Subgroups — C₄₃×#28772 #24948789

Order 11008.

742 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#2442	C₂	1	2	yes	maximal
2	C₄₃×#2085	C₂	1	2	yes	maximal, complemented
3	C₄₃×#2059	C₂	1	2	yes	maximal, complemented
4	C₄₃×#1991	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1942	C₂	1	2	yes	maximal, complemented
6	C₄₃×#1852	C₂	1	2	yes	maximal
7	C₄₃×#1807	C₂	1	1	yes	maximal, characteristic
8	C₈×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₃×#340	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×#125×C₄₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₂×#107×C₄₃	C₂²	1	2	yes	—
15	C₂×#106×C₄₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
18	C₄₃×#400	—	2	2	no	—
19	C₄₃×#398	—	2	2	no	—
20	C₄₃×#389	C₂²	1	2	yes	—
21	C₄₃×#389	C₂²	1	2	yes	—
22	C₄₃×#385	—	2	2	no	—
23	C₄₃×#382	C₂²	1	2	yes	—
24	C₄₃×#380	—	2	2	no	—
25	C₄₃×#380	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#340	C₂²	1	1	yes	characteristic
28	C₄₃×#340	C₂²	1	1	yes	characteristic
29	C₄₃×#340	C₂²	1	1	yes	characteristic
30	C₄₃×#340	C₂²	1	2	yes	—
31	C₄₃×#340	C₂²	1	2	yes	—
32	C₄₃×#340	C₂²	1	2	yes	—
33	C₄₃×#340	C₂²	1	1	yes	characteristic
34	C₄₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₄₃	—	2	2	no	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	—	4	2	no	—
38	C₂²×C₈×C₄₃	Q₈	1	4	yes	—
39	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
42	#125×C₄₃	—	2	4	no	—
43	#125×C₄₃	—	4	2	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
58	#107×C₄₃	—	2	4	no	—
59	#107×C₄₃	—	4	2	no	—
60	#106×C₄₃	—	4	2	no	—
61	C₂²⋊C₈×C₄₃	Q₈	1	4	yes	—
62	C₄×C₈×C₄₃	—	4	2	no	—
63	#95×C₄₃	—	2	2	no	—
64	#95×C₄₃	—	2	2	no	—
65	#95×C₄₃	—	2	2	no	—
66	#95×C₄₃	—	2	2	no	—
67	#95×C₄₃	—	2	2	no	—
68	#95×C₄₃	D₄	1	1	yes	characteristic
69	#95×C₄₃	D₄	1	1	yes	characteristic
70	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	—	2	2	no	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	—	4	4	no	—
78	C₂²×C₄×C₄₃	—	4	4	no	—
79	C₂²×C₄×C₄₃	—	8	2	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	4	4	no	—
91	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
92	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₄⋊C₄×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	4	4	no	—
102	C₄⋊C₄×C₄₃	—	4	4	no	—
103	C₄⋊C₄×C₄₃	—	4	4	no	—
104	C₄⋊C₄×C₄₃	—	8	2	no	—
105	C₄⋊C₄×C₄₃	—	8	2	no	—
106	C₄⋊C₄×C₄₃	#41	1	4	yes	—
107	C₂²⋊C₄×C₄₃	—	2	4	no	—
108	C₂²⋊C₄×C₄₃	—	4	4	no	—
109	C₄²×C₄₃	—	2	2	no	—
110	C₄²×C₄₃	—	2	2	no	—
111	C₂³×C₄₃	—	2	2	no	—
112	C₂³×C₄₃	—	2	1	no	—
113	C₂³×C₄₃	—	2	1	no	—
114	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
115	C₂³×C₄₃	#135	1	4	yes	—
116	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	4	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	4	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
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	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	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	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	8	8	no	—
140	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
145	C₈×C₄₃	—	2	2	no	—
146	C₈×C₄₃	—	2	2	no	—
147	C₈×C₄₃	—	2	4	no	—
148	#28772	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₄₃	—	2	2	no	—
150	C₂²×C₄₃	—	2	4	no	—
151	C₂²×C₄₃	—	2	4	no	—
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	#499	1	4	yes	—
154	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄₃	#546	1	2	yes	—
156	C₂²×C₄₃	C₂×#135	1	2	yes	—
157	C₂²×C₄₃	#576	1	1	yes	characteristic
158	C₂²×C₄₃	#578	1	1	yes	characteristic
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	—	4	4	no	—
162	C₄×C₄₃	—	4	2	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	—	8	2	no	—
165	C₄×C₄₃	—	8	2	no	—
166	C₄×C₄₃	#533	1	1	yes	characteristic
167	C₄×C₄₃	#543	1	1	yes	characteristic
168	C₄×C₄₃	#554	1	2	yes	—
169	#2442	C₂×C₄₃	1	2	yes	—
170	#2085	C₂×C₄₃	1	2	yes	complemented
171	#2059	C₂×C₄₃	1	2	yes	complemented
172	#1991	C₂×C₄₃	1	2	yes	complemented
173	#1942	C₂×C₄₃	1	2	yes	complemented
174	#1852	C₂×C₄₃	1	2	yes	—
175	#1807	C₂×C₄₃	1	1	yes	characteristic
176	C₈×C₂²⋊C₄	C₂×C₄₃	1	1	yes	characteristic
177	C₂×#340	C₂×C₄₃	1	1	yes	characteristic
178	C₂×C₄₃	—	2	4	no	—
179	C₂×C₄₃	#2549	1	1	yes	characteristic
180	C₂×C₄₃	C₂×#499	1	1	yes	characteristic
181	C₂×C₄₃	#3164	1	1	yes	characteristic
182	C₂×C₄₃	#3297	1	2	yes	—
183	C₂×C₄₃	#3300	1	2	yes	—
184	C₂³×C₈	C₂²×C₄₃	1	1	yes	characteristic
185	C₂×#125	C₂²×C₄₃	1	2	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
187	C₂×#107	C₂²×C₄₃	1	2	yes	—
188	C₂×#107	C₂²×C₄₃	1	2	yes	—
189	C₂×#106	C₂²×C₄₃	1	2	yes	—
190	C₂×C₂²⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂²×C₄₃	1	2	yes	—
192	#400	—	2	2	no	—
193	#398	—	2	2	no	—
194	#389	C₂²×C₄₃	1	2	yes	—
195	#389	C₂²×C₄₃	1	2	yes	—
196	#385	—	2	2	no	—
197	#382	C₂²×C₄₃	1	2	yes	—
198	#380	—	2	2	no	—
199	#380	C₂²×C₄₃	1	2	yes	—
200	C₄×C₂²⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
201	#340	C₂²×C₄₃	1	1	yes	characteristic
202	#340	C₂²×C₄₃	1	2	yes	—
203	#340	C₂²×C₄₃	1	1	yes	characteristic
204	#340	C₂²×C₄₃	1	1	yes	characteristic
205	#340	C₂²×C₄₃	1	1	yes	characteristic
206	#340	C₂²×C₄₃	1	2	yes	—
207	#340	C₂²×C₄₃	1	2	yes	—
208	#336	C₂²×C₄₃	1	2	yes	—
209	C₄₃	#28772	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³×C₄	—	2	2	no	—
211	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
212	C₂²×C₈	—	4	2	no	—
213	C₂²×C₈	Q₈×C₄₃	1	4	yes	—
214	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
217	#125	—	2	4	no	—
218	#125	—	4	2	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	4	2	no	—
226	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
232	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
233	#107	—	2	4	no	—
234	#107	—	4	2	no	—
235	#106	—	4	2	no	—
236	C₂²⋊C₈	Q₈×C₄₃	1	4	yes	—
237	C₄×C₈	—	4	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	D₄×C₄₃	1	1	yes	characteristic
244	#95	D₄×C₄₃	1	1	yes	characteristic
245	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	4	no	—
253	C₂²×C₄	—	4	4	no	—
254	C₂²×C₄	—	8	2	no	—
255	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×Q₈×C₄₃	1	2	yes	—
259	C₂²×C₄	C₂×Q₈×C₄₃	1	2	yes	—
260	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	C₂×Q₈×C₄₃	1	4	yes	—
267	C₂×C₈	C₂×Q₈×C₄₃	1	2	yes	—
268	C₂×C₈	C₂×Q₈×C₄₃	1	2	yes	—
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	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₄⋊C₄	#41×C₄₃	1	4	yes	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₄⋊Q₈×C₄₃	1	2	yes	—
290	C₂³	#135×C₄₃	1	4	yes	—
291	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	8	no	—
315	C₂×C₄	C₄⋊Q₈×C₄₃	1	2	yes	—
316	C₂×C₄	C₂²×Q₈×C₄₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×Q₈×C₄₃	1	1	yes	characteristic
318	C₂×C₄	C₂×#41×C₄₃	1	2	yes	—
319	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	C₄₃×#499	1	4	yes	—
328	C₂²	C₂×C₄⋊Q₈×C₄₃	1	1	yes	characteristic
329	C₂²	C₄₃×#546	1	2	yes	—
330	C₂²	C₂×#135×C₄₃	1	2	yes	—
331	C₂²	C₄₃×#576	1	1	yes	characteristic
332	C₂²	C₄₃×#578	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	4	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₄₃×#533	1	1	yes	characteristic
341	C₄	C₄₃×#543	1	1	yes	characteristic
342	C₄	C₄₃×#554	1	2	yes	—
343	C₂	—	2	4	no	—
344	C₂	C₄₃×#2549	1	1	yes	characteristic
345	C₂	C₂×C₄₃×#499	1	1	yes	characteristic
346	C₂	C₄₃×#3164	1	1	yes	characteristic
347	C₂	C₄₃×#3297	1	2	yes	—
348	C₂	C₄₃×#3300	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂⁴×C₄₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₄×C₂²⋊C₄×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#2085	C₂	semidirect
3. C₄₃×#2059	C₂	semidirect
4. C₄₃×#1991	C₂	semidirect
5. C₄₃×#1942	C₂	semidirect
148. #28772	C₄₃	direct (N × H)
170. #2085	C₂×C₄₃	semidirect
171. #2059	C₂×C₄₃	semidirect
172. #1991	C₂×C₄₃	semidirect
173. #1942	C₂×C₄₃	semidirect
209. C₄₃	#28772	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×#2442	2
C₄₃×#2085	2
C₄₃×#2059	2
C₄₃×#1991	2
C₄₃×#1942	2
C₄₃×#1852	2
C₄₃×#1807	2
C₈×C₂²⋊C₄×C₄₃	2
C₂×C₄₃×#340	2
#28772	43

Subgroups — C43×#28772 #24948789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#28772 #24948789