Subgroups — C₄₃×#29777 #24949794

Order 11008.

754 conjugacy classes of subgroups.

The subgroup lattice has 316 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
2	C₂×C₄₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2625	C₂	1	2	yes	maximal
4	C₄₃×#2087	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2066	C₂	1	4	yes	maximal, complemented
6	C₄₃×#2056	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1894	C₂	1	2	yes	maximal
8	C₄₃×#1807	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
13	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#467	—	2	4	no	—
15	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#106×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#424	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
20	C₂×#103×C₄₃	C₂²	1	2	yes	—
21	C₂×#103×C₄₃	C₂²	1	4	yes	—
22	C₄₃×#400	—	2	2	no	—
23	C₄₃×#398	—	2	2	no	—
24	C₄₃×#397	—	2	4	no	—
25	C₄₃×#391	C₂²	1	4	yes	—
26	C₄₃×#389	C₂²	1	4	yes	—
27	C₄₃×#380	C₂²	1	2	yes	—
28	C₄₃×#340	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×Q₁₆×C₄₃	—	2	2	no	—
33	C₂×Q₁₆×C₄₃	—	4	2	no	—
34	C₂×M₁₆×C₄₃	—	2	2	no	—
35	C₂×M₁₆×C₄₃	—	4	2	no	—
36	C₂²×C₈×C₄₃	—	4	2	no	—
37	C₂²×C₈×C₄₃	D₄	1	4	yes	—
38	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²⋊Q₈×C₄₃	—	2	4	no	—
42	C₂²⋊Q₈×C₄₃	—	2	4	no	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
55	#107×C₄₃	—	2	2	no	—
56	#106×C₄₃	—	4	2	no	—
57	C₄⋊C₈×C₄₃	—	2	4	no	—
58	C₄⋊C₈×C₄₃	—	2	4	no	—
59	#103×C₄₃	—	2	4	no	—
60	#103×C₄₃	—	4	4	no	—
61	#95×C₄₃	—	2	2	no	—
62	#95×C₄₃	—	2	4	no	—
63	#95×C₄₃	—	2	4	no	—
64	#95×C₄₃	—	2	4	no	—
65	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₄₃	—	2	2	no	—
67	C₂×Q₈×C₄₃	—	2	2	no	—
68	C₂×Q₈×C₄₃	—	2	4	no	—
69	C₂×Q₈×C₄₃	—	2	4	no	—
70	C₂²×C₄×C₄₃	—	2	2	no	—
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	4	no	—
73	C₂²×C₄×C₄₃	—	2	2	no	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	#41	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₄₃	—	4	4	no	—
82	M₁₆×C₄₃	—	4	4	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	4	no	—
87	C₂×C₈×C₄₃	—	4	4	no	—
88	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	2	no	—
94	C₄⋊C₄×C₄₃	—	2	4	no	—
95	C₄⋊C₄×C₄₃	—	2	4	no	—
96	C₄⋊C₄×C₄₃	—	4	4	no	—
97	C₄⋊C₄×C₄₃	—	8	4	no	—
98	C₂²⋊C₄×C₄₃	—	2	4	no	—
99	C₂²⋊C₄×C₄₃	—	4	4	no	—
100	C₂²⋊C₄×C₄₃	—	8	2	no	—
101	C₄²×C₄₃	—	2	2	no	—
102	C₄²×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	#121	1	2	yes	—
107	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₄₃	—	4	2	no	—
109	Q₈×C₄₃	—	4	2	no	—
110	Q₈×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	—	2	4	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	4	no	—
122	C₂×C₄×C₄₃	—	4	4	no	—
123	C₂×C₄×C₄₃	—	4	4	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	4	4	no	—
127	C₂×C₄×C₄₃	#121	1	2	yes	—
128	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
132	C₈×C₄₃	—	2	2	no	—
133	C₈×C₄₃	—	2	2	no	—
134	C₈×C₄₃	—	4	2	no	—
135	#29777	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	2	2	no	—
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	4	4	no	—
140	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
141	C₂²×C₄₃	#537	1	2	yes	—
142	C₂²×C₄₃	#577	1	2	yes	—
143	C₂²×C₄₃	#578	1	2	yes	—
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	4	2	no	—
146	C₄×C₄₃	—	4	2	no	—
147	C₄×C₄₃	—	8	2	no	—
148	C₄×C₄₃	—	8	4	no	—
149	C₄×C₄₃	—	8	2	no	—
150	C₄×C₄₃	#534	1	1	yes	characteristic
151	C₄×C₄₃	#536	1	1	yes	characteristic
152	C₄×C₄₃	#539	1	2	yes	—
153	C₂×#467	C₂×C₄₃	1	1	yes	characteristic
154	C₂×#424	C₂×C₄₃	1	1	yes	characteristic
155	#2625	C₂×C₄₃	1	2	yes	—
156	#2087	C₂×C₄₃	1	2	yes	complemented
157	#2066	C₂×C₄₃	1	4	yes	complemented
158	#2056	C₂×C₄₃	1	2	yes	complemented
159	#1894	C₂×C₄₃	1	2	yes	—
160	#1807	C₂×C₄₃	1	1	yes	characteristic
161	C₂×C₄₃	—	4	2	no	—
162	C₂×C₄₃	#2618	1	1	yes	characteristic
163	C₂×C₄₃	#3059	1	1	yes	characteristic
164	C₂×C₄₃	#3060	1	1	yes	characteristic
165	C₂×C₄₃	#3258	1	4	yes	—
166	C₂²×Q₁₆	C₂²×C₄₃	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₄₃	1	1	yes	characteristic
169	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
170	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
171	#467	—	2	4	no	—
172	C₂×#107	C₂²×C₄₃	1	1	yes	characteristic
173	C₂×#106	C₂²×C₄₃	1	2	yes	—
174	#424	—	2	4	no	—
175	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
177	C₂×#103	C₂²×C₄₃	1	2	yes	—
178	C₂×#103	C₂²×C₄₃	1	4	yes	—
179	#400	—	2	2	no	—
180	#398	—	2	2	no	—
181	#397	—	2	4	no	—
182	#391	C₂²×C₄₃	1	4	yes	—
183	#389	C₂²×C₄₃	1	4	yes	—
184	#380	C₂²×C₄₃	1	2	yes	—
185	#340	C₂²×C₄₃	1	2	yes	—
186	#340	C₂²×C₄₃	1	4	yes	—
187	C₄₃	#29777	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
189	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	4	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	D₄×C₄₃	1	4	yes	—
196	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
199	C₂²⋊Q₈	—	2	4	no	—
200	C₂²⋊Q₈	—	2	4	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₄₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
213	#107	—	2	2	no	—
214	#106	—	4	2	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	#103	—	2	4	no	—
218	#103	—	4	4	no	—
219	#95	—	2	2	no	—
220	#95	—	2	4	no	—
221	#95	—	2	4	no	—
222	#95	—	2	4	no	—
223	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
237	C₂²×C₄	#41×C₄₃	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
239	Q₁₆	—	4	4	no	—
240	M₁₆	—	4	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	4	no	—
246	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	8	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	8	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	#121×C₄₃	1	2	yes	—
265	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
266	Q₈	—	4	2	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	4	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	#121×C₄₃	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₄₃	1	1	yes	characteristic
288	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
289	C₂×C₄	2₋¹⁺⁴×C₄₃	1	2	yes	—
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	4	no	—
297	C₂²	C₂×#121×C₄₃	1	1	yes	characteristic
298	C₂²	C₄₃×#537	1	2	yes	—
299	C₂²	C₄₃×#577	1	2	yes	—
300	C₂²	C₄₃×#578	1	2	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	4	no	—
306	C₄	—	8	2	no	—
307	C₄	C₄₃×#534	1	1	yes	characteristic
308	C₄	C₄₃×#536	1	1	yes	characteristic
309	C₄	C₄₃×#539	1	2	yes	—
310	C₂	—	4	2	no	—
311	C₂	C₄₃×#2618	1	1	yes	characteristic
312	C₂	C₄₃×#3059	1	1	yes	characteristic
313	C₂	C₄₃×#3060	1	1	yes	characteristic
314	C₂	C₄₃×#3258	1	4	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
4. C₄₃×#2087	C₂	semidirect
5. C₄₃×#2066	C₂	semidirect
6. C₄₃×#2056	C₂	semidirect
135. #29777	C₄₃	direct (N × H)
156. #2087	C₂×C₄₃	semidirect
157. #2066	C₂×C₄₃	semidirect
158. #2056	C₂×C₄₃	semidirect
187. C₄₃	#29777	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×#424	2
C₄₃×#2625	2
C₄₃×#2087	2
C₄₃×#2066	2
C₄₃×#2056	2
C₄₃×#1894	2
C₄₃×#1807	2
#29777	43

Subgroups — C43×#29777 #24949794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29777 #24949794