Subgroups — C₂×C₂₃×#1570 #19165426

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 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×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1570	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#334	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₂₃	—	2	1	no	—
7	C₂³×C₈×C₂₃	—	2	1	no	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#431	C₄	1	4	yes	—
10	C₂₃×#431	C₂²	1	4	yes	—
11	C₂₃×#423	—	2	2	no	—
12	C₂₃×#423	C₂²	1	4	yes	—
13	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
15	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
16	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
20	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#334	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	—	4	4	no	—
28	C₂²×C₈×C₂₃	—	2	4	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	—	4	4	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	D₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	4	no	—
43	C₂₃×C₄⋊C₈	—	2	4	no	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
46	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
47	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
48	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₂₃	C₂³	1	4	yes	—
51	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
61	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
68	M₁₆×C₂₃	—	4	8	no	—
69	C₂×C₈×C₂₃	—	2	8	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	4	8	no	—
73	C₂×C₈×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	4	4	no	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
76	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₂₃	C₂×D₄	1	2	yes	—
82	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₂×#1570	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	#106	1	4	yes	—
90	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	#106	1	8	yes	—
100	C₂×C₄×C₂₃	#106	1	4	yes	—
101	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#131	1	4	yes	—
108	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
109	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
112	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₂₃	—	2	8	no	—
116	C₈×C₂₃	—	4	4	no	—
117	C₈×C₂₃	—	4	4	no	—
118	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
120	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
121	#1570	C₂×C₂₃	1	8	yes	direct product factor
122	C₂×#334	C₂×C₂₃	1	4	yes	complemented
123	C₂²×C₂₃	—	2	8	no	—
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	#424	1	2	yes	—
127	C₂²×C₂₃	C₂×#106	1	4	yes	—
128	C₂²×C₂₃	C₂×#106	1	2	yes	—
129	C₂²×C₂₃	#430	1	2	yes	—
130	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	#424	1	2	yes	—
136	C₄×C₂₃	C₂×#106	1	4	yes	—
137	C₄×C₂₃	C₂×#106	1	2	yes	—
138	C₄×C₂₃	C₂×#131	1	2	yes	—
139	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
140	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
141	C₂²×M₁₆	—	2	1	no	—
142	C₂³×C₈	—	2	1	no	—
143	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
144	#431	C₄×C₂₃	1	4	yes	—
145	#431	C₂²×C₂₃	1	4	yes	—
146	#423	—	2	2	no	—
147	#423	C₂²×C₂₃	1	4	yes	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	—	2	2	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
155	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
157	#334	C₂²×C₂₃	1	16	yes	complemented
158	C₂×C₂₃	—	2	4	no	—
159	C₂×C₂₃	#1570	1	4	yes	direct product factor
160	C₂×C₂₃	C₂×#424	1	1	yes	characteristic
161	C₂×C₂₃	C₂²×#106	1	1	yes	characteristic
162	C₂×C₂₃	C₂×#430	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
165	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	4	4	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	4	4	no	—
172	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
175	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₂₃	1	2	yes	—
177	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
186	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
187	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
188	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
189	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
190	C₄×C₈	C₂³×C₂₃	1	4	yes	—
191	C₂₃	C₂×#1570	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
198	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
200	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
202	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
209	M₁₆	—	4	8	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
217	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
218	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
219	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
220	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
222	C₄²	C₂×D₄×C₂₃	1	2	yes	—
223	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
225	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
226	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₂₃×#106	1	4	yes	—
230	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
231	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂₃×#106	1	8	yes	—
240	C₂×C₄	C₂₃×#106	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
247	C₂×C₄	C₂₃×#131	1	4	yes	—
248	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
249	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
250	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
251	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
252	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	2	2	no	—
261	C₂²	C₂₃×#424	1	2	yes	—
262	C₂²	C₂×C₂₃×#106	1	4	yes	—
263	C₂²	C₂×C₂₃×#106	1	2	yes	—
264	C₂²	C₂₃×#430	1	2	yes	—
265	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	C₂₃×#424	1	2	yes	—
271	C₄	C₂×C₂₃×#106	1	2	yes	—
272	C₄	C₂×C₂₃×#106	1	4	yes	—
273	C₄	C₂×C₂₃×#131	1	2	yes	—
274	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
275	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	C₂₃×#1570	1	4	yes	direct product factor
278	C₂	C₂×C₂₃×#424	1	1	yes	characteristic
279	C₂	C₂²×C₂₃×#106	1	1	yes	characteristic
280	C₂	C₂×C₂₃×#430	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	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₄⋊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×#1570	C₂	direct (N × H)
5. C₂×C₂₃×#334	C₂	semidirect
22. C₂₃×#334	C₂²	semidirect
86. C₂×#1570	C₂₃	direct (N × H)
121. #1570	C₂×C₂₃	direct (N × H)
122. C₂×#334	C₂×C₂₃	semidirect
157. #334	C₂²×C₂₃	semidirect
159. C₂×C₂₃	#1570	direct (N × H)
191. C₂₃	C₂×#1570	direct (N × H)
277. C₂	C₂₃×#1570	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×C₂₃×#423	2
C₂²×C₂₃×C₄⋊C₈	2
C₂₃×#1570	2
C₂×C₂₃×#334	2
C₂×#1570	23

Subgroups — C2×C23×#1570 #19165426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1570 #19165426