Subgroups — C₄₃×#8385 #24928402

Order 11008.

1050 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#550	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1800	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1545	C₂	1	2	yes	maximal, complemented
4	C₄₃×#1487	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1460	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₄₃	C₄	1	2	yes	—
7	C₄₃×#550	—	2	4	no	—
8	C₄₃×#550	—	2	4	no	—
9	C₂×#127×C₄₃	—	2	1	no	—
10	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×#121×C₄₃	—	2	2	no	—
12	C₂×#121×C₄₃	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₃	—	2	1	no	—
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×#100×C₄₃	—	2	1	no	—
19	C₄₃×#343	—	2	1	no	—
20	C₄₃×#331	—	2	4	no	—
21	C₄₃×Q₈⋊C₈	—	2	4	no	—
22	C₄₃×#323	—	2	2	no	—
23	C₂×#41×C₄₃	—	2	4	no	—
24	C₂×#41×C₄₃	—	2	4	no	—
25	C₂×#41×C₄₃	—	4	8	no	—
26	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₄₃	—	2	2	no	—
28	C₂²×D₄×C₄₃	—	2	2	no	—
29	C₂²×D₄×C₄₃	—	2	1	no	—
30	C₂²×D₄×C₄₃	—	2	1	no	—
31	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
32	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₃	—	2	2	no	—
34	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	#127×C₄₃	—	2	8	no	—
37	#127×C₄₃	—	2	2	no	—
38	#127×C₄₃	D₄	1	4	yes	—
39	#121×C₄₃	—	2	16	no	—
40	#121×C₄₃	—	2	4	no	—
41	#121×C₄₃	—	2	4	no	—
42	C₄×Q₈×C₄₃	—	2	2	no	—
43	C₄×Q₈×C₄₃	D₄	1	4	yes	—
44	C₄×D₄×C₄₃	—	2	8	no	—
45	C₄×D₄×C₄₃	—	2	2	no	—
46	C₄×D₄×C₄₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	—	2	1	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	—	2	2	no	—
55	C₄⋊C₈×C₄₃	—	2	2	no	—
56	C₄⋊C₈×C₄₃	—	2	2	no	—
57	#100×C₄₃	—	4	2	no	—
58	C₂²⋊C₈×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	—	2	2	no	—
60	C₂⁴×C₄₃	—	2	2	no	—
61	C₂⁴×C₄₃	C₂²⋊C₄	1	2	yes	—
62	#41×C₄₃	—	4	16	no	—
63	#41×C₄₃	—	4	8	no	—
64	#41×C₄₃	—	4	8	no	—
65	C₂×Q₈×C₄₃	—	2	4	no	—
66	C₂×Q₈×C₄₃	—	2	1	no	—
67	C₂×Q₈×C₄₃	C₂²⋊C₄	1	2	yes	—
68	C₂×D₄×C₄₃	—	2	8	no	—
69	C₂×D₄×C₄₃	—	2	8	no	—
70	C₂×D₄×C₄₃	—	2	2	no	—
71	C₂×D₄×C₄₃	—	2	2	no	—
72	C₂×D₄×C₄₃	—	2	4	no	—
73	C₂×D₄×C₄₃	—	2	4	no	—
74	C₂×D₄×C₄₃	—	2	2	no	—
75	C₂×D₄×C₄₃	—	2	1	no	—
76	C₂×D₄×C₄₃	—	2	1	no	—
77	C₂×D₄×C₄₃	—	4	4	no	—
78	C₂×D₄×C₄₃	—	4	8	no	—
79	C₂×D₄×C₄₃	—	4	2	no	—
80	C₂×D₄×C₄₃	—	4	1	no	—
81	C₂×D₄×C₄₃	—	4	1	no	—
82	C₂×D₄×C₄₃	—	8	2	no	—
83	C₂×D₄×C₄₃	—	8	4	no	—
84	C₂×D₄×C₄₃	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₄₃	—	2	8	no	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	1	no	—
89	C₂²×C₄×C₄₃	—	2	1	no	—
90	C₂²×C₄×C₄₃	—	4	4	no	—
91	C₂²×C₄×C₄₃	—	4	8	no	—
92	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	M₁₆×C₄₃	—	4	2	no	—
97	M₁₆×C₄₃	—	4	2	no	—
98	C₂×C₈×C₄₃	—	4	2	no	—
99	C₂×C₈×C₄₃	—	4	2	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	4	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₄₃	—	2	8	no	—
105	C₂²⋊C₄×C₄₃	—	2	4	no	—
106	C₄²×C₄₃	—	2	4	no	—
107	C₄²×C₄₃	C₂×D₄	1	2	yes	—
108	C₄²×C₄₃	C₂×D₄	1	2	yes	—
109	C₂³×C₄₃	—	2	2	no	—
110	C₂³×C₄₃	—	2	4	no	—
111	C₂³×C₄₃	—	2	2	no	—
112	C₂³×C₄₃	—	2	2	no	—
113	C₂³×C₄₃	—	4	4	no	—
114	C₂³×C₄₃	—	4	4	no	—
115	C₂³×C₄₃	—	4	2	no	—
116	C₂³×C₄₃	—	4	2	no	—
117	C₂³×C₄₃	—	4	2	no	—
118	C₂³×C₄₃	—	4	2	no	—
119	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	Q₈×C₄₃	—	2	4	no	—
121	Q₈×C₄₃	—	2	2	no	—
122	Q₈×C₄₃	#100	1	4	yes	complemented
123	D₄×C₄₃	—	4	4	no	—
124	D₄×C₄₃	—	4	16	no	—
125	D₄×C₄₃	—	4	4	no	—
126	D₄×C₄₃	—	4	2	no	—
127	D₄×C₄₃	—	4	4	no	—
128	D₄×C₄₃	—	4	2	no	—
129	D₄×C₄₃	—	4	4	no	—
130	D₄×C₄₃	—	8	4	no	—
131	D₄×C₄₃	—	8	8	no	—
132	C₂×C₄×C₄₃	—	2	4	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	16	no	—
139	C₂×C₄×C₄₃	—	4	4	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	4	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#120	1	2	yes	—
146	C₂×C₄×C₄₃	#120	1	2	yes	—
147	C₂×C₄×C₄₃	#135	1	2	yes	—
148	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
150	C₈×C₄₃	—	8	2	no	—
151	C₈×C₄₃	—	8	2	no	—
152	#8385	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	—	2	1	no	—
155	C₂²×C₄₃	—	4	4	no	—
156	C₂²×C₄₃	—	4	2	no	—
157	C₂²×C₄₃	—	4	4	no	—
158	C₂²×C₄₃	—	4	2	no	—
159	C₂²×C₄₃	—	4	2	no	—
160	C₂²×C₄₃	—	4	4	no	—
161	C₂²×C₄₃	—	8	2	no	—
162	C₂²×C₄₃	—	8	2	no	—
163	C₂²×C₄₃	—	8	4	no	—
164	C₂²×C₄₃	—	8	4	no	—
165	C₂²×C₄₃	#379	1	1	yes	characteristic
166	C₂²×C₄₃	#416	1	1	yes	characteristic
167	C₂²×C₄₃	#417	1	1	yes	characteristic
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	2	4	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	4	2	no	—
173	C₄×C₄₃	C₂×#100	1	2	yes	—
174	C₄×C₄₃	#449	1	2	yes	—
175	C₄×C₄₃	#452	1	2	yes	—
176	C₂×#550	C₂×C₄₃	1	1	yes	characteristic
177	#1800	C₂×C₄₃	1	1	yes	characteristic, complemented
178	#1545	C₂×C₄₃	1	2	yes	complemented
179	#1487	C₂×C₄₃	1	2	yes	complemented
180	#1460	C₂×C₄₃	1	1	yes	characteristic, complemented
181	C₂×C₄₃	—	2	1	no	—
182	C₂×C₄₃	—	2	1	no	—
183	C₂×C₄₃	—	8	2	no	—
184	C₂×C₄₃	—	8	4	no	—
185	C₂×C₄₃	#1783	1	1	yes	characteristic
186	C₂×C₄₃	#1788	1	1	yes	characteristic
187	C₂×C₄₃	#1795	1	1	yes	characteristic
188	C₂²×#41	C₄×C₄₃	1	2	yes	—
189	#550	—	2	4	no	—
190	#550	—	2	4	no	—
191	C₂×#127	—	2	1	no	—
192	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
193	C₂×#121	—	2	2	no	—
194	C₂×#121	C₄×C₄₃	1	2	yes	—
195	C₂×C₄×Q₈	C₂²×C₄₃	1	1	yes	characteristic
196	C₂×C₄×D₄	—	2	1	no	—
197	C₂×C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
198	#431	C₂²×C₄₃	1	2	yes	—
199	#423	C₂²×C₄₃	1	2	yes	—
200	C₂×#100	—	2	1	no	—
201	#343	—	2	1	no	—
202	#331	—	2	4	no	—
203	Q₈⋊C₈	—	2	4	no	—
204	#323	—	2	2	no	—
205	C₄₃	#8385	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×#41	—	2	4	no	—
207	C₂×#41	—	2	4	no	—
208	C₂×#41	—	4	8	no	—
209	C₂²×Q₈	C₂×C₄×C₄₃	1	1	yes	characteristic
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	—	2	2	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	2	1	no	—
214	C₂²×D₄	C₂×C₄×C₄₃	1	2	yes	—
215	C₂²×D₄	D₄×C₄₃	1	1	yes	characteristic
216	C₂³×C₄	—	2	2	no	—
217	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
218	C₂×M₁₆	—	2	2	no	—
219	#127	—	2	8	no	—
220	#127	—	2	2	no	—
221	#127	D₄×C₄₃	1	4	yes	—
222	#121	—	2	16	no	—
223	#121	—	2	4	no	—
224	#121	—	2	4	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	D₄×C₄₃	1	4	yes	—
227	C₄×D₄	—	2	8	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	#100	—	4	2	no	—
241	C₂²⋊C₈	—	2	2	no	—
242	C₄×C₈	—	2	2	no	—
243	C₂⁴	—	2	2	no	—
244	C₂⁴	C₂²⋊C₄×C₄₃	1	2	yes	—
245	#41	—	4	16	no	—
246	#41	—	4	8	no	—
247	#41	—	4	8	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	C₂²⋊C₄×C₄₃	1	2	yes	—
251	C₂×D₄	—	2	8	no	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	8	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	8	2	no	—
266	C₂×D₄	—	8	4	no	—
267	C₂×D₄	C₂²⋊C₄×C₄₃	1	4	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	8	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	8	no	—
274	C₂²×C₄	—	4	4	no	—
275	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
276	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	2	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
287	C₂²⋊C₄	—	2	8	no	—
288	C₂²⋊C₄	—	2	4	no	—
289	C₄²	—	2	4	no	—
290	C₄²	C₂×D₄×C₄₃	1	2	yes	—
291	C₄²	C₂×D₄×C₄₃	1	2	yes	—
292	C₂³	—	2	2	no	—
293	C₂³	—	2	4	no	—
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
303	Q₈	—	2	4	no	—
304	Q₈	—	2	2	no	—
305	Q₈	#100×C₄₃	1	4	yes	complemented
306	D₄	—	4	4	no	—
307	D₄	—	4	16	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	8	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	16	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
328	C₂×C₄	#120×C₄₃	1	2	yes	—
329	C₂×C₄	#120×C₄₃	1	2	yes	—
330	C₂×C₄	#135×C₄₃	1	2	yes	—
331	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	1	yes	characteristic
332	C₂×C₄	#137×C₄₃	1	1	yes	characteristic
333	C₈	—	8	2	no	—
334	C₈	—	8	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	4	no	—
346	C₂²	—	8	4	no	—
347	C₂²	C₄₃×#379	1	1	yes	characteristic
348	C₂²	C₄₃×#416	1	1	yes	characteristic
349	C₂²	C₄₃×#417	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	C₂×#100×C₄₃	1	2	yes	—
356	C₄	C₄₃×#449	1	2	yes	—
357	C₄	C₄₃×#452	1	2	yes	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	8	2	no	—
362	C₂	C₄₃×#1783	1	1	yes	characteristic
363	C₂	C₄₃×#1788	1	1	yes	characteristic
364	C₂	C₄₃×#1795	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	C₂×C₄₃×#550	C₂
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×#1800	C₂	semidirect
3. C₄₃×#1545	C₂	semidirect
4. C₄₃×#1487	C₂	semidirect
5. C₄₃×#1460	C₂	semidirect
122. Q₈×C₄₃	#100	semidirect
152. #8385	C₄₃	direct (N × H)
177. #1800	C₂×C₄₃	semidirect
178. #1545	C₂×C₄₃	semidirect
179. #1487	C₂×C₄₃	semidirect
180. #1460	C₂×C₄₃	semidirect
205. C₄₃	#8385	direct (N × H)
305. Q₈	#100×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₃×#550	2
C₄₃×#1800	2
C₄₃×#1545	2
C₄₃×#1487	2
C₄₃×#1460	2
#8385	43

Subgroups — C43×#8385 #24928402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8385 #24928402