Subgroups — C₄₃×#28624 #24948641

Order 11008.

986 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#2641	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#2328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2054	C₂	1	4	yes	maximal, complemented
6	C₄₃×#1991	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1961	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#1879	C₂	1	4	yes	maximal, complemented
9	C₄₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₄₃	C₂²	1	4	yes	—
11	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#493	—	2	4	no	—
16	C₄₃×#490	—	2	4	no	—
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
18	C₂×#102×C₄₃	C₂²	1	4	yes	—
19	C₂×#102×C₄₃	C₂²	1	4	yes	—
20	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#394	—	2	4	no	—
23	C₄₃×#390	C₂²	1	2	yes	—
24	C₄₃×#390	C₂²	1	2	yes	—
25	C₄₃×#390	C₂²	1	2	yes	—
26	C₄₃×#386	—	2	4	no	—
27	C₄₃×#382	C₂²	1	1	yes	characteristic
28	C₄₃×#376	C₂²	1	1	yes	characteristic
29	C₄₃×#340	C₂²	1	4	yes	—
30	C₄₃×#336	C₂²	1	2	yes	—
31	C₂²×D₄×C₄₃	—	2	2	no	—
32	C₂²×D₄×C₄₃	—	2	2	no	—
33	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
34	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
35	C₂³×C₄×C₄₃	—	2	2	no	—
36	C₂×D₈×C₄₃	—	2	8	no	—
37	C₂×D₈×C₄₃	—	2	8	no	—
38	C₂×D₈×C₄₃	—	4	4	no	—
39	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
41	#127×C₄₃	—	2	2	no	—
42	#127×C₄₃	—	2	2	no	—
43	#127×C₄₃	—	2	2	no	—
44	#127×C₄₃	D₄	1	4	yes	—
45	#125×C₄₃	—	2	2	no	—
46	#125×C₄₃	—	2	2	no	—
47	C₄×D₄×C₄₃	—	4	4	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	—	2	4	no	—
59	#102×C₄₃	—	2	8	no	—
60	#102×C₄₃	—	4	4	no	—
61	#97×C₄₃	—	2	2	no	—
62	C₄×C₈×C₄₃	—	2	2	no	—
63	#95×C₄₃	—	2	2	no	—
64	#95×C₄₃	D₄	1	4	yes	—
65	C₂⁴×C₄₃	—	2	2	no	—
66	C₂⁴×C₄₃	—	2	2	no	—
67	C₂×D₄×C₄₃	—	2	8	no	—
68	C₂×D₄×C₄₃	—	2	2	no	—
69	C₂×D₄×C₄₃	—	2	2	no	—
70	C₂×D₄×C₄₃	—	2	4	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	4	no	—
75	C₂×D₄×C₄₃	—	4	4	no	—
76	C₂×D₄×C₄₃	—	8	4	no	—
77	C₂×D₄×C₄₃	—	8	4	no	—
78	C₂×D₄×C₄₃	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	4	4	no	—
82	C₂²×C₄×C₄₃	—	8	4	no	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	D₈×C₄₃	—	4	16	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	#41	1	4	yes	—
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₄⋊C₄×C₄₃	—	2	4	no	—
97	C₄⋊C₄×C₄₃	—	4	4	no	—
98	C₂²⋊C₄×C₄₃	—	4	4	no	—
99	C₂²⋊C₄×C₄₃	—	8	4	no	—
100	C₂²⋊C₄×C₄₃	—	8	4	no	—
101	C₂²⋊C₄×C₄₃	—	8	4	no	—
102	C₄²×C₄₃	—	2	2	no	—
103	C₄²×C₄₃	—	2	2	no	—
104	C₄²×C₄₃	C₂×D₄	1	2	yes	—
105	C₄²×C₄₃	#41	1	2	yes	—
106	C₂³×C₄₃	—	2	4	no	—
107	C₂³×C₄₃	—	4	2	no	—
108	C₂³×C₄₃	—	4	2	no	—
109	C₂³×C₄₃	—	4	4	no	—
110	C₂³×C₄₃	—	4	4	no	—
111	C₂³×C₄₃	—	4	2	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	—	4	2	no	—
114	C₂³×C₄₃	—	4	4	no	—
115	C₂³×C₄₃	—	4	4	no	—
116	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	D₄×C₄₃	—	2	8	no	—
118	D₄×C₄₃	—	4	4	no	—
119	D₄×C₄₃	—	4	4	no	—
120	D₄×C₄₃	—	4	4	no	—
121	D₄×C₄₃	—	8	8	no	—
122	D₄×C₄₃	—	8	8	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	4	no	—
127	C₂×C₄×C₄₃	—	2	4	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	4	4	no	—
131	C₂×C₄×C₄₃	—	4	4	no	—
132	C₂×C₄×C₄₃	—	8	8	no	—
133	C₂×C₄×C₄₃	#121	1	4	yes	—
134	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
138	C₈×C₄₃	—	2	4	no	—
139	C₈×C₄₃	—	4	2	no	—
140	#28624	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₄₃	—	4	8	no	—
142	C₂²×C₄₃	—	4	4	no	—
143	C₂²×C₄₃	—	8	4	no	—
144	C₂²×C₄₃	—	8	2	no	—
145	C₂²×C₄₃	—	8	2	no	—
146	C₂²×C₄₃	—	8	4	no	—
147	C₂²×C₄₃	—	8	4	no	—
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	—	8	2	no	—
150	C₂²×C₄₃	—	8	2	no	—
151	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
152	C₂²×C₄₃	#532	1	1	yes	characteristic
153	C₂²×C₄₃	#535	1	1	yes	characteristic
154	C₂²×C₄₃	#576	1	4	yes	—
155	C₄×C₄₃	—	2	4	no	—
156	C₄×C₄₃	—	4	2	no	—
157	C₄×C₄₃	—	4	2	no	—
158	C₄×C₄₃	—	8	2	no	—
159	C₄×C₄₃	—	8	2	no	—
160	C₄×C₄₃	#546	1	2	yes	—
161	C₄×C₄₃	#559	1	2	yes	—
162	C₂×#493	C₂×C₄₃	1	1	yes	characteristic
163	C₂×#490	C₂×C₄₃	1	1	yes	characteristic, complemented
164	#2641	C₂×C₄₃	1	1	yes	characteristic, complemented
165	#2328	C₂×C₄₃	1	1	yes	characteristic, complemented
166	#2054	C₂×C₄₃	1	4	yes	complemented
167	#1991	C₂×C₄₃	1	1	yes	characteristic, complemented
168	#1961	C₂×C₄₃	1	1	yes	characteristic, complemented
169	#1879	C₂×C₄₃	1	4	yes	complemented
170	#1768	C₂×C₄₃	1	1	yes	characteristic, complemented
171	C₂×C₄₃	—	8	4	no	—
172	C₂×C₄₃	—	16	2	no	—
173	C₂×C₄₃	#2544	1	1	yes	characteristic
174	C₂×C₄₃	#3041	1	2	yes	—
175	C₂×C₄₃	#3176	1	2	yes	—
176	C₂×C₄₃	#3220	1	2	yes	—
177	C₂²×D₈	C₂²×C₄₃	1	4	yes	—
178	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
179	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
180	C₂×#125	C₂²×C₄₃	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₄₃	1	2	yes	—
182	#493	—	2	4	no	—
183	#490	—	2	4	no	—
184	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
185	C₂×#102	C₂²×C₄₃	1	4	yes	—
186	C₂×#102	C₂²×C₄₃	1	4	yes	—
187	C₂×#97	C₂²×C₄₃	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
189	#394	—	2	4	no	—
190	#390	C₂²×C₄₃	1	2	yes	—
191	#390	C₂²×C₄₃	1	2	yes	—
192	#390	C₂²×C₄₃	1	2	yes	—
193	#386	—	2	4	no	—
194	#382	C₂²×C₄₃	1	1	yes	characteristic
195	#376	C₂²×C₄₃	1	1	yes	characteristic
196	#340	C₂²×C₄₃	1	4	yes	—
197	#336	C₂²×C₄₃	1	2	yes	—
198	C₄₃	#28624	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	C₂³×C₄₃	1	2	yes	—
202	C₂²×D₄	C₂³×C₄₃	1	2	yes	—
203	C₂³×C₄	—	2	2	no	—
204	C₂×D₈	—	2	8	no	—
205	C₂×D₈	—	2	8	no	—
206	C₂×D₈	—	4	4	no	—
207	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
208	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
209	#127	—	2	2	no	—
210	#127	—	2	2	no	—
211	#127	—	2	2	no	—
212	#127	D₄×C₄₃	1	4	yes	—
213	#125	—	2	2	no	—
214	#125	—	2	2	no	—
215	C₄×D₄	—	4	4	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	4	no	—
221	C₂×C₂²⋊C₄	—	2	4	no	—
222	C₂×C₂²⋊C₄	—	2	4	no	—
223	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	4	no	—
227	#102	—	2	8	no	—
228	#102	—	4	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	D₄×C₄₃	1	4	yes	—
233	C₂⁴	—	2	2	no	—
234	C₂⁴	—	2	2	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	4	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	8	4	no	—
245	C₂×D₄	—	8	4	no	—
246	C₂×D₄	C₂×D₄×C₄₃	1	4	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	8	4	no	—
251	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
254	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
255	D₈	—	4	16	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	#41×C₄₃	1	4	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	8	4	no	—
268	C₂²⋊C₄	—	8	4	no	—
269	C₂²⋊C₄	—	8	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	C₂×D₄×C₄₃	1	2	yes	—
273	C₄²	#41×C₄₃	1	2	yes	—
274	C₂³	—	2	4	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	4	4	no	—
284	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
285	D₄	—	2	8	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	8	8	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	8	no	—
301	C₂×C₄	#121×C₄₃	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₄₃	1	2	yes	—
304	C₂×C₄	C₂×#41×C₄₃	1	1	yes	characteristic
305	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
306	C₈	—	2	4	no	—
307	C₈	—	4	2	no	—
308	C₂²	—	4	8	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂×#121×C₄₃	1	1	yes	characteristic
319	C₂²	C₄₃×#532	1	1	yes	characteristic
320	C₂²	C₄₃×#535	1	1	yes	characteristic
321	C₂²	C₄₃×#576	1	4	yes	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₄₃×#546	1	2	yes	—
328	C₄	C₄₃×#559	1	2	yes	—
329	C₂	—	8	4	no	—
330	C₂	—	16	2	no	—
331	C₂	C₄₃×#2544	1	1	yes	characteristic
332	C₂	C₄₃×#3041	1	2	yes	—
333	C₂	C₄₃×#3176	1	2	yes	—
334	C₂	C₄₃×#3220	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂×C₄₃×#493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₄₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃×#490	C₂	semidirect
3. C₄₃×#2641	C₂	semidirect
4. C₄₃×#2328	C₂	semidirect
5. C₄₃×#2054	C₂	semidirect
6. C₄₃×#1991	C₂	semidirect
7. C₄₃×#1961	C₂	semidirect
8. C₄₃×#1879	C₂	semidirect
9. C₄₃×#1768	C₂	semidirect
140. #28624	C₄₃	direct (N × H)
163. C₂×#490	C₂×C₄₃	semidirect
164. #2641	C₂×C₄₃	semidirect
165. #2328	C₂×C₄₃	semidirect
166. #2054	C₂×C₄₃	semidirect
167. #1991	C₂×C₄₃	semidirect
168. #1961	C₂×C₄₃	semidirect
169. #1879	C₂×C₄₃	semidirect
170. #1768	C₂×C₄₃	semidirect
198. C₄₃	#28624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₃×#493	2
C₂×C₄₃×#490	2
C₄₃×#2641	2
C₄₃×#2328	2
C₄₃×#2054	2
C₄₃×#1991	2
C₄₃×#1961	2
C₄₃×#1879	2
C₄₃×#1768	2
#28624	43

Subgroups — C43×#28624 #24948641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#28624 #24948641