Subgroups — C₂₃×#7891 #19155517

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 312 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×C₂₃×C₄⋊C₈	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	—	2	1	no	—
11	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
13	C₂×C₂₃×C₄⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
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	4	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	4	4	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	1	yes	characteristic
42	C₂×C₄²×C₂₃	D₄	1	2	yes	—
43	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×C₄⋊C₈	—	2	4	no	—
46	C₂₃×C₄⋊C₈	—	2	4	no	—
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₂₃×C₂²⋊C₈	—	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	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₄⋊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	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₂₃	M₁₆	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	4	4	no	—
76	C₂×C₈×C₂₃	—	4	8	no	—
77	C₂×C₈×C₂₃	—	4	4	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	C₂×C₈	1	8	yes	complemented
80	C₂²⋊C₄×C₂₃	—	2	8	no	—
81	C₄²×C₂₃	—	2	8	no	—
82	C₄²×C₂₃	—	4	4	no	—
83	C₄²×C₂₃	—	4	2	no	—
84	C₄²×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	C₄²	1	2	yes	—
86	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
95	#7891	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	—	2	2	no	—
98	C₂³×C₂₃	#95	1	2	yes	—
99	C₂³×C₂₃	C₂²⋊C₈	1	4	yes	—
100	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	8	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	4	4	no	—
112	C₂×C₄×C₂₃	#95	1	2	yes	—
113	C₂×C₄×C₂₃	#95	1	2	yes	—
114	C₂×C₄×C₂₃	#95	1	2	yes	—
115	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
116	C₂×C₄×C₂₃	#97	1	4	yes	—
117	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
118	C₂×C₄×C₂₃	C₄⋊C₈	1	8	yes	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₂₃	#137	1	2	yes	—
129	C₈×C₂₃	—	4	4	no	—
130	C₈×C₂₃	—	4	4	no	—
131	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
132	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
133	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
134	#1279	C₂×C₂₃	1	4	yes	complemented
135	C₂²×C₂₃	—	2	8	no	—
136	C₂²×C₂₃	—	2	2	no	—
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	#336	1	4	yes	—
139	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
141	C₂²×C₂₃	#417	1	2	yes	—
142	C₂²×C₂₃	#421	1	2	yes	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	4	8	no	—
148	C₄×C₂₃	#336	1	4	yes	—
149	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
150	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
151	C₄×C₂₃	#418	1	1	yes	characteristic
152	C₄×C₂₃	#419	1	1	yes	characteristic
153	C₄×C₂₃	C₂×C₄⋊C₈	1	2	yes	—
154	C₄×C₂₃	#428	1	2	yes	—
155	C₂³×C₈	—	2	1	no	—
156	C₂×#117	C₄×C₂₃	1	2	yes	—
157	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
158	#431	—	2	2	no	—
159	#431	C₄×C₂₃	1	4	yes	complemented
160	C₂×C₄⋊C₈	—	2	1	no	—
161	C₂×C₄⋊C₈	—	2	2	no	—
162	C₂×C₄⋊C₈	—	2	1	no	—
163	C₂×C₄⋊C₈	C₄×C₂₃	1	4	yes	complemented
164	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂²⋊C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
167	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
169	C₄×C₂²⋊C₄	—	2	1	no	—
170	C₄×C₂²⋊C₄	—	2	1	no	—
171	#376	—	2	1	no	—
172	C₄³	—	2	1	no	—
173	C₂×C₂₃	—	2	4	no	—
174	C₂×C₂₃	#1477	1	4	yes	—
175	C₂×C₂₃	C₂×#336	1	1	yes	characteristic
176	C₂×C₂₃	#1733	1	1	yes	characteristic
177	C₂×C₂₃	#1736	1	1	yes	characteristic
178	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
180	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	4	4	no	—
185	#117	—	2	4	no	—
186	#117	C₈×C₂₃	1	8	yes	complemented
187	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	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	D₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₂₃	1	2	yes	—
198	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
204	C₂²⋊C₈	—	2	4	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂₃	#7891	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
225	C₂²×C₄	M₁₆×C₂₃	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	8	no	—
232	C₂×C₈	—	4	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂×C₈×C₂₃	1	8	yes	complemented
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	4	4	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₄²×C₂₃	1	2	yes	—
242	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
243	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
244	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
245	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
246	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
247	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
248	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂₃×#95	1	2	yes	—
254	C₂³	C₂₃×C₂²⋊C₈	1	4	yes	—
255	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₂₃×#95	1	2	yes	—
268	C₂×C₄	C₂₃×#95	1	2	yes	—
269	C₂×C₄	C₂₃×#95	1	2	yes	—
270	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
271	C₂×C₄	C₂₃×#97	1	4	yes	—
272	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
273	C₂×C₄	C₂₃×C₄⋊C₈	1	8	yes	—
274	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
282	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
283	C₂×C₄	C₂₃×#137	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	4	4	no	—
286	C₂²	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	C₂₃×#336	1	4	yes	—
290	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
291	C₂²	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
292	C₂²	C₂₃×#417	1	2	yes	—
293	C₂²	C₂₃×#421	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	8	no	—
299	C₄	C₂₃×#336	1	4	yes	—
300	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
301	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
302	C₄	C₂₃×#418	1	1	yes	characteristic
303	C₄	C₂₃×#419	1	1	yes	characteristic
304	C₄	C₂×C₂₃×C₄⋊C₈	1	2	yes	—
305	C₄	C₂₃×#428	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	C₂₃×#1477	1	4	yes	—
308	C₂	C₂×C₂₃×#336	1	1	yes	characteristic
309	C₂	C₂₃×#1733	1	1	yes	characteristic
310	C₂	C₂₃×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×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 ◁ 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
9. C₂₃×#431	C₄	semidirect
13. C₂×C₂₃×C₄⋊C₈	C₄	semidirect
31. C₂₃×#117	C₈	semidirect
79. C₄⋊C₄×C₂₃	C₂×C₈	semidirect
95. #7891	C₂₃	direct (N × H)
134. #1279	C₂×C₂₃	semidirect
159. #431	C₄×C₂₃	semidirect
163. C₂×C₄⋊C₈	C₄×C₂₃	semidirect
186. #117	C₈×C₂₃	semidirect
209. C₂₃	#7891	direct (N × H)
235. 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×C₂₃×C₄⋊C₈	2
C₄×C₂₃×#117	2
C₂₃×#1279	2
#7891	23

Subgroups — C23×#7891 #19155517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7891 #19155517