Subgroups — C₄₃×#29627 #24949644

Order 11008.

1326 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#451	C₂	1	2	yes	maximal
3	C₄₃×#2855	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2043	C₂	1	4	yes	maximal, complemented
5	C₄₃×#2016	C₂	1	4	yes	maximal, complemented
6	C₄₃×#1817	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1787	C₂	1	2	yes	maximal
8	C₂³×Q₈×C₄₃	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₄₃	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₄₃	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#451	—	2	8	no	—
18	C₂×#103×C₄₃	C₂²	1	4	yes	—
19	C₂×#103×C₄₃	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
21	C₄₃×#391	C₂²	1	2	yes	—
22	C₄₃×#389	C₂²	1	2	yes	—
23	C₄₃×#387	—	2	2	no	—
24	C₄₃×#385	—	2	1	no	—
25	C₄₃×#381	—	2	1	no	—
26	C₄₃×#380	C₂²	1	1	yes	characteristic
27	C₄₃×#336	C₂²	1	4	yes	—
28	C₂²×Q₈×C₄₃	—	2	4	no	—
29	C₂²×Q₈×C₄₃	—	2	2	no	—
30	C₂²×Q₈×C₄₃	—	4	8	no	—
31	C₂²×Q₈×C₄₃	D₄	1	8	yes	—
32	C₂²×Q₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₄₃	—	2	2	no	—
35	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×Q₁₆×C₄₃	—	2	8	no	—
39	C₂×Q₁₆×C₄₃	—	4	8	no	—
40	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₄₃	—	2	16	no	—
42	C₄⋊Q₈×C₄₃	—	2	16	no	—
43	C₂²⋊Q₈×C₄₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
46	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
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×C₄²×C₄₃	—	2	4	no	—
55	C₂×C₄²×C₄₃	D₄	1	2	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
57	#103×C₄₃	—	2	8	no	—
58	#103×C₄₃	—	4	8	no	—
59	C₂²⋊C₈×C₄₃	—	2	4	no	—
60	#95×C₄₃	—	2	1	no	—
61	#95×C₄₃	—	2	2	no	—
62	#95×C₄₃	—	2	2	no	—
63	#95×C₄₃	—	2	2	no	—
64	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₄₃	—	2	8	no	—
66	C₂×Q₈×C₄₃	—	2	4	no	—
67	C₂×Q₈×C₄₃	—	2	2	no	—
68	C₂×Q₈×C₄₃	—	2	8	no	—
69	C₂×Q₈×C₄₃	—	2	8	no	—
70	C₂×Q₈×C₄₃	—	2	4	no	—
71	C₂×Q₈×C₄₃	—	4	16	no	—
72	C₂×Q₈×C₄₃	—	4	8	no	—
73	C₂×Q₈×C₄₃	—	4	16	no	—
74	C₂×Q₈×C₄₃	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₃	—	2	4	no	—
76	C₂²×C₄×C₄₃	—	2	4	no	—
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	—	2	8	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	2	4	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	2	1	no	—
84	C₂²×C₄×C₄₃	—	2	1	no	—
85	C₂²×C₄×C₄₃	—	4	8	no	—
86	C₂²×C₄×C₄₃	Q₁₆	1	4	yes	complemented
87	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
92	Q₁₆×C₄₃	—	4	16	no	—
93	C₂×C₈×C₄₃	—	2	4	no	—
94	C₂×C₈×C₄₃	—	2	8	no	—
95	C₄⋊C₄×C₄₃	—	2	16	no	—
96	C₄⋊C₄×C₄₃	—	2	16	no	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	2	4	no	—
99	C₄⋊C₄×C₄₃	—	4	2	no	—
100	C₄⋊C₄×C₄₃	—	4	4	no	—
101	C₂²⋊C₄×C₄₃	—	4	2	no	—
102	C₂²⋊C₄×C₄₃	—	4	2	no	—
103	C₄²×C₄₃	—	2	4	no	—
104	C₄²×C₄₃	—	2	4	no	—
105	C₂³×C₄₃	—	2	2	no	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	#120	1	2	yes	—
108	C₂³×C₄₃	C₂×Q₁₆	1	2	yes	—
109	C₂³×C₄₃	#137	1	2	yes	—
110	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₄₃	—	2	8	no	—
112	Q₈×C₄₃	—	2	8	no	—
113	Q₈×C₄₃	—	4	16	no	—
114	Q₈×C₄₃	—	4	8	no	—
115	Q₈×C₄₃	—	4	4	no	—
116	C₂×C₄×C₄₃	—	2	8	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	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	8	no	—
123	C₂×C₄×C₄₃	—	2	8	no	—
124	C₂×C₄×C₄₃	—	2	8	no	—
125	C₂×C₄×C₄₃	—	4	16	no	—
126	C₂×C₄×C₄₃	—	4	8	no	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	4	1	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	#120	1	2	yes	—
135	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
136	C₂×C₄×C₄₃	#137	1	4	yes	—
137	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
139	C₈×C₄₃	—	4	4	no	—
140	#29627	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	4	no	—
143	C₂²×C₄₃	—	2	4	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	#451	1	4	yes	—
146	C₂²×C₄₃	C₂×#120	1	1	yes	characteristic
147	C₂²×C₄₃	#535	1	2	yes	—
148	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂²×C₄₃	C₂×#137	1	2	yes	—
150	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	2	4	no	—
153	C₄×C₄₃	—	2	4	no	—
154	C₄×C₄₃	—	4	8	no	—
155	C₄×C₄₃	—	8	1	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	—	8	1	no	—
158	C₄×C₄₃	#534	1	4	yes	—
159	C₂²×C₄⋊Q₈	C₂×C₄₃	1	1	yes	characteristic
160	C₂×#451	C₂×C₄₃	1	2	yes	—
161	#2855	C₂×C₄₃	1	1	yes	characteristic
162	#2043	C₂×C₄₃	1	4	yes	complemented
163	#2016	C₂×C₄₃	1	4	yes	complemented
164	#1817	C₂×C₄₃	1	1	yes	characteristic
165	#1787	C₂×C₄₃	1	2	yes	—
166	C₂×C₄₃	—	2	4	no	—
167	C₂×C₄₃	#2613	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#451	1	1	yes	characteristic
169	C₂×C₄₃	#3004	1	1	yes	characteristic
170	C₂×C₄₃	#3209	1	2	yes	—
171	C₂×C₄₃	#3218	1	2	yes	—
172	C₂³×Q₈	C₂²×C₄₃	1	2	yes	—
173	C₂²×Q₁₆	C₂²×C₄₃	1	4	yes	—
174	C₂×C₄⋊Q₈	—	2	8	no	—
175	C₂×C₄⋊Q₈	—	2	2	no	—
176	C₂×C₄⋊Q₈	C₂²×C₄₃	1	4	yes	—
177	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	—	2	1	no	—
179	C₂²×C₄⋊C₄	—	2	1	no	—
180	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
181	#451	—	2	8	no	—
182	C₂×#103	C₂²×C₄₃	1	4	yes	—
183	C₂×#103	C₂²×C₄₃	1	8	yes	—
184	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
185	#391	C₂²×C₄₃	1	2	yes	—
186	#389	C₂²×C₄₃	1	2	yes	—
187	#387	—	2	2	no	—
188	#385	—	2	1	no	—
189	#381	—	2	1	no	—
190	#380	C₂²×C₄₃	1	1	yes	characteristic
191	#336	C₂²×C₄₃	1	4	yes	—
192	C₄₃	#29627	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	—	4	8	no	—
196	C₂²×Q₈	D₄×C₄₃	1	8	yes	—
197	C₂²×Q₈	C₂³×C₄₃	1	1	yes	characteristic
198	C₂²×Q₈	C₂³×C₄₃	1	4	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
203	C₂×Q₁₆	—	2	8	no	—
204	C₂×Q₁₆	—	4	8	no	—
205	C₂²×C₈	C₂³×C₄₃	1	4	yes	—
206	C₄⋊Q₈	—	2	16	no	—
207	C₄⋊Q₈	—	2	16	no	—
208	C₂²⋊Q₈	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	8	no	—
210	C₂×C₄⋊C₄	—	2	8	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₄²	—	2	4	no	—
220	C₂×C₄²	D₄×C₄₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
222	#103	—	2	8	no	—
223	#103	—	4	8	no	—
224	C₂²⋊C₈	—	2	4	no	—
225	#95	—	2	1	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
230	C₂×Q₈	—	2	8	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	8	no	—
234	C₂×Q₈	—	2	8	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	16	no	—
237	C₂×Q₈	—	4	8	no	—
238	C₂×Q₈	—	4	16	no	—
239	C₂×Q₈	C₂×D₄×C₄₃	1	8	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	8	no	—
251	C₂²×C₄	Q₁₆×C₄₃	1	4	yes	complemented
252	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
254	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
257	Q₁₆	—	4	16	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	8	no	—
260	C₄⋊C₄	—	2	16	no	—
261	C₄⋊C₄	—	2	16	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	4	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	#120×C₄₃	1	2	yes	—
273	C₂³	C₂×Q₁₆×C₄₃	1	2	yes	—
274	C₂³	#137×C₄₃	1	2	yes	—
275	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
276	Q₈	—	2	8	no	—
277	Q₈	—	2	8	no	—
278	Q₈	—	4	16	no	—
279	Q₈	—	4	8	no	—
280	Q₈	—	4	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	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	#120×C₄₃	1	2	yes	—
300	C₂×C₄	C₂×Q₁₆×C₄₃	1	4	yes	—
301	C₂×C₄	#137×C₄₃	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
303	C₂×C₄	H(5,𝔽₂)×C₄₃	1	4	yes	—
304	C₈	—	4	4	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₄₃×#451	1	4	yes	—
310	C₂²	C₂×#120×C₄₃	1	1	yes	characteristic
311	C₂²	C₄₃×#535	1	2	yes	—
312	C₂²	C₂²×Q₁₆×C₄₃	1	1	yes	characteristic
313	C₂²	C₂×#137×C₄₃	1	2	yes	—
314	C₂²	C₂×#137×C₄₃	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	4	no	—
318	C₄	—	4	8	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	C₄₃×#534	1	4	yes	—
323	C₂	—	2	4	no	—
324	C₂	C₄₃×#2613	1	1	yes	characteristic
325	C₂	C₂×C₄₃×#451	1	1	yes	characteristic
326	C₂	C₄₃×#3004	1	1	yes	characteristic
327	C₂	C₄₃×#3209	1	2	yes	—
328	C₂	C₄₃×#3218	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
4. C₄₃×#2043	C₂	semidirect
5. C₄₃×#2016	C₂	semidirect
86. C₂²×C₄×C₄₃	Q₁₆	semidirect
140. #29627	C₄₃	direct (N × H)
162. #2043	C₂×C₄₃	semidirect
163. #2016	C₂×C₄₃	semidirect
192. C₄₃	#29627	direct (N × H)
251. C₂²×C₄	Q₁₆×C₄₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊Q₈×C₄₃	2
C₂×C₄₃×#451	2
C₄₃×#2855	2
C₄₃×#2043	2
C₄₃×#2016	2
C₄₃×#1817	2
C₄₃×#1787	2
#29627	43

Subgroups — C43×#29627 #24949644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29627 #24949644