Subgroups — C₂₃×#7890 #19155516

Order 5888.

830 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	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₂₃	—	2	1	no	—
6	C₂×C₂₃×#117	C₄	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#431	—	2	2	no	—
9	C₂₃×#431	C₄	1	4	yes	complemented
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	—	2	2	no	—
14	C₂×C₄×C₈×C₂₃	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
21	C₂₃×#376	—	2	1	no	—
22	C₄³×C₂₃	—	2	1	no	—
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	C₂²×C₈×C₂₃	—	2	8	no	—
27	C₂²×C₈×C₂₃	—	2	4	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂₃×#117	—	2	4	no	—
31	C₂₃×#117	C₈	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄²×C₂₃	—	2	2	no	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₂₃×C₂²⋊C₈	—	2	4	no	—
48	C₄×C₈×C₂₃	—	2	4	no	—
49	C₄×C₈×C₂₃	—	2	4	no	—
50	C₄×C₈×C₂₃	—	2	2	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
52	C₂₃×#95	—	2	2	no	—
53	C₂₃×#95	—	2	2	no	—
54	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₂₃	M₁₆	1	4	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	—	2	8	no	—
73	C₂×C₈×C₂₃	—	2	16	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	C₂×C₈	1	8	yes	complemented
79	C₂²⋊C₄×C₂₃	—	2	8	no	—
80	C₄²×C₂₃	—	2	8	no	—
81	C₄²×C₂₃	—	4	4	no	—
82	C₄²×C₂₃	—	4	4	no	—
83	C₄²×C₂₃	C₄²	1	2	yes	—
84	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
93	#7890	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	#95	1	2	yes	—
97	C₂³×C₂₃	C₂²⋊C₈	1	4	yes	—
98	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	#95	1	2	yes	—
109	C₂×C₄×C₂₃	#95	1	2	yes	—
110	C₂×C₄×C₂₃	#95	1	2	yes	—
111	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
112	C₂×C₄×C₂₃	#97	1	4	yes	—
113	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₂₃	#104	1	4	yes	—
115	C₂×C₄×C₂₃	C₄⋊C₈	1	8	yes	—
116	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₂₃	#135	1	4	yes	—
125	C₈×C₂₃	—	2	8	no	—
126	C₈×C₂₃	—	4	4	no	—
127	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
128	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
129	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
130	#1279	C₂×C₂₃	1	4	yes	complemented
131	C₂²×C₂₃	—	2	8	no	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	#336	1	4	yes	—
135	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
136	C₂²×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₂₃	#416	1	2	yes	—
138	C₂²×C₂₃	C₂×#104	1	2	yes	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	4	8	no	—
143	C₄×C₂₃	#336	1	4	yes	—
144	C₄×C₂₃	#339	1	4	yes	—
145	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
147	C₄×C₂₃	#416	1	2	yes	—
148	C₄×C₂₃	C₂×C₄⋊C₈	1	2	yes	—
149	C₄×C₂₃	#427	1	2	yes	—
150	C₂³×C₈	—	2	1	no	—
151	C₂×#117	C₄×C₂₃	1	2	yes	—
152	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
153	#431	—	2	2	no	—
154	#431	C₄×C₂₃	1	4	yes	complemented
155	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₄×C₈	—	2	1	no	—
158	C₂×C₄×C₈	—	2	2	no	—
159	C₂×C₄×C₈	—	2	1	no	—
160	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
161	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
162	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
164	C₄×C₂²⋊C₄	—	2	1	no	—
165	C₄×C₂²⋊C₄	—	2	1	no	—
166	#376	—	2	1	no	—
167	C₄³	—	2	1	no	—
168	C₂×C₂₃	—	2	4	no	—
169	C₂×C₂₃	#1476	1	4	yes	—
170	C₂×C₂₃	C₂×#336	1	1	yes	characteristic
171	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
172	C₂×C₂₃	#1735	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	#117	—	2	4	no	—
181	#117	C₈×C₂₃	1	8	yes	complemented
182	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
189	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
190	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₂₃	1	2	yes	—
193	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
194	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
197	C₂²⋊C₈	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₂₃	#7890	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
216	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
217	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
219	C₂²×C₄	M₁₆×C₂₃	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	16	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂×C₈×C₂₃	1	8	yes	complemented
230	C₂²⋊C₄	—	2	8	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₄²×C₂₃	1	2	yes	—
235	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
236	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
237	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
238	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
239	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
240	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₂₃×#95	1	2	yes	—
247	C₂³	C₂₃×C₂²⋊C₈	1	4	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂₃×#95	1	2	yes	—
259	C₂×C₄	C₂₃×#95	1	2	yes	—
260	C₂×C₄	C₂₃×#95	1	2	yes	—
261	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
262	C₂×C₄	C₂₃×#97	1	4	yes	—
263	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
264	C₂×C₄	C₂₃×#104	1	4	yes	—
265	C₂×C₄	C₂₃×C₄⋊C₈	1	8	yes	—
266	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
273	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
274	C₂×C₄	C₂₃×#135	1	4	yes	—
275	C₈	—	2	8	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	C₂₃×#336	1	4	yes	—
281	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
282	C₂²	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
283	C₂²	C₂₃×#416	1	2	yes	—
284	C₂²	C₂×C₂₃×#104	1	2	yes	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	8	no	—
289	C₄	C₂₃×#336	1	4	yes	—
290	C₄	C₂₃×#339	1	4	yes	—
291	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
292	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
293	C₄	C₂₃×#416	1	2	yes	—
294	C₄	C₂×C₂₃×C₄⋊C₈	1	2	yes	—
295	C₄	C₂₃×#427	1	2	yes	—
296	C₂	—	2	4	no	—
297	C₂	C₂₃×#1476	1	4	yes	—
298	C₂	C₂×C₂₃×#336	1	1	yes	characteristic
299	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
300	C₂	C₂₃×#1735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₂₃	D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×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
9. C₂₃×#431	C₄	semidirect
15. C₂×C₄×C₈×C₂₃	C₄	semidirect
31. C₂₃×#117	C₈	semidirect
78. C₄⋊C₄×C₂₃	C₂×C₈	semidirect
93. #7890	C₂₃	direct (N × H)
130. #1279	C₂×C₂₃	semidirect
154. #431	C₄×C₂₃	semidirect
160. C₂×C₄×C₈	C₄×C₂₃	semidirect
181. #117	C₈×C₂₃	semidirect
204. C₂₃	#7890	direct (N × H)
229. C₄⋊C₄	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#431	2
C₂²×C₄×C₈×C₂₃	2
C₄×C₂₃×#117	2
C₂₃×#1279	2
#7890	23

Subgroups — C23×#7890 #19155516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7890 #19155516