Subgroups — #25012884

Order 11008.

470 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₄₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#19134621	C₂	1	1	yes	maximal, characteristic
4	#19132687	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×D₄₃	—	2	1	no	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#13063259	C₂²	1	1	yes	characteristic
12	C₂×#13063259	C₂²	1	1	yes	characteristic
13	C₂×#13063258	—	2	1	no	—
14	#107×D₄₃	—	2	2	no	—
15	#107×D₄₃	C₄	1	4	yes	complemented
16	#14582756	—	2	1	no	—
17	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
18	#14582661	—	2	1	no	—
19	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₄₃	—	4	1	no	—
23	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₈×D₄₃	—	2	1	no	—
26	C₂×C₈×D₄₃	—	2	1	no	—
27	C₂×C₈×D₄₃	—	2	1	no	—
28	C₂×C₈×D₄₃	—	2	1	no	—
29	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₄₃	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₄₃	—	2	1	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×Dic₄₃⋊C₄	—	4	1	no	—
34	C₄⋊C₄×D₄₃	—	2	4	no	—
35	C₄⋊C₄×D₄₃	—	4	2	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
37	C₂×C₄₃⋊C₁₆	—	2	1	no	—
38	C₂×C₄₃⋊C₁₆	—	2	1	no	—
39	C₂×C₄₃⋊C₁₆	—	2	1	no	—
40	#13063259	—	2	1	no	—
41	#13063259	—	2	1	no	—
42	#13063259	—	2	1	no	—
43	#13063259	C₂×C₄	1	2	yes	—
44	#13063258	—	2	4	no	—
45	C₄₃⋊#95	—	4	1	no	—
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
49	C₂×C₁₆×C₄₃	—	2	1	no	—
50	C₂×C₁₆×C₄₃	—	2	1	no	—
51	C₂×C₁₆×C₄₃	—	2	1	no	—
52	#107×C₄₃	—	2	1	no	—
53	#107×C₄₃	—	2	1	no	—
54	#107×C₄₃	—	2	1	no	—
55	#107×C₄₃	C₂×C₄	1	2	yes	—
56	C₂³×D₄₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	4	1	no	—
58	C₂²×C₄×C₄₃	—	4	1	no	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	4	1	no	—
61	C₂²×Dic₄₃	—	4	1	no	—
62	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	4	4	no	—
68	C₂×C₄×D₄₃	—	8	2	no	—
69	C₂×C₄×D₄₃	—	8	2	no	—
70	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	—	4	1	no	—
80	C₄⋊C₄×C₄₃	—	4	1	no	—
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	(C₂×D₄₃)⋊C₄	—	8	2	no	—
83	C₄⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	4	1	no	—
85	C₄⋊Dic₄₃	—	4	1	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	Dic₄₃⋊C₄	—	4	4	no	—
88	C₂×C₄₃⋊C₈	—	2	1	no	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₈×D₄₃	—	2	2	no	—
92	C₈×D₄₃	—	2	2	no	—
93	C₈×D₄₃	—	2	2	no	—
94	C₈×D₄₃	—	2	2	no	—
95	C₁₆×C₄₃	—	4	1	no	—
96	C₄₃⋊C₁₆	—	4	1	no	—
97	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₄₃	—	2	1	no	—
99	C₂²×D₄₃	—	2	1	no	—
100	C₂²×D₄₃	—	2	1	no	—
101	C₂²×D₄₃	—	2	1	no	—
102	C₂²×D₄₃	#102	1	2	yes	—
103	C₂²×D₄₃	D₁₆	1	2	yes	—
104	C₂²×D₄₃	SD₃₂	1	2	yes	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	8	1	no	—
107	C₂×C₄×C₄₃	—	8	1	no	—
108	C₂×C₄×C₄₃	—	8	1	no	—
109	C₂×C₄×C₄₃	—	8	1	no	—
110	C₂×C₄×C₄₃	—	8	1	no	—
111	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	4	2	no	—
120	C₂×Dic₄₃	—	8	1	no	—
121	C₂×Dic₄₃	—	8	1	no	—
122	C₂×Dic₄₃	—	8	1	no	—
123	C₂×Dic₄₃	—	8	1	no	—
124	C₂×Dic₄₃	—	8	1	no	—
125	C₂×Dic₄₃	#102	1	2	yes	—
126	C₄×D₄₃	—	2	2	no	—
127	C₄×D₄₃	—	2	2	no	—
128	C₄×D₄₃	—	2	2	no	—
129	C₄×D₄₃	—	2	2	no	—
130	C₄×D₄₃	—	8	8	no	—
131	C₈×C₄₃	#117	1	1	yes	characteristic
132	C₈×C₄₃	#117	1	1	yes	characteristic
133	C₈×C₄₃	#123	1	1	yes	characteristic
134	C₈×C₄₃	#123	1	1	yes	characteristic
135	C₄₃⋊C₈	—	2	2	no	—
136	#12866	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₄₃	#480	1	1	yes	characteristic
139	C₂²×C₄₃	#480	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
142	C₂²×C₄₃	#510	1	1	yes	characteristic
143	C₂²×C₄₃	#510	1	1	yes	characteristic
144	C₂×D₄₃	—	2	2	no	—
145	C₂×D₄₃	—	2	2	no	—
146	C₂×D₄₃	—	2	2	no	—
147	C₂×D₄₃	—	2	2	no	—
148	C₂×D₄₃	—	2	2	no	—
149	C₂×D₄₃	—	2	2	no	—
150	C₂×D₄₃	#357	1	4	yes	—
151	C₄×C₄₃	—	8	2	no	—
152	C₄×C₄₃	—	16	1	no	—
153	C₄×C₄₃	#381	1	1	yes	characteristic
154	C₄×C₄₃	#419	1	1	yes	characteristic
155	C₄×C₄₃	#483	1	1	yes	characteristic
156	C₄×C₄₃	#483	1	1	yes	characteristic
157	Dic₄₃	—	2	2	no	—
158	Dic₄₃	—	8	2	no	—
159	Dic₄₃	—	16	1	no	—
160	C₂²×#107	—	43	1	no	—
161	C₂×C₂²⋊C₁₆	—	43	1	no	—
162	#1808	—	43	1	no	—
163	#1380	—	43	1	no	—
164	#1380	—	43	1	no	—
165	#1380	—	43	1	no	—
166	#1380	D₄₃	1	1	yes	characteristic, complemented
167	C₂×C₄₃	#1823	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
169	C₂×C₄₃	#2141	1	1	yes	characteristic
170	C₂×C₄₃	#2232	1	1	yes	characteristic
171	C₂×C₄₃	#2232	1	1	yes	characteristic
172	C₂×C₄₃	#2235	1	1	yes	characteristic
173	C₂×C₄₃	#2235	1	1	yes	characteristic
174	D₄₃	—	2	4	no	—
175	C₂³×C₈	—	43	1	no	—
176	C₂²×C₄⋊C₄	—	86	1	no	—
177	C₂²×C₁₆	—	43	1	no	—
178	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
179	C₂×#107	—	43	1	no	—
180	C₂×#107	—	43	4	no	—
181	C₂×#107	—	43	1	no	—
182	C₂×#107	—	86	2	no	—
183	C₂×#107	—	86	1	no	—
184	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
185	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
186	#380	—	86	1	no	—
187	C₂²⋊C₁₆	—	86	2	no	—
188	#340	—	86	1	no	—
189	C₄₃	#12866	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	—	43	1	no	—
191	C₂³×C₄	—	172	1	no	—
192	C₂²×C₈	—	43	1	no	—
193	C₂²×C₈	—	43	2	no	—
194	C₂²×C₈	—	43	2	no	—
195	C₂²×C₈	—	86	1	no	—
196	C₂²×C₈	—	86	1	no	—
197	C₂²×C₈	—	86	1	no	—
198	C₂²×C₈	—	86	1	no	—
199	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	86	4	no	—
203	C₂×C₄⋊C₄	—	86	1	no	—
204	C₂×C₄⋊C₄	—	86	1	no	—
205	C₂×C₄⋊C₄	—	172	2	no	—
206	C₂×C₄⋊C₄	—	172	1	no	—
207	C₂×C₂²⋊C₄	—	172	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	86	1	no	—
212	C₂×C₁₆	—	86	1	no	—
213	C₂×C₁₆	—	86	1	no	—
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#107	—	43	2	no	—
218	#107	—	86	4	no	—
219	#107	—	86	1	no	—
220	#107	—	86	1	no	—
221	#107	—	86	1	no	—
222	#107	C₄×D₄₃	1	2	yes	—
223	#95	—	172	1	no	—
224	C₂⁴	—	43	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	43	1	no	—
228	C₂²×C₄	—	43	2	no	—
229	C₂²×C₄	—	43	2	no	—
230	C₂²×C₄	—	86	1	no	—
231	C₂²×C₄	—	86	1	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	—	86	1	no	—
234	C₂²×C₄	—	344	2	no	—
235	C₂²×C₄	—	344	2	no	—
236	C₂²×C₄	—	172	4	no	—
237	C₂²×C₄	—	172	1	no	—
238	C₂²×C₄	—	172	1	no	—
239	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
240	C₂×C₈	—	43	2	no	—
241	C₂×C₈	—	86	1	no	—
242	C₂×C₈	—	86	1	no	—
243	C₂×C₈	—	86	2	no	—
244	C₂×C₈	—	86	2	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	—	86	2	no	—
247	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
248	C₂×C₈	#10621898	1	1	yes	characteristic
249	C₂×C₈	#10621898	1	1	yes	characteristic
250	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
251	C₂×C₈	#10621900	1	1	yes	characteristic
252	C₂×C₈	#10621902	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	86	2	no	—
258	C₄⋊C₄	—	172	4	no	—
259	C₄⋊C₄	—	172	1	no	—
260	C₄⋊C₄	—	172	1	no	—
261	C₄⋊C₄	—	172	1	no	—
262	C₂²⋊C₄	—	344	2	no	—
263	C₁₆	—	4	1	no	—
264	C₁₆	—	172	1	no	—
265	C₂³	—	43	2	no	—
266	C₂³	—	43	2	no	—
267	C₂³	—	43	2	no	—
268	C₂³	—	86	1	no	—
269	C₂³	—	86	1	no	—
270	C₂³	—	86	1	no	—
271	C₂³	—	86	1	no	—
272	C₂³	D₈×D₄₃	1	1	yes	characteristic
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	43	2	no	—
280	C₂×C₄	—	86	1	no	—
281	C₂×C₄	—	86	1	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	86	2	no	—
284	C₂×C₄	—	86	2	no	—
285	C₂×C₄	—	86	2	no	—
286	C₂×C₄	—	344	8	no	—
287	C₂×C₄	—	344	1	no	—
288	C₂×C₄	—	344	1	no	—
289	C₂×C₄	—	344	1	no	—
290	C₂×C₄	—	344	1	no	—
291	C₂×C₄	—	344	1	no	—
292	C₂×C₄	—	172	2	no	—
293	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
294	C₂×C₄	#13063307	1	1	yes	characteristic
295	C₂×C₄	#13063310	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
297	C₂×C₄	#13063354	1	1	yes	characteristic
298	C₂×C₄	#13063357	1	1	yes	characteristic
299	C₈	—	86	2	no	—
300	C₈	#13063282	1	1	yes	characteristic
301	C₈	#13063283	1	1	yes	characteristic
302	C₈	#13063307	1	1	yes	characteristic
303	C₈	#13063310	1	1	yes	characteristic
304	C₂²	—	43	4	no	—
305	C₂²	—	86	2	no	—
306	C₂²	—	86	2	no	—
307	C₂²	—	86	2	no	—
308	C₂²	—	86	2	no	—
309	C₂²	—	86	2	no	—
310	C₂²	—	86	2	no	—
311	C₂²	#102×D₄₃	1	1	yes	characteristic
312	C₂²	#14583211	1	1	yes	characteristic
313	C₂²	#14583214	1	1	yes	characteristic
314	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
315	C₂²	SD₃₂×D₄₃	1	1	yes	characteristic
316	C₂²	#14583338	1	1	yes	characteristic
317	C₂²	#14583341	1	1	yes	characteristic
318	C₄	—	8	2	no	—
319	C₄	—	16	1	no	—
320	C₄	—	86	2	no	—
321	C₄	—	344	2	no	—
322	C₄	—	688	1	no	—
323	C₄	#14582761	1	1	yes	characteristic
324	C₄	#14582906	1	1	yes	characteristic
325	C₄	#14583228	1	1	yes	characteristic
326	C₄	#14583231	1	1	yes	characteristic
327	C₂	—	86	4	no	—
328	C₂	#19134680	1	1	yes	characteristic
329	C₂	#357×D₄₃	1	1	yes	characteristic
330	C₂	#19136313	1	1	yes	characteristic
331	C₂	#19136774	1	1	yes	characteristic
332	C₂	#19136777	1	1	yes	characteristic
333	C₂	#19136791	1	1	yes	characteristic
334	C₂	#19136794	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂³×D₄₃	D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#12866
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. #19132687	C₂	semidirect
5. #19132686	C₂	semidirect
6. #19132686	C₂	semidirect
7. C₄₃×#1380	C₂	semidirect
15. #107×D₄₃	C₄	semidirect
166. #1380	D₄₃	semidirect
189. C₄₃	#12866	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107×D₄₃	2
C₂×(C₂×D₄₃)⋊C₁₆	2
#19134621	2
#19132687	2
#19132686	2
C₄₃×#1380	2
#12866	43