Subgroups — C₂₃×#20575 #19168201

Order 5888.

798 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	C₂×C₂₃×#446	C₂	1	2	yes	maximal
2	C₂₃×#2295	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1771	C₂	1	2	yes	maximal
4	C₂₃×#1771	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
6	C₂₃×#1768	C₂	1	2	yes	maximal
7	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1746	C₂	1	2	yes	maximal
9	C₂₃×#1745	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#446	C₄	1	16	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#391	C₄	1	2	yes	—
23	C₂₃×#389	C₄	1	2	yes	—
24	C₂₃×#387	C₄	1	4	yes	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#376	C₂²	1	2	yes	—
28	C₄³×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#336	C₂²	1	1	yes	characteristic
30	C₂₃×#336	C₂²	1	1	yes	characteristic
31	C₂₃×#336	C₂²	1	2	yes	—
32	C₂₃×#336	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
52	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₂₃×#97	—	4	2	no	—
56	C₂₃×#97	C₂×C₄	1	8	yes	—
57	C₂₃×#97	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₂₃	—	4	1	no	—
59	C₄×C₈×C₂₃	—	4	1	no	—
60	C₄×C₈×C₂₃	—	4	2	no	—
61	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
62	C₂₃×#95	C₂×C₄	1	4	yes	—
63	C₂₃×#95	C₂×C₄	1	4	yes	—
64	C₂₃×#95	C₂×C₄	1	2	yes	—
65	C₂₃×#95	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₂₃	—	4	2	no	—
67	C₂×Q₈×C₂₃	C₄²	1	4	yes	—
68	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
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	4	yes	—
88	C₄⋊C₄×C₂₃	—	2	8	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	C₄²	1	8	yes	—
91	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
92	C₄²×C₂₃	—	2	4	no	—
93	C₄²×C₂₃	—	2	1	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	1	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
100	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
102	#20575	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	Q₈×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	4	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	4	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₂₃	#131	1	4	yes	—
122	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₂₃	—	2	4	no	—
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	—	2	4	no	—
129	C₈×C₂₃	—	2	4	no	—
130	C₂×#446	C₂×C₂₃	1	2	yes	—
131	#2295	C₂×C₂₃	1	1	yes	characteristic
132	#1771	C₂×C₂₃	1	2	yes	—
133	#1771	C₂×C₂₃	1	2	yes	—
134	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
135	#1768	C₂×C₂₃	1	2	yes	—
136	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
137	#1746	C₂×C₂₃	1	2	yes	—
138	#1745	C₂×C₂₃	1	1	yes	characteristic
139	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
140	C₂²×C₂₃	#519	1	1	yes	characteristic
141	C₂²×C₂₃	#520	1	1	yes	characteristic
142	C₂²×C₂₃	C₂×#131	1	2	yes	—
143	C₂²×C₂₃	#568	1	2	yes	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	2	4	no	—
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	2	4	no	—
148	C₄×C₂₃	—	2	4	no	—
149	C₄×C₂₃	—	4	2	no	—
150	C₄×C₂₃	—	4	4	no	—
151	C₄×C₂₃	#519	1	2	yes	—
152	C₄×C₂₃	#520	1	2	yes	—
153	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
154	#446	C₄×C₂₃	1	16	yes	complemented
155	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
158	C₂×#97	C₂²×C₂₃	1	2	yes	—
159	C₂×#97	C₂²×C₂₃	1	2	yes	—
160	C₂×#97	C₂²×C₂₃	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
165	#391	C₄×C₂₃	1	2	yes	—
166	#389	C₄×C₂₃	1	2	yes	—
167	#387	C₄×C₂₃	1	4	yes	—
168	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
170	#376	C₂²×C₂₃	1	2	yes	—
171	C₄³	C₂²×C₂₃	1	1	yes	characteristic
172	#336	C₂²×C₂₃	1	1	yes	characteristic
173	#336	C₂²×C₂₃	1	1	yes	characteristic
174	#336	C₂²×C₂₃	1	2	yes	—
175	#336	C₂²×C₂₃	1	2	yes	—
176	#336	C₂²×C₂₃	1	2	yes	—
177	#336	C₂²×C₂₃	1	2	yes	—
178	C₂×C₂₃	#2276	1	1	yes	characteristic
179	C₂×C₂₃	#2873	1	1	yes	characteristic
180	C₂×C₂₃	C₄×#131	1	1	yes	characteristic
181	C₂×C₂₃	#2985	1	2	yes	—
182	C₂×C₂₃	#2992	1	2	yes	—
183	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
185	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
188	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
200	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
201	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
202	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
203	#97	—	4	2	no	—
204	#97	C₂×C₄×C₂₃	1	8	yes	—
205	#97	C₂×C₄×C₂₃	1	8	yes	—
206	C₄×C₈	—	4	1	no	—
207	C₄×C₈	—	4	1	no	—
208	C₄×C₈	—	4	2	no	—
209	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
210	#95	C₂×C₄×C₂₃	1	4	yes	—
211	#95	C₂×C₄×C₂₃	1	4	yes	—
212	#95	C₂×C₄×C₂₃	1	2	yes	—
213	#95	C₂×C₄×C₂₃	1	4	yes	—
214	C₂₃	#20575	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	C₄²×C₂₃	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
234	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₄²×C₂₃	1	8	yes	—
240	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
249	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
250	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
251	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
266	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
267	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
268	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
269	C₂×C₄	C₂₃×#131	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
271	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
272	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
273	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
279	C₂²	C₂₃×#519	1	1	yes	characteristic
280	C₂²	C₂₃×#520	1	1	yes	characteristic
281	C₂²	C₂×C₂₃×#131	1	2	yes	—
282	C₂²	C₂₃×#568	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂₃×#519	1	2	yes	—
291	C₄	C₂₃×#520	1	2	yes	—
292	C₂	C₂₃×#2276	1	1	yes	characteristic
293	C₂	C₂₃×#2873	1	1	yes	characteristic
294	C₂	C₄×C₂₃×#131	1	1	yes	characteristic
295	C₂	C₂₃×#2985	1	2	yes	—
296	C₂	C₂₃×#2992	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂³×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
11. C₂₃×#446	C₄	semidirect
102. #20575	C₂₃	direct (N × H)
154. #446	C₄×C₂₃	semidirect
214. C₂₃	#20575	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×#446	2
C₂₃×#2295	2
C₂₃×#1771	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1768	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1746	2
C₂₃×#1745	2
#20575	23

Subgroups — C23×#20575 #19168201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20575 #19168201