Subgroups — C₂×C₂₃×#1589 #19165509

Order 5888.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#436	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1589	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₂₃	—	2	2	no	—
9	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
10	C₄×D₈×C₂₃	C₄	1	4	yes	—
11	C₄×D₈×C₂₃	C₂²	1	4	yes	—
12	C₂₃×#436	—	2	4	no	—
13	C₂₃×#436	C₂²	1	8	yes	complemented
14	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×#102	C₄	1	2	yes	—
17	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#335	C₂²	1	4	yes	complemented
22	C₂₃×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₈²×C₂₃	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₂₃	—	2	2	no	—
26	C₂×D₈×C₂₃	—	2	4	no	—
27	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₂₃	—	2	8	no	—
29	C₂×M₁₆×C₂₃	—	4	8	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₄×D₄×C₂₃	—	2	2	no	—
37	C₄×D₄×C₂₃	—	2	2	no	—
38	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₂₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×#107	C₂×C₄	1	4	yes	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
46	C₂₃×#102	C₂×C₄	1	8	yes	—
47	C₂₃×C₂²⋊C₈	—	2	8	no	—
48	C₄×C₈×C₂₃	—	2	1	no	—
49	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₂₃	D₄	1	2	yes	—
51	C₄×C₈×C₂₃	C₂³	1	2	yes	—
52	C₄×C₈×C₂₃	C₂³	1	2	yes	—
53	C₄×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂⁴×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	2	4	no	—
57	C₂×D₄×C₂₃	—	2	4	no	—
58	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	2	4	no	—
62	C₂²×C₄×C₂₃	—	4	4	no	—
63	C₂²×C₄×C₂₃	—	4	4	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
67	D₈×C₂₃	—	2	8	no	—
68	M₁₆×C₂₃	—	4	16	no	—
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	4	4	no	—
75	C₂×C₈×C₂₃	M₁₆	1	4	yes	complemented
76	C₂×C₈×C₂₃	D₈	1	4	yes	complemented
77	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₂₃	—	2	4	no	—
84	C₂²⋊C₄×C₂₃	—	2	4	no	—
85	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₂₃	#41	1	1	yes	characteristic
88	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
89	C₂×#1589	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	4	no	—
91	C₂³×C₂₃	—	4	4	no	—
92	C₂³×C₂₃	—	4	4	no	—
93	C₂³×C₂₃	—	4	4	no	—
94	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
95	D₄×C₂₃	—	2	4	no	—
96	D₄×C₂₃	—	2	4	no	—
97	D₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	8	no	—
104	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₂₃	#131	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₂₃	#135	1	2	yes	—
111	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₈×C₂₃	—	2	4	no	—
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	2	2	no	—
117	C₈×C₂₃	—	4	4	no	—
118	C₈×C₂₃	C₂×M₁₆	1	4	yes	complemented
119	C₈×C₂₃	C₂×D₈	1	4	yes	complemented
120	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
121	C₂×#436	C₂×C₂₃	1	2	yes	complemented
122	#1589	C₂×C₂₃	1	8	yes	direct product factor
123	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂×D₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
125	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂²×C₂₃	—	4	4	no	—
127	C₂²×C₂₃	—	4	4	no	—
128	C₂²×C₂₃	—	4	4	no	—
129	C₂²×C₂₃	—	8	8	no	—
130	C₂²×C₂₃	#436	1	2	yes	—
131	C₂²×C₂₃	C₄×D₈	1	2	yes	—
132	C₂²×C₂₃	#443	1	2	yes	—
133	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	4	4	no	—
137	C₄×C₂₃	#436	1	2	yes	—
138	C₄×C₂₃	C₄×D₈	1	2	yes	—
139	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
140	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
141	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
143	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic
144	C₂²×M₁₆	—	2	2	no	—
145	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
146	C₄×D₈	C₄×C₂₃	1	4	yes	—
147	C₄×D₈	C₂²×C₂₃	1	4	yes	—
148	#436	—	2	4	no	—
149	#436	C₂²×C₂₃	1	8	yes	complemented
150	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×#102	C₄×C₂₃	1	2	yes	—
153	C₂×C₂²⋊C₈	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	#335	C₂²×C₂₃	1	4	yes	complemented
158	D₄⋊C₈	C₂²×C₂₃	1	8	yes	complemented
159	C₈²	C₂²×C₂₃	1	4	yes	complemented
160	C₂×C₂₃	—	8	4	no	—
161	C₂×C₂₃	#1589	1	4	yes	direct product factor
162	C₂×C₂₃	C₂×#436	1	1	yes	characteristic
163	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
164	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
165	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
166	C₂³×C₄	—	2	2	no	—
167	C₂×D₈	—	2	4	no	—
168	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	—	4	8	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
180	C₄×D₄	C₂³×C₂₃	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
184	#107	C₂×C₄×C₂₃	1	4	yes	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
187	#102	C₂×C₄×C₂₃	1	8	yes	—
188	C₂²⋊C₈	—	2	8	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
191	C₄×C₈	D₄×C₂₃	1	2	yes	—
192	C₄×C₈	C₂³×C₂₃	1	2	yes	—
193	C₄×C₈	C₂³×C₂₃	1	2	yes	—
194	C₄×C₈	C₂³×C₂₃	1	2	yes	—
195	C₂₃	C₂×#1589	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
209	D₈	—	2	8	no	—
210	M₁₆	—	4	16	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	M₁₆×C₂₃	1	4	yes	complemented
218	C₂×C₈	D₈×C₂₃	1	4	yes	complemented
219	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₄²	#41×C₂₃	1	1	yes	characteristic
230	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
249	C₂×C₄	C₂₃×#131	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
251	C₂×C₄	C₂₃×#135	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	C₂×M₁₆×C₂₃	1	4	yes	complemented
260	C₈	C₂×D₈×C₂₃	1	4	yes	complemented
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	8	no	—
265	C₂²	C₂₃×#436	1	2	yes	—
266	C₂²	C₄×D₈×C₂₃	1	2	yes	—
267	C₂²	C₂₃×#443	1	2	yes	—
268	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	C₂₃×#436	1	2	yes	—
273	C₄	C₄×D₈×C₂₃	1	2	yes	—
274	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
275	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
276	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
277	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	C₂₃×#1589	1	4	yes	direct product factor
280	C₂	C₂×C₂₃×#436	1	1	yes	characteristic
281	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic
282	C₂	C₂×C₂₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₄×D₄
prime radical	C₂²×D₈×C₂₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#436	C₂	semidirect
3. C₂₃×#1589	C₂	direct (N × H)
4. C₂×C₂₃×#335	C₂	semidirect
5. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
6. C₂×C₈²×C₂₃	C₂	semidirect
13. C₂₃×#436	C₂²	semidirect
21. C₂₃×#335	C₂²	semidirect
22. C₂₃×D₄⋊C₈	C₂²	semidirect
23. C₈²×C₂₃	C₂²	semidirect
75. C₂×C₈×C₂₃	M₁₆	semidirect
76. C₂×C₈×C₂₃	D₈	semidirect
89. C₂×#1589	C₂₃	direct (N × H)
118. C₈×C₂₃	C₂×M₁₆	semidirect
119. C₈×C₂₃	C₂×D₈	semidirect
121. C₂×#436	C₂×C₂₃	semidirect
122. #1589	C₂×C₂₃	direct (N × H)
123. C₂×#335	C₂×C₂₃	semidirect
124. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
125. C₂×C₈²	C₂×C₂₃	semidirect
149. #436	C₂²×C₂₃	semidirect
157. #335	C₂²×C₂₃	semidirect
158. D₄⋊C₈	C₂²×C₂₃	semidirect
159. C₈²	C₂²×C₂₃	semidirect
161. C₂×C₂₃	#1589	direct (N × H)
195. C₂₃	C₂×#1589	direct (N × H)
217. C₂×C₈	M₁₆×C₂₃	semidirect
218. C₂×C₈	D₈×C₂₃	semidirect
259. C₈	C₂×M₁₆×C₂₃	semidirect
260. C₈	C₂×D₈×C₂₃	semidirect
279. C₂	C₂₃×#1589	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×C₄×D₈×C₂₃	2
C₂×C₂₃×#436	2
C₂₃×#1589	2
C₂×C₂₃×#335	2
C₂×C₂₃×D₄⋊C₈	2
C₂×C₈²×C₂₃	2
C₂×#1589	23

Subgroups — C2×C23×#1589 #19165509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1589 #19165509