Subgroups — C₄₃×#8388 #24928405

Order 11008.

966 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1801	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1546	C₂	1	2	yes	maximal, complemented
4	C₄₃×#1486	C₂	1	2	yes	maximal
5	C₄₃×#1461	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₄₃	—	2	1	no	—
7	D₄×Q₈×C₄₃	—	2	4	no	—
8	D₄×Q₈×C₄₃	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₄₃	C₄	1	2	yes	—
10	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₄₃	—	2	1	no	—
12	C₂×C₂²⋊Q₈×C₄₃	—	2	2	no	—
13	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₃	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#431	C₂²	1	2	yes	—
17	C₄₃×#423	C₂²	1	2	yes	—
18	C₂×#101×C₄₃	—	2	1	no	—
19	C₄₃×#343	—	2	1	no	—
20	C₄₃×#332	—	2	4	no	—
21	C₄₃×D₄⋊C₈	—	2	4	no	—
22	C₄₃×#324	—	2	2	no	—
23	C₂²×Q₈×C₄₃	—	2	8	no	—
24	C₂²×Q₈×C₄₃	—	2	2	no	—
25	C₂²×Q₈×C₄₃	—	2	1	no	—
26	C₂²×Q₈×C₄₃	—	4	4	no	—
27	C₂²×Q₈×C₄₃	—	4	2	no	—
28	C₂²×Q₈×C₄₃	—	4	2	no	—
29	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₃	—	2	2	no	—
32	C₂³×C₄×C₄₃	—	2	1	no	—
33	C₂×M₁₆×C₄₃	—	2	2	no	—
34	C₄⋊Q₈×C₄₃	—	2	4	no	—
35	C₄⋊Q₈×C₄₃	—	2	4	no	—
36	C₄⋊Q₈×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	D₄	1	4	yes	—
38	C₂²⋊Q₈×C₄₃	—	2	8	no	—
39	C₂²⋊Q₈×C₄₃	—	2	16	no	—
40	C₄×Q₈×C₄₃	—	2	2	no	—
41	C₄×Q₈×C₄₃	—	2	2	no	—
42	C₄×D₄×C₄₃	—	2	4	no	—
43	C₄×D₄×C₄₃	—	2	4	no	—
44	C₄×D₄×C₄₃	—	2	2	no	—
45	C₄×D₄×C₄₃	D₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	—	2	2	no	—
57	C₄⋊C₈×C₄₃	—	2	2	no	—
58	C₄⋊C₈×C₄₃	—	2	2	no	—
59	#101×C₄₃	—	4	2	no	—
60	C₂²⋊C₈×C₄₃	—	2	2	no	—
61	C₄×C₈×C₄₃	—	2	2	no	—
62	C₂⁴×C₄₃	—	2	1	no	—
63	C₂×Q₈×C₄₃	—	2	8	no	—
64	C₂×Q₈×C₄₃	—	2	2	no	—
65	C₂×Q₈×C₄₃	—	2	4	no	—
66	C₂×Q₈×C₄₃	—	2	2	no	—
67	C₂×Q₈×C₄₃	—	4	4	no	—
68	C₂×Q₈×C₄₃	—	4	16	no	—
69	C₂×Q₈×C₄₃	—	4	16	no	—
70	C₂×Q₈×C₄₃	—	4	4	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂×Q₈×C₄₃	—	4	1	no	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂×Q₈×C₄₃	D₈	1	2	yes	—
75	C₂×Q₈×C₄₃	SD₁₆	1	2	yes	—
76	C₂×D₄×C₄₃	—	2	4	no	—
77	C₂×D₄×C₄₃	—	2	1	no	—
78	C₂×D₄×C₄₃	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₄₃	—	2	8	no	—
80	C₂²×C₄×C₄₃	—	2	4	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	1	no	—
83	C₂²×C₄×C₄₃	—	4	4	no	—
84	C₂²×C₄×C₄₃	—	4	4	no	—
85	C₂²×C₄×C₄₃	—	4	1	no	—
86	C₂²×C₄×C₄₃	—	4	1	no	—
87	C₂²×C₄×C₄₃	—	4	2	no	—
88	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	M₁₆×C₄₃	—	4	2	no	—
94	M₁₆×C₄₃	—	4	2	no	—
95	C₂×C₈×C₄₃	—	4	2	no	—
96	C₂×C₈×C₄₃	—	4	2	no	—
97	C₄⋊C₄×C₄₃	—	2	8	no	—
98	C₄⋊C₄×C₄₃	—	2	1	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	4	no	—
101	C₄⋊C₄×C₄₃	—	2	4	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
104	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₄₃	—	2	8	no	—
106	C₂²⋊C₄×C₄₃	—	2	4	no	—
107	C₄²×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	—	2	2	no	—
109	C₄²×C₄₃	C₂×D₄	1	2	yes	—
110	C₄²×C₄₃	C₂×D₄	1	2	yes	—
111	C₂³×C₄₃	—	2	4	no	—
112	C₂³×C₄₃	—	2	2	no	—
113	C₂³×C₄₃	—	4	2	no	—
114	C₂³×C₄₃	—	4	1	no	—
115	C₂³×C₄₃	—	4	1	no	—
116	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	Q₈×C₄₃	—	2	4	no	—
118	Q₈×C₄₃	—	2	4	no	—
119	Q₈×C₄₃	—	4	16	no	—
120	Q₈×C₄₃	—	4	4	no	—
121	Q₈×C₄₃	—	4	4	no	—
122	D₄×C₄₃	—	2	4	no	—
123	D₄×C₄₃	—	2	2	no	—
124	D₄×C₄₃	#101	1	4	yes	complemented
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	16	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	4	no	—
138	C₂×C₄×C₄₃	—	4	4	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	#102	1	4	yes	—
141	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#120	1	2	yes	—
144	C₂×C₄×C₄₃	#120	1	2	yes	—
145	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₄₃	—	8	2	no	—
149	C₈×C₄₃	—	8	2	no	—
150	#8388	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₄₃	—	2	1	no	—
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	4	no	—
154	C₂²×C₄₃	—	4	4	no	—
155	C₂²×C₄₃	—	4	4	no	—
156	C₂²×C₄₃	—	4	2	no	—
157	C₂²×C₄₃	—	4	2	no	—
158	C₂²×C₄₃	#379	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
160	C₂²×C₄₃	#418	1	1	yes	characteristic
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	2	4	no	—
164	C₄×C₄₃	—	4	2	no	—
165	C₄×C₄₃	—	4	4	no	—
166	C₄×C₄₃	—	4	2	no	—
167	C₄×C₄₃	C₂×#101	1	2	yes	—
168	C₄×C₄₃	#447	1	2	yes	—
169	C₄×C₄₃	#450	1	2	yes	—
170	C₂×D₄×Q₈	C₂×C₄₃	1	1	yes	characteristic
171	#1801	C₂×C₄₃	1	1	yes	characteristic, complemented
172	#1546	C₂×C₄₃	1	2	yes	complemented
173	#1486	C₂×C₄₃	1	2	yes	—
174	#1461	C₂×C₄₃	1	1	yes	characteristic, complemented
175	C₂×C₄₃	—	2	1	no	—
176	C₂×C₄₃	—	2	1	no	—
177	C₂×C₄₃	—	4	4	no	—
178	C₂×C₄₃	#1784	1	1	yes	characteristic
179	C₂×C₄₃	#1786	1	1	yes	characteristic
180	C₂×C₄₃	#1796	1	1	yes	characteristic
181	C₂³×Q₈	—	2	1	no	—
182	D₄×Q₈	—	2	4	no	—
183	D₄×Q₈	—	2	4	no	—
184	C₂×C₄⋊Q₈	C₄×C₄₃	1	2	yes	—
185	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
186	C₂×C₂²⋊Q₈	—	2	1	no	—
187	C₂×C₂²⋊Q₈	—	2	2	no	—
188	C₂×C₄×Q₈	C₂²×C₄₃	1	1	yes	characteristic
189	C₂×C₄×D₄	C₄×C₄₃	1	2	yes	—
190	C₂×C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
191	#431	C₂²×C₄₃	1	2	yes	—
192	#423	C₂²×C₄₃	1	2	yes	—
193	C₂×#101	—	2	1	no	—
194	#343	—	2	1	no	—
195	#332	—	2	4	no	—
196	D₄⋊C₈	—	2	4	no	—
197	#324	—	2	2	no	—
198	C₄₃	#8388	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	—	2	8	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×Q₈	—	4	4	no	—
203	C₂²×Q₈	—	4	2	no	—
204	C₂²×Q₈	—	4	2	no	—
205	C₂²×Q₈	D₄×C₄₃	1	1	yes	characteristic
206	C₂²×D₄	C₂×C₄×C₄₃	1	1	yes	characteristic
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	—	2	1	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	2	4	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	D₄×C₄₃	1	4	yes	—
214	C₂²⋊Q₈	—	2	8	no	—
215	C₂²⋊Q₈	—	2	16	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×D₄	—	2	4	no	—
219	C₄×D₄	—	2	4	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	D₄×C₄₃	1	4	yes	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
227	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
231	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	—	2	2	no	—
235	#101	—	4	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₂⁴	—	2	1	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	16	no	—
245	C₂×Q₈	—	4	16	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	D₈×C₄₃	1	2	yes	—
251	C₂×Q₈	SD₁₆×C₄₃	1	2	yes	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	C₂²⋊C₄×C₄₃	1	2	yes	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
265	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
266	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	4	yes	—
280	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
281	C₂²⋊C₄	—	2	8	no	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂×D₄×C₄₃	1	2	yes	—
286	C₄²	C₂×D₄×C₄₃	1	2	yes	—
287	C₂³	—	2	4	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
293	Q₈	—	2	4	no	—
294	Q₈	—	2	4	no	—
295	Q₈	—	4	16	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	2	2	no	—
300	D₄	#101×C₄₃	1	4	yes	complemented
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	16	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	#102×C₄₃	1	4	yes	—
317	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
318	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
319	C₂×C₄	#120×C₄₃	1	2	yes	—
320	C₂×C₄	#120×C₄₃	1	2	yes	—
321	C₂×C₄	C₂×D₈×C₄₃	1	1	yes	characteristic
322	C₂×C₄	C₂×SD₁₆×C₄₃	1	1	yes	characteristic
323	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	2	yes	—
324	C₈	—	8	2	no	—
325	C₈	—	8	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	C₄₃×#379	1	1	yes	characteristic
334	C₂²	C₂×#102×C₄₃	1	1	yes	characteristic
335	C₂²	C₄₃×#418	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	C₂×#101×C₄₃	1	2	yes	—
343	C₄	C₄₃×#447	1	2	yes	—
344	C₄	C₄₃×#450	1	2	yes	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	4	no	—
348	C₂	C₄₃×#1784	1	1	yes	characteristic
349	C₂	C₄₃×#1786	1	1	yes	characteristic
350	C₂	C₄₃×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₂²×D₄×C₄₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1801	C₂	semidirect
3. C₄₃×#1546	C₂	semidirect
5. C₄₃×#1461	C₂	semidirect
124. D₄×C₄₃	#101	semidirect
150. #8388	C₄₃	direct (N × H)
171. #1801	C₂×C₄₃	semidirect
172. #1546	C₂×C₄₃	semidirect
174. #1461	C₂×C₄₃	semidirect
198. C₄₃	#8388	direct (N × H)
300. D₄	#101×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₄₃	2
C₄₃×#1801	2
C₄₃×#1546	2
C₄₃×#1486	2
C₄₃×#1461	2
#8388	43

Subgroups — C43×#8388 #24928405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8388 #24928405