Subgroups — C₄₃×#28391 #24948408

Order 11008.

846 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#479	C₂	1	2	yes	maximal
2	C₂×C₄₃×#462	C₂	1	2	yes	maximal
3	C₄₃×#2642	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2327	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#2097	C₂	1	2	yes	maximal
6	C₄₃×#2016	C₂	1	2	yes	maximal
7	C₄₃×#1971	C₂	1	2	yes	maximal
8	C₄₃×#1969	C₂	1	2	yes	maximal
9	C₄₃×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#479	—	2	8	no	—
17	C₄₃×#462	—	2	8	no	—
18	C₂×#107×C₄₃	C₂²	1	2	yes	—
19	C₂×#106×C₄₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
21	C₂×#103×C₄₃	C₂²	1	2	yes	—
22	C₂×#103×C₄₃	C₂²	1	2	yes	—
23	C₂×#103×C₄₃	C₂²	1	2	yes	—
24	C₂×#103×C₄₃	C₂²	1	2	yes	—
25	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#400	—	2	4	no	—
28	C₄₃×#391	C₂²	1	2	yes	—
29	C₄₃×#391	C₂²	1	2	yes	—
30	C₄₃×#391	C₂²	1	2	yes	—
31	C₄₃×#387	—	2	4	no	—
32	C₄₃×#382	C₂²	1	1	yes	characteristic
33	C₄₃×#376	C₂²	1	1	yes	characteristic
34	C₄₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
36	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₄₃	—	2	4	no	—
38	C₂×Q₁₆×C₄₃	—	2	4	no	—
39	C₂×Q₁₆×C₄₃	—	2	4	no	—
40	C₂×Q₁₆×C₄₃	—	2	4	no	—
41	C₂×Q₁₆×C₄₃	—	4	2	no	—
42	C₂×Q₁₆×C₄₃	—	4	2	no	—
43	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₄₃	—	2	4	no	—
46	C₄⋊Q₈×C₄₃	—	2	4	no	—
47	#125×C₄₃	—	2	4	no	—
48	C₄×Q₈×C₄₃	—	2	4	no	—
49	C₄×Q₈×C₄₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₃	—	2	2	no	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#107×C₄₃	—	2	4	no	—
61	#106×C₄₃	—	2	4	no	—
62	C₄⋊C₈×C₄₃	D₄	1	8	yes	—
63	#103×C₄₃	—	2	4	no	—
64	#103×C₄₃	—	2	4	no	—
65	#103×C₄₃	—	2	4	no	—
66	#103×C₄₃	—	2	4	no	—
67	#97×C₄₃	—	4	1	no	—
68	C₄×C₈×C₄₃	—	4	1	no	—
69	#95×C₄₃	—	2	2	no	—
70	#95×C₄₃	—	2	4	no	—
71	#95×C₄₃	—	2	4	no	—
72	#95×C₄₃	—	2	4	no	—
73	#95×C₄₃	D₄	1	4	yes	—
74	C₂×Q₈×C₄₃	—	2	2	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂×Q₈×C₄₃	—	2	2	no	—
78	C₂×Q₈×C₄₃	—	2	2	no	—
79	C₂×Q₈×C₄₃	—	2	2	no	—
80	C₂×Q₈×C₄₃	—	2	2	no	—
81	C₂×Q₈×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×Q₈×C₄₃	—	2	2	no	—
85	C₂×Q₈×C₄₃	#41	1	4	yes	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	2	no	—
89	C₂²×C₄×C₄₃	—	2	2	no	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
94	Q₁₆×C₄₃	—	4	8	no	—
95	Q₁₆×C₄₃	—	4	8	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
102	C₄⋊C₄×C₄₃	—	2	8	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	4	no	—
108	C₄⋊C₄×C₄₃	—	4	4	no	—
109	C₄⋊C₄×C₄₃	—	4	4	no	—
110	C₄²×C₄₃	—	2	2	no	—
111	C₄²×C₄₃	—	2	1	no	—
112	C₄²×C₄₃	—	2	1	no	—
113	C₄²×C₄₃	—	4	4	no	—
114	C₄²×C₄₃	C₂×D₄	1	4	yes	—
115	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₄₃	—	2	4	no	—
117	Q₈×C₄₃	—	2	4	no	—
118	Q₈×C₄₃	—	4	2	no	—
119	Q₈×C₄₃	—	4	2	no	—
120	Q₈×C₄₃	—	4	2	no	—
121	Q₈×C₄₃	—	4	2	no	—
122	Q₈×C₄₃	—	4	2	no	—
123	Q₈×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	2	4	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	4	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	—	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	4	no	—
138	C₂×C₄×C₄₃	—	4	4	no	—
139	C₂×C₄×C₄₃	#127	1	4	yes	—
140	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
142	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₄₃	—	2	4	no	—
145	C₈×C₄₃	—	2	4	no	—
146	#28391	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₄₃	#525	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
149	C₂²×C₄₃	#537	1	1	yes	characteristic
150	C₂²×C₄₃	#578	1	4	yes	—
151	C₄×C₄₃	—	2	4	no	—
152	C₄×C₄₃	—	4	2	no	—
153	C₄×C₄₃	—	4	2	no	—
154	C₄×C₄₃	—	4	4	no	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	—	8	2	no	—
158	C₄×C₄₃	#546	1	2	yes	—
159	C₄×C₄₃	#548	1	2	yes	—
160	C₂×#479	C₂×C₄₃	1	2	yes	—
161	C₂×#462	C₂×C₄₃	1	2	yes	—
162	#2642	C₂×C₄₃	1	1	yes	characteristic
163	#2327	C₂×C₄₃	1	1	yes	characteristic
164	#2097	C₂×C₄₃	1	2	yes	—
165	#2016	C₂×C₄₃	1	2	yes	—
166	#1971	C₂×C₄₃	1	2	yes	—
167	#1969	C₂×C₄₃	1	2	yes	—
168	#1768	C₂×C₄₃	1	1	yes	characteristic
169	C₂×C₄₃	#2540	1	1	yes	characteristic
170	C₂×C₄₃	#3127	1	2	yes	—
171	C₂×C₄₃	#3156	1	2	yes	—
172	C₂×C₄₃	#3261	1	2	yes	—
173	C₂²×Q₁₆	C₂²×C₄₃	1	2	yes	—
174	C₂²×Q₁₆	C₂²×C₄₃	1	2	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
177	C₂×#125	C₂²×C₄₃	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂²×C₄₃	1	2	yes	—
179	#479	—	2	8	no	—
180	#462	—	2	8	no	—
181	C₂×#107	C₂²×C₄₃	1	2	yes	—
182	C₂×#106	C₂²×C₄₃	1	2	yes	—
183	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
184	C₂×#103	C₂²×C₄₃	1	2	yes	—
185	C₂×#103	C₂²×C₄₃	1	2	yes	—
186	C₂×#103	C₂²×C₄₃	1	2	yes	—
187	C₂×#103	C₂²×C₄₃	1	2	yes	—
188	C₂×#97	C₂²×C₄₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
190	#400	—	2	4	no	—
191	#391	C₂²×C₄₃	1	2	yes	—
192	#391	C₂²×C₄₃	1	2	yes	—
193	#391	C₂²×C₄₃	1	2	yes	—
194	#387	—	2	4	no	—
195	#382	C₂²×C₄₃	1	1	yes	characteristic
196	#376	C₂²×C₄₃	1	1	yes	characteristic
197	#336	C₂²×C₄₃	1	2	yes	—
198	C₄₃	#28391	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
200	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
201	C₂×Q₁₆	—	2	4	no	—
202	C₂×Q₁₆	—	2	4	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	—	4	2	no	—
206	C₂×Q₁₆	—	4	2	no	—
207	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
208	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	#125	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
224	#107	—	2	4	no	—
225	#106	—	2	4	no	—
226	C₄⋊C₈	D₄×C₄₃	1	8	yes	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	#103	—	2	4	no	—
231	#97	—	4	1	no	—
232	C₄×C₈	—	4	1	no	—
233	#95	—	2	2	no	—
234	#95	—	2	4	no	—
235	#95	—	2	4	no	—
236	#95	—	2	4	no	—
237	#95	D₄×C₄₃	1	4	yes	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	#41×C₄₃	1	4	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
258	Q₁₆	—	4	8	no	—
259	Q₁₆	—	4	8	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₄×C₄₃	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₄₃	1	4	yes	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	4	4	no	—
278	C₄²	C₂×D₄×C₄₃	1	4	yes	—
279	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
280	Q₈	—	2	4	no	—
281	Q₈	—	2	4	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	#127×C₄₃	1	4	yes	—
304	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
305	C₂×C₄	C₂×#41×C₄₃	1	2	yes	—
306	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
307	C₂×C₄	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	2	4	no	—
310	C₂²	C₄₃×#525	1	1	yes	characteristic
311	C₂²	C₂×#127×C₄₃	1	1	yes	characteristic
312	C₂²	C₄₃×#537	1	1	yes	characteristic
313	C₂²	C₄₃×#578	1	4	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₄₃×#546	1	2	yes	—
322	C₄	C₄₃×#548	1	2	yes	—
323	C₂	C₄₃×#2540	1	1	yes	characteristic
324	C₂	C₄₃×#3127	1	2	yes	—
325	C₂	C₄₃×#3156	1	2	yes	—
326	C₂	C₄₃×#3261	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
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
146. #28391	C₄₃	direct (N × H)
198. C₄₃	#28391	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×#479	2
C₂×C₄₃×#462	2
C₄₃×#2642	2
C₄₃×#2327	2
C₄₃×#2097	2
C₄₃×#2016	2
C₄₃×#1971	2
C₄₃×#1969	2
C₄₃×#1768	2
#28391	43

Subgroups — C43×#28391 #24948408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#28391 #24948408