Subgroups — C₂×C₂₃×#1572 #19165428

Order 5888.

670 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×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1572	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#335	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#334	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#432	—	2	2	no	—
9	C₂₃×#432	C₂²	1	4	yes	—
10	C₂₃×#431	—	2	2	no	—
11	C₂₃×#431	C₄	1	4	yes	—
12	C₂₃×#431	C₂²	1	4	yes	—
13	C₂₃×#431	C₂²	1	4	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
20	C₂×C₂₃×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₂₃	—	2	1	no	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#335	C₂²	1	8	yes	complemented
25	C₂₃×#334	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	2	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	2	no	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
44	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
45	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
46	C₂₃×C₂²⋊C₈	—	2	4	no	—
47	C₂₃×C₂²⋊C₈	—	2	4	no	—
48	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
49	C₂₃×#97	—	2	4	no	—
50	C₄×C₈×C₂₃	—	2	4	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₂₃	C₂³	1	2	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	4	4	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
79	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	2	yes	—
86	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
87	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
89	C₂×#1572	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
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	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#131	1	4	yes	—
112	C₂×C₄×C₂₃	#135	1	4	yes	—
113	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
116	C₈×C₂₃	—	2	4	no	—
117	C₈×C₂₃	—	2	4	no	—
118	C₈×C₂₃	—	4	4	no	—
119	C₈×C₂₃	—	4	4	no	—
120	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
121	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
122	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
123	#1572	C₂×C₂₃	1	8	yes	direct product factor
124	C₂×#335	C₂×C₂₃	1	2	yes	complemented
125	C₂×#334	C₂×C₂₃	1	2	yes	complemented
126	C₂²×C₂₃	—	2	2	no	—
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	4	4	no	—
130	C₂²×C₂₃	#424	1	2	yes	—
131	C₂²×C₂₃	#427	1	2	yes	—
132	C₂²×C₂₃	#430	1	2	yes	—
133	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	#424	1	2	yes	—
140	C₄×C₂₃	#427	1	2	yes	—
141	C₄×C₂₃	C₂×#131	1	2	yes	—
142	C₄×C₂₃	C₂×#135	1	2	yes	—
143	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
144	#432	—	2	2	no	—
145	#432	C₂²×C₂₃	1	4	yes	—
146	#431	—	2	2	no	—
147	#431	C₄×C₂₃	1	4	yes	—
148	#431	C₂²×C₂₃	1	4	yes	—
149	#431	C₂²×C₂₃	1	4	yes	—
150	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂²⋊C₈	—	2	1	no	—
154	C₂×C₂²⋊C₈	—	2	1	no	—
155	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
156	C₂×#97	—	2	1	no	—
157	C₂×C₄×C₈	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
160	#335	C₂²×C₂₃	1	8	yes	complemented
161	#334	C₂²×C₂₃	1	8	yes	complemented
162	C₂×C₂₃	—	4	2	no	—
163	C₂×C₂₃	#1572	1	4	yes	direct product factor
164	C₂×C₂₃	C₂×#424	1	1	yes	characteristic
165	C₂×C₂₃	C₂×#427	1	1	yes	characteristic
166	C₂×C₂₃	C₂×#430	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
169	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
176	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₂₃	1	2	yes	—
178	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
185	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
186	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
187	C₂²⋊C₈	—	2	4	no	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
193	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
194	C₄×C₈	C₂³×C₂₃	1	2	yes	—
195	C₄×C₈	C₂³×C₂₃	1	2	yes	—
196	C₂₃	C₂×#1572	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
220	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
221	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
224	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
226	C₄²	C₂×D₄×C₂₃	1	2	yes	—
227	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
229	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
230	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
235	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
250	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#131	1	4	yes	—
253	C₂×C₄	C₂₃×#135	1	4	yes	—
254	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	—	2	4	no	—
259	C₈	—	4	4	no	—
260	C₈	—	4	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	4	no	—
265	C₂²	C₂₃×#424	1	2	yes	—
266	C₂²	C₂₃×#427	1	2	yes	—
267	C₂²	C₂₃×#430	1	2	yes	—
268	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	C₂₃×#424	1	2	yes	—
275	C₄	C₂₃×#427	1	2	yes	—
276	C₄	C₂×C₂₃×#131	1	2	yes	—
277	C₄	C₂×C₂₃×#135	1	2	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₂₃×#1572	1	4	yes	direct product factor
280	C₂	C₂×C₂₃×#424	1	1	yes	characteristic
281	C₂	C₂×C₂₃×#427	1	1	yes	characteristic
282	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×#1572	C₂	direct (N × H)
5. C₂×C₂₃×#335	C₂	semidirect
6. C₂×C₂₃×#334	C₂	semidirect
24. C₂₃×#335	C₂²	semidirect
25. C₂₃×#334	C₂²	semidirect
89. C₂×#1572	C₂₃	direct (N × H)
123. #1572	C₂×C₂₃	direct (N × H)
124. C₂×#335	C₂×C₂₃	semidirect
125. C₂×#334	C₂×C₂₃	semidirect
160. #335	C₂²×C₂₃	semidirect
161. #334	C₂²×C₂₃	semidirect
163. C₂×C₂₃	#1572	direct (N × H)
196. C₂₃	C₂×#1572	direct (N × H)
279. C₂	C₂₃×#1572	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×#432	2
C₂×C₂₃×#431	2
C₂₃×#1572	2
C₂×C₂₃×#335	2
C₂×C₂₃×#334	2
C₂×#1572	23

Subgroups — C2×C23×#1572 #19165428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1572 #19165428