	C₂²×C₂₃	#493	1	2	yes	—
145	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	8	2	no	—
149	C₄×C₂₃	—	8	4	no	—
150	C₄×C₂₃	#493	1	2	yes	—
151	C₄×C₂₃	C₂²×D₈	1	2	yes	—
152	C₄×C₂₃	C₂×#137	1	2	yes	—
153	C₄×C₂₃	#578	1	2	yes	—
154	C₂×#137	—	2	4	no	—
155	#532	C₂²×C₂₃	1	2	yes	—
156	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
157	#500	—	2	2	no	—
158	#500	C₂²×C₂₃	1	2	yes	—
159	#500	C₂²×C₂₃	1	2	yes	—
160	#500	C₂²×C₂₃	1	1	yes	characteristic
161	#500	C₂²×C₂₃	1	4	yes	—
162	#497	—	2	4	no	—
163	#494	C₂²×C₂₃	1	4	yes	—
164	#493	C₂²×C₂₃	1	1	yes	characteristic
165	#492	C₂²×C₂₃	1	4	yes	—
166	#486	C₂²×C₂₃	1	2	yes	—
167	#486	C₂²×C₂₃	1	4	yes	—
168	#480	—	2	4	no	—
169	#474	—	2	4	no	—
170	#466	—	2	2	no	—
171	C₄×D₈	—	2	2	no	—
172	#431	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×#107	—	2	2	no	—
174	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
176	#322	C₂²×C₂₃	1	2	yes	—
177	C₈²	C₂²×C₂₃	1	2	yes	—
178	C₂×C₂₃	—	2	2	no	—
179	C₂×C₂₃	—	16	2	no	—
180	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
181	C₂×C₂₃	#3152	1	1	yes	characteristic
182	C₂×C₂₃	#3156	1	1	yes	characteristic
183	C₂×#41	—	2	2	no	—
184	C₂²×Q₈	—	2	2	no	—
185	#137	—	4	16	no	—
186	C₂×Q₁₆	—	2	8	no	—
187	C₂×SD₁₆	—	2	8	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
190	C₂²×C₈	D₄×C₂₃	1	2	yes	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
195	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
196	#127	C₂³×C₂₃	1	2	yes	—
197	#124	—	2	2	no	—
198	#121	—	2	2	no	—
199	#121	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
205	#107	—	2	4	no	—
206	#107	—	2	4	no	—
207	#107	—	2	2	no	—
208	#107	—	2	8	no	—
209	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
210	#102	—	2	4	no	—
211	#102	—	2	8	no	—
212	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
213	#97	D₄×C₂₃	1	2	yes	—
214	C₄×C₈	D₄×C₂₃	1	2	yes	—
215	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₂₃	1	2	yes	—
217	C₄×C₈	C₂³×C₂₃	1	2	yes	—
218	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
219	C₂₃	#19630	1	1	yes	characteristic, direct product factor, Sylow
220	#41	—	4	8	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
234	Q₁₆	—	4	16	no	—
235	SD₁₆	—	4	16	no	—
236	D₈	—	4	4	no	—
237	M₁₆	D₈×C₂₃	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
241	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
255	C₂³	—	4	2	no	—
256	C₂³	C₂₃×#127	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	8	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
272	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
274	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	C₂×D₈×C₂₃	1	8	yes	—
280	C₈	C₂₃×#137	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	C₂₃×#493	1	2	yes	—
287	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	4	no	—
292	C₄	C₂₃×#493	1	2	yes	—
293	C₄	C₂²×D₈×C₂₃	1	2	yes	—
294	C₄	C₂×C₂₃×#137	1	2	yes	—
295	C₄	C₂₃×#578	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
299	C₂	C₂₃×#3152	1	1	yes	characteristic
300	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3169	C₂
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
1. C₂₃×#3226	C₂	semidirect
3. C₂×C₂₃×#500	C₂	semidirect
4. C₂₃×#3152	C₂	semidirect
5. C₂₃×#1722	C₂	semidirect
6. C₂₃×#1713	C₂	semidirect
7. C₂₃×#1706	C₂	semidirect
8. C₈×M₁₆×C₂₃	C₂	semidirect
86. M₁₆×C₂₃	D₈	semidirect
104. #19630	C₂₃	direct (N × H)
130. C₈×C₂₃	#137	semidirect
131. #3226	C₂×C₂₃	semidirect
133. C₂×#500	C₂×C₂₃	semidirect
134. #3152	C₂×C₂₃	semidirect
135. #1722	C₂×C₂₃	semidirect
136. #1713	C₂×C₂₃	semidirect
137. #1706	C₂×C₂₃	semidirect
138. C₈×M₁₆	C₂×C₂₃	semidirect
219. C₂₃	#19630	direct (N × H)
237. M₁₆	D₈×C₂₃	semidirect
280. C₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3226	2
C₂₃×#3169	2
C₂×C₂₃×#500	2
C₂₃×#3152	2
C₂₃×#1722	2
C₂₃×#1713	2
C₂₃×#1706	2
C₈×M₁₆×C₂₃	2
#19630	23

Subgroups — C23×#19630 #19167256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19630 #19167256