Subgroups — #25251282

Order 11008.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145131	C₂	1	1	yes	maximal, characteristic, complemented
2	#19143447	C₂	1	1	yes	maximal, characteristic
3	#19136754	C₂	1	2	yes	maximal, complemented
4	#19136750	C₂	1	2	yes	maximal, complemented
5	#19136740	C₂	1	2	yes	maximal, complemented
6	#19136730	C₂	1	2	yes	maximal, complemented
7	#19136381	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136353	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136311	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄₃×#357	C₂	1	1	yes	maximal, characteristic
12	#14583530	—	2	1	no	—
13	#14583374	—	2	1	no	—
14	C₂×#13063265	C₂²	1	1	yes	characteristic
15	#14583307	—	2	2	no	—
16	#14583303	C₂²	1	2	yes	complemented
17	#14583300	C₂²	1	2	yes	complemented
18	#14583011	—	2	2	no	—
19	D₈×Dic₄₃	C₂²	1	2	yes	—
20	D₈×Dic₄₃	C₂²	1	2	yes	—
21	#14582939	—	2	1	no	—
22	#14582937	C₂²	1	1	yes	characteristic
23	#14582937	C₂²	1	1	yes	characteristic
24	#14582904	—	2	1	no	—
25	#14582854	C₂²	1	1	yes	characteristic
26	#14582727	C₂²	1	1	yes	characteristic, complemented
27	#14582727	C₂²	1	1	yes	characteristic, complemented
28	#14582726	C₂²	1	2	yes	complemented
29	#14582725	C₂²	1	1	yes	characteristic, complemented
30	#14582725	C₂²	1	1	yes	characteristic, complemented
31	#14582724	C₂²	1	2	yes	complemented
32	Dic₄₃⋊C₁₆	C₂²	1	2	yes	complemented
33	Dic₄₃⋊C₁₆	C₂²	1	2	yes	complemented
34	#14582700	C₂²	1	2	yes	—
35	#14582700	C₂²	1	2	yes	—
36	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#357	C₂²	1	2	yes	—
40	C₄₃×#357	C₂²	1	2	yes	—
41	C₂×#10621877	D₄	1	1	yes	characteristic
42	#13063324	—	2	2	no	—
43	#13063324	—	4	1	no	—
44	#13063316	—	2	2	no	—
45	#13063316	—	4	1	no	—
46	#13063304	—	4	2	no	—
47	D₄×Dic₄₃	—	2	2	no	—
48	D₄×Dic₄₃	—	2	2	no	—
49	D₄×Dic₄₃	—	4	1	no	—
50	D₄×Dic₄₃	—	4	1	no	—
51	#13063284	—	2	1	no	—
52	#13063284	—	2	1	no	—
53	#13063284	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊Dic₄₃	—	4	2	no	—
55	#13063265	—	2	2	no	—
56	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
57	#13063259	C₂³	1	1	yes	characteristic
58	#13063259	C₂³	1	1	yes	characteristic
59	#13063259	C₂³	1	1	yes	characteristic
60	#13063259	C₂³	1	1	yes	characteristic
61	#13063258	—	2	2	no	—
62	#13063257	—	2	1	no	—
63	#13063257	—	2	1	no	—
64	#13063256	—	2	2	no	—
65	#13063246	—	2	2	no	—
66	#13063246	—	2	2	no	—
67	#13063235	D₄	1	2	yes	—
68	C₈×Dic₄₃	C₂³	1	2	yes	—
69	C₂²×D₄×C₄₃	—	2	1	no	—
70	C₂×D₈×C₄₃	—	2	1	no	—
71	C₂×D₈×C₄₃	—	2	2	no	—
72	C₂×D₈×C₄₃	Q₈	1	4	yes	complemented
73	C₂×D₈×C₄₃	C₂³	1	2	yes	—
74	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₂×C₁₆×C₄₃	—	2	2	no	—
77	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	—	2	1	no	—
80	#107×C₄₃	C₂³	1	2	yes	—
81	C₂⁴×C₄₃	—	4	1	no	—
82	C₂×D₄×C₄₃	—	2	4	no	—
83	C₂×D₄×C₄₃	—	2	2	no	—
84	C₂×D₄×C₄₃	—	4	1	no	—
85	C₂×D₄×C₄₃	—	4	2	no	—
86	C₂²×C₄×C₄₃	—	4	1	no	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	4	2	no	—
89	C₂²×Dic₄₃	—	4	2	no	—
90	C₂×Dic₈₆	—	4	2	no	—
91	D₈×C₄₃	—	2	4	no	—
92	D₈×C₄₃	—	2	2	no	—
93	D₈×C₄₃	C₂×Q₈	1	4	yes	—
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₄₃	#41	1	2	yes	—
97	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	4	1	no	—
100	C₂²⋊Dic₄₃	—	4	2	no	—
101	C₂²⋊Dic₄₃	—	4	2	no	—
102	C₂²⋊Dic₄₃	—	4	1	no	—
103	C₂²⋊Dic₄₃	—	4	1	no	—
104	C₂²⋊Dic₄₃	D₈	1	2	yes	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	—	2	1	no	—
107	C₄⋊Dic₄₃	—	2	1	no	—
108	C₄⋊Dic₄₃	—	2	1	no	—
109	C₄⋊Dic₄₃	—	4	2	no	—
110	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
111	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	C₄×Dic₄₃	—	4	1	no	—
115	C₄×Dic₄₃	—	4	1	no	—
116	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
117	#10621877	—	2	2	no	—
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	C₁₆×C₄₃	—	2	1	no	—
120	C₁₆×C₄₃	—	2	1	no	—
121	C₄₃⋊C₁₆	—	2	2	no	—
122	C₂³×C₄₃	—	4	2	no	—
123	C₂³×C₄₃	—	4	4	no	—
124	C₂³×C₄₃	—	16	1	no	—
125	C₂³×C₄₃	—	16	1	no	—
126	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
127	D₄×C₄₃	—	2	4	no	—
128	D₄×C₄₃	—	4	2	no	—
129	D₄×C₄₃	—	4	4	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	16	1	no	—
132	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#137	1	2	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	—	4	1	no	—
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	4	1	no	—
140	C₂×Dic₄₃	—	16	2	no	—
141	C₂×Dic₄₃	—	16	2	no	—
142	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
143	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
144	Dic₈₆	—	16	2	no	—
145	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₄₃⋊C₈	—	2	2	no	—
149	#33631	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	2	1	no	—
152	C₂²×C₄₃	—	4	4	no	—
153	C₂²×C₄₃	—	8	2	no	—
154	C₂²×C₄₃	—	8	2	no	—
155	C₂²×C₄₃	—	16	4	no	—
156	C₂²×C₄₃	#474	1	2	yes	—
157	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₄₃	—	8	2	no	—
159	C₄×C₄₃	#474	1	2	yes	—
160	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
162	Dic₄₃	—	2	2	no	—
163	Dic₄₃	—	2	2	no	—
164	Dic₄₃	—	8	1	no	—
165	Dic₄₃	—	8	1	no	—
166	Dic₄₃	—	8	1	no	—
167	Dic₄₃	—	8	1	no	—
168	#3164	—	43	1	no	—
169	#2948	—	43	1	no	—
170	#2228	—	43	2	no	—
171	#2228	—	43	2	no	—
172	#2226	—	43	2	no	—
173	#2224	—	43	2	no	—
174	#2159	—	43	1	no	—
175	#2158	—	43	1	no	—
176	#2151	—	43	1	no	—
177	#2140	—	43	1	no	—
178	C₂×#357	D₄₃	1	1	yes	characteristic
179	C₂×C₄₃	—	2	2	no	—
180	C₂×C₄₃	—	8	4	no	—
181	C₂×C₄₃	C₂×#474	1	1	yes	characteristic
182	C₂×C₄₃	#3415	1	1	yes	characteristic
183	C₂×C₄₃	#3416	1	1	yes	characteristic
184	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
185	#544	—	86	1	no	—
186	#518	—	86	1	no	—
187	C₂×M₃₂	—	43	1	no	—
188	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
189	#501	—	86	2	no	—
190	#500	—	43	2	no	—
191	#499	—	43	2	no	—
192	#442	—	86	2	no	—
193	C₄×D₈	—	43	2	no	—
194	C₄×D₈	—	43	2	no	—
195	#428	—	86	1	no	—
196	#427	—	43	1	no	—
197	#427	—	43	1	no	—
198	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
199	#418	—	86	1	no	—
200	#405	—	43	1	no	—
201	#367	—	43	1	no	—
202	#367	—	43	1	no	—
203	#367	—	43	2	no	—
204	#366	—	43	1	no	—
205	#366	—	43	1	no	—
206	#366	—	43	2	no	—
207	C₄⋊C₁₆	—	43	2	no	—
208	C₄⋊C₁₆	—	43	2	no	—
209	#357	—	43	2	no	—
210	#357	—	43	2	no	—
211	#357	C₂×D₄₃	1	2	yes	—
212	#357	C₂×D₄₃	1	2	yes	—
213	C₄₃	#33631	1	1	yes	characteristic, complemented, Sylow
214	C₂²×D₄	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	Dic₈₆	1	4	yes	complemented
218	C₂×D₈	C₂²×D₄₃	1	2	yes	—
219	C₂×M₁₆	—	43	1	no	—
220	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	86	2	no	—
222	C₄⋊Q₈	—	172	1	no	—
223	#125	—	86	2	no	—
224	#125	—	172	1	no	—
225	C₂²⋊Q₈	—	172	2	no	—
226	C₄×D₄	—	86	2	no	—
227	C₄×D₄	—	86	2	no	—
228	C₄×D₄	—	172	1	no	—
229	C₄×D₄	—	172	1	no	—
230	#117	—	43	1	no	—
231	#117	—	86	1	no	—
232	#117	—	86	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	172	2	no	—
235	M₃₂	—	86	2	no	—
236	C₂×C₁₆	—	2	2	no	—
237	C₂×C₁₆	—	43	2	no	—
238	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
240	#107	—	2	1	no	—
241	#107	—	43	1	no	—
242	#107	—	43	1	no	—
243	#107	—	43	1	no	—
244	#107	—	43	1	no	—
245	#107	—	86	2	no	—
246	#107	C₂²×D₄₃	1	2	yes	—
247	#106	—	86	2	no	—
248	#106	—	86	1	no	—
249	#106	—	86	1	no	—
250	#102	—	86	2	no	—
251	#102	—	86	2	no	—
252	#97	—	43	2	no	—
253	C₄×C₈	—	43	2	no	—
254	C₂⁴	—	4	1	no	—
255	C₂×Q₈	—	172	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	172	2	no	—
262	C₂²×C₄	—	172	2	no	—
263	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
264	D₈	—	2	4	no	—
265	D₈	—	2	2	no	—
266	D₈	C₂×Dic₈₆	1	4	yes	—
267	M₁₆	—	86	2	no	—
268	C₂×C₈	—	43	2	no	—
269	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
270	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
271	C₂×C₈	#10621900	1	2	yes	—
272	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	43	1	no	—
276	C₄⋊C₄	—	43	1	no	—
277	C₄⋊C₄	—	86	1	no	—
278	C₄⋊C₄	—	86	1	no	—
279	C₄⋊C₄	—	86	1	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	—	172	2	no	—
282	C₄⋊C₄	—	172	2	no	—
283	C₄⋊C₄	—	172	2	no	—
284	C₂²⋊C₄	—	43	2	no	—
285	C₂²⋊C₄	—	172	2	no	—
286	C₂²⋊C₄	—	172	2	no	—
287	C₂²⋊C₄	—	172	1	no	—
288	C₂²⋊C₄	—	172	1	no	—
289	C₄²	—	43	2	no	—
290	C₄²	—	172	1	no	—
291	C₄²	—	172	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	86	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	16	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	D₈×D₄₃	1	1	yes	characteristic
300	Q₈	—	688	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	16	1	no	—
306	C₂×C₄	—	43	2	no	—
307	C₂×C₄	—	43	2	no	—
308	C₂×C₄	—	172	1	no	—
309	C₂×C₄	—	172	1	no	—
310	C₂×C₄	—	172	1	no	—
311	C₂×C₄	—	172	1	no	—
312	C₂×C₄	—	688	2	no	—
313	C₂×C₄	—	688	2	no	—
314	C₂×C₄	#13063302	1	2	yes	—
315	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
316	C₂×C₄	#13063354	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
318	C₈	—	86	2	no	—
319	C₈	#13063302	1	2	yes	—
320	C₈	C₂²×Dic₈₆	1	1	yes	characteristic
321	C₈	C₂×#10621900	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#14583177	1	2	yes	—
329	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	86	2	no	—
332	C₄	—	86	2	no	—
333	C₄	—	344	1	no	—
334	C₄	—	344	1	no	—
335	C₄	—	344	1	no	—
336	C₄	—	344	1	no	—
337	C₄	#14583177	1	2	yes	—
338	C₄	C₂×#13063302	1	1	yes	characteristic
339	C₄	C₂×#13063354	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#14583177	1	1	yes	characteristic
343	C₂	#19147732	1	1	yes	characteristic
344	C₂	#19147737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#357	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₈×C₄₃	C₂²
hypercenter	C₂×#357	D₄₃
hypocenter	C₄₃	#33631
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. #19145131	C₂	semidirect
3. #19136754	C₂	semidirect
4. #19136750	C₂	semidirect
5. #19136740	C₂	semidirect
6. #19136730	C₂	semidirect
7. #19136381	C₂	semidirect
8. #19136379	C₂	semidirect
9. #19136353	C₂	semidirect
16. #14583303	C₂²	semidirect
17. #14583300	C₂²	semidirect
26. #14582727	C₂²	semidirect
27. #14582727	C₂²	semidirect
28. #14582726	C₂²	semidirect
29. #14582725	C₂²	semidirect
30. #14582725	C₂²	semidirect
31. #14582724	C₂²	semidirect
32. Dic₄₃⋊C₁₆	C₂²	semidirect
33. Dic₄₃⋊C₁₆	C₂²	semidirect
72. C₂×D₈×C₄₃	Q₈	semidirect
213. C₄₃	#33631	semidirect
217. C₂×D₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145131	2
#19143447	2
#19136754	2
#19136750	2
#19136740	2
#19136730	2
#19136381	2
#19136379	2
#19136353	2
#19136311	2
C₂×C₄₃×#357	2
#33631	43