Subgroups — C₄₃×#8471 #24928488

Order 11008.

924 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#3499	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1997	C₂	1	2	yes	maximal, complemented
3	C₄₃×#1560	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1489	C₂	1	2	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×C₄₃	C₄	1	2	yes	—
7	D₄×Q₈×C₄₃	—	2	2	no	—
8	C₄₃×#547	—	2	2	no	—
9	C₄₃×#547	—	2	4	no	—
10	C₄₃×#536	—	2	2	no	—
11	C₄₃×#536	C₄	1	2	yes	—
12	C₄₃×#532	—	2	1	no	—
13	C₄₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₄₃	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#423	C₂²	1	2	yes	—
18	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
19	C₄₃×#343	—	2	4	no	—
20	C₄₃×#333	—	2	1	no	—
21	C₄₃×#331	—	2	1	no	—
22	C₄₃×#329	—	2	2	no	—
23	C₄₃×Q₈⋊C₈	—	2	2	no	—
24	C₄₃×D₄⋊C₈	—	2	2	no	—
25	2₋¹⁺⁴×C₄₃	—	4	4	no	—
26	2₋¹⁺⁴×C₄₃	—	4	4	no	—
27	C₂×#41×C₄₃	—	2	2	no	—
28	C₂×#41×C₄₃	—	2	2	no	—
29	C₂×#41×C₄₃	—	2	4	no	—
30	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#41×C₄₃	C₂×C₄	1	2	yes	—
32	C₂²×Q₈×C₄₃	—	2	2	no	—
33	C₂²×Q₈×C₄₃	—	2	2	no	—
34	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	C₂×M₁₆×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	—	2	4	no	—
38	C₄⋊Q₈×C₄₃	—	2	2	no	—
39	C₄⋊Q₈×C₄₃	—	2	1	no	—
40	C₄⋊Q₈×C₄₃	—	2	1	no	—
41	C₄⋊Q₈×C₄₃	—	2	2	no	—
42	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
43	#127×C₄₃	D₄	1	1	yes	characteristic
44	#124×C₄₃	—	2	2	no	—
45	#124×C₄₃	D₄	1	2	yes	—
46	#123×C₄₃	—	2	8	no	—
47	#123×C₄₃	—	2	4	no	—
48	C₂²⋊Q₈×C₄₃	—	2	8	no	—
49	C₂²⋊Q₈×C₄₃	—	2	4	no	—
50	#121×C₄₃	—	2	4	no	—
51	#121×C₄₃	—	2	2	no	—
52	C₄×Q₈×C₄₃	D₄	1	2	yes	—
53	C₄×D₄×C₄₃	—	2	4	no	—
54	C₄×D₄×C₄₃	—	2	2	no	—
55	C₄×D₄×C₄₃	—	2	2	no	—
56	C₄×D₄×C₄₃	D₄	1	2	yes	—
57	#117×C₄₃	—	2	1	no	—
58	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
60	C₂×C₄⋊C₄×C₄₃	D₄	1	4	yes	—
61	C₂×C₄²×C₄₃	—	2	1	no	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₃	—	2	2	no	—
64	C₄⋊C₈×C₄₃	—	2	2	no	—
65	#97×C₄₃	—	2	2	no	—
66	C₄×C₈×C₄₃	—	2	2	no	—
67	#41×C₄₃	—	2	2	no	—
68	#41×C₄₃	—	2	2	no	—
69	#41×C₄₃	—	4	8	no	—
70	#41×C₄₃	—	4	2	no	—
71	#41×C₄₃	—	4	2	no	—
72	#41×C₄₃	—	4	16	no	—
73	#41×C₄₃	—	4	4	no	—
74	#41×C₄₃	—	4	4	no	—
75	C₂×Q₈×C₄₃	—	2	4	no	—
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂×Q₈×C₄₃	—	2	2	no	—
78	C₂×Q₈×C₄₃	—	4	2	no	—
79	C₂×Q₈×C₄₃	—	4	2	no	—
80	C₂×Q₈×C₄₃	—	4	8	no	—
81	C₂×Q₈×C₄₃	—	4	4	no	—
82	C₂×Q₈×C₄₃	—	4	4	no	—
83	C₂×Q₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Q₈×C₄₃	C₂²⋊C₄	1	2	yes	—
85	C₂×D₄×C₄₃	—	2	4	no	—
86	C₂×D₄×C₄₃	—	2	2	no	—
87	C₂×D₄×C₄₃	—	2	1	no	—
88	C₂×D₄×C₄₃	—	2	2	no	—
89	C₂×D₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×D₄×C₄₃	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₄₃	—	2	1	no	—
92	C₂²×C₄×C₄₃	—	2	4	no	—
93	C₂²×C₄×C₄₃	—	2	4	no	—
94	C₂²×C₄×C₄₃	—	2	2	no	—
95	C₂²×C₄×C₄₃	—	2	2	no	—
96	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
97	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
99	M₁₆×C₄₃	—	2	4	no	—
100	M₁₆×C₄₃	—	4	2	no	—
101	M₁₆×C₄₃	—	4	4	no	—
102	C₂×C₈×C₄₃	—	4	2	no	—
103	C₂×C₈×C₄₃	—	4	2	no	—
104	C₄⋊C₄×C₄₃	—	2	8	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	4	no	—
107	C₄⋊C₄×C₄₃	—	2	4	no	—
108	C₄⋊C₄×C₄₃	—	2	2	no	—
109	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
110	C₂²⋊C₄×C₄₃	—	2	8	no	—
111	C₂²⋊C₄×C₄₃	—	2	2	no	—
112	C₂²⋊C₄×C₄₃	—	2	4	no	—
113	C₂²⋊C₄×C₄₃	—	2	1	no	—
114	C₄²×C₄₃	—	2	1	no	—
115	C₄²×C₄₃	—	2	2	no	—
116	C₄²×C₄₃	C₂×D₄	1	2	yes	—
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂³×C₄₃	—	2	1	no	—
120	C₂³×C₄₃	—	2	2	no	—
121	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	Q₈×C₄₃	—	2	1	no	—
123	Q₈×C₄₃	—	4	2	no	—
124	Q₈×C₄₃	—	4	4	no	—
125	Q₈×C₄₃	—	4	8	no	—
126	Q₈×C₄₃	—	4	2	no	—
127	Q₈×C₄₃	—	4	2	no	—
128	Q₈×C₄₃	#100	1	2	yes	—
129	D₄×C₄₃	—	2	4	no	—
130	D₄×C₄₃	—	2	1	no	—
131	D₄×C₄₃	—	4	2	no	—
132	D₄×C₄₃	—	4	4	no	—
133	D₄×C₄₃	—	4	8	no	—
134	D₄×C₄₃	—	4	2	no	—
135	D₄×C₄₃	—	4	2	no	—
136	D₄×C₄₃	#100	1	2	yes	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	2	1	no	—
139	C₂×C₄×C₄₃	—	2	1	no	—
140	C₂×C₄×C₄₃	—	2	1	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	2	2	no	—
145	C₂×C₄×C₄₃	—	2	4	no	—
146	C₂×C₄×C₄₃	—	4	8	no	—
147	C₂×C₄×C₄₃	—	4	8	no	—
148	C₂×C₄×C₄₃	—	4	2	no	—
149	C₂×C₄×C₄₃	—	4	2	no	—
150	C₂×C₄×C₄₃	—	4	2	no	—
151	C₂×C₄×C₄₃	—	4	2	no	—
152	C₂×C₄×C₄₃	—	4	4	no	—
153	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₄₃	#120	1	2	yes	—
156	C₂×C₄×C₄₃	#120	1	2	yes	—
157	C₈×C₄₃	—	4	4	no	—
158	C₈×C₄₃	—	8	2	no	—
159	#8471	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	—	2	2	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	4	2	no	—
164	C₂²×C₄₃	—	4	4	no	—
165	C₂²×C₄₃	—	4	2	no	—
166	C₂²×C₄₃	#379	1	1	yes	characteristic
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	4	2	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	4	4	no	—
173	C₄×C₄₃	C₂×#100	1	1	yes	characteristic
174	C₄×C₄₃	C₂×#100	1	1	yes	characteristic
175	C₄×C₄₃	#454	1	2	yes	—
176	C₄×C₄₃	#456	1	2	yes	—
177	#3499	C₂×C₄₃	1	1	yes	characteristic
178	#1997	C₂×C₄₃	1	2	yes	complemented
179	#1560	C₂×C₄₃	1	1	yes	characteristic, complemented
180	#1525	C₂×C₄₃	1	1	yes	characteristic, complemented
181	#1489	C₂×C₄₃	1	2	yes	complemented
182	C₂×C₄₃	—	4	1	no	—
183	C₂×C₄₃	—	4	2	no	—
184	C₂×C₄₃	—	8	2	no	—
185	C₂×C₄₃	#1783	1	1	yes	characteristic
186	C₂×C₄₃	#1793	1	2	yes	—
187	C₂×2₋¹⁺⁴	C₄×C₄₃	1	2	yes	—
188	D₄×Q₈	—	2	2	no	—
189	#547	—	2	2	no	—
190	#547	—	2	4	no	—
191	#536	—	2	2	no	—
192	#536	C₄×C₄₃	1	2	yes	—
193	#532	—	2	1	no	—
194	#532	C₂²×C₄₃	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	—	2	1	no	—
196	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
197	C₄×#41	C₂²×C₄₃	1	1	yes	characteristic
198	#423	C₂²×C₄₃	1	2	yes	—
199	C₄×M₁₆	C₂²×C₄₃	1	2	yes	—
200	#343	—	2	4	no	—
201	#333	—	2	1	no	—
202	#331	—	2	1	no	—
203	#329	—	2	2	no	—
204	Q₈⋊C₈	—	2	2	no	—
205	D₄⋊C₈	—	2	2	no	—
206	C₄₃	#8471	1	1	yes	characteristic, direct product factor, Sylow
207	2₋¹⁺⁴	—	4	4	no	—
208	2₋¹⁺⁴	—	4	4	no	—
209	C₂×#41	—	2	2	no	—
210	C₂×#41	—	2	2	no	—
211	C₂×#41	—	2	4	no	—
212	C₂×#41	C₂×C₄×C₄₃	1	1	yes	characteristic
213	C₂×#41	C₂×C₄×C₄₃	1	2	yes	—
214	C₂²×Q₈	—	2	2	no	—
215	C₂²×Q₈	—	2	2	no	—
216	C₂²×Q₈	C₂×C₄×C₄₃	1	2	yes	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₄⋊Q₈	—	2	4	no	—
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	D₄×C₄₃	1	1	yes	characteristic
225	#127	D₄×C₄₃	1	1	yes	characteristic
226	#124	—	2	2	no	—
227	#124	D₄×C₄₃	1	2	yes	—
228	#123	—	2	8	no	—
229	#123	—	2	4	no	—
230	C₂²⋊Q₈	—	2	8	no	—
231	C₂²⋊Q₈	—	2	4	no	—
232	#121	—	2	4	no	—
233	#121	—	2	2	no	—
234	C₄×Q₈	D₄×C₄₃	1	2	yes	—
235	C₄×D₄	—	2	4	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	D₄×C₄₃	1	2	yes	—
239	#117	—	2	1	no	—
240	#117	C₂×C₄×C₄₃	1	1	yes	characteristic
241	C₂×C₄⋊C₄	—	2	4	no	—
242	C₂×C₄⋊C₄	D₄×C₄₃	1	4	yes	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#97	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	#41	—	2	2	no	—
250	#41	—	2	2	no	—
251	#41	—	4	8	no	—
252	#41	—	4	2	no	—
253	#41	—	4	2	no	—
254	#41	—	4	16	no	—
255	#41	—	4	4	no	—
256	#41	—	4	4	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	—	4	8	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	4	no	—
265	C₂×Q₈	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
266	C₂×Q₈	C₂²⋊C₄×C₄₃	1	2	yes	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	4	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
272	C₂×D₄	C₂²⋊C₄×C₄₃	1	2	yes	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	C₂²⋊C₄×C₄₃	1	4	yes	—
279	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
281	M₁₆	—	2	4	no	—
282	M₁₆	—	4	2	no	—
283	M₁₆	—	4	4	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
292	C₂²⋊C₄	—	2	8	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	C₂×D₄×C₄₃	1	2	yes	—
299	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
301	C₂³	—	2	1	no	—
302	C₂³	—	2	2	no	—
303	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	4	no	—
307	Q₈	—	4	8	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	#100×C₄₃	1	2	yes	—
311	D₄	—	2	4	no	—
312	D₄	—	2	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	8	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	#100×C₄₃	1	2	yes	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	4	8	no	—
329	C₂×C₄	—	4	8	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
337	C₂×C₄	#120×C₄₃	1	2	yes	—
338	C₂×C₄	#120×C₄₃	1	2	yes	—
339	C₈	—	4	4	no	—
340	C₈	—	8	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	4	no	—
347	C₂²	C₄₃×#379	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	4	2	no	—
354	C₄	C₂×#100×C₄₃	1	1	yes	characteristic
355	C₄	C₂×#100×C₄₃	1	1	yes	characteristic
356	C₄	C₄₃×#454	1	2	yes	—
357	C₄	C₄₃×#456	1	2	yes	—
358	C₂	—	4	1	no	—
359	C₂	—	4	2	no	—
360	C₂	—	8	2	no	—
361	C₂	C₄₃×#1783	1	1	yes	characteristic
362	C₂	C₄₃×#1793	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₄₃×#3499	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×#1997	C₂	semidirect
3. C₄₃×#1560	C₂	semidirect
4. C₄₃×#1525	C₂	semidirect
5. C₄₃×#1489	C₂	semidirect
159. #8471	C₄₃	direct (N × H)
178. #1997	C₂×C₄₃	semidirect
179. #1560	C₂×C₄₃	semidirect
180. #1525	C₂×C₄₃	semidirect
181. #1489	C₂×C₄₃	semidirect
206. C₄₃	#8471	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×#3499	2
C₄₃×#1997	2
C₄₃×#1560	2
C₄₃×#1525	2
C₄₃×#1489	2
#8471	43

Subgroups — C43×#8471 #24928488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8471 #24928488