Subgroups — C₂₃×#10857 #19158483

Order 5888.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 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	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#343	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
8	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#423	C₄	1	4	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
13	C₄×M₁₆×C₂₃	—	2	2	no	—
14	C₄×M₁₆×C₂₃	C₄	1	4	yes	complemented
15	C₂×C₂₃×#97	—	2	1	no	—
16	C₂×C₄×C₈×C₂₃	—	2	1	no	—
17	C₂₃×#384	—	2	1	no	—
18	C₂₃×#384	—	2	1	no	—
19	C₂₃×#384	—	2	2	no	—
20	C₂₃×#343	—	2	8	no	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	4	no	—
29	C₂×M₁₆×C₂₃	—	2	4	no	—
30	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
33	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	D₄	1	2	yes	—
49	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
50	C₂₃×C₄⋊C₈	—	2	8	no	—
51	C₂₃×#97	—	2	4	no	—
52	C₄×C₈×C₂₃	—	2	4	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	4	no	—
57	C₂²×C₄×C₂₃	—	2	8	no	—
58	C₂²×C₄×C₂₃	—	4	8	no	—
59	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	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	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₂₃	#41	1	2	yes	—
72	C₂²×C₄×C₂₃	#41	1	2	yes	—
73	M₁₆×C₂₃	—	2	4	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	2	8	no	—
77	M₁₆×C₂₃	—	4	4	no	—
78	M₁₆×C₂₃	C₄⋊C₄	1	8	yes	complemented
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	4	2	no	—
84	C₂×C₈×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	8	no	—
86	C₄⋊C₄×C₂₃	—	4	8	no	—
87	C₄⋊C₄×C₂₃	—	4	4	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
91	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
95	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
96	#10857	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	2	no	—
98	C₂³×C₂₃	—	2	2	no	—
99	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
100	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₂₃	C₄×D₄	1	2	yes	—
103	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	8	no	—
107	C₂×C₄×C₂₃	—	2	8	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	4	8	no	—
110	C₂×C₄×C₂₃	—	8	8	no	—
111	C₂×C₄×C₂₃	#95	1	4	yes	—
112	C₂×C₄×C₂₃	#95	1	4	yes	—
113	C₂×C₄×C₂₃	#100	1	4	yes	—
114	C₂×C₄×C₂₃	#101	1	4	yes	—
115	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#117	1	2	yes	—
121	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₂₃	#121	1	2	yes	—
124	C₂×C₄×C₂₃	#121	1	2	yes	—
125	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₂₃	#125	1	2	yes	—
130	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
133	C₈×C₂₃	—	2	8	no	—
134	C₈×C₂₃	—	4	4	no	—
135	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
136	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
137	C₂×#384	C₂×C₂₃	1	1	yes	characteristic
138	C₂×#343	C₂×C₂₃	1	4	yes	—
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	2	2	no	—
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	2	4	no	—
144	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
145	C₂²×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
146	C₂²×C₂₃	#389	1	2	yes	—
147	C₂²×C₂₃	#390	1	1	yes	characteristic
148	C₂²×C₂₃	#391	1	1	yes	characteristic
149	C₂²×C₂₃	C₂×#100	1	2	yes	—
150	C₂²×C₂₃	C₂×#101	1	2	yes	—
151	C₄×C₂₃	—	2	4	no	—
152	C₄×C₂₃	—	8	4	no	—
153	C₄×C₂₃	#343	1	8	yes	complemented
154	C₄×C₂₃	#380	1	2	yes	—
155	C₄×C₂₃	#382	1	1	yes	characteristic
156	C₄×C₂₃	#384	1	1	yes	characteristic
157	C₄×C₂₃	#389	1	2	yes	—
158	C₄×C₂₃	#390	1	1	yes	characteristic
159	C₄×C₂₃	#391	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
162	C₂²×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
163	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
164	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
165	#423	—	2	2	no	—
166	#423	C₄×C₂₃	1	4	yes	complemented
167	C₂×C₄⋊C₈	—	2	2	no	—
168	C₄×M₁₆	—	2	2	no	—
169	C₄×M₁₆	C₄×C₂₃	1	4	yes	complemented
170	C₂×#97	—	2	1	no	—
171	C₂×C₄×C₈	—	2	1	no	—
172	#384	—	2	1	no	—
173	#384	—	2	1	no	—
174	#384	—	2	2	no	—
175	#343	—	2	8	no	—
176	C₂×C₂₃	—	2	2	no	—
177	C₂×C₂₃	—	2	2	no	—
178	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
179	C₂×C₂₃	C₂×#343	1	2	yes	—
180	C₂×C₂₃	#1965	1	1	yes	characteristic
181	C₂×C₂₃	#1966	1	1	yes	characteristic
182	C₂×C₂₃	#1997	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
185	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
193	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
194	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
195	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	4	8	no	—
205	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
206	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
207	C₂×C₄²	D₄×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	C₄⋊C₈	—	2	8	no	—
213	#97	—	2	4	no	—
214	C₄×C₈	—	2	4	no	—
215	C₂₃	#10857	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	8	no	—
222	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
225	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
232	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
234	C₂²×C₄	#41×C₂₃	1	2	yes	—
235	C₂²×C₄	#41×C₂₃	1	2	yes	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	8	no	—
240	M₁₆	—	4	4	no	—
241	M₁₆	C₄⋊C₄×C₂₃	1	8	yes	complemented
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
254	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
255	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
256	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
257	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
258	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
263	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
264	C₂³	C₄×D₄×C₂₃	1	2	yes	—
265	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	C₂₃×#95	1	4	yes	—
274	C₂×C₄	C₂₃×#95	1	4	yes	—
275	C₂×C₄	C₂₃×#100	1	4	yes	—
276	C₂×C₄	C₂₃×#101	1	4	yes	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂₃×#117	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
284	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
285	C₂×C₄	C₂₃×#121	1	2	yes	—
286	C₂×C₄	C₂₃×#121	1	2	yes	—
287	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
289	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
290	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
291	C₂×C₄	C₂₃×#125	1	2	yes	—
292	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
293	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
294	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₈	—	4	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
303	C₂²	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
304	C₂²	C₂₃×#389	1	2	yes	—
305	C₂²	C₂₃×#390	1	1	yes	characteristic
306	C₂²	C₂₃×#391	1	1	yes	characteristic
307	C₂²	C₂×C₂₃×#100	1	2	yes	—
308	C₂²	C₂×C₂₃×#101	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	8	4	no	—
311	C₄	C₂₃×#343	1	8	yes	complemented
312	C₄	C₂₃×#380	1	2	yes	—
313	C₄	C₂₃×#382	1	1	yes	characteristic
314	C₄	C₂₃×#384	1	1	yes	characteristic
315	C₄	C₂₃×#389	1	2	yes	—
316	C₄	C₂₃×#390	1	1	yes	characteristic
317	C₄	C₂₃×#391	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
321	C₂	C₂×C₂₃×#343	1	2	yes	—
322	C₂	C₂₃×#1965	1	1	yes	characteristic
323	C₂	C₂₃×#1966	1	1	yes	characteristic
324	C₂	C₂₃×#1997	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂₃×#423	C₄	semidirect
14. C₄×M₁₆×C₂₃	C₄	semidirect
78. M₁₆×C₂₃	C₄⋊C₄	semidirect
96. #10857	C₂₃	direct (N × H)
153. C₄×C₂₃	#343	semidirect
166. #423	C₄×C₂₃	semidirect
169. C₄×M₁₆	C₄×C₂₃	semidirect
215. C₂₃	#10857	direct (N × H)
241. M₁₆	C₄⋊C₄×C₂₃	semidirect
311. C₄	C₂₃×#343	semidirect

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
C₂×C₄×M₁₆×C₂₃	2
C₂×C₂₃×#384	2
C₂×C₂₃×#343	2
#10857	23

Subgroups — C23×#10857 #19158483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10857 #19158483