Subgroups — C₄₃×#20306 #24940323

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#3367	C₂	1	2	yes	maximal
2	C₄₃×#3328	C₂	1	2	yes	maximal, complemented
3	C₄₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1700	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1699	C₂	1	2	yes	maximal, complemented
6	C₄₃×#1650	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1649	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1589	C₂	1	1	yes	maximal, characteristic
10	C₄₃×#556	C₂²	1	2	yes	—
11	D₄×Q₈×C₄₃	C₂²	1	2	yes	—
12	C₄₃×#500	C₂²	1	1	yes	characteristic
13	C₄₃×#500	C₂²	1	1	yes	characteristic
14	C₄₃×#489	—	2	2	no	—
15	C₄₃×#487	C₂²	1	2	yes	—
16	C₄₃×#487	C₂²	1	2	yes	—
17	C₄₃×#485	—	2	2	no	—
18	C₄₃×#477	—	2	2	no	—
19	C₄₃×#475	C₂²	1	2	yes	—
20	C₄₃×#474	C₂²	1	2	yes	—
21	C₄₃×#470	—	2	2	no	—
22	C₄₃×#463	—	2	2	no	—
23	C₄₃×#462	C₂²	1	2	yes	—
24	C₄₃×#461	C₂²	1	2	yes	—
25	C₄₃×#451	—	2	2	no	—
26	C₄₃×#446	C₂²	1	2	yes	—
27	C₄₃×#440	—	2	2	no	—
28	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#436	C₂²	1	2	yes	—
31	C₄₃×#419	—	2	2	no	—
32	C₄₃×#335	C₂²	1	1	yes	characteristic
33	C₄₃×#335	C₂²	1	1	yes	characteristic
34	C₄₃×#332	C₂²	1	2	yes	—
35	C₄₃×#332	C₂²	1	2	yes	—
36	C₄₃×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₃×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₃	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₄₃	—	2	2	no	—
40	C₂×Q₁₆×C₄₃	—	2	2	no	—
41	C₂×Q₁₆×C₄₃	D₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₄₃	—	2	2	no	—
43	C₂×D₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₄₃	—	2	2	no	—
45	C₄⋊Q₈×C₄₃	—	2	2	no	—
46	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
48	#126×C₄₃	—	4	2	no	—
49	#124×C₄₃	—	2	2	no	—
50	C₂²⋊Q₈×C₄₃	—	2	2	no	—
51	C₂²⋊Q₈×C₄₃	—	2	2	no	—
52	C₂²⋊Q₈×C₄₃	—	4	2	no	—
53	C₄×Q₈×C₄₃	—	2	2	no	—
54	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
55	C₄×D₄×C₄₃	—	2	2	no	—
56	C₄×D₄×C₄₃	C₂³	1	2	yes	—
57	#117×C₄₃	—	2	2	no	—
58	#107×C₄₃	—	2	2	no	—
59	#107×C₄₃	—	4	1	no	—
60	#107×C₄₃	—	4	1	no	—
61	#107×C₄₃	D₄	1	1	yes	characteristic
62	#107×C₄₃	D₄	1	1	yes	characteristic
63	#106×C₄₃	—	2	2	no	—
64	C₄⋊C₈×C₄₃	—	2	2	no	—
65	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
66	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
67	#103×C₄₃	—	2	2	no	—
68	#103×C₄₃	—	2	2	no	—
69	#103×C₄₃	—	2	2	no	—
70	#103×C₄₃	—	2	2	no	—
71	#103×C₄₃	—	2	2	no	—
72	#103×C₄₃	—	2	2	no	—
73	#103×C₄₃	—	2	2	no	—
74	#103×C₄₃	D₄	1	2	yes	—
75	#102×C₄₃	—	2	2	no	—
76	#102×C₄₃	D₄	1	2	yes	—
77	C₂²⋊C₈×C₄₃	—	2	2	no	—
78	#97×C₄₃	—	2	2	no	—
79	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×Q₈×C₄₃	—	4	2	no	—
85	C₂×Q₈×C₄₃	—	4	4	no	—
86	C₂×Q₈×C₄₃	—	8	2	no	—
87	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×D₄×C₄₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₃	—	2	2	no	—
90	C₂²×C₄×C₄₃	—	4	2	no	—
91	Q₁₆×C₄₃	—	4	1	no	—
92	Q₁₆×C₄₃	—	8	2	no	—
93	SD₁₆×C₄₃	—	4	4	no	—
94	D₈×C₄₃	—	4	1	no	—
95	M₁₆×C₄₃	—	8	2	no	—
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	4	2	no	—
109	C₄⋊C₄×C₄₃	—	4	2	no	—
110	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₄₃	—	2	2	no	—
113	C₂²⋊C₄×C₄₃	—	4	2	no	—
114	C₂²⋊C₄×C₄₃	—	4	2	no	—
115	C₂²⋊C₄×C₄₃	—	4	2	no	—
116	C₄²×C₄₃	—	2	2	no	—
117	C₄²×C₄₃	—	4	2	no	—
118	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
119	C₂³×C₄₃	—	2	2	no	—
120	Q₈×C₄₃	—	2	2	no	—
121	Q₈×C₄₃	—	2	2	no	—
122	Q₈×C₄₃	—	4	2	no	—
123	Q₈×C₄₃	—	4	2	no	—
124	Q₈×C₄₃	—	4	2	no	—
125	Q₈×C₄₃	—	4	2	no	—
126	Q₈×C₄₃	—	8	4	no	—
127	D₄×C₄₃	—	2	2	no	—
128	D₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	8	4	no	—
137	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	—	2	1	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₈×C₄₃	—	4	2	no	—
146	C₈×C₄₃	—	4	2	no	—
147	#20306	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₃	—	4	2	no	—
149	C₂²×C₄₃	—	4	2	no	—
150	C₂²×C₄₃	—	4	2	no	—
151	C₂²×C₄₃	D₄²	1	1	yes	characteristic
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	4	2	no	—
154	C₄×C₄₃	—	4	2	no	—
155	C₄×C₄₃	—	4	2	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	—	8	2	no	—
158	C₄×C₄₃	#575	1	1	yes	characteristic
159	C₄×C₄₃	#575	1	1	yes	characteristic
160	C₄×C₄₃	#576	1	1	yes	characteristic
161	C₄×C₄₃	#578	1	1	yes	characteristic
162	#3367	C₂×C₄₃	1	2	yes	—
163	#3328	C₂×C₄₃	1	2	yes	complemented
164	#1710	C₂×C₄₃	1	1	yes	characteristic, complemented
165	#1700	C₂×C₄₃	1	2	yes	complemented
166	#1699	C₂×C₄₃	1	2	yes	complemented
167	#1650	C₂×C₄₃	1	2	yes	complemented
168	#1649	C₂×C₄₃	1	2	yes	complemented
169	#1591	C₂×C₄₃	1	1	yes	characteristic, complemented
170	#1589	C₂×C₄₃	1	1	yes	characteristic
171	C₂×C₄₃	—	8	2	no	—
172	C₂×C₄₃	#3273	1	1	yes	characteristic
173	C₂×C₄₃	#3277	1	1	yes	characteristic
174	C₂×C₄₃	#3293	1	1	yes	characteristic
175	#556	C₂²×C₄₃	1	2	yes	—
176	D₄×Q₈	C₂²×C₄₃	1	2	yes	—
177	#500	C₂²×C₄₃	1	1	yes	characteristic
178	#500	C₂²×C₄₃	1	1	yes	characteristic
179	#489	—	2	2	no	—
180	#487	C₂²×C₄₃	1	2	yes	—
181	#487	C₂²×C₄₃	1	2	yes	—
182	#485	—	2	2	no	—
183	#477	—	2	2	no	—
184	#475	C₂²×C₄₃	1	2	yes	—
185	#474	C₂²×C₄₃	1	2	yes	—
186	#470	—	2	2	no	—
187	#463	—	2	2	no	—
188	#462	C₂²×C₄₃	1	2	yes	—
189	#461	C₂²×C₄₃	1	2	yes	—
190	#451	—	2	2	no	—
191	#446	C₂²×C₄₃	1	2	yes	—
192	#440	—	2	2	no	—
193	C₄×Q₁₆	C₂²×C₄₃	1	1	yes	characteristic
194	C₄×D₈	C₂²×C₄₃	1	1	yes	characteristic
195	#436	C₂²×C₄₃	1	2	yes	—
196	#419	—	2	2	no	—
197	#335	C₂²×C₄₃	1	1	yes	characteristic
198	#335	C₂²×C₄₃	1	1	yes	characteristic
199	#332	C₂²×C₄₃	1	2	yes	—
200	#332	C₂²×C₄₃	1	2	yes	—
201	Q₈⋊C₈	C₂²×C₄₃	1	2	yes	—
202	D₄⋊C₈	C₂²×C₄₃	1	2	yes	—
203	C₈²	C₂²×C₄₃	1	1	yes	characteristic
204	C₄₃	#20306	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×Q₈	—	2	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	D₄×C₄₃	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×D₈	D₄×C₄₃	1	1	yes	characteristic
210	C₂×M₁₆	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂³×C₄₃	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₄₃	1	2	yes	—
214	#126	—	4	2	no	—
215	#124	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	4	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	C₂³×C₄₃	1	2	yes	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₄₃	1	2	yes	—
223	#117	—	2	2	no	—
224	#107	—	2	2	no	—
225	#107	—	4	1	no	—
226	#107	—	4	1	no	—
227	#107	D₄×C₄₃	1	1	yes	characteristic
228	#107	D₄×C₄₃	1	1	yes	characteristic
229	#106	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	C₂³×C₄₃	1	2	yes	—
232	C₄⋊C₈	C₂³×C₄₃	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	#103	D₄×C₄₃	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	D₄×C₄₃	1	2	yes	—
243	C₂²⋊C₈	—	2	2	no	—
244	#97	—	2	2	no	—
245	C₄×C₈	C₂³×C₄₃	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₄₃	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₄₃	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	8	2	no	—
253	C₂×Q₈	C₂×D₄×C₄₃	1	2	yes	—
254	C₂×D₄	C₂×D₄×C₄₃	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	Q₁₆	—	4	1	no	—
258	Q₁₆	—	8	2	no	—
259	SD₁₆	—	4	4	no	—
260	D₈	—	4	1	no	—
261	M₁₆	—	8	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₄₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₄₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₄₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₄₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	C₂×D₄×C₄₃	1	2	yes	—
277	C₄⋊C₄	C₂×D₄×C₄₃	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂⁴×C₄₃	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	4	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
305	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	D₄²×C₄₃	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₄₃×#575	1	1	yes	characteristic
324	C₄	C₄₃×#575	1	1	yes	characteristic
325	C₄	C₄₃×#576	1	1	yes	characteristic
326	C₄	C₄₃×#578	1	1	yes	characteristic
327	C₂	—	8	2	no	—
328	C₂	C₄₃×#3273	1	1	yes	characteristic
329	C₂	C₄₃×#3277	1	1	yes	characteristic
330	C₂	C₄₃×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₃
prime radical	C₂×D₈×C₄₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃ ◁ C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#3328	C₂	semidirect
3. C₄₃×#1710	C₂	semidirect
4. C₄₃×#1700	C₂	semidirect
5. C₄₃×#1699	C₂	semidirect
6. C₄₃×#1650	C₂	semidirect
7. C₄₃×#1649	C₂	semidirect
8. C₄₃×#1591	C₂	semidirect
147. #20306	C₄₃	direct (N × H)
163. #3328	C₂×C₄₃	semidirect
164. #1710	C₂×C₄₃	semidirect
165. #1700	C₂×C₄₃	semidirect
166. #1699	C₂×C₄₃	semidirect
167. #1650	C₂×C₄₃	semidirect
168. #1649	C₂×C₄₃	semidirect
169. #1591	C₂×C₄₃	semidirect
204. C₄₃	#20306	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₃×#3367	2
C₄₃×#3328	2
C₄₃×#1710	2
C₄₃×#1700	2
C₄₃×#1699	2
C₄₃×#1650	2
C₄₃×#1649	2
C₄₃×#1591	2
C₄₃×#1589	2
#20306	43

Subgroups — C43×#20306 #24940323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#20306 #24940323