Subgroups — C₄₃×#29674 #24949691

Order 11008.

1032 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#541	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#462	C₂	1	2	yes	maximal, complemented
3	C₄₃×#2621	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2026	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2016	C₂	1	2	yes	maximal, complemented
6	C₄₃×#2001	C₂	1	4	yes	maximal
7	C₄₃×#1805	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1798	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₄₃	C₂²	1	4	yes	—
10	C₄₃×#541	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#126×C₄₃	—	2	2	no	—
14	C₂×#125×C₄₃	—	2	2	no	—
15	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
19	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#462	—	2	8	no	—
21	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
22	C₂×#103×C₄₃	C₂²	1	4	yes	—
23	C₂×#103×C₄₃	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
25	C₄₃×#398	—	2	2	no	—
26	C₄₃×#397	—	2	2	no	—
27	C₄₃×#391	C₂²	1	2	yes	—
28	C₄₃×#382	C₂²	1	1	yes	characteristic
29	C₄₃×#340	C₂²	1	4	yes	—
30	C₄₃×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
32	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₄₃	—	2	8	no	—
35	C₂×Q₁₆×C₄₃	—	4	4	no	—
36	C₂×Q₁₆×C₄₃	—	4	4	no	—
37	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₄₃	—	2	4	no	—
39	C₄⋊Q₈×C₄₃	—	2	4	no	—
40	#126×C₄₃	—	2	16	no	—
41	#125×C₄₃	—	2	16	no	—
42	#125×C₄₃	—	2	4	no	—
43	C₂²⋊Q₈×C₄₃	—	2	4	no	—
44	C₂²⋊Q₈×C₄₃	—	2	4	no	—
45	C₂²⋊Q₈×C₄₃	—	4	4	no	—
46	C₄×Q₈×C₄₃	—	2	4	no	—
47	C₄×Q₈×C₄₃	D₄	1	8	yes	—
48	#117×C₄₃	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	2	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	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
61	C₂×C₄²×C₄₃	—	2	2	no	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₃	—	2	4	no	—
65	#103×C₄₃	—	2	8	no	—
66	#103×C₄₃	—	4	4	no	—
67	C₂²⋊C₈×C₄₃	—	4	2	no	—
68	#95×C₄₃	—	2	2	no	—
69	#95×C₄₃	—	2	4	no	—
70	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	2	4	no	—
73	C₂×Q₈×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	—	2	2	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	2	4	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	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	1	no	—
83	C₂²×C₄×C₄₃	—	2	2	no	—
84	C₂²×C₄×C₄₃	—	2	1	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	2	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₄₃	—	4	16	no	—
94	C₂×C₈×C₄₃	—	2	4	no	—
95	C₂×C₈×C₄₃	—	2	4	no	—
96	C₂×C₈×C₄₃	—	2	4	no	—
97	C₄⋊C₄×C₄₃	—	2	8	no	—
98	C₄⋊C₄×C₄₃	—	2	8	no	—
99	C₄⋊C₄×C₄₃	—	2	8	no	—
100	C₄⋊C₄×C₄₃	—	2	8	no	—
101	C₄⋊C₄×C₄₃	—	2	4	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	4	no	—
104	C₄⋊C₄×C₄₃	—	4	4	no	—
105	C₄⋊C₄×C₄₃	—	4	4	no	—
106	C₄⋊C₄×C₄₃	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₄₃	—	2	8	no	—
108	C₂²⋊C₄×C₄₃	—	2	4	no	—
109	C₂²⋊C₄×C₄₃	—	4	4	no	—
110	C₄²×C₄₃	—	2	8	no	—
111	C₄²×C₄₃	—	2	2	no	—
112	C₄²×C₄₃	C₂×D₄	1	4	yes	—
113	C₂³×C₄₃	—	2	1	no	—
114	C₂³×C₄₃	—	2	1	no	—
115	C₂³×C₄₃	—	2	1	no	—
116	C₂³×C₄₃	—	2	4	no	—
117	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₄₃	—	2	4	no	—
119	Q₈×C₄₃	—	2	4	no	—
120	Q₈×C₄₃	—	4	2	no	—
121	Q₈×C₄₃	—	4	2	no	—
122	Q₈×C₄₃	—	4	4	no	—
123	Q₈×C₄₃	—	4	4	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	4	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	4	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	4	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	4	no	—
142	C₂×C₄×C₄₃	#120	1	4	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
146	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₈×C₄₃	—	4	4	no	—
148	#29674	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	2	no	—
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	—	4	2	no	—
153	C₂²×C₄₃	C₂×#120	1	1	yes	characteristic
154	C₂²×C₄₃	#535	1	1	yes	characteristic
155	C₂²×C₄₃	#537	1	1	yes	characteristic
156	C₂²×C₄₃	#578	1	4	yes	—
157	C₄×C₄₃	—	2	4	no	—
158	C₄×C₄₃	—	4	1	no	—
159	C₄×C₄₃	—	4	4	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	—	4	4	no	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	#535	1	2	yes	—
165	C₄×C₄₃	#537	1	2	yes	—
166	C₂×#541	C₂×C₄₃	1	1	yes	characteristic
167	C₂×#462	C₂×C₄₃	1	2	yes	complemented
168	#2621	C₂×C₄₃	1	1	yes	characteristic
169	#2026	C₂×C₄₃	1	2	yes	complemented
170	#2016	C₂×C₄₃	1	2	yes	complemented
171	#2001	C₂×C₄₃	1	4	yes	—
172	#1805	C₂×C₄₃	1	2	yes	complemented
173	#1798	C₂×C₄₃	1	1	yes	characteristic
174	C₂×C₄₃	—	8	1	no	—
175	C₂×C₄₃	#2615	1	1	yes	characteristic
176	C₂×C₄₃	#3011	1	2	yes	—
177	C₂×C₄₃	#3223	1	2	yes	—
178	C₂×C₄₃	#3261	1	2	yes	—
179	C₂²×Q₁₆	C₂²×C₄₃	1	4	yes	—
180	#541	—	2	8	no	—
181	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
182	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
183	C₂×#126	—	2	2	no	—
184	C₂×#125	—	2	2	no	—
185	C₂×#125	C₂²×C₄₃	1	1	yes	characteristic
186	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
187	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
188	C₂×C₄×Q₈	C₂²×C₄₃	1	2	yes	—
189	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
190	#462	—	2	8	no	—
191	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
192	C₂×#103	C₂²×C₄₃	1	4	yes	—
193	C₂×#103	C₂²×C₄₃	1	4	yes	—
194	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
195	#398	—	2	2	no	—
196	#397	—	2	2	no	—
197	#391	C₂²×C₄₃	1	2	yes	—
198	#382	C₂²×C₄₃	1	1	yes	characteristic
199	#340	C₂²×C₄₃	1	4	yes	—
200	#336	C₂²×C₄₃	1	2	yes	—
201	C₄₃	#29674	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
203	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
204	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	8	no	—
206	C₂×Q₁₆	—	4	4	no	—
207	C₂×Q₁₆	—	4	4	no	—
208	C₂²×C₈	C₂³×C₄₃	1	4	yes	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	#126	—	2	16	no	—
212	#125	—	2	16	no	—
213	#125	—	2	4	no	—
214	C₂²⋊Q₈	—	2	4	no	—
215	C₂²⋊Q₈	—	2	4	no	—
216	C₂²⋊Q₈	—	4	4	no	—
217	C₄×Q₈	—	2	4	no	—
218	C₄×Q₈	D₄×C₄₃	1	8	yes	—
219	#117	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	4	no	—
236	#103	—	2	8	no	—
237	#103	—	4	4	no	—
238	C₂²⋊C₈	—	4	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	4	no	—
241	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	C₂×D₄×C₄₃	1	4	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
263	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
264	Q₁₆	—	4	16	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	8	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	C₂×D₄×C₄₃	1	4	yes	—
278	C₂²⋊C₄	—	2	8	no	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	2	8	no	—
282	C₄²	—	2	2	no	—
283	C₄²	C₂×D₄×C₄₃	1	4	yes	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	4	no	—
288	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
289	Q₈	—	2	4	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	4	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	#120×C₄₃	1	4	yes	—
314	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
315	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
316	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
317	C₂×C₄	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
318	C₈	—	4	4	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂×#120×C₄₃	1	1	yes	characteristic
324	C₂²	C₄₃×#535	1	1	yes	characteristic
325	C₂²	C₄₃×#537	1	1	yes	characteristic
326	C₂²	C₄₃×#578	1	4	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	C₄₃×#535	1	2	yes	—
335	C₄	C₄₃×#537	1	2	yes	—
336	C₂	—	8	1	no	—
337	C₂	C₄₃×#2615	1	1	yes	characteristic
338	C₂	C₄₃×#3011	1	2	yes	—
339	C₂	C₄₃×#3223	1	2	yes	—
340	C₂	C₄₃×#3261	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 ◁ C₂×#117×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃×#462	C₂	semidirect
4. C₄₃×#2026	C₂	semidirect
5. C₄₃×#2016	C₂	semidirect
7. C₄₃×#1805	C₂	semidirect
148. #29674	C₄₃	direct (N × H)
167. C₂×#462	C₂×C₄₃	semidirect
169. #2026	C₂×C₄₃	semidirect
170. #2016	C₂×C₄₃	semidirect
172. #1805	C₂×C₄₃	semidirect
201. C₄₃	#29674	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×#541	2
C₂×C₄₃×#462	2
C₄₃×#2621	2
C₄₃×#2026	2
C₄₃×#2016	2
C₄₃×#2001	2
C₄₃×#1805	2
C₄₃×#1798	2
#29674	43

Subgroups — C43×#29674 #24949691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29674 #24949691