Subgroups — C₂×C₄₃×#1461 #24937365

Order 11008.

1282 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#431	C₂	1	2	yes	maximal
3	C₄₃×#1461	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₃×#324	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₄₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₄₃	—	2	2	no	—
7	C₂×C₄×Q₈×C₄₃	—	2	4	no	—
8	C₂×C₄×Q₈×C₄₃	—	2	2	no	—
9	C₂×C₄×Q₈×C₄₃	C₄	1	4	yes	—
10	C₂×C₄×Q₈×C₄₃	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₃	C₄	1	2	yes	—
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#431	—	2	4	no	—
16	C₄₃×#431	C₂²	1	8	yes	—
17	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
18	C₂×C₂²⋊C₈×C₄₃	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₄₃	—	2	2	no	—
20	C₄₃×#324	C₂²	1	16	yes	—
21	C₂²×Q₈×C₄₃	—	2	2	no	—
22	C₂²×Q₈×C₄₃	—	2	2	no	—
23	C₂²×Q₈×C₄₃	—	2	4	no	—
24	C₂²×Q₈×C₄₃	C₈	1	8	yes	complemented
25	C₂²×Q₈×C₄₃	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	—	2	4	no	—
32	C₄×Q₈×C₄₃	—	2	8	no	—
33	C₄×Q₈×C₄₃	—	2	8	no	—
34	C₄×Q₈×C₄₃	—	2	8	no	—
35	C₄×Q₈×C₄₃	—	2	8	no	—
36	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₃	—	2	2	no	—
44	C₂×C₄²×C₄₃	—	2	4	no	—
45	C₂×C₄²×C₄₃	—	2	2	no	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	D₄	1	2	yes	—
49	C₂×C₄²×C₄₃	D₄	1	4	yes	—
50	C₂×C₄²×C₄₃	D₄	1	2	yes	—
51	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
52	C₄⋊C₈×C₄₃	—	2	8	no	—
53	C₂²⋊C₈×C₄₃	—	2	4	no	—
54	C₂²⋊C₈×C₄₃	C₂³	1	8	yes	—
55	C₄×C₈×C₄₃	—	2	8	no	—
56	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×Q₈×C₄₃	—	2	16	no	—
58	C₂×Q₈×C₄₃	—	2	16	no	—
59	C₂×Q₈×C₄₃	—	2	8	no	—
60	C₂×Q₈×C₄₃	—	2	8	no	—
61	C₂×Q₈×C₄₃	C₂×C₈	1	16	yes	complemented
62	C₂²×C₄×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	4	no	—
68	C₂²×C₄×C₄₃	—	4	2	no	—
69	C₂²×C₄×C₄₃	—	4	4	no	—
70	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×C₈	1	8	yes	—
76	C₂²×C₄×C₄₃	C₂×C₈	1	4	yes	—
77	C₂²×C₄×C₄₃	M₁₆	1	4	yes	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₄₃	—	2	8	no	—
87	C₂×C₈×C₄₃	—	4	8	no	—
88	C₄⋊C₄×C₄₃	—	2	8	no	—
89	C₄⋊C₄×C₄₃	—	2	8	no	—
90	C₄⋊C₄×C₄₃	—	2	8	no	—
91	C₄²×C₄₃	—	2	8	no	—
92	C₄²×C₄₃	—	2	8	no	—
93	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
94	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
95	C₄²×C₄₃	C₂×D₄	1	4	yes	—
96	C₄²×C₄₃	C₂×D₄	1	4	yes	—
97	C₂³×C₄₃	—	2	2	no	—
98	C₂³×C₄₃	—	2	2	no	—
99	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
102	C₂³×C₄₃	C₂×M₁₆	1	2	yes	—
103	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	Q₈×C₄₃	—	2	32	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	8	no	—
108	C₂×C₄×C₄₃	—	2	8	no	—
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	1	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	4	4	no	—
119	C₂×C₄×C₄₃	—	4	4	no	—
120	C₂×C₄×C₄₃	C₂²⋊C₈	1	8	yes	—
121	C₂×C₄×C₄₃	C₂²⋊C₈	1	8	yes	—
122	C₂×C₄×C₄₃	#101	1	4	yes	—
123	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₂²×C₈	1	8	yes	—
128	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
129	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
130	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
131	C₈×C₄₃	—	4	8	no	—
132	C₂×#1461	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₄₃	—	2	1	no	—
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	—	2	1	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	4	no	—
139	C₂²×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
140	C₂²×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
141	C₂²×C₄₃	#410	1	2	yes	—
142	C₂²×C₄₃	C₂×#101	1	2	yes	—
143	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
145	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
146	C₄×C₄₃	—	2	8	no	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	4	4	no	—
151	C₄×C₄₃	C₂×C₂²⋊C₈	1	8	yes	—
152	C₄×C₄₃	C₂×#101	1	4	yes	—
153	C₂²×C₄×Q₈	C₂×C₄₃	1	1	yes	characteristic
154	C₂×#431	C₂×C₄₃	1	2	yes	—
155	#1461	C₂×C₄₃	1	8	yes	direct product factor
156	C₂×#324	C₂×C₄₃	1	4	yes	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	—	2	1	no	—
159	C₂×C₄₃	—	2	2	no	—
160	C₂×C₄₃	#1461	1	4	yes	direct product factor
161	C₂×C₄₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₄₃	C₂×#410	1	1	yes	characteristic
163	C₂×C₄₃	C₂²×#101	1	1	yes	characteristic
164	C₂³×Q₈	C₄×C₄₃	1	1	yes	characteristic
165	C₂×C₄×Q₈	—	2	2	no	—
166	C₂×C₄×Q₈	—	2	2	no	—
167	C₂×C₄×Q₈	—	2	4	no	—
168	C₂×C₄×Q₈	C₄×C₄₃	1	4	yes	—
169	C₂×C₄×Q₈	C₂²×C₄₃	1	4	yes	—
170	C₂²×C₄⋊C₄	C₄×C₄₃	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
172	C₂²×C₄²	C₄×C₄₃	1	2	yes	—
173	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
174	#431	—	2	4	no	—
175	#431	C₂²×C₄₃	1	8	yes	—
176	C₂×C₄⋊C₈	—	2	2	no	—
177	C₂×C₂²⋊C₈	C₂²×C₄₃	1	4	yes	—
178	C₂×C₄×C₈	—	2	2	no	—
179	#324	C₂²×C₄₃	1	16	yes	—
180	C₄₃	C₂×#1461	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×Q₈	—	2	2	no	—
183	C₂²×Q₈	—	2	4	no	—
184	C₂²×Q₈	C₈×C₄₃	1	8	yes	complemented
185	C₂²×Q₈	C₂×C₄×C₄₃	1	4	yes	—
186	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
187	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
188	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
189	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
190	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
191	C₂²×C₈	—	2	4	no	—
192	C₄×Q₈	—	2	8	no	—
193	C₄×Q₈	—	2	8	no	—
194	C₄×Q₈	—	2	8	no	—
195	C₄×Q₈	—	2	8	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	4	yes	—
201	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₄₃	1	4	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂×C₄×C₄₃	1	8	yes	—
207	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
208	C₂×C₄²	D₄×C₄₃	1	2	yes	—
209	C₂×C₄²	D₄×C₄₃	1	2	yes	—
210	C₂×C₄²	D₄×C₄₃	1	4	yes	—
211	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
212	C₄⋊C₈	—	2	8	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₂²⋊C₈	C₂³×C₄₃	1	8	yes	—
215	C₄×C₈	—	2	8	no	—
216	C₂⁴	C₂²×C₄×C₄₃	1	1	yes	characteristic
217	C₂×Q₈	—	2	16	no	—
218	C₂×Q₈	—	2	16	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	8	no	—
221	C₂×Q₈	C₂×C₈×C₄₃	1	16	yes	complemented
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
232	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
234	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₈×C₄₃	1	8	yes	—
236	C₂²×C₄	C₂×C₈×C₄₃	1	4	yes	—
237	C₂²×C₄	M₁₆×C₄₃	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
240	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
241	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	4	8	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄²	—	2	8	no	—
252	C₄²	—	2	8	no	—
253	C₄²	C₂²⋊C₄×C₄₃	1	4	yes	—
254	C₄²	C₂²⋊C₄×C₄₃	1	4	yes	—
255	C₄²	C₂×D₄×C₄₃	1	4	yes	—
256	C₄²	C₂×D₄×C₄₃	1	4	yes	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
260	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
261	C₂³	C₂²×C₈×C₄₃	1	2	yes	—
262	C₂³	C₂×M₁₆×C₄₃	1	2	yes	—
263	C₂³	C₂³×C₄×C₄₃	1	1	yes	characteristic
264	Q₈	—	2	32	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂²⋊C₈×C₄₃	1	8	yes	—
281	C₂×C₄	C₂²⋊C₈×C₄₃	1	8	yes	—
282	C₂×C₄	#101×C₄₃	1	4	yes	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	4	yes	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
287	C₂×C₄	C₂²×C₈×C₄₃	1	8	yes	—
288	C₂×C₄	C₂²×C₈×C₄₃	1	4	yes	—
289	C₂×C₄	C₂×M₁₆×C₄₃	1	4	yes	—
290	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
291	C₈	—	4	8	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	C₂×C₂²⋊C₈×C₄₃	1	2	yes	—
299	C₂²	C₂×C₂²⋊C₈×C₄₃	1	2	yes	—
300	C₂²	C₄₃×#410	1	2	yes	—
301	C₂²	C₂×#101×C₄₃	1	2	yes	—
302	C₂²	C₂²×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
303	C₂²	C₂³×C₈×C₄₃	1	1	yes	characteristic
304	C₂²	C₂²×M₁₆×C₄₃	1	1	yes	characteristic
305	C₄	—	2	8	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	4	no	—
310	C₄	C₂×C₂²⋊C₈×C₄₃	1	8	yes	—
311	C₄	C₂×#101×C₄₃	1	4	yes	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	C₄₃×#1461	1	4	yes	direct product factor
316	C₂	C₂²×C₂²⋊C₈×C₄₃	1	1	yes	characteristic
317	C₂	C₂×C₄₃×#410	1	1	yes	characteristic
318	C₂	C₂²×#101×C₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₃	C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₄₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₃ ◁ C₂²×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₃×#1461	C₂	direct (N × H)
24. C₂²×Q₈×C₄₃	C₈	semidirect
61. C₂×Q₈×C₄₃	C₂×C₈	semidirect
132. C₂×#1461	C₄₃	direct (N × H)
155. #1461	C₂×C₄₃	direct (N × H)
160. C₂×C₄₃	#1461	direct (N × H)
180. C₄₃	C₂×#1461	direct (N × H)
184. C₂²×Q₈	C₈×C₄₃	semidirect
221. C₂×Q₈	C₂×C₈×C₄₃	semidirect
315. C₂	C₄₃×#1461	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×C₄×Q₈×C₄₃	2
C₂×C₄₃×#431	2
C₄₃×#1461	2
C₂×C₄₃×#324	2
C₂×#1461	43

Subgroups — C2×C43×#1461 #24937365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₃×#1461 #24937365