Subgroups — C₄₃×#29930 #24949947

Order 11008.

668 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×#2621	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#2314	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2097	C₂	1	2	yes	maximal, complemented
4	C₄₃×#2072	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2036	C₂	1	2	yes	maximal
6	C₄₃×#2035	C₂	1	2	yes	maximal
7	C₄₃×#1981	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1949	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1798	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
12	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₂×#106×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
16	C₂×#103×C₄₃	C₂²	1	2	yes	—
17	C₂×#103×C₄₃	C₂²	1	2	yes	—
18	C₂×#103×C₄₃	C₂²	1	2	yes	—
19	C₂×#103×C₄₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#398	—	2	2	no	—
23	C₄₃×#397	—	2	2	no	—
24	C₄₃×#391	C₂²	1	2	yes	—
25	C₄₃×#388	—	2	2	no	—
26	C₄₃×#387	—	2	2	no	—
27	C₄₃×#384	C₂²	1	1	yes	characteristic
28	C₄₃×#382	C₂²	1	1	yes	characteristic
29	C₄₃×#382	C₂²	1	1	yes	characteristic
30	C₄₃×#380	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₄₃	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	C₂²×Q₈×C₄₃	C₂³	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	C₄⋊Q₈×C₄₃	—	2	4	no	—
40	C₂²⋊Q₈×C₄₃	—	4	2	no	—
41	C₂²⋊Q₈×C₄₃	—	4	2	no	—
42	#117×C₄₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
53	C₂×C₄²×C₄₃	—	2	2	no	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	#107×C₄₃	—	4	2	no	—
57	#106×C₄₃	—	4	2	no	—
58	C₄⋊C₈×C₄₃	—	2	4	no	—
59	#103×C₄₃	—	4	2	no	—
60	#103×C₄₃	—	4	2	no	—
61	#103×C₄₃	—	4	2	no	—
62	#103×C₄₃	—	4	2	no	—
63	C₂²⋊C₈×C₄₃	—	4	1	no	—
64	C₂²⋊C₈×C₄₃	—	4	1	no	—
65	#95×C₄₃	—	2	2	no	—
66	#95×C₄₃	—	2	2	no	—
67	#95×C₄₃	—	2	2	no	—
68	#95×C₄₃	—	2	2	no	—
69	#95×C₄₃	—	2	4	no	—
70	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	2	2	no	—
73	C₂×Q₈×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	—	2	2	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	1	no	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	1	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	2	2	no	—
84	C₂²×C₄×C₄₃	—	2	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	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	4	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	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	4	4	no	—
103	C₄⋊C₄×C₄₃	—	4	4	no	—
104	C₄⋊C₄×C₄₃	—	8	2	no	—
105	C₂²⋊C₄×C₄₃	—	2	4	no	—
106	C₂²⋊C₄×C₄₃	—	4	4	no	—
107	C₂²⋊C₄×C₄₃	—	8	2	no	—
108	C₄²×C₄₃	—	2	2	no	—
109	C₄²×C₄₃	—	8	2	no	—
110	C₄²×C₄₃	#41	1	4	yes	—
111	C₂³×C₄₃	—	2	1	no	—
112	C₂³×C₄₃	—	2	1	no	—
113	C₂³×C₄₃	—	2	1	no	—
114	C₂³×C₄₃	—	2	2	no	—
115	C₂³×C₄₃	—	2	2	no	—
116	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₄₃	—	4	2	no	—
118	Q₈×C₄₃	—	4	2	no	—
119	Q₈×C₄₃	—	4	2	no	—
120	Q₈×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	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	—	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	#123	1	4	yes	—
144	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
147	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₄₃	—	4	2	no	—
149	C₈×C₄₃	—	4	2	no	—
150	#29930	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	—	4	2	no	—
154	C₂²×C₄₃	—	4	1	no	—
155	C₂²×C₄₃	—	4	1	no	—
156	C₂²×C₄₃	C₂×#123	1	1	yes	characteristic
157	C₂²×C₄₃	#535	1	1	yes	characteristic
158	C₂²×C₄₃	#537	1	1	yes	characteristic
159	C₂²×C₄₃	#577	1	2	yes	—
160	C₂²×C₄₃	#578	1	2	yes	—
161	C₄×C₄₃	—	2	4	no	—
162	C₄×C₄₃	—	4	2	no	—
163	C₄×C₄₃	—	4	1	no	—
164	C₄×C₄₃	—	8	2	no	—
165	C₄×C₄₃	—	8	2	no	—
166	C₄×C₄₃	—	8	2	no	—
167	C₄×C₄₃	—	8	2	no	—
168	C₄×C₄₃	#540	1	2	yes	—
169	C₄×C₄₃	#542	1	2	yes	—
170	#2621	C₂×C₄₃	1	1	yes	characteristic
171	#2314	C₂×C₄₃	1	1	yes	characteristic
172	#2097	C₂×C₄₃	1	2	yes	complemented
173	#2072	C₂×C₄₃	1	2	yes	complemented
174	#2036	C₂×C₄₃	1	2	yes	—
175	#2035	C₂×C₄₃	1	2	yes	—
176	#1981	C₂×C₄₃	1	2	yes	complemented
177	#1949	C₂×C₄₃	1	2	yes	complemented
178	#1798	C₂×C₄₃	1	1	yes	characteristic
179	C₂×C₄₃	—	8	1	no	—
180	C₂×C₄₃	#2624	1	1	yes	characteristic
181	C₂×C₄₃	#3097	1	2	yes	—
182	C₂×C₄₃	#3225	1	2	yes	—
183	C₂×C₄₃	#3260	1	1	yes	characteristic
184	C₂×C₄₃	#3261	1	1	yes	characteristic
185	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
186	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
187	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
188	C₂×#107	C₂²×C₄₃	1	2	yes	—
189	C₂×#106	C₂²×C₄₃	1	2	yes	—
190	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
191	C₂×#103	C₂²×C₄₃	1	2	yes	—
192	C₂×#103	C₂²×C₄₃	1	2	yes	—
193	C₂×#103	C₂²×C₄₃	1	2	yes	—
194	C₂×#103	C₂²×C₄₃	1	2	yes	—
195	C₂×C₂²⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
196	C₂×C₂²⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
197	#398	—	2	2	no	—
198	#397	—	2	2	no	—
199	#391	C₂²×C₄₃	1	2	yes	—
200	#388	—	2	2	no	—
201	#387	—	2	2	no	—
202	#384	C₂²×C₄₃	1	1	yes	characteristic
203	#382	C₂²×C₄₃	1	1	yes	characteristic
204	#382	C₂²×C₄₃	1	1	yes	characteristic
205	#380	C₂²×C₄₃	1	2	yes	—
206	C₄×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
207	#340	C₂²×C₄₃	1	2	yes	—
208	#340	C₂²×C₄₃	1	2	yes	—
209	#336	C₂²×C₄₃	1	2	yes	—
210	C₄₃	#29930	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
212	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
214	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
215	C₄⋊Q₈	—	2	4	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	#117	—	2	4	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
232	#107	—	4	2	no	—
233	#106	—	4	2	no	—
234	C₄⋊C₈	—	2	4	no	—
235	#103	—	4	2	no	—
236	#103	—	4	2	no	—
237	#103	—	4	2	no	—
238	#103	—	4	2	no	—
239	C₂²⋊C₈	—	4	1	no	—
240	C₂²⋊C₈	—	4	1	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	4	no	—
246	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
264	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₄⋊C₄	—	2	4	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	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	8	2	no	—
286	C₄²	#41×C₄₃	1	4	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	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	—	4	2	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	#123×C₄₃	1	4	yes	—
320	C₂×C₄	C₂×#41×C₄₃	1	2	yes	—
321	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
322	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
323	C₂×C₄	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
324	C₈	—	4	2	no	—
325	C₈	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	C₂×#123×C₄₃	1	1	yes	characteristic
332	C₂²	C₄₃×#535	1	1	yes	characteristic
333	C₂²	C₄₃×#537	1	1	yes	characteristic
334	C₂²	C₄₃×#577	1	2	yes	—
335	C₂²	C₄₃×#578	1	2	yes	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₄₃×#540	1	2	yes	—
344	C₄	C₄₃×#542	1	2	yes	—
345	C₂	—	8	1	no	—
346	C₂	C₄₃×#2624	1	1	yes	characteristic
347	C₂	C₄₃×#3097	1	2	yes	—
348	C₂	C₄₃×#3225	1	2	yes	—
349	C₂	C₄₃×#3260	1	1	yes	characteristic
350	C₂	C₄₃×#3261	1	1	yes	characteristic

	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 ◁ C₂×#117×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₃×#2097	C₂	semidirect
4. C₄₃×#2072	C₂	semidirect
7. C₄₃×#1981	C₂	semidirect
8. C₄₃×#1949	C₂	semidirect
150. #29930	C₄₃	direct (N × H)
172. #2097	C₂×C₄₃	semidirect
173. #2072	C₂×C₄₃	semidirect
176. #1981	C₂×C₄₃	semidirect
177. #1949	C₂×C₄₃	semidirect
210. C₄₃	#29930	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×#2621	2
C₄₃×#2314	2
C₄₃×#2097	2
C₄₃×#2072	2
C₄₃×#2036	2
C₄₃×#2035	2
C₄₃×#1981	2
C₄₃×#1949	2
C₄₃×#1798	2
#29930	43

Subgroups — C43×#29930 #24949947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29930 #24949947