Subgroups — C₄₃×#8314 #24928331

Order 11008.

836 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₃×Q₈⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₄₃×#533	—	2	8	no	—
6	C₂×C₄⋊Q₈×C₄₃	C₄	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₄₃	C₄	1	1	yes	characteristic
8	C₂×#125×C₄₃	—	2	1	no	—
9	C₂×C₂²⋊Q₈×C₄₃	—	2	1	no	—
10	C₂×C₂²⋊Q₈×C₄₃	C₄	1	2	yes	—
11	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
12	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
13	C₂×#117×C₄₃	—	2	1	no	—
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
17	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
19	C₄₃×#336	—	2	1	no	—
20	C₄₃×#336	—	2	1	no	—
21	C₄₃×Q₈⋊C₈	—	2	8	no	—
22	C₂²×Q₈×C₄₃	—	2	1	no	—
23	C₂²×Q₈×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	C₄⋊Q₈×C₄₃	—	2	4	no	—
30	C₄⋊Q₈×C₄₃	—	2	4	no	—
31	#125×C₄₃	—	2	8	no	—
32	C₂²⋊Q₈×C₄₃	—	2	8	no	—
33	C₂²⋊Q₈×C₄₃	—	2	4	no	—
34	C₂²⋊Q₈×C₄₃	—	2	4	no	—
35	C₄×Q₈×C₄₃	—	2	8	no	—
36	C₄×Q₈×C₄₃	—	2	4	no	—
37	C₄×Q₈×C₄₃	D₄	1	8	yes	—
38	#117×C₄₃	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄²×C₄₃	—	2	2	no	—
47	C₂×C₄²×C₄₃	—	2	4	no	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₃	—	2	4	no	—
53	C₂²⋊C₈×C₄₃	—	2	4	no	—
54	C₂²⋊C₈×C₄₃	—	4	2	no	—
55	C₄×C₈×C₄₃	—	2	4	no	—
56	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Q₈×C₄₃	—	2	4	no	—
58	C₂×Q₈×C₄₃	—	2	2	no	—
59	C₂×Q₈×C₄₃	—	2	4	no	—
60	C₂×Q₈×C₄₃	—	2	4	no	—
61	C₂×Q₈×C₄₃	—	4	2	no	—
62	C₂×Q₈×C₄₃	—	4	2	no	—
63	C₂×Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
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	—	2	2	no	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	2	2	no	—
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	—	4	1	no	—
74	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	4	2	no	—
84	C₂×C₈×C₄₃	—	4	2	no	—
85	C₂×C₈×C₄₃	—	4	2	no	—
86	C₂×C₈×C₄₃	—	4	2	no	—
87	C₂×C₈×C₄₃	—	4	2	no	—
88	C₄⋊C₄×C₄₃	—	2	4	no	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₄⋊C₄×C₄₃	—	2	8	no	—
91	C₄⋊C₄×C₄₃	—	2	4	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	8	no	—
94	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
95	C₂²⋊C₄×C₄₃	—	2	8	no	—
96	C₄²×C₄₃	—	2	2	no	—
97	C₄²×C₄₃	—	2	2	no	—
98	C₄²×C₄₃	—	2	2	no	—
99	C₄²×C₄₃	—	2	8	no	—
100	C₄²×C₄₃	C₂×D₄	1	4	yes	—
101	C₂³×C₄₃	—	2	1	no	—
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	—	2	1	no	—
104	C₂³×C₄₃	—	2	1	no	—
105	C₂³×C₄₃	—	2	1	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	#103	1	2	yes	—
108	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	Q₈×C₄₃	—	2	8	no	—
110	Q₈×C₄₃	—	4	8	no	—
111	Q₈×C₄₃	—	4	4	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	4	no	—
119	C₂×C₄×C₄₃	—	2	4	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	4	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	2	no	—
127	C₂×C₄×C₄₃	—	4	2	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	#103	1	2	yes	—
132	C₂×C₄×C₄₃	#104	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₃	#120	1	2	yes	—
135	C₂×C₄×C₄₃	#120	1	2	yes	—
136	C₂×C₄×C₄₃	#131	1	2	yes	—
137	C₂×C₄×C₄₃	#131	1	2	yes	—
138	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#137	1	2	yes	—
141	C₈×C₄₃	—	4	4	no	—
142	C₈×C₄₃	—	8	2	no	—
143	#8314	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	#379	1	1	yes	characteristic
152	C₂²×C₄₃	#408	1	1	yes	characteristic
153	C₂²×C₄₃	#408	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
155	C₂²×C₄₃	#419	1	1	yes	characteristic
156	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
157	C₂²×C₄₃	#421	1	1	yes	characteristic
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	4	1	no	—
162	C₄×C₄₃	—	4	4	no	—
163	C₄×C₄₃	—	4	4	no	—
164	C₄×C₄₃	#408	1	4	yes	—
165	C₄×C₄₃	#448	1	2	yes	—
166	C₄×C₄₃	#451	1	2	yes	—
167	C₂×#533	C₂×C₄₃	1	1	yes	characteristic
168	#1797	C₂×C₄₃	1	1	yes	characteristic
169	C₄×C₂²⋊C₈	C₂×C₄₃	1	1	yes	characteristic
170	C₂×Q₈⋊C₈	C₂×C₄₃	1	4	yes	complemented
171	C₂×C₄₃	—	4	2	no	—
172	C₂×C₄₃	#1481	1	1	yes	characteristic
173	C₂×C₄₃	#1485	1	1	yes	characteristic
174	C₂×C₄₃	#1493	1	1	yes	characteristic
175	C₂×C₄₃	#1497	1	1	yes	characteristic
176	C₂×C₄₃	#1780	1	1	yes	characteristic
177	C₂×C₄₃	#1787	1	1	yes	characteristic
178	C₂×C₄₃	#1789	1	1	yes	characteristic
179	#533	—	2	8	no	—
180	C₂×C₄⋊Q₈	C₄×C₄₃	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	C₄×C₄₃	1	1	yes	characteristic
182	C₂×#125	—	2	1	no	—
183	C₂×C₂²⋊Q₈	—	2	1	no	—
184	C₂×C₂²⋊Q₈	C₄×C₄₃	1	2	yes	—
185	C₂×C₄×Q₈	—	2	1	no	—
186	C₂×C₄×Q₈	C₂²×C₄₃	1	2	yes	—
187	C₂×#117	—	2	1	no	—
188	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
190	C₂×C₂²⋊C₈	—	2	1	no	—
191	C₂×C₂²⋊C₈	—	2	1	no	—
192	C₂×C₄×C₈	C₂²×C₄₃	1	2	yes	—
193	#336	—	2	1	no	—
194	#336	—	2	1	no	—
195	Q₈⋊C₈	—	2	8	no	—
196	C₄₃	#8314	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×Q₈	C₂×C₄×C₄₃	1	2	yes	—
199	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
200	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	—	2	4	no	—
206	#125	—	2	8	no	—
207	C₂²⋊Q₈	—	2	8	no	—
208	C₂²⋊Q₈	—	2	4	no	—
209	C₂²⋊Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	8	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	D₄×C₄₃	1	8	yes	—
213	#117	—	2	8	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
224	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
225	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
227	C₄⋊C₈	—	2	4	no	—
228	C₂²⋊C₈	—	2	4	no	—
229	C₂²⋊C₈	—	4	2	no	—
230	C₄×C₈	—	2	4	no	—
231	C₂⁴	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	C₂²⋊C₄×C₄₃	1	4	yes	—
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	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
252	C₂²×C₄	SD₁₆×C₄₃	1	2	yes	—
253	C₂²×C₄	Q₁₆×C₄₃	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	4	yes	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	8	no	—
275	C₄²	C₂×D₄×C₄₃	1	4	yes	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	#103×C₄₃	1	2	yes	—
283	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	Q₈	—	4	8	no	—
286	Q₈	—	4	4	no	—
287	C₂×C₄	—	2	1	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	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	#103×C₄₃	1	2	yes	—
307	C₂×C₄	#104×C₄₃	1	2	yes	—
308	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
309	C₂×C₄	#120×C₄₃	1	2	yes	—
310	C₂×C₄	#120×C₄₃	1	2	yes	—
311	C₂×C₄	#131×C₄₃	1	2	yes	—
312	C₂×C₄	#131×C₄₃	1	2	yes	—
313	C₂×C₄	C₂×SD₁₆×C₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂×Q₁₆×C₄₃	1	1	yes	characteristic
315	C₂×C₄	#137×C₄₃	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₈	—	8	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₄₃×#379	1	1	yes	characteristic
326	C₂²	C₄₃×#408	1	1	yes	characteristic
327	C₂²	C₄₃×#408	1	1	yes	characteristic
328	C₂²	C₂×#103×C₄₃	1	1	yes	characteristic
329	C₂²	C₄₃×#419	1	1	yes	characteristic
330	C₂²	C₂×#104×C₄₃	1	1	yes	characteristic
331	C₂²	C₄₃×#421	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	4	no	—
337	C₄	—	4	4	no	—
338	C₄	C₄₃×#408	1	4	yes	—
339	C₄	C₄₃×#448	1	2	yes	—
340	C₄	C₄₃×#451	1	2	yes	—
341	C₂	—	4	2	no	—
342	C₂	C₄₃×#1481	1	1	yes	characteristic
343	C₂	C₄₃×#1485	1	1	yes	characteristic
344	C₂	C₄₃×#1493	1	1	yes	characteristic
345	C₂	C₄₃×#1497	1	1	yes	characteristic
346	C₂	C₄₃×#1780	1	1	yes	characteristic
347	C₂	C₄₃×#1787	1	1	yes	characteristic
348	C₂	C₄₃×#1789	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₃ ◁ C₂²×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₄₃×Q₈⋊C₈	C₂	semidirect
143. #8314	C₄₃	direct (N × H)
170. C₂×Q₈⋊C₈	C₂×C₄₃	semidirect
196. C₄₃	#8314	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×#533	2
C₄₃×#1797	2
C₄×C₂²⋊C₈×C₄₃	2
C₂×C₄₃×Q₈⋊C₈	2
#8314	43

Subgroups — C43×#8314 #24928331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8314 #24928331