Subgroups — C₂₃×#10861 #19158487

Order 5888.

982 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#106	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#340	C₂	1	2	yes	maximal
4	C₂₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#106	—	2	4	no	—
10	C₂×C₂₃×#106	—	2	4	no	—
11	C₂×C₂₃×#106	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₂₃	—	2	4	no	—
13	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
14	C₂×C₂₃×#95	—	2	4	no	—
15	C₂₃×#340	—	2	4	no	—
16	C₂³×C₄×C₂₃	—	2	4	no	—
17	C₂³×C₄×C₂₃	D₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂²×C₈×C₂₃	D₄	1	4	yes	—
26	C₂²×C₈×C₂₃	D₄	1	2	yes	—
27	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
39	C₂₃×#106	—	2	16	no	—
40	C₄×C₈×C₂₃	—	2	8	no	—
41	C₂₃×#95	—	4	8	no	—
42	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₃	—	2	8	no	—
46	C₂²×C₄×C₂₃	—	4	4	no	—
47	C₂²×C₄×C₂₃	—	4	4	no	—
48	C₂²×C₄×C₂₃	—	4	8	no	—
49	C₂²×C₄×C₂₃	—	4	8	no	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	#41	1	2	yes	—
61	C₂²×C₄×C₂₃	#41	1	2	yes	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	C₄²	1	4	yes	—
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
72	C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₄⋊C₄×C₂₃	—	2	8	no	—
74	C₄⋊C₄×C₂₃	—	4	4	no	—
75	C₄⋊C₄×C₂₃	—	4	4	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	4	no	—
79	#10861	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₂₃	C₄×D₄	1	2	yes	—
82	C₂³×C₂₃	#121	1	2	yes	—
83	C₂³×C₂₃	#127	1	1	yes	characteristic
84	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
86	C₂³×C₂₃	#135	1	4	yes	—
87	C₂×C₄×C₂₃	—	2	4	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	2	16	no	—
90	C₂×C₄×C₂₃	—	4	8	no	—
91	C₂×C₄×C₂₃	—	4	16	no	—
92	C₂×C₄×C₂₃	—	8	8	no	—
93	C₂×C₄×C₂₃	—	8	4	no	—
94	C₂×C₄×C₂₃	—	8	4	no	—
95	C₂×C₄×C₂₃	#106	1	8	yes	complemented
96	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₂₃	#121	1	2	yes	—
108	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₂₃	#124	1	2	yes	—
111	C₂×C₄×C₂₃	#125	1	2	yes	—
112	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
114	C₈×C₂₃	—	2	4	no	—
115	C₈×C₂₃	#95	1	8	yes	complemented
116	C₂²×#106	C₂×C₂₃	1	2	yes	—
117	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
118	C₂×#340	C₂×C₂₃	1	2	yes	—
119	C₄⋊#95	C₂×C₂₃	1	2	yes	—
120	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₂₃	#380	1	1	yes	characteristic
122	C₂²×C₂₃	#384	1	1	yes	characteristic
123	C₂²×C₂₃	#389	1	2	yes	—
124	C₂²×C₂₃	#390	1	2	yes	—
125	C₂²×C₂₃	C₂×#106	1	2	yes	—
126	C₂²×C₂₃	#427	1	2	yes	—
127	C₂²×C₂₃	C₄×SD₁₆	1	8	yes	—
128	C₂²×C₂₃	#465	1	8	yes	—
129	C₂²×C₂₃	#492	1	2	yes	—
130	C₂²×C₂₃	#495	1	2	yes	—
131	C₂²×C₂₃	#498	1	2	yes	—
132	C₂²×C₂₃	#499	1	2	yes	—
133	C₄×C₂₃	—	2	8	no	—
134	C₄×C₂₃	—	8	8	no	—
135	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
136	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₂₃	#380	1	1	yes	characteristic
138	C₄×C₂₃	#382	1	1	yes	characteristic
139	C₄×C₂₃	#389	1	2	yes	—
140	C₄×C₂₃	#391	1	2	yes	—
141	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
144	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×#106	—	2	4	no	—
146	C₂×#106	—	2	4	no	—
147	C₂×#106	C₄×C₂₃	1	8	yes	complemented
148	C₂×C₄×C₈	—	2	4	no	—
149	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
150	C₂×#95	—	2	4	no	—
151	#340	—	2	4	no	—
152	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
153	C₂×C₂₃	C₄×#106	1	2	yes	—
154	C₂×C₂₃	#1807	1	2	yes	—
155	C₂×C₂₃	#1844	1	1	yes	characteristic
156	C₂×C₂₃	#1846	1	1	yes	characteristic
157	C₂×C₂₃	#1941	1	4	yes	—
158	C₂×C₂₃	#1968	1	4	yes	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	D₄×C₂₃	1	2	yes	—
161	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
167	C₂²×C₈	D₄×C₂₃	1	2	yes	—
168	C₂²×C₈	D₄×C₂₃	1	4	yes	—
169	C₂²×C₈	D₄×C₂₃	1	2	yes	—
170	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
171	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
181	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
182	#106	—	2	16	no	—
183	C₄×C₈	—	2	8	no	—
184	#95	—	4	8	no	—
185	C₂₃	#10861	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	8	no	—
193	C₂²×C₄	—	4	8	no	—
194	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
195	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	#41×C₂₃	1	2	yes	—
205	C₂²×C₄	#41×C₂₃	1	2	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	C₄²×C₂₃	1	4	yes	—
211	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
212	C₂×C₈	C₂²⋊C₄×C₂₃	1	8	yes	—
213	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
214	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
215	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	4	no	—
223	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂³	C₄×D₄×C₂₃	1	2	yes	—
225	C₂³	C₂₃×#121	1	2	yes	—
226	C₂³	C₂₃×#127	1	1	yes	characteristic
227	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
228	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
229	C₂³	C₂₃×#135	1	4	yes	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	16	no	—
235	C₂×C₄	—	8	8	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₂₃×#106	1	8	yes	complemented
239	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
246	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
248	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
249	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
250	C₂×C₄	C₂₃×#121	1	2	yes	—
251	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
252	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
253	C₂×C₄	C₂₃×#124	1	2	yes	—
254	C₂×C₄	C₂₃×#125	1	2	yes	—
255	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	C₂₃×#95	1	8	yes	complemented
259	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
260	C₂²	C₂₃×#380	1	1	yes	characteristic
261	C₂²	C₂₃×#384	1	1	yes	characteristic
262	C₂²	C₂₃×#389	1	2	yes	—
263	C₂²	C₂₃×#390	1	2	yes	—
264	C₂²	C₂×C₂₃×#106	1	2	yes	—
265	C₂²	C₂₃×#427	1	2	yes	—
266	C₂²	C₄×SD₁₆×C₂₃	1	8	yes	—
267	C₂²	C₂₃×#465	1	8	yes	—
268	C₂²	C₂₃×#492	1	2	yes	—
269	C₂²	C₂₃×#495	1	2	yes	—
270	C₂²	C₂₃×#498	1	2	yes	—
271	C₂²	C₂₃×#499	1	2	yes	—
272	C₄	—	2	8	no	—
273	C₄	—	8	8	no	—
274	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
275	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
276	C₄	C₂₃×#380	1	1	yes	characteristic
277	C₄	C₂₃×#382	1	1	yes	characteristic
278	C₄	C₂₃×#389	1	2	yes	—
279	C₄	C₂₃×#391	1	2	yes	—
280	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
281	C₂	C₄×C₂₃×#106	1	2	yes	—
282	C₂	C₂₃×#1807	1	2	yes	—
283	C₂	C₂₃×#1844	1	1	yes	characteristic
284	C₂	C₂₃×#1846	1	1	yes	characteristic
285	C₂	C₂₃×#1941	1	4	yes	—
286	C₂	C₂₃×#1968	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₂₃×#106	C₄	semidirect
13. C₂×C₄×C₈×C₂₃	C₄	semidirect
79. #10861	C₂₃	direct (N × H)
95. C₂×C₄×C₂₃	#106	semidirect
115. C₈×C₂₃	#95	semidirect
147. C₂×#106	C₄×C₂₃	semidirect
149. C₂×C₄×C₈	C₄×C₂₃	semidirect
185. C₂₃	#10861	direct (N × H)
238. C₂×C₄	C₂₃×#106	semidirect
258. C₈	C₂₃×#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#106	2
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#340	2
C₂₃×C₄⋊#95	2
#10861	23

Subgroups — C23×#10861 #19158487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10861 #19158487