Subgroups — C₂₃×#20470 #19168096

Order 5888.

958 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1767	C₂	1	2	yes	maximal, complemented
5	C₄×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1750	C₂	1	4	yes	maximal
7	C₂₃×#1746	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1726	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#432	C₄	1	8	yes	complemented
12	C₂₃×#431	C₄	1	8	yes	complemented
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	4	yes	—
19	C₂×C₂₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
26	C₂₃×#376	C₂²	1	1	yes	characteristic
27	C₂₃×#376	C₂²	1	1	yes	characteristic
28	C₄³×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#336	C₂²	1	4	yes	—
30	C₂₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂₃×#117	—	2	4	no	—
38	C₂₃×#117	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
51	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
55	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₂₃×#97	—	2	8	no	—
57	C₂₃×#97	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₂₃	—	2	8	no	—
59	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
60	C₂₃×#95	C₂×C₄	1	2	yes	—
61	C₂₃×#95	C₂×C₄	1	2	yes	—
62	C₂₃×#95	C₂×C₄	1	2	yes	—
63	C₂₃×#95	C₂×C₄	1	2	yes	—
64	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	—	2	4	no	—
66	C₂²×C₄×C₂₃	—	2	2	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	4	2	no	—
69	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
70	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	4	no	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
88	C₄⋊C₄×C₂₃	—	2	8	no	—
89	C₂²⋊C₄×C₂₃	—	2	8	no	—
90	C₄²×C₂₃	—	2	8	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	4	4	no	—
94	C₄²×C₂₃	C₄²	1	4	yes	—
95	C₄²×C₂₃	C₄²	1	4	yes	—
96	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₂₃	#41	1	4	yes	—
101	C₄²×C₂₃	#41	1	4	yes	—
102	#20470	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	—	2	2	no	—
104	C₂³×C₂₃	—	2	2	no	—
105	C₂³×C₂₃	—	2	2	no	—
106	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
107	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	8	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	2	4	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₂₃	#117	1	8	yes	—
128	C₂×C₄×C₂₃	#131	1	4	yes	—
129	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
133	C₈×C₂₃	—	2	4	no	—
134	C₈×C₂₃	—	2	8	no	—
135	C₈×C₂₃	—	2	4	no	—
136	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
137	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
138	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
139	#1767	C₂×C₂₃	1	2	yes	complemented
140	C₄×C₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
141	#1750	C₂×C₂₃	1	4	yes	—
142	#1746	C₂×C₂₃	1	2	yes	complemented
143	#1726	C₂×C₂₃	1	2	yes	complemented
144	C₂²×C₂₃	—	2	2	no	—
145	C₂²×C₂₃	—	2	2	no	—
146	C₂²×C₂₃	—	2	2	no	—
147	C₂²×C₂₃	—	4	4	no	—
148	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₂₃	C₂×#117	1	2	yes	—
150	C₂²×C₂₃	C₂×#131	1	2	yes	—
151	C₂²×C₂₃	#568	1	2	yes	—
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	2	2	no	—
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	2	8	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	4	4	no	—
158	C₄×C₂₃	#405	1	4	yes	—
159	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
160	C₄×C₂₃	C₄×#41	1	2	yes	—
161	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
162	C₂×#117	C₂²×C₂₃	1	2	yes	—
163	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
164	#432	C₄×C₂₃	1	8	yes	complemented
165	#431	C₄×C₂₃	1	8	yes	complemented
166	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
170	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
171	C₂×#97	C₂²×C₂₃	1	4	yes	—
172	C₂×#97	C₂²×C₂₃	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
174	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
175	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
176	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
177	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
178	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
179	#376	C₂²×C₂₃	1	1	yes	characteristic
180	#376	C₂²×C₂₃	1	1	yes	characteristic
181	C₄³	C₂²×C₂₃	1	2	yes	—
182	#336	C₂²×C₂₃	1	4	yes	—
183	#336	C₂²×C₂₃	1	4	yes	—
184	C₂×C₂₃	—	4	2	no	—
185	C₂×C₂₃	C₄×#117	1	1	yes	characteristic
186	C₂×C₂₃	C₂×#405	1	1	yes	characteristic
187	C₂×C₂₃	#2873	1	1	yes	characteristic
188	C₂×C₂₃	#2925	1	4	yes	—
189	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
194	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
195	#117	—	2	4	no	—
196	#117	—	2	4	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
198	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
199	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
200	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
204	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
205	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
206	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
209	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
210	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
211	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
212	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
213	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
214	#97	—	2	8	no	—
215	#97	C₂×C₄×C₂₃	1	8	yes	—
216	C₄×C₈	—	2	8	no	—
217	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
218	#95	C₂×C₄×C₂₃	1	2	yes	—
219	#95	C₂×C₄×C₂₃	1	2	yes	—
220	#95	C₂×C₄×C₂₃	1	2	yes	—
221	#95	C₂×C₄×C₂₃	1	2	yes	—
222	C₂₃	#20470	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
229	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
245	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
246	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₄²×C₂₃	1	4	yes	—
254	C₄²	C₄²×C₂₃	1	4	yes	—
255	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
256	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
257	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
258	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
259	C₄²	#41×C₂₃	1	4	yes	—
260	C₄²	#41×C₂₃	1	4	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
265	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
281	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
282	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
283	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
284	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
285	C₂×C₄	C₂₃×#117	1	8	yes	—
286	C₂×C₄	C₂₃×#131	1	4	yes	—
287	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
290	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	8	no	—
293	C₈	—	2	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	4	no	—
298	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
299	C₂²	C₂×C₂₃×#117	1	2	yes	—
300	C₂²	C₂×C₂₃×#131	1	2	yes	—
301	C₂²	C₂₃×#568	1	2	yes	—
302	C₄	—	2	8	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	4	no	—
308	C₄	C₂₃×#405	1	4	yes	—
309	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
310	C₄	C₄×#41×C₂₃	1	2	yes	—
311	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
312	C₂	—	4	2	no	—
313	C₂	C₄×C₂₃×#117	1	1	yes	characteristic
314	C₂	C₂×C₂₃×#405	1	1	yes	characteristic
315	C₂	C₂₃×#2873	1	1	yes	characteristic
316	C₂	C₂₃×#2925	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1767	C₂	semidirect
5. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
7. C₂₃×#1746	C₂	semidirect
8. C₂₃×#1726	C₂	semidirect
11. C₂₃×#432	C₄	semidirect
12. C₂₃×#431	C₄	semidirect
102. #20470	C₂₃	direct (N × H)
139. #1767	C₂×C₂₃	semidirect
140. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
142. #1746	C₂×C₂₃	semidirect
143. #1726	C₂×C₂₃	semidirect
164. #432	C₄×C₂₃	semidirect
165. #431	C₄×C₂₃	semidirect
222. C₂₃	#20470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#432	2
C₂×C₂₃×#431	2
C₄×C₂₃×#117	2
C₂₃×#1767	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1750	2
C₂₃×#1746	2
C₂₃×#1726	2
#20470	23

Subgroups — C23×#20470 #19168096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20470 #19168096