Subgroups — C₄₃×#29300 #24949317

Order 11008.

738 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#2851	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#2296	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2085	C₂	1	2	yes	maximal, complemented
4	C₄₃×#2059	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1942	C₂	1	2	yes	maximal, complemented
6	C₄₃×#1941	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1854	C₂	1	2	yes	maximal
8	C₄₃×#1854	C₂	1	2	yes	maximal
9	C₄×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₄₃	C₂²	1	2	yes	—
15	C₂×#107×C₄₃	C₂²	1	2	yes	—
16	C₂×#106×C₄₃	C₂²	1	2	yes	—
17	C₂×#106×C₄₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#389	C₂²	1	2	yes	—
22	C₄₃×#389	C₂²	1	2	yes	—
23	C₄₃×#389	C₂²	1	2	yes	—
24	C₄₃×#387	—	2	2	no	—
25	C₄₃×#385	—	2	2	no	—
26	C₄₃×#385	—	2	2	no	—
27	C₄₃×#382	C₂²	1	1	yes	characteristic
28	C₄₃×#380	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
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	2	yes	—
34	C₄₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₄₃	D₄	1	2	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
39	#125×C₄₃	—	4	1	no	—
40	#125×C₄₃	—	4	1	no	—
41	#117×C₄₃	—	2	4	no	—
42	#117×C₄₃	—	4	1	no	—
43	#117×C₄₃	—	4	1	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	2	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	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₄²×C₄₃	—	2	1	no	—
56	C₂×C₄²×C₄₃	—	2	1	no	—
57	C₂×C₄²×C₄₃	—	4	1	no	—
58	C₂×C₄²×C₄₃	—	4	1	no	—
59	C₂×C₄²×C₄₃	D₄	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	#107×C₄₃	—	2	4	no	—
63	#107×C₄₃	—	4	2	no	—
64	#106×C₄₃	—	2	4	no	—
65	#106×C₄₃	—	4	2	no	—
66	C₂²⋊C₈×C₄₃	Q₈	1	8	yes	—
67	C₄×C₈×C₄₃	—	4	1	no	—
68	C₄×C₈×C₄₃	—	4	1	no	—
69	#95×C₄₃	—	2	2	no	—
70	#95×C₄₃	—	2	2	no	—
71	#95×C₄₃	—	2	2	no	—
72	#95×C₄₃	—	2	2	no	—
73	#95×C₄₃	—	2	2	no	—
74	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	—	2	1	no	—
77	C₂²×C₄×C₄₃	—	2	1	no	—
78	C₂²×C₄×C₄₃	—	2	4	no	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	4	2	no	—
84	C₂²×C₄×C₄₃	—	4	2	no	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
94	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
95	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
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	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	4	4	no	—
104	C₄⋊C₄×C₄₃	—	4	4	no	—
105	C₄⋊C₄×C₄₃	—	4	4	no	—
106	C₄⋊C₄×C₄₃	—	4	4	no	—
107	C₄⋊C₄×C₄₃	#41	1	4	yes	—
108	C₂²⋊C₄×C₄₃	—	2	4	no	—
109	C₂²⋊C₄×C₄₃	—	4	4	no	—
110	C₂²⋊C₄×C₄₃	—	4	2	no	—
111	C₄²×C₄₃	—	2	4	no	—
112	C₄²×C₄₃	—	2	1	no	—
113	C₄²×C₄₃	—	2	1	no	—
114	C₄²×C₄₃	—	2	1	no	—
115	C₄²×C₄₃	—	2	1	no	—
116	C₄²×C₄₃	—	4	2	no	—
117	C₄²×C₄₃	—	4	2	no	—
118	C₂³×C₄₃	—	2	1	no	—
119	C₂³×C₄₃	—	2	1	no	—
120	C₂³×C₄₃	—	2	2	no	—
121	C₂³×C₄₃	—	2	2	no	—
122	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
123	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	4	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	4	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	4	8	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	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
147	C₂×C₄×C₄₃	#135	1	4	yes	—
148	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
149	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
150	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
152	C₈×C₄₃	—	2	4	no	—
153	C₈×C₄₃	—	2	4	no	—
154	#29300	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	4	2	no	—
159	C₂²×C₄₃	—	4	2	no	—
160	C₂²×C₄₃	#525	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄₃	#534	1	1	yes	characteristic
163	C₂²×C₄₃	C₂×#135	1	2	yes	—
164	C₂²×C₄₃	#575	1	2	yes	—
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	2	4	no	—
167	C₄×C₄₃	—	4	4	no	—
168	C₄×C₄₃	—	4	2	no	—
169	C₄×C₄₃	—	8	2	no	—
170	C₄×C₄₃	—	8	2	no	—
171	C₄×C₄₃	—	8	2	no	—
172	C₄×C₄₃	#554	1	2	yes	—
173	C₄×C₄₃	#554	1	2	yes	—
174	#2851	C₂×C₄₃	1	1	yes	characteristic
175	#2296	C₂×C₄₃	1	1	yes	characteristic
176	#2085	C₂×C₄₃	1	2	yes	complemented
177	#2059	C₂×C₄₃	1	2	yes	complemented
178	#1942	C₂×C₄₃	1	2	yes	complemented
179	#1941	C₂×C₄₃	1	2	yes	complemented
180	#1854	C₂×C₄₃	1	2	yes	—
181	#1854	C₂×C₄₃	1	2	yes	—
182	C₄×C₂²⋊C₈	C₂×C₄₃	1	1	yes	characteristic
183	C₂×C₄₃	—	4	2	no	—
184	C₂×C₄₃	#2583	1	1	yes	characteristic
185	C₂×C₄₃	#3102	1	2	yes	—
186	C₂×C₄₃	#3160	1	1	yes	characteristic
187	C₂×C₄₃	#3166	1	1	yes	characteristic
188	C₂×C₄₃	#3191	1	2	yes	—
189	C₂×#125	C₂²×C₄₃	1	1	yes	characteristic
190	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
191	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
192	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
193	C₂×#107	C₂²×C₄₃	1	2	yes	—
194	C₂×#107	C₂²×C₄₃	1	2	yes	—
195	C₂×#106	C₂²×C₄₃	1	2	yes	—
196	C₂×#106	C₂²×C₄₃	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
198	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
200	#389	C₂²×C₄₃	1	2	yes	—
201	#389	C₂²×C₄₃	1	2	yes	—
202	#389	C₂²×C₄₃	1	2	yes	—
203	#387	—	2	2	no	—
204	#385	—	2	2	no	—
205	#385	—	2	2	no	—
206	#382	C₂²×C₄₃	1	1	yes	characteristic
207	#380	C₂²×C₄₃	1	2	yes	—
208	C₄×C₄⋊C₄	—	2	2	no	—
209	#340	C₂²×C₄₃	1	2	yes	—
210	#340	C₂²×C₄₃	1	2	yes	—
211	#340	C₂²×C₄₃	1	2	yes	—
212	#340	C₂²×C₄₃	1	2	yes	—
213	#336	C₂²×C₄₃	1	2	yes	—
214	C₄₃	#29300	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³×C₄	D₄×C₄₃	1	2	yes	—
216	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
218	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
219	#125	—	4	1	no	—
220	#125	—	4	1	no	—
221	#117	—	2	4	no	—
222	#117	—	4	1	no	—
223	#117	—	4	1	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
232	C₂×C₂²⋊C₄	—	2	1	no	—
233	C₂×C₂²⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	—	2	1	no	—
237	C₂×C₄²	—	4	1	no	—
238	C₂×C₄²	—	4	1	no	—
239	C₂×C₄²	D₄×C₄₃	1	2	yes	—
240	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
242	#107	—	2	4	no	—
243	#107	—	4	2	no	—
244	#106	—	2	4	no	—
245	#106	—	4	2	no	—
246	C₂²⋊C₈	Q₈×C₄₃	1	8	yes	—
247	C₄×C₈	—	4	1	no	—
248	C₄×C₈	—	4	1	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
268	C₂²×C₄	C₂×Q₈×C₄₃	1	4	yes	—
269	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂×Q₈×C₄₃	1	4	yes	—
275	C₂×C₈	C₂×Q₈×C₄₃	1	4	yes	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	#41×C₄₃	1	4	yes	—
288	C₂²⋊C₄	—	2	4	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₄²	—	2	4	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	C₄⋊Q₈×C₄₃	1	2	yes	—
303	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	8	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	C₄⋊Q₈×C₄₃	1	2	yes	—
327	C₂×C₄	#135×C₄₃	1	4	yes	—
328	C₂×C₄	C₂²×Q₈×C₄₃	1	2	yes	—
329	C₂×C₄	C₂×#41×C₄₃	1	2	yes	—
330	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
331	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
332	C₈	—	2	4	no	—
333	C₈	—	2	4	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₄₃×#525	1	1	yes	characteristic
340	C₂²	C₂×C₄⋊Q₈×C₄₃	1	1	yes	characteristic
341	C₂²	C₄₃×#534	1	1	yes	characteristic
342	C₂²	C₂×#135×C₄₃	1	2	yes	—
343	C₂²	C₄₃×#575	1	2	yes	—
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	4	4	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	C₄₃×#554	1	2	yes	—
352	C₄	C₄₃×#554	1	2	yes	—
353	C₂	—	4	2	no	—
354	C₂	C₄₃×#2583	1	1	yes	characteristic
355	C₂	C₄₃×#3102	1	2	yes	—
356	C₂	C₄₃×#3160	1	1	yes	characteristic
357	C₂	C₄₃×#3166	1	1	yes	characteristic
358	C₂	C₄₃×#3191	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 ◁ C₂²×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₃×#2085	C₂	semidirect
4. C₄₃×#2059	C₂	semidirect
5. C₄₃×#1942	C₂	semidirect
6. C₄₃×#1941	C₂	semidirect
154. #29300	C₄₃	direct (N × H)
176. #2085	C₂×C₄₃	semidirect
177. #2059	C₂×C₄₃	semidirect
178. #1942	C₂×C₄₃	semidirect
179. #1941	C₂×C₄₃	semidirect
214. C₄₃	#29300	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×#2851	2
C₄₃×#2296	2
C₄₃×#2085	2
C₄₃×#2059	2
C₄₃×#1942	2
C₄₃×#1941	2
C₄₃×#1854	2
C₄×C₂²⋊C₈×C₄₃	2
#29300	43

Subgroups — C43×#29300 #24949317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29300 #24949317