Subgroups — #25251110

Order 11008.

682 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	C₂²×#13063268	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583316	C₂	1	1	yes	maximal, characteristic, complemented
3	#19144243	C₂	1	1	yes	maximal, characteristic
4	#19136710	C₂	1	4	yes	maximal, complemented
5	#19136693	C₂	1	4	yes	maximal, complemented
6	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136315	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136310	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
11	#14583463	—	2	1	no	—
12	C₂×#13063268	—	2	4	no	—
13	C₂×#13063268	C₂²	1	2	yes	—
14	C₂×#13063268	C₂²	1	2	yes	—
15	#14583316	—	2	4	no	—
16	C₂×#13063265	C₂²	1	1	yes	characteristic
17	C₂×#13063265	C₂²	1	2	yes	complemented
18	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
20	#14583152	—	2	2	no	—
21	#14583135	C₂²	1	4	yes	—
22	#14582937	C₂²	1	1	yes	characteristic
23	#14582905	—	2	1	no	—
24	#14582726	C₂²	1	2	yes	complemented
25	#14582724	C₂²	1	2	yes	complemented
26	#14582703	C₂²	1	4	yes	complemented
27	#14582699	C₂²	1	4	yes	—
28	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#536	—	2	1	no	—
31	C₄₃×#470	—	2	2	no	—
32	C₄₃×#467	C₂²	1	4	yes	complemented
33	C₄₃×#427	C₂²	1	1	yes	characteristic
34	C₄₃×#419	—	2	1	no	—
35	C₂²×D₁₇₂	—	2	1	no	—
36	C₂×D₃₄₄	—	2	2	no	—
37	C₂×D₃₄₄	—	2	2	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×D₃₄₄	C₂³	1	2	yes	—
40	#13063321	—	4	1	no	—
41	#13063314	—	4	1	no	—
42	#13063299	—	2	4	no	—
43	#13063293	—	4	2	no	—
44	#13063284	—	2	1	no	—
45	#13063268	—	2	4	no	—
46	#13063268	—	2	4	no	—
47	#13063265	—	2	2	no	—
48	#13063265	—	2	4	no	—
49	C₂×C₄₃⋊C₁₆	—	2	4	no	—
50	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	complemented
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13063259	C₂³	1	2	yes	—
55	#13063257	—	2	1	no	—
56	#13063247	—	2	4	no	—
57	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₄₃	—	2	1	no	—
59	C₂×Q₁₆×C₄₃	—	2	2	no	—
60	C₂×Q₁₆×C₄₃	—	2	1	no	—
61	C₂×Q₁₆×C₄₃	—	2	2	no	—
62	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
63	#131×C₄₃	—	2	2	no	—
64	#131×C₄₃	—	2	2	no	—
65	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
66	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₄₃	—	4	1	no	—
70	#124×C₄₃	—	4	1	no	—
71	#123×C₄₃	—	4	2	no	—
72	C₂²⋊Q₈×C₄₃	—	2	4	no	—
73	#117×C₄₃	—	2	1	no	—
74	#107×C₄₃	C₂³	1	2	yes	—
75	#106×C₄₃	—	2	1	no	—
76	#103×C₄₃	—	2	4	no	—
77	C₂³×D₄₃	—	4	1	no	—
78	#41×C₄₃	—	2	2	no	—
79	#41×C₄₃	—	2	2	no	—
80	C₂×Q₈×C₄₃	—	2	2	no	—
81	C₂×Q₈×C₄₃	—	4	1	no	—
82	C₂×Q₈×C₄₃	—	4	2	no	—
83	C₂×Q₈×C₄₃	—	4	2	no	—
84	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
85	C₂×D₄×C₄₃	D₈	1	2	yes	—
86	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	—	4	1	no	—
88	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₃	—	4	4	no	—
91	C₂×D₁₇₂	—	2	2	no	—
92	C₂×D₁₇₂	—	4	1	no	—
93	C₂×D₁₇₂	—	4	2	no	—
94	C₂×D₁₇₂	—	4	2	no	—
95	Q₁₆×C₄₃	—	2	2	no	—
96	Q₁₆×C₄₃	—	2	2	no	—
97	Q₁₆×C₄₃	—	4	2	no	—
98	M₁₆×C₄₃	—	2	2	no	—
99	M₁₆×C₄₃	—	2	4	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	#41	1	2	yes	—
106	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	4	4	no	—
109	C₂²⋊C₄×C₄₃	—	4	1	no	—
110	C₂²⋊C₄×C₄₃	—	4	4	no	—
111	C₄²×C₄₃	—	4	1	no	—
112	C₂²⋊Dic₄₃	—	4	1	no	—
113	(C₂×D₄₃)⋊C₄	—	4	4	no	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄×Dic₄₃	—	4	1	no	—
116	D₃₄₄	—	2	2	no	—
117	D₃₄₄	—	2	2	no	—
118	D₃₄₄	—	4	2	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
121	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
122	C₂³×C₄₃	C₂×D₈	1	2	yes	—
123	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂²×D₄₃	—	4	2	no	—
125	C₂²×D₄₃	—	8	2	no	—
126	C₂²×D₄₃	—	16	2	no	—
127	Q₈×C₄₃	—	2	2	no	—
128	Q₈×C₄₃	—	4	2	no	—
129	Q₈×C₄₃	—	4	2	no	—
130	Q₈×C₄₃	—	8	2	no	—
131	D₄×C₄₃	—	2	4	no	—
132	D₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	8	2	no	—
138	C₂×C₄×C₄₃	#121	1	2	yes	—
139	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#135	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₄₃	—	16	4	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	D₁₇₂	—	4	2	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	8	2	no	—
148	C₈×C₄₃	—	2	2	no	—
149	C₈×C₄₃	#121	1	2	yes	—
150	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
152	#33609	—	43	1	no	maximal, Sylow
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	—	2	4	no	—
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	#466	1	2	yes	—
157	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₄₃	—	8	4	no	—
159	C₂×D₄₃	—	8	2	no	—
160	C₂×D₄₃	—	16	4	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	#466	1	2	yes	—
165	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
166	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
167	Dic₄₃	—	8	2	no	—
168	C₂²×SD₃₂	—	43	1	no	—
169	C₂×#504	—	43	1	no	—
170	#3059	D₄₃	1	1	yes	characteristic, complemented
171	#3058	—	43	1	no	—
172	#2220	—	43	4	no	—
173	#2217	—	43	4	no	—
174	#2158	—	43	1	no	—
175	#2141	—	43	1	no	—
176	#2140	—	43	1	no	—
177	C₂×C₄₃	—	2	2	no	—
178	C₂×C₄₃	—	4	2	no	—
179	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
180	C₂×C₄₃	#3416	1	2	yes	—
181	D₄₃	—	16	2	no	—
182	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
183	C₂²×D₈	—	43	1	no	—
184	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
185	#536	—	2	1	no	—
186	#535	—	86	1	no	—
187	C₂×SD₃₂	—	43	2	no	—
188	C₂×SD₃₂	—	43	2	no	—
189	C₂×SD₃₂	—	86	4	no	—
190	#504	—	86	4	no	—
191	C₂×M₃₂	—	43	1	no	—
192	C₂×M₃₂	—	43	2	no	—
193	C₂²×C₁₆	—	43	1	no	—
194	C₂²×C₁₆	—	43	2	no	—
195	#470	—	2	2	no	—
196	#469	—	86	2	no	—
197	#467	C₂×D₄₃	1	4	yes	complemented
198	#466	—	43	4	no	—
199	#427	—	43	1	no	—
200	#427	C₂×D₄₃	1	1	yes	characteristic
201	#419	—	2	1	no	—
202	#418	—	86	1	no	—
203	#367	—	43	2	no	—
204	#366	—	43	2	no	—
205	#358	—	43	4	no	—
206	#357	—	43	4	no	—
207	C₄₃	#33609	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×D₄	—	86	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
215	C₂×D₈	—	43	2	no	—
216	C₂×D₈	—	86	2	no	—
217	C₂×D₈	—	86	2	no	—
218	C₂×D₈	—	86	1	no	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
223	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
224	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
225	C₄⋊Q₈	—	4	1	no	—
226	#127	—	172	1	no	—
227	#124	—	4	1	no	—
228	#124	—	172	1	no	—
229	#123	—	4	2	no	—
230	C₂²⋊Q₈	—	2	4	no	—
231	#121	—	86	4	no	—
232	#120	—	172	2	no	—
233	#117	—	2	1	no	—
234	#117	—	86	1	no	—
235	SD₃₂	—	86	4	no	—
236	SD₃₂	—	86	4	no	—
237	M₃₂	—	86	4	no	—
238	M₃₂	—	86	2	no	—
239	C₂×C₁₆	—	43	4	no	—
240	C₂×C₁₆	—	43	1	no	—
241	C₂×C₁₆	—	43	1	no	—
242	C₂×C₁₆	—	43	2	no	—
243	C₂×C₁₆	—	86	4	no	—
244	#107	—	43	2	no	—
245	#107	C₂²×D₄₃	1	2	yes	—
246	#106	—	2	1	no	—
247	#106	—	86	1	no	—
248	#103	—	2	4	no	—
249	#102	—	86	4	no	—
250	C₂⁴	—	172	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	#10621882	1	1	yes	characteristic
258	C₂×D₄	—	86	2	no	—
259	C₂×D₄	—	172	4	no	—
260	C₂×D₄	—	172	2	no	—
261	C₂×D₄	—	172	1	no	—
262	C₂×D₄	—	172	2	no	—
263	C₂×D₄	#10621882	1	2	yes	—
264	C₂×D₄	#10621882	1	1	yes	characteristic
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	4	2	no	—
271	D₈	—	86	2	no	—
272	D₈	—	86	2	no	—
273	D₈	—	172	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	D₄×D₄₃	1	2	yes	—
279	C₂×C₈	#10621900	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
281	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	86	2	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	172	1	no	—
289	C₂²⋊C₄	—	172	4	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	172	1	no	—
292	C₁₆	—	43	2	no	—
293	C₁₆	—	43	2	no	—
294	C₁₆	—	86	2	no	—
295	C₂³	—	344	2	no	—
296	C₂³	—	172	2	no	—
297	C₂³	—	688	2	no	—
298	C₂³	C₂×#10621882	1	2	yes	—
299	C₂³	C₂×#10621882	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	344	2	no	—
307	D₄	—	172	2	no	—
308	D₄	—	172	2	no	—
309	D₄	—	688	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	172	2	no	—
316	C₂×C₄	#13063297	1	2	yes	—
317	C₂×C₄	C₂×#10621882	1	2	yes	—
318	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
319	C₂×C₄	#13063345	1	2	yes	—
320	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#13063297	1	2	yes	—
323	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
324	C₈	C₂×#10621900	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	344	4	no	—
329	C₂²	—	344	2	no	—
330	C₂²	—	688	4	no	—
331	C₂²	#14583133	1	2	yes	—
332	C₂²	C₂²×#10621882	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	344	2	no	—
337	C₄	#14583133	1	2	yes	—
338	C₄	C₂×#13063297	1	1	yes	characteristic
339	C₄	C₂×#13063345	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	688	2	no	—
343	C₂	C₂×#14583133	1	1	yes	characteristic
344	C₂	#19147739	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3059	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19144243	C₂
hypercenter	#3059	D₄₃
hypocenter	C₄₃	#33609
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3059

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063268	C₂	semidirect
2. C₂×#14583316	C₂	semidirect
4. #19136710	C₂	semidirect
5. #19136693	C₂	semidirect
6. #19136377	C₂	semidirect
7. #19136315	C₂	semidirect
8. #19136310	C₂	semidirect
9. C₄₃×#3059	C₂	semidirect
17. C₂×#13063265	C₂²	semidirect
19. C₂²×C₄₃⋊C₁₆	C₂²	semidirect
24. #14582726	C₂²	semidirect
25. #14582724	C₂²	semidirect
26. #14582703	C₂²	semidirect
32. C₄₃×#467	C₂²	semidirect
50. C₂×C₄₃⋊C₁₆	D₄	semidirect
170. #3059	D₄₃	semidirect
197. #467	C₂×D₄₃	semidirect
207. C₄₃	#33609	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063268	2
C₂×#14583316	2
#19144243	2
#19136710	2
#19136693	2
#19136377	2
#19136315	2
#19136310	2
C₄₃×#3059	2
#33609	43