Subgroups — C₂₃×#10759 #19158385

Order 5888.

1202 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#410	C₂	1	2	yes	maximal
3	C₂₃×#1735	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
5	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×#131	—	2	4	no	—
7	C₂×C₂₃×#131	—	2	4	no	—
8	C₂×C₂₃×#131	—	2	4	no	—
9	C₂²×M₁₆×C₂₃	C₄	1	2	yes	complemented
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₄	1	2	yes	complemented
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#117	C₂²	1	4	yes	—
14	C₂₃×#410	—	2	4	no	—
15	C₂₃×#410	C₄	1	8	yes	complemented
16	C₂×C₂₃×#99	—	2	4	no	—
17	C₂₃×#340	—	2	4	no	—
18	C₂₃×H(3,ℤ/4)	—	2	4	no	—
19	C₂×#41×C₂₃	—	2	2	no	—
20	C₂×#41×C₂₃	—	2	2	no	—
21	C₂×#41×C₂₃	—	2	2	no	—
22	C₂×#41×C₂₃	—	2	2	no	—
23	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×#41×C₂₃	D₄	1	4	yes	—
25	C₂²×Q₈×C₂₃	Q₈	1	1	yes	characteristic
26	C₂²×D₄×C₂₃	D₄	1	2	yes	—
27	C₂²×D₄×C₂₃	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×#131	—	2	32	no	—
31	C₂×M₁₆×C₂₃	—	2	4	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	—	2	2	no	—
34	C₂×M₁₆×C₂₃	—	2	4	no	—
35	C₂×M₁₆×C₂₃	—	2	4	no	—
36	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
37	C₂²×C₈×C₂₃	—	2	4	no	—
38	C₂²×C₈×C₂₃	—	2	2	no	—
39	C₂²×C₈×C₂₃	—	2	2	no	—
40	C₂²×C₈×C₂₃	—	2	4	no	—
41	C₂²×C₈×C₂₃	—	2	4	no	—
42	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	D₄	1	4	yes	—
46	C₂₃×#117	—	2	4	no	—
47	C₂₃×#117	—	2	4	no	—
48	C₂₃×#117	—	2	4	no	—
49	C₂₃×#117	C₂×C₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂×C₄²×C₂₃	—	2	4	no	—
53	C₂₃×#99	—	2	16	no	—
54	C₂⁴×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
56	#41×C₂₃	—	2	8	no	—
57	#41×C₂₃	—	2	8	no	—
58	#41×C₂₃	—	2	8	no	—
59	#41×C₂₃	—	2	8	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×Q₈×C₂₃	—	2	2	no	—
62	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
63	C₂×D₄×C₂₃	—	2	4	no	—
64	C₂×D₄×C₂₃	—	2	4	no	—
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	C₂²⋊C₄	1	8	yes	—
68	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	2	2	no	—
73	C₂²×C₄×C₂₃	—	4	4	no	—
74	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
75	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	#41	1	2	yes	—
84	C₂²×C₄×C₂₃	#41	1	2	yes	—
85	M₁₆×C₂₃	—	2	16	no	—
86	M₁₆×C₂₃	—	2	8	no	—
87	C₂×C₈×C₂₃	—	2	16	no	—
88	C₂×C₈×C₂₃	—	2	1	no	—
89	C₂×C₈×C₂₃	—	2	1	no	—
90	C₂×C₈×C₂₃	—	2	1	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	1	no	—
94	C₂×C₈×C₂₃	—	2	8	no	—
95	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
97	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
98	C₄⋊C₄×C₂₃	—	2	8	no	—
99	C₄⋊C₄×C₂₃	—	4	4	no	—
100	C₂²⋊C₄×C₂₃	—	2	8	no	—
101	C₂²⋊C₄×C₂₃	—	4	4	no	—
102	C₄²×C₂₃	—	2	8	no	—
103	C₄²×C₂₃	—	4	4	no	—
104	#10759	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	2	2	no	—
106	C₂³×C₂₃	—	2	2	no	—
107	C₂³×C₂₃	—	4	2	no	—
108	C₂³×C₂₃	—	4	4	no	—
109	C₂³×C₂₃	#95	1	4	yes	—
110	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₂₃	#121	1	2	yes	—
113	C₂³×C₂₃	#121	1	2	yes	—
114	Q₈×C₂₃	—	2	8	no	—
115	D₄×C₂₃	—	2	16	no	—
116	D₄×C₂₃	—	2	8	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	2	4	no	—
121	C₂×C₄×C₂₃	—	2	4	no	—
122	C₂×C₄×C₂₃	—	2	4	no	—
123	C₂×C₄×C₂₃	—	4	8	no	—
124	C₂×C₄×C₂₃	—	8	8	no	—
125	C₂×C₄×C₂₃	#95	1	4	yes	—
126	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₂₃	#120	1	4	yes	—
136	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₂₃	#123	1	4	yes	—
139	C₈×C₂₃	—	2	2	no	—
140	C₈×C₂₃	—	2	4	no	—
141	C₈×C₂₃	—	2	2	no	—
142	C₂²×#131	C₂×C₂₃	1	1	yes	characteristic
143	C₂×#410	C₂×C₂₃	1	2	yes	—
144	#1735	C₂×C₂₃	1	2	yes	complemented
145	C₂×H(3,ℤ/4)	C₂×C₂₃	1	2	yes	complemented
146	C₂²×C₂₃	—	2	4	no	—
147	C₂²×C₂₃	—	2	2	no	—
148	C₂²×C₂₃	—	2	4	no	—
149	C₂²×C₂₃	—	4	2	no	—
150	C₂²×C₂₃	—	4	2	no	—
151	C₂²×C₂₃	—	4	4	no	—
152	C₂²×C₂₃	—	4	4	no	—
153	C₂²×C₂₃	—	4	4	no	—
154	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
155	C₂²×C₂₃	#380	1	1	yes	characteristic
156	C₂²×C₂₃	#380	1	1	yes	characteristic
157	C₂²×C₂₃	#386	1	4	yes	—
158	C₄×C₂₃	—	2	4	no	—
159	C₄×C₂₃	—	8	8	no	—
160	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₂₃	#379	1	1	yes	characteristic
163	C₄×C₂₃	#381	1	1	yes	characteristic
164	C₄×C₂₃	#385	1	4	yes	—
165	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#131	—	2	4	no	—
167	C₂×#131	—	2	4	no	—
168	C₂×#131	—	2	4	no	—
169	C₂²×M₁₆	C₄×C₂₃	1	2	yes	complemented
170	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
171	C₂³×C₈	C₄×C₂₃	1	2	yes	complemented
172	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×#117	C₂²×C₂₃	1	4	yes	—
174	#410	—	2	4	no	—
175	#410	C₄×C₂₃	1	8	yes	complemented
176	C₂×#99	—	2	4	no	—
177	#340	—	2	4	no	—
178	H(3,ℤ/4)	—	2	4	no	—
179	C₂×C₂₃	—	2	4	no	—
180	C₂×C₂₃	—	2	1	no	—
181	C₂×C₂₃	—	2	1	no	—
182	C₂×C₂₃	—	4	4	no	—
183	C₂×C₂₃	C₂²⋊#95	1	1	yes	characteristic
184	C₂×C₂₃	#1812	1	2	yes	—
185	C₂×#41	—	2	2	no	—
186	C₂×#41	—	2	2	no	—
187	C₂×#41	—	2	2	no	—
188	C₂×#41	—	2	2	no	—
189	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
190	C₂×#41	D₄×C₂₃	1	4	yes	—
191	C₂²×Q₈	Q₈×C₂₃	1	1	yes	characteristic
192	C₂²×D₄	D₄×C₂₃	1	2	yes	—
193	C₂²×D₄	Q₈×C₂₃	1	1	yes	characteristic
194	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
195	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
196	#131	—	2	32	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
203	C₂²×C₈	—	2	4	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
210	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₈	D₄×C₂₃	1	4	yes	—
212	#117	—	2	4	no	—
213	#117	—	2	4	no	—
214	#117	—	2	4	no	—
215	#117	C₂×C₄×C₂₃	1	8	yes	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₂²⋊C₄	—	2	4	no	—
218	C₂×C₄²	—	2	4	no	—
219	#99	—	2	16	no	—
220	C₂₃	#10759	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
223	#41	—	2	8	no	—
224	#41	—	2	8	no	—
225	#41	—	2	8	no	—
226	#41	—	2	8	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₂²⋊C₄×C₂₃	1	8	yes	—
235	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
242	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
243	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
249	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
250	C₂²×C₄	#41×C₂₃	1	2	yes	—
251	C₂²×C₄	#41×C₂₃	1	2	yes	—
252	M₁₆	—	2	16	no	—
253	M₁₆	—	2	8	no	—
254	C₂×C₈	—	2	16	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
263	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
264	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₄²	—	2	8	no	—
270	C₄²	—	4	4	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	C₂₃×#95	1	4	yes	—
276	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
277	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂³	C₂₃×#121	1	2	yes	—
279	C₂³	C₂₃×#121	1	2	yes	—
280	Q₈	—	2	8	no	—
281	D₄	—	2	16	no	—
282	D₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	8	8	no	—
291	C₂×C₄	C₂₃×#95	1	4	yes	—
292	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
297	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
298	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
300	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
301	C₂×C₄	C₂₃×#120	1	4	yes	—
302	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
303	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
304	C₂×C₄	C₂₃×#123	1	4	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
317	C₂²	C₂₃×#380	1	1	yes	characteristic
318	C₂²	C₂₃×#380	1	1	yes	characteristic
319	C₂²	C₂₃×#386	1	4	yes	—
320	C₄	—	2	4	no	—
321	C₄	—	8	8	no	—
322	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
323	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
324	C₄	C₂₃×#379	1	1	yes	characteristic
325	C₄	C₂₃×#381	1	1	yes	characteristic
326	C₄	C₂₃×#385	1	4	yes	—
327	C₂	—	2	4	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	4	no	—
331	C₂	C₂₃×C₂²⋊#95	1	1	yes	characteristic
332	C₂	C₂₃×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1735	C₂	semidirect
4. C₂×C₂₃×H(3,ℤ/4)	C₂	semidirect
9. C₂²×M₁₆×C₂₃	C₄	semidirect
11. C₂³×C₈×C₂₃	C₄	semidirect
15. C₂₃×#410	C₄	semidirect
104. #10759	C₂₃	direct (N × H)
144. #1735	C₂×C₂₃	semidirect
145. C₂×H(3,ℤ/4)	C₂×C₂₃	semidirect
169. C₂²×M₁₆	C₄×C₂₃	semidirect
171. C₂³×C₈	C₄×C₂₃	semidirect
175. #410	C₄×C₂₃	semidirect
220. C₂₃	#10759	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#131	2
C₂×C₂₃×#410	2
C₂₃×#1735	2
C₂×C₂₃×H(3,ℤ/4)	2
#10759	23

Subgroups — C23×#10759 #19158385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10759 #19158385