Subgroups — C₂₃×#7896 #19155522

Order 5888.

722 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₂×C₄×M₁₆×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	C₂×C₂₃×#117	—	2	1	no	—
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	C₄×M₁₆×C₂₃	—	2	2	no	—
13	C₄×M₁₆×C₂₃	C₄	1	4	yes	complemented
14	C₂×C₂₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	—	2	1	no	—
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	8	no	—
27	C₂×M₁₆×C₂₃	—	2	4	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂₃×#117	—	4	4	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₂₃×C₂²⋊C₈	—	2	4	no	—
47	C₂₃×#97	—	2	4	no	—
48	C₄×C₈×C₂₃	—	2	4	no	—
49	C₄×C₈×C₂₃	—	2	2	no	—
50	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
51	C₂₃×#95	—	2	2	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	2	4	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	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	M₁₆×C₂₃	—	2	16	no	—
72	C₂×C₈×C₂₃	—	2	8	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₄⋊C₄×C₂₃	—	2	8	no	—
78	C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₂²⋊C₄×C₂₃	—	4	4	no	—
81	C₄²×C₂₃	—	4	4	no	—
82	C₄²×C₂₃	C₄²	1	2	yes	—
83	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
92	#7896	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	#95	1	2	yes	—
98	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	4	4	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	#104	1	4	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	#131	1	4	yes	—
123	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₂₃	#137	1	2	yes	—
125	C₈×C₂₃	—	2	8	no	—
126	C₈×C₂₃	—	4	4	no	—
127	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
128	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
129	#2285	C₂×C₂₃	1	1	yes	characteristic
130	#1279	C₂×C₂₃	1	4	yes	complemented
131	C₂²×C₂₃	—	2	2	no	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	—	4	2	no	—
135	C₂²×C₂₃	—	4	2	no	—
136	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
137	C₂²×C₂₃	#408	1	2	yes	—
138	C₂²×C₂₃	#418	1	1	yes	characteristic
139	C₂²×C₂₃	#419	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×#104	1	2	yes	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	8	4	no	—
146	C₄×C₂₃	#339	1	4	yes	—
147	C₄×C₂₃	#405	1	2	yes	—
148	C₄×C₂₃	#417	1	2	yes	—
149	C₄×C₂₃	#424	1	2	yes	—
150	C₄×C₂₃	#428	1	2	yes	—
151	C₂²×M₁₆	—	2	1	no	—
152	C₂×#117	—	2	1	no	—
153	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	#431	—	2	2	no	—
155	#431	C₄×C₂₃	1	4	yes	complemented
156	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂²⋊C₈	—	2	1	no	—
158	C₄×M₁₆	—	2	2	no	—
159	C₄×M₁₆	C₄×C₂₃	1	4	yes	complemented
160	C₂×#97	—	2	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
163	#384	C₄×C₂₃	1	2	yes	—
164	#382	—	2	1	no	—
165	#380	—	2	1	no	—
166	#380	C₄×C₂₃	1	2	yes	—
167	C₄×C₄⋊C₄	—	2	1	no	—
168	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂₃	—	4	2	no	—
170	C₂×C₂₃	#1484	1	2	yes	—
171	C₂×C₂₃	#1485	1	2	yes	—
172	C₂×C₂₃	#1729	1	1	yes	characteristic
173	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
174	C₂×C₂₃	#1736	1	1	yes	characteristic
175	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
178	C₂×M₁₆	—	2	8	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	#117	—	4	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₂₃	1	2	yes	—
194	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
198	C₂²⋊C₈	—	2	4	no	—
199	#97	—	2	4	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂₃	#7896	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
219	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
224	M₁₆	—	2	16	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₄²	C₄²×C₂₃	1	2	yes	—
236	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
237	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
238	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
239	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
240	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
241	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
242	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
244	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
245	C₂³	—	2	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂₃×#95	1	2	yes	—
250	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂₃×#95	1	2	yes	—
265	C₂×C₄	C₂₃×#95	1	2	yes	—
266	C₂×C₄	C₂₃×#95	1	2	yes	—
267	C₂×C₄	C₂₃×#104	1	4	yes	—
268	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂₃×#131	1	4	yes	—
275	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
276	C₂×C₄	C₂₃×#137	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	4	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
285	C₂²	C₂₃×#408	1	2	yes	—
286	C₂²	C₂₃×#418	1	1	yes	characteristic
287	C₂²	C₂₃×#419	1	1	yes	characteristic
288	C₂²	C₂×C₂₃×#104	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	8	4	no	—
294	C₄	C₂₃×#339	1	4	yes	—
295	C₄	C₂₃×#405	1	2	yes	—
296	C₄	C₂₃×#417	1	2	yes	—
297	C₄	C₂₃×#424	1	2	yes	—
298	C₄	C₂₃×#428	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₂₃×#1484	1	2	yes	—
301	C₂	C₂₃×#1485	1	2	yes	—
302	C₂	C₂₃×#1729	1	1	yes	characteristic
303	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
304	C₂	C₂₃×#1736	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
9. C₂₃×#431	C₄	semidirect
13. C₄×M₁₆×C₂₃	C₄	semidirect
92. #7896	C₂₃	direct (N × H)
130. #1279	C₂×C₂₃	semidirect
155. #431	C₄×C₂₃	semidirect
159. C₄×M₁₆	C₄×C₂₃	semidirect
205. C₂₃	#7896	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×#431	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#2285	2
C₂₃×#1279	2
#7896	23

Subgroups — C23×#7896 #19155522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7896 #19155522