Subgroups — C₄₃×#9589 #24929606

Order 11008.

326 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#1986	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1950	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1542	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1493	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1303	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1279	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#533	C₂²	1	1	yes	characteristic
9	C₄₃×#431	C₂²	1	1	yes	characteristic
10	C₂×#108×C₄₃	—	2	1	no	—
11	C₄₃×#423	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#341	—	2	1	no	—
17	C₄₃×#339	—	2	1	no	—
18	C₄₃×#339	—	2	1	no	—
19	C₄₃×#333	C₄	1	1	yes	characteristic, complemented
20	C₄₃×#333	C₄	1	1	yes	characteristic, complemented
21	C₄₃×#332	C₄	1	1	yes	characteristic, complemented
22	C₄₃×#332	C₄	1	1	yes	characteristic, complemented
23	C₄₃×Q₈⋊C₈	—	2	1	no	—
24	C₄₃×Q₈⋊C₈	—	2	1	no	—
25	C₄₃×#322	—	2	1	no	—
26	C₈²×C₄₃	—	2	1	no	—
27	C₂×M₁₆×C₄₃	—	2	1	no	—
28	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	—	2	1	no	—
31	C₂²×C₈×C₄₃	—	2	1	no	—
32	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
34	#125×C₄₃	—	2	1	no	—
35	C₂²⋊Q₈×C₄₃	—	2	1	no	—
36	C₂²⋊Q₈×C₄₃	—	2	1	no	—
37	C₄×Q₈×C₄₃	—	2	1	no	—
38	C₄×Q₈×C₄₃	—	2	1	no	—
39	#117×C₄₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄²×C₄₃	—	2	1	no	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#108×C₄₃	—	4	2	no	—
44	C₄⋊C₈×C₄₃	—	2	1	no	—
45	C₄⋊C₈×C₄₃	—	2	1	no	—
46	C₄⋊C₈×C₄₃	—	2	1	no	—
47	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₄₃	—	2	1	no	—
52	#97×C₄₃	—	2	1	no	—
53	C₄×C₈×C₄₃	—	2	1	no	—
54	C₄×C₈×C₄₃	—	2	1	no	—
55	C₄×C₈×C₄₃	—	2	1	no	—
56	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
58	#95×C₄₃	—	2	1	no	—
59	C₂×Q₈×C₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	M₁₆×C₄₃	—	2	1	no	—
66	M₁₆×C₄₃	—	2	1	no	—
67	M₁₆×C₄₃	—	4	1	no	—
68	M₁₆×C₄₃	—	4	1	no	—
69	M₁₆×C₄₃	SD₁₆	1	2	yes	—
70	M₁₆×C₄₃	Q₁₆	1	2	yes	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	—	2	1	no	—
73	C₂×C₈×C₄₃	—	2	1	no	—
74	C₂×C₈×C₄₃	—	2	1	no	—
75	C₂×C₈×C₄₃	—	2	1	no	—
76	C₂×C₈×C₄₃	—	2	1	no	—
77	C₂×C₈×C₄₃	—	2	1	no	—
78	C₂×C₈×C₄₃	—	2	1	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₂×C₈×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	2	2	no	—
84	C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₄⋊C₄×C₄₃	—	2	1	no	—
86	C₄⋊C₄×C₄₃	—	2	1	no	—
87	C₄⋊C₄×C₄₃	—	2	1	no	—
88	C₄⋊C₄×C₄₃	—	4	1	no	—
89	C₄⋊C₄×C₄₃	—	4	1	no	—
90	C₄⋊C₄×C₄₃	—	4	1	no	—
91	C₂²⋊C₄×C₄₃	—	4	1	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	—	4	1	no	—
94	C₄²×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	#41	1	1	yes	characteristic
98	C₄²×C₄₃	#41	1	1	yes	characteristic
99	C₂³×C₄₃	#124	1	1	yes	characteristic
100	Q₈×C₄₃	—	4	1	no	—
101	Q₈×C₄₃	—	4	1	no	—
102	Q₈×C₄₃	—	8	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	4	1	no	—
106	C₂×C₄×C₄₃	—	4	2	no	—
107	C₂×C₄×C₄₃	—	4	1	no	—
108	C₂×C₄×C₄₃	—	4	1	no	—
109	C₂×C₄×C₄₃	—	8	1	no	—
110	C₂×C₄×C₄₃	—	8	1	no	—
111	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
121	C₈×C₄₃	—	2	2	no	—
122	C₈×C₄₃	—	2	2	no	—
123	C₈×C₄₃	—	2	2	no	—
124	C₈×C₄₃	—	4	1	no	—
125	C₈×C₄₃	—	4	1	no	—
126	C₈×C₄₃	—	4	1	no	—
127	C₈×C₄₃	—	4	1	no	—
128	C₈×C₄₃	—	4	1	no	—
129	C₈×C₄₃	—	8	1	no	—
130	#9589	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	#388	1	1	yes	characteristic
134	C₂²×C₄₃	#487	1	1	yes	characteristic
135	C₂²×C₄₃	#488	1	1	yes	characteristic
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	8	2	no	—
140	C₄×C₄₃	—	8	1	no	—
141	C₄×C₄₃	—	8	1	no	—
142	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
143	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
144	C₄×C₄₃	#443	1	1	yes	characteristic
145	C₄×C₄₃	#444	1	1	yes	characteristic
146	C₄×C₄₃	#465	1	1	yes	characteristic
147	C₄×C₄₃	#467	1	1	yes	characteristic
148	#1986	C₂×C₄₃	1	1	yes	characteristic
149	#1950	C₂×C₄₃	1	1	yes	characteristic
150	#1542	C₂×C₄₃	1	1	yes	characteristic
151	#1493	C₂×C₄₃	1	1	yes	characteristic
152	C₈×M₁₆	C₂×C₄₃	1	1	yes	characteristic
153	#1303	C₂×C₄₃	1	1	yes	characteristic
154	#1279	C₂×C₄₃	1	1	yes	characteristic
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	#1929	1	1	yes	characteristic
158	C₂×C₄₃	#1932	1	1	yes	characteristic
159	C₂×C₄₃	#1933	1	1	yes	characteristic
160	#533	C₂²×C₄₃	1	1	yes	characteristic
161	#431	C₂²×C₄₃	1	1	yes	characteristic
162	C₂×#108	—	2	1	no	—
163	#423	C₂²×C₄₃	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
165	C₄×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
167	C₄×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
168	#341	—	2	1	no	—
169	#339	—	2	1	no	—
170	#339	—	2	1	no	—
171	#333	C₄×C₄₃	1	1	yes	characteristic, complemented
172	#333	C₄×C₄₃	1	1	yes	characteristic, complemented
173	#332	C₄×C₄₃	1	1	yes	characteristic, complemented
174	#332	C₄×C₄₃	1	1	yes	characteristic, complemented
175	Q₈⋊C₈	—	2	1	no	—
176	Q₈⋊C₈	—	2	1	no	—
177	#322	—	2	1	no	—
178	C₈²	—	2	1	no	—
179	C₄₃	#9589	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	D₄×C₄₃	1	1	yes	characteristic
182	C₂×M₁₆	D₄×C₄₃	1	1	yes	characteristic
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₄⋊Q₈	C₂×C₄×C₄₃	1	1	yes	characteristic
186	C₄⋊Q₈	C₂×C₄×C₄₃	1	1	yes	characteristic
187	#125	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	—	2	1	no	—
192	#117	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
196	#108	—	4	2	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
201	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
203	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	D₄×C₄₃	1	1	yes	characteristic
210	C₄×C₈	D₄×C₄₃	1	1	yes	characteristic
211	#95	—	2	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
216	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
217	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	SD₁₆×C₄₃	1	2	yes	—
223	M₁₆	Q₁₆×C₄₃	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	C₂²×C₄×C₄₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
250	C₄²	#41×C₄₃	1	1	yes	characteristic
251	C₄²	#41×C₄₃	1	1	yes	characteristic
252	C₂³	#124×C₄₃	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	8	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
267	C₂×C₄	#121×C₄₃	1	1	yes	characteristic
268	C₂×C₄	#123×C₄₃	1	1	yes	characteristic
269	C₂×C₄	#126×C₄₃	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₄₃	1	1	yes	characteristic
271	C₂×C₄	C₂×Q₁₆×C₄₃	1	1	yes	characteristic
272	C₂×C₄	#135×C₄₃	1	1	yes	characteristic
273	C₂×C₄	#135×C₄₃	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	4	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	4	1	no	—
282	C₈	—	8	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₄₃×#388	1	1	yes	characteristic
286	C₂²	C₄₃×#487	1	1	yes	characteristic
287	C₂²	C₄₃×#488	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	1	no	—
293	C₄	—	8	1	no	—
294	C₄	C₄×SD₁₆×C₄₃	1	1	yes	characteristic
295	C₄	C₄×Q₁₆×C₄₃	1	1	yes	characteristic
296	C₄	C₄₃×#443	1	1	yes	characteristic
297	C₄	C₄₃×#444	1	1	yes	characteristic
298	C₄	C₄₃×#465	1	1	yes	characteristic
299	C₄	C₄₃×#467	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	C₄₃×#1929	1	1	yes	characteristic
303	C₂	C₄₃×#1932	1	1	yes	characteristic
304	C₂	C₄₃×#1933	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₃	#121
commutator	C₄²	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
socle	C₂²×C₄₃	#388
prime radical	C₂³×C₄₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₄₃×#333	C₄	semidirect
20. C₄₃×#333	C₄	semidirect
21. C₄₃×#332	C₄	semidirect
22. C₄₃×#332	C₄	semidirect
130. #9589	C₄₃	direct (N × H)
171. #333	C₄×C₄₃	semidirect
172. #333	C₄×C₄₃	semidirect
173. #332	C₄×C₄₃	semidirect
174. #332	C₄×C₄₃	semidirect
179. C₄₃	#9589	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×#1986	2
C₄₃×#1950	2
C₄₃×#1542	2
C₄₃×#1493	2
C₈×M₁₆×C₄₃	2
C₄₃×#1303	2
C₄₃×#1279	2
#9589	43

Subgroups — C43×#9589 #24929606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#9589 #24929606