Subgroups — C₂₃×#61818 #19209444

Order 5888.

1930 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	C₂₃×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	Q₈×SD₁₆×C₂₃	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3359	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3229	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3101	C₂	1	6	yes	maximal
7	C₂₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₂₃	—	2	1	no	—
9	Q₈²×C₂₃	C₂²	1	4	yes	complemented
10	C₂₃×#557	—	2	4	no	—
11	C₂₃×#557	C₂²	1	4	yes	complemented
12	D₄×Q₈×C₂₃	—	2	4	no	—
13	D₄×Q₈×C₂₃	C₂²	1	8	yes	—
14	C₂₃×#544	—	2	3	no	—
15	C₂₃×#533	C₂²	1	6	yes	—
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₂×C₂₃×C₂²⋊Q₈	—	2	3	no	—
18	C₂₃×#519	—	2	1	no	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	—
21	C₂₃×#498	C₂²	1	12	yes	complemented
22	C₂₃×#487	C₂²	1	12	yes	complemented
23	C₂₃×#483	C₂²	1	12	yes	—
24	C₂₃×#476	C₂²	1	12	yes	—
25	C₂₃×#475	C₂²	1	24	yes	complemented
26	C₂₃×#461	C₂²	1	12	yes	—
27	C₂₃×#450	C₂²	1	4	yes	—
28	C₈×Q₈×C₂₃	C₂²	1	8	yes	complemented
29	C₄×SD₁₆×C₂₃	C₂²	1	24	yes	—
30	C₂₃×#431	C₂²	1	6	yes	—
31	C₂²×Q₈×C₂₃	—	2	8	no	—
32	C₂²×Q₈×C₂₃	—	2	3	no	—
33	C₂²×Q₈×C₂₃	—	4	8	no	—
34	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₂₃	—	2	3	no	—
37	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
38	C₂₃×C₄⋊Q₈	—	2	12	no	—
39	C₂₃×C₄⋊Q₈	—	2	6	no	—
40	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₂₃×C₄⋊Q₈	C₂³	1	12	yes	—
42	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
43	C₂₃×#125	—	2	12	no	—
44	C₂₃×#125	—	2	6	no	—
45	C₂₃×C₂²⋊Q₈	—	2	6	no	—
46	C₂₃×C₂²⋊Q₈	—	2	24	no	—
47	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
48	C₄×Q₈×C₂₃	—	2	4	no	—
49	C₄×Q₈×C₂₃	—	2	4	no	—
50	C₄×Q₈×C₂₃	—	2	6	no	—
51	C₄×Q₈×C₂₃	D₄	1	4	yes	—
52	C₄×Q₈×C₂₃	C₂³	1	12	yes	—
53	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
54	C₄×D₄×C₂₃	—	2	6	no	—
55	C₄×D₄×C₂₃	C₂³	1	12	yes	—
56	C₂₃×#117	—	2	6	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
58	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
60	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
61	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
62	C₂₃×#106	C₂³	1	24	yes	—
63	C₂₃×C₄⋊C₈	C₂³	1	12	yes	—
64	C₂₃×#103	C₂³	1	24	yes	—
65	C₂₃×#102	C₂³	1	8	yes	—
66	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
67	C₄×C₈×C₂₃	C₂³	1	12	yes	—
68	C₂⁴×C₂₃	—	2	1	no	—
69	C₂×Q₈×C₂₃	—	2	12	no	—
70	C₂×Q₈×C₂₃	—	2	6	no	—
71	C₂×Q₈×C₂₃	—	4	32	no	—
72	C₂×Q₈×C₂₃	—	4	12	no	—
73	C₂×Q₈×C₂₃	SD₁₆	1	8	yes	complemented
74	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
75	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
76	C₂×D₄×C₂₃	—	2	4	no	—
77	C₂×D₄×C₂₃	—	2	1	no	—
78	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₂₃	—	2	3	no	—
80	C₂²×C₄×C₂₃	—	2	12	no	—
81	C₂²×C₄×C₂₃	—	4	12	no	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
85	SD₁₆×C₂₃	—	2	16	no	—
86	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₂₃	—	2	12	no	—
88	C₄⋊C₄×C₂₃	—	2	3	no	—
89	C₄⋊C₄×C₂₃	—	2	6	no	—
90	C₄⋊C₄×C₂₃	—	2	6	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	2	6	no	—
93	C₄⋊C₄×C₂₃	—	2	12	no	—
94	C₄⋊C₄×C₂₃	C₂×D₄	1	12	yes	—
95	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
97	C₂²⋊C₄×C₂₃	—	2	4	no	—
98	C₂²⋊C₄×C₂₃	—	2	12	no	—
99	C₄²×C₂₃	—	2	12	no	—
100	C₄²×C₂₃	C₂×D₄	1	6	yes	—
101	C₄²×C₂₃	C₂⁴	1	6	yes	—
102	#61818	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	—	2	4	no	—
104	C₂³×C₂₃	—	2	2	no	—
105	C₂³×C₂₃	—	4	2	no	—
106	C₂³×C₂₃	—	4	2	no	—
107	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₂₃	—	2	8	no	—
109	Q₈×C₂₃	—	4	24	no	—
110	Q₈×C₂₃	—	8	6	no	—
111	Q₈×C₂₃	C₂×SD₁₆	1	16	yes	—
112	D₄×C₂₃	—	2	4	no	—
113	D₄×C₂₃	—	2	2	no	—
114	D₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	4	no	—
118	C₂×C₄×C₂₃	—	2	6	no	—
119	C₂×C₄×C₂₃	—	4	3	no	—
120	C₂×C₄×C₂₃	—	4	6	no	—
121	C₂×C₄×C₂₃	—	4	24	no	—
122	C₂×C₄×C₂₃	C₂×SD₁₆	1	12	yes	—
123	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
126	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
127	C₈×C₂₃	—	2	8	no	—
128	#3502	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
130	Q₈×SD₁₆	C₂×C₂₃	1	8	yes	complemented
131	#3359	C₂×C₂₃	1	8	yes	complemented
132	#3229	C₂×C₂₃	1	6	yes	complemented
133	#3101	C₂×C₂₃	1	6	yes	—
134	#2982	C₂×C₂₃	1	1	yes	characteristic, complemented
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	4	no	—
138	C₂²×C₂₃	—	4	8	no	—
139	C₂²×C₂₃	—	4	2	no	—
140	C₂²×C₂₃	—	4	2	no	—
141	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
142	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₂₃	—	2	6	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	4	4	no	—
146	C₄×C₂₃	—	4	4	no	—
147	C₄×C₂₃	—	4	6	no	—
148	C₄×C₂₃	C₂²×SD₁₆	1	12	yes	—
149	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
150	C₂³×Q₈	—	2	1	no	—
151	Q₈²	C₂²×C₂₃	1	4	yes	complemented
152	#557	—	2	4	no	—
153	#557	C₂²×C₂₃	1	4	yes	complemented
154	D₄×Q₈	—	2	4	no	—
155	D₄×Q₈	C₂²×C₂₃	1	8	yes	—
156	#544	—	2	3	no	—
157	#533	C₂²×C₂₃	1	6	yes	—
158	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
159	C₂×C₂²⋊Q₈	—	2	3	no	—
160	#519	—	2	1	no	—
161	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	—
163	#498	C₂²×C₂₃	1	12	yes	complemented
164	#487	C₂²×C₂₃	1	12	yes	complemented
165	#483	C₂²×C₂₃	1	12	yes	—
166	#476	C₂²×C₂₃	1	12	yes	—
167	#475	C₂²×C₂₃	1	24	yes	complemented
168	#461	C₂²×C₂₃	1	12	yes	—
169	#450	C₂²×C₂₃	1	4	yes	—
170	C₈×Q₈	C₂²×C₂₃	1	8	yes	complemented
171	C₄×SD₁₆	C₂²×C₂₃	1	24	yes	—
172	#431	C₂²×C₂₃	1	6	yes	—
173	C₂×C₂₃	—	2	2	no	—
174	C₂×C₂₃	—	4	4	no	—
175	C₂×C₂₃	#3486	1	1	yes	characteristic
176	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
177	C₂×C₂₃	#3586	1	1	yes	characteristic
178	C₂²×Q₈	—	2	8	no	—
179	C₂²×Q₈	—	2	3	no	—
180	C₂²×Q₈	—	4	8	no	—
181	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
182	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
183	C₂³×C₄	—	2	3	no	—
184	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
185	C₄⋊Q₈	—	2	12	no	—
186	C₄⋊Q₈	—	2	6	no	—
187	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
188	C₄⋊Q₈	C₂³×C₂₃	1	12	yes	—
189	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
190	#125	—	2	12	no	—
191	#125	—	2	6	no	—
192	C₂²⋊Q₈	—	2	6	no	—
193	C₂²⋊Q₈	—	2	24	no	—
194	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
195	C₄×Q₈	—	2	4	no	—
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	6	no	—
198	C₄×Q₈	D₄×C₂₃	1	4	yes	—
199	C₄×Q₈	C₂³×C₂₃	1	12	yes	—
200	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
201	C₄×D₄	—	2	6	no	—
202	C₄×D₄	C₂³×C₂₃	1	12	yes	—
203	#117	—	2	6	no	—
204	C₂×C₄⋊C₄	—	2	3	no	—
205	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
206	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
207	C₂×C₂²⋊C₄	—	2	3	no	—
208	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
209	#106	C₂³×C₂₃	1	24	yes	—
210	C₄⋊C₈	C₂³×C₂₃	1	12	yes	—
211	#103	C₂³×C₂₃	1	24	yes	—
212	#102	C₂³×C₂₃	1	8	yes	—
213	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
214	C₄×C₈	C₂³×C₂₃	1	12	yes	—
215	C₂₃	#61818	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	—	2	1	no	—
217	C₂×Q₈	—	2	12	no	—
218	C₂×Q₈	—	2	6	no	—
219	C₂×Q₈	—	4	32	no	—
220	C₂×Q₈	—	4	12	no	—
221	C₂×Q₈	SD₁₆×C₂₃	1	8	yes	complemented
222	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
223	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	—	2	12	no	—
229	C₂²×C₄	—	4	12	no	—
230	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
231	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
232	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
233	SD₁₆	—	2	16	no	—
234	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	3	no	—
237	C₄⋊C₄	—	2	6	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	—	2	12	no	—
242	C₄⋊C₄	C₂×D₄×C₂₃	1	12	yes	—
243	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
244	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	12	no	—
247	C₄²	—	2	12	no	—
248	C₄²	C₂×D₄×C₂₃	1	6	yes	—
249	C₄²	C₂⁴×C₂₃	1	6	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
255	Q₈	—	2	8	no	—
256	Q₈	—	4	24	no	—
257	Q₈	—	8	6	no	—
258	Q₈	C₂×SD₁₆×C₂₃	1	16	yes	—
259	D₄	—	2	4	no	—
260	D₄	—	2	2	no	—
261	D₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	4	6	no	—
268	C₂×C₄	—	4	24	no	—
269	C₂×C₄	C₂×SD₁₆×C₂₃	1	12	yes	—
270	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
272	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
273	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	8	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
282	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	6	no	—
285	C₄	—	4	4	no	—
286	C₄	—	4	6	no	—
287	C₄	—	4	4	no	—
288	C₄	C₂²×SD₁₆×C₂₃	1	12	yes	—
289	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
290	C₂	—	2	2	no	—
291	C₂	—	4	4	no	—
292	C₂	C₂₃×#3486	1	1	yes	characteristic
293	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
294	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3502	C₂	semidirect
3. Q₈×SD₁₆×C₂₃	C₂	semidirect
4. C₂₃×#3359	C₂	semidirect
5. C₂₃×#3229	C₂	semidirect
7. C₂₃×#2982	C₂	semidirect
9. Q₈²×C₂₃	C₂²	semidirect
11. C₂₃×#557	C₂²	semidirect
21. C₂₃×#498	C₂²	semidirect
22. C₂₃×#487	C₂²	semidirect
25. C₂₃×#475	C₂²	semidirect
28. C₈×Q₈×C₂₃	C₂²	semidirect
73. C₂×Q₈×C₂₃	SD₁₆	semidirect
102. #61818	C₂₃	direct (N × H)
128. #3502	C₂×C₂₃	semidirect
130. Q₈×SD₁₆	C₂×C₂₃	semidirect
131. #3359	C₂×C₂₃	semidirect
132. #3229	C₂×C₂₃	semidirect
134. #2982	C₂×C₂₃	semidirect
151. Q₈²	C₂²×C₂₃	semidirect
153. #557	C₂²×C₂₃	semidirect
163. #498	C₂²×C₂₃	semidirect
164. #487	C₂²×C₂₃	semidirect
167. #475	C₂²×C₂₃	semidirect
170. C₈×Q₈	C₂²×C₂₃	semidirect
215. C₂₃	#61818	direct (N × H)
221. C₂×Q₈	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3502	2
C₂×D₄×Q₈×C₂₃	2
Q₈×SD₁₆×C₂₃	2
C₂₃×#3359	2
C₂₃×#3229	2
C₂₃×#3101	2
C₂₃×#2982	2
#61818	23

Subgroups — C23×#61818 #19209444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61818 #19209444