Subgroups — C₄₃×#29737 #24949754

Order 11008.

966 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₃×#408	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2616	C₂	1	2	yes	maximal
4	C₄₃×#2076	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2035	C₂	1	4	yes	maximal, complemented
6	C₄₃×#2001	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1894	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	—
15	C₄₃×#467	—	2	4	no	—
16	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#106×C₄₃	C₂²	1	2	yes	—
18	C₂×#103×C₄₃	C₂²	1	2	yes	—
19	C₂×#103×C₄₃	C₂²	1	4	yes	—
20	C₄₃×#408	—	2	4	no	—
21	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
23	C₄₃×#395	—	2	2	no	—
24	C₄₃×#394	—	2	4	no	—
25	C₄₃×#394	—	2	2	no	—
26	C₄₃×#380	C₂²	1	2	yes	—
27	C₄₃×#380	C₂²	1	4	yes	—
28	C₄₃×#340	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	4	yes	—
30	C₂×#41×C₄₃	—	2	2	no	—
31	C₂×#41×C₄₃	—	2	2	no	—
32	C₂×#41×C₄₃	—	4	2	no	—
33	C₂×#41×C₄₃	—	4	2	no	—
34	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
35	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
36	C₂²×D₄×C₄₃	D₄	1	2	yes	—
37	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
39	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×Q₁₆×C₄₃	—	2	2	no	—
41	C₂×Q₁₆×C₄₃	—	4	2	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	—	4	2	no	—
44	C₂²×C₈×C₄₃	—	4	2	no	—
45	C₂²×C₈×C₄₃	D₄	1	4	yes	—
46	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²⋊Q₈×C₄₃	—	2	4	no	—
50	C₂²⋊Q₈×C₄₃	—	2	4	no	—
51	C₂²⋊Q₈×C₄₃	—	4	4	no	—
52	C₂²⋊Q₈×C₄₃	—	4	4	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
61	#107×C₄₃	—	2	2	no	—
62	#106×C₄₃	—	4	2	no	—
63	#103×C₄₃	—	2	4	no	—
64	#103×C₄₃	—	4	4	no	—
65	C₂²⋊C₈×C₄₃	—	2	4	no	—
66	C₂²⋊C₈×C₄₃	—	2	4	no	—
67	#95×C₄₃	—	2	4	no	—
68	#95×C₄₃	—	2	2	no	—
69	C₂⁴×C₄₃	C₂×D₄	1	2	yes	—
70	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
71	#41×C₄₃	—	4	4	no	—
72	#41×C₄₃	—	4	4	no	—
73	#41×C₄₃	—	4	8	no	—
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	4	no	—
78	C₂×Q₈×C₄₃	—	2	4	no	—
79	C₂×Q₈×C₄₃	—	4	2	no	—
80	C₂×D₄×C₄₃	—	2	2	no	—
81	C₂×D₄×C₄₃	—	2	4	no	—
82	C₂×D₄×C₄₃	—	4	4	no	—
83	C₂×D₄×C₄₃	—	4	2	no	—
84	C₂²×C₄×C₄₃	—	2	2	no	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	4	no	—
89	C₂²×C₄×C₄₃	—	2	4	no	—
90	C₂²×C₄×C₄₃	—	2	2	no	—
91	C₂²×C₄×C₄₃	—	2	4	no	—
92	C₂²×C₄×C₄₃	—	2	2	no	—
93	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₃	#41	1	2	yes	—
97	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₄₃	—	4	4	no	—
99	M₁₆×C₄₃	—	4	4	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	4	no	—
104	C₂×C₈×C₄₃	—	4	4	no	—
105	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	2	no	—
109	C₄⋊C₄×C₄₃	—	2	4	no	—
110	C₄⋊C₄×C₄₃	—	2	4	no	—
111	C₄⋊C₄×C₄₃	—	4	8	no	—
112	C₄⋊C₄×C₄₃	—	4	4	no	—
113	C₂²⋊C₄×C₄₃	—	4	8	no	—
114	C₂²⋊C₄×C₄₃	—	4	4	no	—
115	C₂²⋊C₄×C₄₃	—	8	2	no	—
116	C₂²⋊C₄×C₄₃	—	8	4	no	—
117	C₂³×C₄₃	—	2	2	no	—
118	C₂³×C₄₃	—	2	2	no	—
119	C₂³×C₄₃	—	2	2	no	—
120	C₂³×C₄₃	—	2	2	no	—
121	C₂³×C₄₃	—	2	4	no	—
122	C₂³×C₄₃	—	2	4	no	—
123	C₂³×C₄₃	—	2	4	no	—
124	C₂³×C₄₃	#121	1	2	yes	—
125	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₄₃	—	4	4	no	—
127	Q₈×C₄₃	—	4	2	no	—
128	Q₈×C₄₃	—	4	2	no	—
129	Q₈×C₄₃	—	4	4	no	—
130	D₄×C₄₃	—	4	4	no	—
131	D₄×C₄₃	—	4	8	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	4	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	4	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	4	no	—
144	C₂×C₄×C₄₃	—	4	4	no	—
145	C₂×C₄×C₄₃	—	4	4	no	—
146	C₂×C₄×C₄₃	—	4	2	no	—
147	C₂×C₄×C₄₃	—	4	2	no	—
148	C₂×C₄×C₄₃	—	4	4	no	—
149	C₂×C₄×C₄₃	#121	1	2	yes	—
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
153	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
154	C₈×C₄₃	—	2	2	no	—
155	C₈×C₄₃	—	2	2	no	—
156	C₈×C₄₃	—	4	2	no	—
157	#29737	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	—	4	4	no	—
162	C₂²×C₄₃	—	4	2	no	—
163	C₂²×C₄₃	—	4	8	no	—
164	C₂²×C₄₃	—	4	2	no	—
165	C₂²×C₄₃	—	4	2	no	—
166	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
167	C₂²×C₄₃	#534	1	2	yes	—
168	C₂²×C₄₃	#577	1	2	yes	—
169	C₂²×C₄₃	#578	1	2	yes	—
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	—	8	4	no	—
174	C₄×C₄₃	—	8	2	no	—
175	C₄×C₄₃	#534	1	1	yes	characteristic
176	C₄×C₄₃	#535	1	1	yes	characteristic
177	C₄×C₄₃	#538	1	2	yes	—
178	C₂×#467	C₂×C₄₃	1	1	yes	characteristic, complemented
179	C₂×#408	C₂×C₄₃	1	1	yes	characteristic
180	#2616	C₂×C₄₃	1	2	yes	—
181	#2076	C₂×C₄₃	1	2	yes	complemented
182	#2035	C₂×C₄₃	1	4	yes	complemented
183	#2001	C₂×C₄₃	1	2	yes	complemented
184	#1894	C₂×C₄₃	1	2	yes	complemented
185	#1807	C₂×C₄₃	1	1	yes	characteristic, complemented
186	C₂×C₄₃	—	4	2	no	—
187	C₂×C₄₃	—	8	2	no	—
188	C₂×C₄₃	#2617	1	1	yes	characteristic
189	C₂×C₄₃	#3059	1	1	yes	characteristic
190	C₂×C₄₃	#3060	1	1	yes	characteristic
191	C₂×C₄₃	#3201	1	4	yes	—
192	C₂²×#41	C₂²×C₄₃	1	1	yes	characteristic
193	C₂²×Q₁₆	C₂²×C₄₃	1	1	yes	characteristic
194	C₂²×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₄₃	1	1	yes	characteristic
196	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
197	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	4	yes	—
198	#467	—	2	4	no	—
199	C₂×#107	C₂²×C₄₃	1	1	yes	characteristic
200	C₂×#106	C₂²×C₄₃	1	2	yes	—
201	C₂×#103	C₂²×C₄₃	1	2	yes	—
202	C₂×#103	C₂²×C₄₃	1	4	yes	—
203	#408	—	2	4	no	—
204	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
205	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
206	#395	—	2	2	no	—
207	#394	—	2	4	no	—
208	#394	—	2	2	no	—
209	#380	C₂²×C₄₃	1	2	yes	—
210	#380	C₂²×C₄₃	1	4	yes	—
211	#340	C₂²×C₄₃	1	2	yes	—
212	#340	C₂²×C₄₃	1	4	yes	—
213	C₄₃	#29737	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×#41	—	2	2	no	—
215	C₂×#41	—	2	2	no	—
216	C₂×#41	—	4	2	no	—
217	C₂×#41	—	4	2	no	—
218	C₂²×Q₈	D₄×C₄₃	1	1	yes	characteristic
219	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
220	C₂²×D₄	D₄×C₄₃	1	2	yes	—
221	C₂²×D₄	D₄×C₄₃	1	1	yes	characteristic
222	C₂³×C₄	C₂³×C₄₃	1	2	yes	—
223	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	—	4	2	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	4	2	no	—
228	C₂²×C₈	—	4	2	no	—
229	C₂²×C₈	D₄×C₄₃	1	4	yes	—
230	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
231	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
233	C₂²⋊Q₈	—	2	4	no	—
234	C₂²⋊Q₈	—	2	4	no	—
235	C₂²⋊Q₈	—	4	4	no	—
236	C₂²⋊Q₈	—	4	4	no	—
237	C₂×C₄⋊C₄	—	2	4	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	C₂³×C₄₃	1	4	yes	—
240	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
241	C₂×C₂²⋊C₄	—	2	4	no	—
242	C₂×C₂²⋊C₄	—	2	2	no	—
243	C₂×C₂²⋊C₄	—	2	4	no	—
244	C₂×C₂²⋊C₄	—	2	2	no	—
245	#107	—	2	2	no	—
246	#106	—	4	2	no	—
247	#103	—	2	4	no	—
248	#103	—	4	4	no	—
249	C₂²⋊C₈	—	2	4	no	—
250	C₂²⋊C₈	—	2	4	no	—
251	#95	—	2	4	no	—
252	#95	—	2	2	no	—
253	C₂⁴	C₂×D₄×C₄₃	1	2	yes	—
254	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
255	#41	—	4	4	no	—
256	#41	—	4	4	no	—
257	#41	—	4	8	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	4	no	—
262	C₂×Q₈	—	2	4	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	4	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
280	C₂²×C₄	#41×C₄₃	1	2	yes	—
281	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
282	Q₁₆	—	4	4	no	—
283	M₁₆	—	4	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
290	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	4	8	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₂²⋊C₄	—	4	8	no	—
298	C₂²⋊C₄	—	4	4	no	—
299	C₂²⋊C₄	—	8	2	no	—
300	C₂²⋊C₄	—	8	4	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	4	no	—
306	C₂³	—	2	4	no	—
307	C₂³	—	2	4	no	—
308	C₂³	#121×C₄₃	1	2	yes	—
309	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
310	Q₈	—	4	4	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	8	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	#121×C₄₃	1	2	yes	—
334	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₄₃	1	1	yes	characteristic
336	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
337	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	4	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	8	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	C₂×#121×C₄₃	1	1	yes	characteristic
350	C₂²	C₄₃×#534	1	2	yes	—
351	C₂²	C₄₃×#577	1	2	yes	—
352	C₂²	C₄₃×#578	1	2	yes	—
353	C₄	—	2	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	4	no	—
357	C₄	—	8	2	no	—
358	C₄	C₄₃×#534	1	1	yes	characteristic
359	C₄	C₄₃×#535	1	1	yes	characteristic
360	C₄	C₄₃×#538	1	2	yes	—
361	C₂	—	4	2	no	—
362	C₂	—	8	2	no	—
363	C₂	C₄₃×#2617	1	1	yes	characteristic
364	C₂	C₄₃×#3059	1	1	yes	characteristic
365	C₂	C₄₃×#3060	1	1	yes	characteristic
366	C₂	C₄₃×#3201	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂²×#41×C₄₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₂²×#41×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₃×#467	C₂	semidirect
4. C₄₃×#2076	C₂	semidirect
5. C₄₃×#2035	C₂	semidirect
6. C₄₃×#2001	C₂	semidirect
7. C₄₃×#1894	C₂	semidirect
8. C₄₃×#1807	C₂	semidirect
157. #29737	C₄₃	direct (N × H)
178. C₂×#467	C₂×C₄₃	semidirect
181. #2076	C₂×C₄₃	semidirect
182. #2035	C₂×C₄₃	semidirect
183. #2001	C₂×C₄₃	semidirect
184. #1894	C₂×C₄₃	semidirect
185. #1807	C₂×C₄₃	semidirect
213. C₄₃	#29737	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×#467	2
C₂×C₄₃×#408	2
C₄₃×#2616	2
C₄₃×#2076	2
C₄₃×#2035	2
C₄₃×#2001	2
C₄₃×#1894	2
C₄₃×#1807	2
#29737	43

Subgroups — C43×#29737 #24949754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29737 #24949754