Subgroups — C₄₃×#7704 #24927721

Order 11008.

640 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#2856	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1270	C₂	1	4	yes	maximal, complemented
5	C₂×#127×C₄₃	C₄	1	1	yes	characteristic
6	C₂×#121×C₄₃	—	2	1	no	—
7	C₂×#121×C₄₃	—	2	1	no	—
8	C₂×#121×C₄₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
12	C₂×#97×C₄₃	—	2	1	no	—
13	C₄₃×#390	—	2	1	no	—
14	C₄₃×#390	—	2	1	no	—
15	C₄₃×#390	C₂²	1	2	yes	—
16	C₄₃×#384	C₄	1	1	yes	characteristic
17	C₄₃×#336	—	2	1	no	—
18	C₄₃×#336	C₂²	1	2	yes	—
19	C₄₃×#336	C₂²	1	2	yes	—
20	C₂²×D₄×C₄₃	—	2	2	no	—
21	C₂²×D₄×C₄₃	—	2	1	no	—
22	C₂²×D₄×C₄₃	—	2	1	no	—
23	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	—	2	2	no	—
28	C₂²×C₈×C₄₃	—	2	2	no	—
29	#127×C₄₃	—	4	2	no	—
30	#121×C₄₃	—	4	4	no	—
31	#121×C₄₃	—	4	4	no	—
32	#121×C₄₃	—	4	4	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
39	C₂×C₄²×C₄₃	—	4	2	no	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²⋊C₈×C₄₃	—	2	4	no	—
45	C₂²⋊C₈×C₄₃	—	4	2	no	—
46	#97×C₄₃	—	4	2	no	—
47	C₂⁴×C₄₃	—	2	2	no	—
48	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₄×C₄₃	—	2	4	no	—
50	C₂×D₄×C₄₃	—	4	4	no	—
51	C₂×D₄×C₄₃	—	4	4	no	—
52	C₂×D₄×C₄₃	—	4	2	no	—
53	C₂×D₄×C₄₃	—	4	2	no	—
54	C₂×D₄×C₄₃	—	4	2	no	—
55	C₂×D₄×C₄₃	—	4	2	no	—
56	C₂×D₄×C₄₃	—	4	2	no	—
57	C₂×D₄×C₄₃	—	4	2	no	—
58	C₂×D₄×C₄₃	—	4	4	no	—
59	C₂×D₄×C₄₃	—	4	2	no	—
60	C₂×D₄×C₄₃	—	4	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	—	4	1	no	—
71	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₂×C₈×C₄₃	—	4	2	no	—
81	C₂×C₈×C₄₃	—	4	2	no	—
82	C₂×C₈×C₄₃	—	4	2	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₄⋊C₄×C₄₃	—	4	2	no	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
86	C₂²⋊C₄×C₄₃	—	4	4	no	—
87	C₂²⋊C₄×C₄₃	—	4	4	no	—
88	C₂²⋊C₄×C₄₃	—	4	4	no	—
89	C₄²×C₄₃	—	2	2	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	4	1	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₂³×C₄₃	—	2	1	no	—
94	C₂³×C₄₃	—	2	1	no	—
95	C₂³×C₄₃	—	2	1	no	—
96	C₂³×C₄₃	—	2	1	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	—	2	1	no	—
99	C₂³×C₄₃	—	4	2	no	—
100	C₂³×C₄₃	—	4	2	no	—
101	C₂³×C₄₃	—	4	2	no	—
102	C₂³×C₄₃	—	4	2	no	—
103	C₂³×C₄₃	—	4	2	no	—
104	C₂³×C₄₃	—	4	2	no	—
105	C₂³×C₄₃	—	4	2	no	—
106	C₂³×C₄₃	#100	1	2	yes	—
107	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	D₄×C₄₃	—	8	4	no	—
109	D₄×C₄₃	—	8	4	no	—
110	D₄×C₄₃	—	8	4	no	—
111	D₄×C₄₃	—	8	8	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	4	4	no	—
126	C₂×C₄×C₄₃	—	4	4	no	—
127	C₂×C₄×C₄₃	—	4	4	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	#99	1	2	yes	—
132	C₂×C₄×C₄₃	#104	1	2	yes	—
133	C₂×C₄×C₄₃	#131	1	2	yes	—
134	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
136	C₈×C₄₃	—	4	4	no	—
137	C₈×C₄₃	—	8	2	no	—
138	#7704	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	4	1	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	8	2	no	—
147	C₂²×C₄₃	—	8	2	no	—
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	—	8	2	no	—
150	C₂²×C₄₃	—	8	2	no	—
151	C₂²×C₄₃	—	8	2	no	—
152	C₂²×C₄₃	—	8	2	no	—
153	C₂²×C₄₃	#408	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#99	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
156	C₂²×C₄₃	#418	1	1	yes	characteristic
157	C₂²×C₄₃	#418	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
159	C₂²×C₄₃	#421	1	1	yes	characteristic
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	2	no	—
165	C₄×C₄₃	—	4	1	no	—
166	C₄×C₄₃	—	8	2	no	—
167	#2856	C₂×C₄₃	1	1	yes	characteristic
168	#1817	C₂×C₄₃	1	1	yes	characteristic, complemented
169	#1749	C₂×C₄₃	1	1	yes	characteristic, complemented
170	#1270	C₂×C₄₃	1	4	yes	complemented
171	C₂×C₄₃	—	4	2	no	—
172	C₂×C₄₃	—	16	2	no	—
173	C₂×C₄₃	#1471	1	1	yes	characteristic
174	C₂×C₄₃	#1484	1	1	yes	characteristic
175	C₂×C₄₃	#1496	1	1	yes	characteristic
176	C₂×C₄₃	#1507	1	1	yes	characteristic
177	C₂×C₄₃	#1510	1	1	yes	characteristic
178	C₂×C₄₃	#1530	1	1	yes	characteristic
179	C₂×C₄₃	#1555	1	1	yes	characteristic
180	C₂×#127	C₄×C₄₃	1	1	yes	characteristic
181	C₂×#121	—	2	1	no	—
182	C₂×#121	—	2	1	no	—
183	C₂×#121	C₄×C₄₃	1	2	yes	—
184	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×C₂²⋊C₈	—	2	1	no	—
187	C₂×#97	—	2	1	no	—
188	#390	—	2	1	no	—
189	#390	—	2	1	no	—
190	#390	C₂²×C₄₃	1	2	yes	—
191	#384	C₄×C₄₃	1	1	yes	characteristic
192	#336	—	2	1	no	—
193	#336	C₂²×C₄₃	1	2	yes	—
194	#336	C₂²×C₄₃	1	2	yes	—
195	C₄₃	#7704	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×D₄	—	2	2	no	—
197	C₂²×D₄	—	2	1	no	—
198	C₂²×D₄	—	2	1	no	—
199	C₂²×D₄	C₂×C₄×C₄₃	1	2	yes	—
200	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
202	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	#127	—	4	2	no	—
206	#121	—	4	4	no	—
207	#121	—	4	4	no	—
208	#121	—	4	4	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	4	2	no	—
216	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
220	C₂²⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	4	2	no	—
222	#97	—	4	2	no	—
223	C₂⁴	—	2	2	no	—
224	C₂⁴	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	#100×C₄₃	1	2	yes	—
283	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
284	D₄	—	8	4	no	—
285	D₄	—	8	4	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	#99×C₄₃	1	2	yes	—
308	C₂×C₄	#104×C₄₃	1	2	yes	—
309	C₂×C₄	#131×C₄₃	1	2	yes	—
310	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	2	yes	—
311	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₈	—	8	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	C₄₃×#408	1	1	yes	characteristic
329	C₂²	C₂×#99×C₄₃	1	1	yes	characteristic
330	C₂²	C₂×#100×C₄₃	1	1	yes	characteristic
331	C₂²	C₄₃×#418	1	1	yes	characteristic
332	C₂²	C₄₃×#418	1	1	yes	characteristic
333	C₂²	C₂×#104×C₄₃	1	1	yes	characteristic
334	C₂²	C₄₃×#421	1	1	yes	characteristic
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	8	2	no	—
342	C₂	—	4	2	no	—
343	C₂	—	16	2	no	—
344	C₂	C₄₃×#1471	1	1	yes	characteristic
345	C₂	C₄₃×#1484	1	1	yes	characteristic
346	C₂	C₄₃×#1496	1	1	yes	characteristic
347	C₂	C₄₃×#1507	1	1	yes	characteristic
348	C₂	C₄₃×#1510	1	1	yes	characteristic
349	C₂	C₄₃×#1530	1	1	yes	characteristic
350	C₂	C₄₃×#1555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₄₃×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₂²×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1817	C₂	semidirect
3. C₄₃×#1749	C₂	semidirect
4. C₄₃×#1270	C₂	semidirect
138. #7704	C₄₃	direct (N × H)
168. #1817	C₂×C₄₃	semidirect
169. #1749	C₂×C₄₃	semidirect
170. #1270	C₂×C₄₃	semidirect
195. C₄₃	#7704	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×#2856	2
C₄₃×#1817	2
C₄₃×#1749	2
C₄₃×#1270	2
#7704	43

Subgroups — C43×#7704 #24927721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#7704 #24927721