Subgroups — #25252195

Order 11008.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145237	C₂	1	2	yes	maximal, complemented
2	#19136856	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136834	C₂	1	2	yes	maximal, complemented
4	#19136709	C₂	1	2	yes	maximal, complemented
5	#19136687	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136682	C₂	1	1	yes	maximal, characteristic
7	#19136458	C₂	1	2	yes	maximal, complemented
8	#19136453	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136441	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136190	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583523	—	2	2	no	—
13	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
14	C₂×#13063270	C₂²	1	2	yes	—
15	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
16	C₂×#13063264	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
18	#14583283	C₂²	1	2	yes	—
19	#14583261	—	2	2	no	—
20	#14583146	—	2	2	no	—
21	#14583141	C₂²	1	2	yes	—
22	#14583136	C₂²	1	2	yes	—
23	#14582998	—	2	2	no	—
24	#14582993	C₂²	1	2	yes	—
25	#14582981	C₂²	1	2	yes	—
26	#14582853	C₂²	1	1	yes	characteristic
27	#14582702	C₂²	1	2	yes	—
28	#14582702	C₂²	1	2	yes	—
29	#14582699	C₂²	1	2	yes	—
30	#14582699	C₂²	1	2	yes	—
31	#14582675	C₂²	1	1	yes	characteristic
32	#14582674	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#500	C₂²	1	2	yes	—
35	C₄₃×#366	C₂²	1	1	yes	characteristic
36	C₄₃×#366	C₂²	1	1	yes	characteristic
37	C₄₃×#366	C₂²	1	2	yes	—
38	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₇₂	C₂³	1	2	yes	—
40	C₂×#10621873	—	2	2	no	—
41	C₂×D₃₄₄	C₂³	1	2	yes	—
42	C₂×#10621872	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
44	#13063320	—	4	2	no	—
45	#13063314	—	4	2	no	—
46	#13063313	—	2	2	no	—
47	#13063310	—	4	2	no	—
48	#13063305	—	2	2	no	—
49	#13063301	—	4	2	no	—
50	#13063300	—	2	2	no	—
51	#13063290	—	2	2	no	—
52	#13063287	—	2	2	no	—
53	#13063282	D₄	1	1	yes	characteristic
54	Dic₃₄₄	—	2	2	no	—
55	#13063270	—	2	4	no	—
56	D₆₈₈	—	2	2	no	—
57	#13063264	—	2	2	no	—
58	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₄₃	—	2	2	no	—
61	#13063248	—	2	2	no	—
62	#13063248	—	2	2	no	—
63	#13063245	—	2	2	no	—
64	#13063245	—	2	2	no	—
65	#13063236	D₄	1	1	yes	characteristic
66	#13063235	D₄	1	1	yes	characteristic
67	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₄₃	—	2	2	no	—
69	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
71	#107×C₄₃	—	2	2	no	—
72	#107×C₄₃	C₂³	1	2	yes	—
73	#107×C₄₃	C₂³	1	2	yes	—
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₄₃	—	4	2	no	—
76	C₂×D₁₇₂	—	2	2	no	—
77	C₂×D₁₇₂	—	4	2	no	—
78	C₂×C₄×D₄₃	—	4	2	no	—
79	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₈₆	—	2	2	no	—
81	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	—	2	2	no	—
84	C₄⋊C₄×C₄₃	—	2	2	no	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	(C₂×D₄₃)⋊C₄	—	4	2	no	—
88	(C₂×D₄₃)⋊C₄	—	4	2	no	—
89	(C₂×D₄₃)⋊C₄	—	4	2	no	—
90	(C₂×D₄₃)⋊C₄	—	4	2	no	—
91	(C₂×D₄₃)⋊C₄	—	4	2	no	—
92	(C₂×D₄₃)⋊C₄	D₈	1	1	yes	characteristic
93	(C₂×D₄₃)⋊C₄	D₈	1	1	yes	characteristic
94	C₄⋊Dic₄₃	—	4	2	no	—
95	Dic₄₃⋊C₄	—	4	2	no	—
96	Dic₄₃⋊C₄	D₈	1	1	yes	characteristic
97	Dic₄₃⋊C₄	D₈	1	1	yes	characteristic
98	C₄×Dic₄₃	—	4	2	no	—
99	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
102	Dic₁₇₂	—	2	2	no	—
103	Dic₁₇₂	—	2	2	no	—
104	D₃₄₄	—	2	2	no	—
105	D₃₄₄	—	2	2	no	—
106	#10621873	—	4	4	no	—
107	#10621872	—	2	2	no	—
108	C₈×D₄₃	—	2	1	no	—
109	C₈×D₄₃	—	2	1	no	—
110	C₁₆×C₄₃	#41	1	2	yes	—
111	C₁₆×C₄₃	#41	1	2	yes	—
112	C₄₃⋊C₁₆	—	2	1	no	—
113	C₄₃⋊C₁₆	—	2	1	no	—
114	C₂²×D₄₃	—	4	2	no	—
115	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₄₃	—	16	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
126	D₁₇₂	—	4	2	no	—
127	D₁₇₂	—	4	2	no	—
128	D₁₇₂	—	16	2	no	—
129	C₄×D₄₃	—	2	1	no	—
130	C₄×D₄₃	—	2	1	no	—
131	C₄×D₄₃	—	16	2	no	—
132	Dic₈₆	—	4	2	no	—
133	Dic₈₆	—	4	2	no	—
134	C₈×C₄₃	#135	1	2	yes	—
135	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
137	C₄₃⋊C₈	—	2	1	no	—
138	C₄₃⋊C₈	—	2	1	no	—
139	#33739	—	43	1	no	maximal, Sylow
140	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₄₃	—	2	1	no	—
142	C₂×D₄₃	—	2	1	no	—
143	C₂×D₄₃	—	2	1	no	—
144	C₂×D₄₃	—	8	2	no	—
145	C₂×D₄₃	—	8	2	no	—
146	C₂×D₄₃	—	8	2	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	—	8	2	no	—
150	C₄×C₄₃	—	8	2	no	—
151	C₄×C₄₃	#532	1	1	yes	characteristic
152	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
153	Dic₄₃	—	2	1	no	—
154	Dic₄₃	—	2	1	no	—
155	Dic₄₃	—	2	1	no	—
156	Dic₄₃	—	8	2	no	—
157	#3175	—	43	2	no	—
158	#2252	D₄₃	1	1	yes	characteristic, complemented
159	#2249	—	43	1	no	—
160	#2243	—	43	2	no	—
161	#2220	—	43	2	no	—
162	#2216	—	43	1	no	—
163	#2215	—	43	1	no	—
164	#2175	—	43	2	no	—
165	C₄×Q₃₂	—	43	1	no	—
166	C₄×D₁₆	—	43	1	no	—
167	#2108	—	43	1	no	—
168	C₂×C₄₃	#3169	1	1	yes	characteristic
169	C₂×C₄₃	#3415	1	1	yes	characteristic
170	C₂×C₄₃	#3417	1	1	yes	characteristic
171	D₄₃	—	4	1	no	—
172	D₄₃	—	16	2	no	—
173	#542	—	86	2	no	—
174	C₂×Q₃₂	—	43	1	no	—
175	C₂×SD₃₂	—	43	2	no	—
176	C₂×D₁₆	—	43	1	no	—
177	C₂×M₃₂	—	43	1	no	—
178	C₂²×C₁₆	—	43	1	no	—
179	#500	C₂×D₄₃	1	2	yes	—
180	#495	—	43	2	no	—
181	#489	—	86	2	no	—
182	#468	—	86	2	no	—
183	#467	—	43	2	no	—
184	#466	—	43	2	no	—
185	#440	—	86	2	no	—
186	C₄×Q₁₆	—	43	2	no	—
187	C₄×D₈	—	43	2	no	—
188	#405	—	43	1	no	—
189	#366	C₂×D₄₃	1	1	yes	characteristic
190	#366	C₂×D₄₃	1	1	yes	characteristic
191	#366	C₂×D₄₃	1	2	yes	—
192	#358	—	43	2	no	—
193	#358	—	43	2	no	—
194	#357	—	43	2	no	—
195	#357	—	43	2	no	—
196	#346	—	43	1	no	—
197	#346	—	43	1	no	—
198	C₄×C₁₆	—	43	1	no	—
199	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
200	C₄₃	#33739	1	1	yes	characteristic, complemented, Sylow
201	C₂×Q₁₆	—	43	2	no	—
202	C₂×SD₁₆	—	86	2	no	—
203	C₂×D₈	—	43	2	no	—
204	C₂×M₁₆	—	43	1	no	—
205	C₂²×C₈	—	43	1	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	#126	—	172	2	no	—
208	#124	—	86	2	no	—
209	#124	—	172	2	no	—
210	#123	—	172	2	no	—
211	C₂²⋊Q₈	—	86	2	no	—
212	C₂²⋊Q₈	—	172	2	no	—
213	#121	—	86	2	no	—
214	C₄×Q₈	—	86	2	no	—
215	C₄×D₄	—	86	2	no	—
216	#117	—	43	1	no	—
217	Q₃₂	—	86	2	no	—
218	SD₃₂	—	86	4	no	—
219	D₁₆	—	86	2	no	—
220	M₃₂	—	86	2	no	—
221	C₂×C₁₆	—	43	1	no	—
222	C₂×C₁₆	—	43	1	no	—
223	C₂×C₁₆	—	86	2	no	—
224	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	C₂²×D₄₃	1	2	yes	—
228	#107	C₂²×D₄₃	1	2	yes	—
229	#103	—	86	2	no	—
230	#103	—	86	2	no	—
231	#102	—	86	2	no	—
232	#102	—	86	2	no	—
233	#97	—	43	1	no	—
234	#97	—	43	1	no	—
235	C₄×C₈	—	43	1	no	—
236	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	86	2	no	—
239	C₂×D₄	—	86	2	no	—
240	C₂×D₄	—	172	2	no	—
241	C₂²×C₄	—	43	1	no	—
242	C₂²×C₄	—	172	2	no	—
243	Q₁₆	—	86	2	no	—
244	Q₁₆	—	86	2	no	—
245	SD₁₆	—	172	4	no	—
246	D₈	—	86	2	no	—
247	D₈	—	86	2	no	—
248	M₁₆	—	86	2	no	—
249	C₂×C₈	—	43	1	no	—
250	C₂×C₈	—	43	1	no	—
251	C₂×C₈	—	86	1	no	—
252	C₂×C₈	—	86	1	no	—
253	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	43	1	no	—
259	C₄⋊C₄	—	43	1	no	—
260	C₄⋊C₄	—	172	2	no	—
261	C₄⋊C₄	—	172	2	no	—
262	C₂²⋊C₄	—	43	1	no	—
263	C₂²⋊C₄	—	43	1	no	—
264	C₂²⋊C₄	—	172	2	no	—
265	C₂²⋊C₄	—	172	2	no	—
266	C₂²⋊C₄	—	172	2	no	—
267	C₂²⋊C₄	—	172	2	no	—
268	C₂²⋊C₄	—	172	2	no	—
269	C₄²	—	43	1	no	—
270	C₄²	—	172	2	no	—
271	C₄²	D₄×D₄₃	1	1	yes	characteristic
272	C₁₆	—	86	1	no	—
273	C₁₆	—	86	1	no	—
274	C₁₆	#10621902	1	2	yes	—
275	C₁₆	#10621902	1	2	yes	—
276	C₂³	—	43	1	no	—
277	C₂³	—	172	2	no	—
278	Q₈	—	16	2	no	—
279	Q₈	—	172	2	no	—
280	Q₈	—	172	2	no	—
281	D₄	—	172	2	no	—
282	D₄	—	172	2	no	—
283	D₄	—	688	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	43	1	no	—
287	C₂×C₄	—	43	1	no	—
288	C₂×C₄	—	43	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	172	2	no	—
292	C₂×C₄	—	688	2	no	—
293	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
294	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
296	C₈	—	86	1	no	—
297	C₈	—	86	1	no	—
298	C₈	#13063348	1	2	yes	—
299	C₈	C₂×#10621902	1	1	yes	characteristic
300	C₈	C₂×#10621902	1	1	yes	characteristic
301	C₂²	—	86	1	no	—
302	C₂²	—	86	1	no	—
303	C₂²	—	86	1	no	—
304	C₂²	—	344	2	no	—
305	C₂²	—	344	2	no	—
306	C₂²	—	344	2	no	—
307	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	86	1	no	—
313	C₄	—	86	1	no	—
314	C₄	—	86	1	no	—
315	C₄	—	344	2	no	—
316	C₄	#14583450	1	1	yes	characteristic
317	C₄	C₂×#13063348	1	1	yes	characteristic
318	C₂	—	172	1	no	—
319	C₂	—	688	2	no	—
320	C₂	#19145175	1	1	yes	characteristic
321	C₂	#19147732	1	1	yes	characteristic
322	C₂	#19147744	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2252	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19136682	C₂
hypercenter	#2252	D₄₃
hypocenter	C₄₃	#33739
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
1. #19145237	C₂	semidirect
2. #19136856	C₂	semidirect
3. #19136834	C₂	semidirect
4. #19136709	C₂	semidirect
5. #19136687	C₂	semidirect
7. #19136458	C₂	semidirect
8. #19136453	C₂	semidirect
9. #19136441	C₂	semidirect
10. #19136190	C₂	semidirect
11. C₄₃×#2252	C₂	semidirect
158. #2252	D₄₃	semidirect
200. C₄₃	#33739	semidirect

Maximal subgroups

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

subgroup	index
#19145237	2
#19136856	2
#19136834	2
#19136709	2
#19136687	2
#19136682	2
#19136458	2
#19136453	2
#19136441	2
#19136190	2
C₄₃×#2252	2
#33739	43