Subgroups — C₂₃×#7897 #19155523

Order 5888.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂³×C₈×C₂₃	—	2	1	no	—
7	C₂×C₂₃×#117	—	2	1	no	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#423	—	2	2	no	—
10	C₂₃×#423	C₄	1	4	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	—	2	2	no	—
15	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#384	C₄	1	2	yes	—
18	C₂₃×#382	—	2	1	no	—
19	C₂₃×#380	—	2	1	no	—
20	C₂₃×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	—	4	2	no	—
28	C₂×M₁₆×C₂₃	—	4	2	no	—
29	C₂²×C₈×C₂₃	—	2	8	no	—
30	C₂²×C₈×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	—	2	2	no	—
33	C₂₃×#117	—	4	2	no	—
34	C₂₃×#117	—	4	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄²×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	—	2	4	no	—
49	C₂₃×C₄⋊C₈	—	2	2	no	—
50	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₂₃	—	2	4	no	—
52	C₄×C₈×C₂₃	—	2	2	no	—
53	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
54	C₂₃×#95	—	2	2	no	—
55	C₂₃×#95	—	2	2	no	—
56	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	2	4	no	—
60	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	D₈	1	2	yes	—
70	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	M₁₆×C₂₃	—	4	8	no	—
77	C₂×C₈×C₂₃	—	2	8	no	—
78	C₂×C₈×C₂₃	—	2	16	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	2	8	no	—
84	C₄⋊C₄×C₂₃	—	4	4	no	—
85	C₄⋊C₄×C₂₃	—	4	4	no	—
86	C₂²⋊C₄×C₂₃	—	4	4	no	—
87	C₄²×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	4	2	no	—
89	C₄²×C₂₃	C₄²	1	2	yes	—
90	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
95	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
99	#7897	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	2	2	no	—
101	C₂³×C₂₃	—	2	2	no	—
102	C₂³×C₂₃	#95	1	2	yes	—
103	C₂³×C₂₃	#102	1	2	yes	—
104	C₂³×C₂₃	#103	1	2	yes	—
105	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	#95	1	2	yes	—
116	C₂×C₄×C₂₃	#95	1	2	yes	—
117	C₂×C₄×C₂₃	#95	1	2	yes	—
118	C₂×C₄×C₂₃	#102	1	2	yes	—
119	C₂×C₄×C₂₃	#102	1	4	yes	—
120	C₂×C₄×C₂₃	#103	1	2	yes	—
121	C₂×C₄×C₂₃	#103	1	4	yes	—
122	C₂×C₄×C₂₃	#104	1	4	yes	—
123	C₂×C₄×C₂₃	#106	1	4	yes	—
124	C₂×C₄×C₂₃	#107	1	4	yes	—
125	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#131	1	4	yes	—
132	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
136	C₈×C₂₃	—	2	8	no	—
137	C₈×C₂₃	—	4	4	no	—
138	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
139	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
140	#2285	C₂×C₂₃	1	1	yes	characteristic
141	#1279	C₂×C₂₃	1	4	yes	complemented
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	2	2	no	—
144	C₂²×C₂₃	—	2	4	no	—
145	C₂²×C₂₃	—	2	4	no	—
146	C₂²×C₂₃	#340	1	4	yes	—
147	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
148	C₂²×C₂₃	#408	1	2	yes	—
149	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
150	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
151	C₂²×C₂₃	C₂×#104	1	2	yes	—
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	—	8	4	no	—
156	C₄×C₂₃	#339	1	4	yes	—
157	C₄×C₂₃	#340	1	4	yes	—
158	C₄×C₂₃	#405	1	2	yes	—
159	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
160	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
161	C₄×C₂₃	#424	1	2	yes	—
162	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
163	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
164	C₂²×M₁₆	—	2	1	no	—
165	C₂³×C₈	—	2	1	no	—
166	C₂×#117	—	2	1	no	—
167	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
168	#423	—	2	2	no	—
169	#423	C₄×C₂₃	1	4	yes	complemented
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
172	C₂×C₄×C₈	—	2	1	no	—
173	C₂×C₄×C₈	—	2	2	no	—
174	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
175	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
176	#384	C₄×C₂₃	1	2	yes	—
177	#382	—	2	1	no	—
178	#380	—	2	1	no	—
179	#380	C₄×C₂₃	1	2	yes	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
182	C₂×C₂₃	—	2	4	no	—
183	C₂×C₂₃	#1492	1	2	yes	—
184	C₂×C₂₃	#1493	1	2	yes	—
185	C₂×C₂₃	#1729	1	1	yes	characteristic
186	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
187	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
188	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	4	2	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂²×C₈	—	2	8	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	#117	—	4	2	no	—
199	#117	—	4	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
207	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
208	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₂₃	1	2	yes	—
211	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	C₂₃	#7897	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
227	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
228	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
230	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
233	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
234	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
235	C₂²×C₄	D₈×C₂₃	1	2	yes	—
236	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
237	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
238	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	M₁₆	—	4	8	no	—
243	C₂×C₈	—	2	8	no	—
244	C₂×C₈	—	2	16	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	C₄²×C₂₃	1	2	yes	—
256	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
257	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
258	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
259	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
260	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
261	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
262	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₂₃×#95	1	2	yes	—
268	C₂³	C₂₃×#102	1	2	yes	—
269	C₂³	C₂₃×#103	1	2	yes	—
270	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂₃×#95	1	2	yes	—
281	C₂×C₄	C₂₃×#95	1	2	yes	—
282	C₂×C₄	C₂₃×#95	1	2	yes	—
283	C₂×C₄	C₂₃×#102	1	2	yes	—
284	C₂×C₄	C₂₃×#102	1	4	yes	—
285	C₂×C₄	C₂₃×#103	1	2	yes	—
286	C₂×C₄	C₂₃×#103	1	4	yes	—
287	C₂×C₄	C₂₃×#104	1	4	yes	—
288	C₂×C₄	C₂₃×#106	1	4	yes	—
289	C₂×C₄	C₂₃×#107	1	4	yes	—
290	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂₃×#131	1	4	yes	—
297	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
298	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
299	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
301	C₈	—	2	8	no	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	C₂₃×#340	1	4	yes	—
308	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
309	C₂²	C₂₃×#408	1	2	yes	—
310	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
311	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
312	C₂²	C₂×C₂₃×#104	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	4	no	—
317	C₄	C₂₃×#339	1	4	yes	—
318	C₄	C₂₃×#340	1	4	yes	—
319	C₄	C₂₃×#405	1	2	yes	—
320	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
321	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
322	C₄	C₂₃×#424	1	2	yes	—
323	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
324	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₂₃×#1492	1	2	yes	—
327	C₂	C₂₃×#1493	1	2	yes	—
328	C₂	C₂₃×#1729	1	1	yes	characteristic
329	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
330	C₂	C₂×C₂₃×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1279	C₂	semidirect
10. C₂₃×#423	C₄	semidirect
15. C₂×C₄×C₈×C₂₃	C₄	semidirect
99. #7897	C₂₃	direct (N × H)
141. #1279	C₂×C₂₃	semidirect
169. #423	C₄×C₂₃	semidirect
174. C₂×C₄×C₈	C₄×C₂₃	semidirect
221. C₂₃	#7897	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×#423	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#2285	2
C₂₃×#1279	2
#7897	23

Subgroups — C23×#7897 #19155523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7897 #19155523