Subgroups — C₄₃×#8397 #24928414

Order 11008.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1800	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1549	C₂	1	2	yes	maximal, complemented
4	C₄₃×#1486	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1460	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#555	—	2	4	no	—
7	C₄₃×#555	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
9	C₂×#124×C₄₃	—	2	2	no	—
10	C₂×#121×C₄₃	—	2	1	no	—
11	C₂×#121×C₄₃	C₄	1	2	yes	—
12	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₃	—	2	1	no	—
15	C₂×#117×C₄₃	C₄	1	2	yes	—
16	C₄₃×#431	C₂²	1	2	yes	—
17	C₄₃×#423	C₂²	1	2	yes	—
18	C₂×#100×C₄₃	—	2	1	no	—
19	C₄₃×#343	—	2	1	no	—
20	C₄₃×#332	—	2	4	no	—
21	C₄₃×D₄⋊C₈	—	2	4	no	—
22	C₄₃×#323	—	2	2	no	—
23	C₂²×Q₈×C₄₃	—	2	1	no	—
24	C₂²×D₄×C₄₃	—	2	2	no	—
25	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₃	—	2	1	no	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₄₃	—	2	2	no	—
30	C₄⋊Q₈×C₄₃	—	2	2	no	—
31	C₄⋊Q₈×C₄₃	D₄	1	4	yes	—
32	#124×C₄₃	—	2	16	no	—
33	#121×C₄₃	—	2	8	no	—
34	#121×C₄₃	—	2	4	no	—
35	#121×C₄₃	—	2	4	no	—
36	C₄×D₄×C₄₃	—	2	2	no	—
37	C₄×D₄×C₄₃	—	2	2	no	—
38	C₄×D₄×C₄₃	—	2	2	no	—
39	C₄×D₄×C₄₃	D₄	1	4	yes	—
40	#117×C₄₃	—	2	8	no	—
41	#117×C₄₃	—	2	4	no	—
42	#117×C₄₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
51	C₂×C₄²×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	—	2	2	no	—
54	C₄⋊C₈×C₄₃	—	2	2	no	—
55	C₄⋊C₈×C₄₃	—	2	2	no	—
56	#100×C₄₃	—	4	2	no	—
57	C₂²⋊C₈×C₄₃	—	2	2	no	—
58	C₄×C₈×C₄₃	—	2	2	no	—
59	C₂⁴×C₄₃	—	2	1	no	—
60	C₂⁴×C₄₃	C₂²⋊C₄	1	2	yes	—
61	C₂×Q₈×C₄₃	—	2	4	no	—
62	C₂×Q₈×C₄₃	—	8	2	no	—
63	C₂×D₄×C₄₃	—	2	8	no	—
64	C₂×D₄×C₄₃	—	2	1	no	—
65	C₂×D₄×C₄₃	—	2	2	no	—
66	C₂×D₄×C₄₃	—	2	2	no	—
67	C₂×D₄×C₄₃	—	2	1	no	—
68	C₂×D₄×C₄₃	—	2	1	no	—
69	C₂×D₄×C₄₃	—	4	2	no	—
70	C₂×D₄×C₄₃	—	8	4	no	—
71	C₂×D₄×C₄₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	—	2	2	no	—
74	C₂²×C₄×C₄₃	—	2	4	no	—
75	C₂²×C₄×C₄₃	—	2	1	no	—
76	C₂²×C₄×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	—	4	4	no	—
78	C₂²×C₄×C₄₃	—	4	2	no	—
79	C₂²×C₄×C₄₃	—	4	1	no	—
80	C₂²×C₄×C₄₃	—	4	1	no	—
81	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₄₃	—	4	2	no	—
86	M₁₆×C₄₃	—	4	2	no	—
87	C₂×C₈×C₄₃	—	4	2	no	—
88	C₂×C₈×C₄₃	—	4	2	no	—
89	C₄⋊C₄×C₄₃	—	2	1	no	—
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	4	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
94	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
95	C₂²⋊C₄×C₄₃	—	2	8	no	—
96	C₂²⋊C₄×C₄₃	—	2	8	no	—
97	C₂²⋊C₄×C₄₃	—	2	4	no	—
98	C₂²⋊C₄×C₄₃	—	2	4	no	—
99	C₄²×C₄₃	—	2	8	no	—
100	C₄²×C₄₃	C₂×D₄	1	2	yes	—
101	C₄²×C₄₃	C₂×D₄	1	2	yes	—
102	C₂³×C₄₃	—	2	2	no	—
103	C₂³×C₄₃	—	2	2	no	—
104	C₂³×C₄₃	—	2	2	no	—
105	C₂³×C₄₃	—	2	2	no	—
106	C₂³×C₄₃	—	4	2	no	—
107	C₂³×C₄₃	—	4	2	no	—
108	C₂³×C₄₃	—	4	2	no	—
109	C₂³×C₄₃	—	4	2	no	—
110	C₂³×C₄₃	—	4	1	no	—
111	C₂³×C₄₃	—	4	1	no	—
112	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	Q₈×C₄₃	—	8	4	no	—
114	D₄×C₄₃	—	2	4	no	—
115	D₄×C₄₃	—	4	2	no	—
116	D₄×C₄₃	—	4	2	no	—
117	D₄×C₄₃	—	4	2	no	—
118	D₄×C₄₃	—	8	8	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	4	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	4	no	—
125	C₂×C₄×C₄₃	—	4	4	no	—
126	C₂×C₄×C₄₃	—	4	1	no	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	4	4	no	—
129	C₂×C₄×C₄₃	—	4	4	no	—
130	C₂×C₄×C₄₃	—	4	4	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#120	1	2	yes	—
135	C₂×C₄×C₄₃	#120	1	2	yes	—
136	C₂×C₄×C₄₃	#135	1	2	yes	—
137	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
139	C₈×C₄₃	—	8	2	no	—
140	C₈×C₄₃	—	8	2	no	—
141	#8397	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	4	2	no	—
145	C₂²×C₄₃	—	4	2	no	—
146	C₂²×C₄₃	—	4	2	no	—
147	C₂²×C₄₃	—	4	2	no	—
148	C₂²×C₄₃	—	4	2	no	—
149	C₂²×C₄₃	—	4	2	no	—
150	C₂²×C₄₃	—	8	2	no	—
151	C₂²×C₄₃	—	8	2	no	—
152	C₂²×C₄₃	—	8	2	no	—
153	C₂²×C₄₃	—	8	2	no	—
154	C₂²×C₄₃	#379	1	1	yes	characteristic
155	C₂²×C₄₃	#416	1	1	yes	characteristic
156	C₂²×C₄₃	#417	1	1	yes	characteristic
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	4	4	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	—	4	2	no	—
162	C₄×C₄₃	#413	1	2	yes	—
163	C₄×C₄₃	#449	1	2	yes	—
164	C₄×C₄₃	#452	1	2	yes	—
165	C₂×#555	C₂×C₄₃	1	1	yes	characteristic
166	#1800	C₂×C₄₃	1	1	yes	characteristic, complemented
167	#1549	C₂×C₄₃	1	2	yes	complemented
168	#1486	C₂×C₄₃	1	2	yes	complemented
169	#1460	C₂×C₄₃	1	1	yes	characteristic, complemented
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	—	8	2	no	—
173	C₂×C₄₃	—	8	2	no	—
174	C₂×C₄₃	#1785	1	1	yes	characteristic
175	C₂×C₄₃	#1788	1	1	yes	characteristic
176	C₂×C₄₃	#1795	1	1	yes	characteristic
177	#555	—	2	4	no	—
178	#555	—	2	4	no	—
179	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
180	C₂×#124	—	2	2	no	—
181	C₂×#121	—	2	1	no	—
182	C₂×#121	C₄×C₄₃	1	2	yes	—
183	C₂×C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
184	C₂×C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
185	C₂×#117	—	2	1	no	—
186	C₂×#117	C₄×C₄₃	1	2	yes	—
187	#431	C₂²×C₄₃	1	2	yes	—
188	#423	C₂²×C₄₃	1	2	yes	—
189	C₂×#100	—	2	1	no	—
190	#343	—	2	1	no	—
191	#332	—	2	4	no	—
192	D₄⋊C₈	—	2	4	no	—
193	#323	—	2	2	no	—
194	C₄₃	#8397	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×D₄	—	2	2	no	—
197	C₂²×D₄	C₂×C₄×C₄₃	1	1	yes	characteristic
198	C₂²×D₄	D₄×C₄₃	1	1	yes	characteristic
199	C₂³×C₄	—	2	1	no	—
200	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
201	C₂×M₁₆	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	D₄×C₄₃	1	4	yes	—
204	#124	—	2	16	no	—
205	#121	—	2	8	no	—
206	#121	—	2	4	no	—
207	#121	—	2	4	no	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	D₄×C₄₃	1	4	yes	—
212	#117	—	2	8	no	—
213	#117	—	2	4	no	—
214	#117	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
218	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	#100	—	4	2	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₂⁴	—	2	1	no	—
232	C₂⁴	C₂²⋊C₄×C₄₃	1	2	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	8	2	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	8	4	no	—
243	C₂×D₄	C₂²⋊C₄×C₄₃	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂²⋊C₄×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	M₁₆	—	4	2	no	—
258	M₁₆	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
266	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	4	yes	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₄²	—	2	8	no	—
272	C₄²	C₂×D₄×C₄₃	1	2	yes	—
273	C₄²	C₂×D₄×C₄₃	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
285	Q₈	—	8	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
306	C₂×C₄	#120×C₄₃	1	2	yes	—
307	C₂×C₄	#120×C₄₃	1	2	yes	—
308	C₂×C₄	#135×C₄₃	1	2	yes	—
309	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	1	yes	characteristic
310	C₂×C₄	#137×C₄₃	1	1	yes	characteristic
311	C₈	—	8	2	no	—
312	C₈	—	8	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₄₃×#379	1	1	yes	characteristic
326	C₂²	C₄₃×#416	1	1	yes	characteristic
327	C₂²	C₄₃×#417	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	4	no	—
333	C₄	C₄₃×#413	1	2	yes	—
334	C₄	C₄₃×#449	1	2	yes	—
335	C₄	C₄₃×#452	1	2	yes	—
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	8	2	no	—
339	C₂	—	8	2	no	—
340	C₂	C₄₃×#1785	1	1	yes	characteristic
341	C₂	C₄₃×#1788	1	1	yes	characteristic
342	C₂	C₄₃×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₂×C₄₃×#555	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1800	C₂	semidirect
3. C₄₃×#1549	C₂	semidirect
4. C₄₃×#1486	C₂	semidirect
5. C₄₃×#1460	C₂	semidirect
141. #8397	C₄₃	direct (N × H)
166. #1800	C₂×C₄₃	semidirect
167. #1549	C₂×C₄₃	semidirect
168. #1486	C₂×C₄₃	semidirect
169. #1460	C₂×C₄₃	semidirect
194. C₄₃	#8397	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×#555	2
C₄₃×#1800	2
C₄₃×#1549	2
C₄₃×#1486	2
C₄₃×#1460	2
#8397	43

Subgroups — C43×#8397 #24928414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8397 #24928414