Subgroups — #25084529

Order 11008.

858 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145712	C₂	1	2	yes	maximal, complemented
2	#19145172	C₂	1	1	yes	maximal, characteristic
3	#500×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19145049	C₂	1	2	yes	maximal, complemented
5	#19134268	C₂	1	2	yes	maximal, complemented
6	#19134264	C₂	1	2	yes	maximal, complemented
7	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134212	C₂	1	1	yes	maximal, characteristic, complemented
9	#19134184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621872	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063354	—	2	4	no	—
13	#14583447	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₄₃	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583304	C₂²	1	2	yes	—
17	#14583304	C₂²	1	1	yes	characteristic
18	#14583303	—	2	2	no	—
19	#14583302	C₂²	1	2	yes	—
20	#14583302	C₂²	1	2	yes	—
21	#14583290	—	2	4	no	—
22	#14583279	C₂²	1	2	yes	—
23	#14583278	C₂²	1	2	yes	—
24	#14583271	C₂²	1	2	yes	—
25	#14583245	C₂²	1	2	yes	—
26	#14583245	C₂²	1	2	yes	—
27	#14583211	—	2	4	no	—
28	#14583177	—	2	4	no	—
29	#14583132	—	2	2	no	—
30	D₈×Dic₄₃	—	2	2	no	—
31	#14582950	C₂²	1	1	yes	characteristic
32	#107×D₄₃	—	2	2	no	—
33	C₄×#10621872	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
35	#14582621	C₂²	1	1	yes	characteristic
36	#14582620	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#493	C₂²	1	1	yes	characteristic
40	C₄₃×#492	C₂²	1	2	yes	—
41	C₄₃×#486	C₂²	1	2	yes	—
42	C₈²×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×#10621900	—	2	2	no	—
44	C₂×Q₈×D₄₃	—	2	2	no	—
45	#13063354	—	4	16	no	—
46	C₂×Dic₁₇₂	—	2	4	no	—
47	C₂×#10621885	—	2	4	no	—
48	C₂×#10621883	—	2	4	no	—
49	C₂×#10621872	D₄	1	2	yes	—
50	C₂×C₈×D₄₃	D₄	1	2	yes	—
51	#13063325	C₂³	1	2	yes	—
52	#13063325	C₂³	1	2	yes	—
53	#13063324	—	2	2	no	—
54	#13063324	—	2	2	no	—
55	#13063323	—	2	2	no	—
56	#13063312	—	2	2	no	—
57	#13063296	—	2	2	no	—
58	#13063296	—	2	2	no	—
59	D₄×Dic₄₃	—	2	2	no	—
60	D₄×Dic₄₃	—	2	2	no	—
61	C₄⋊C₄×D₄₃	—	2	2	no	—
62	C₄⋊C₄×D₄₃	—	2	2	no	—
63	C₄²×D₄₃	C₂³	1	1	yes	characteristic
64	#13063259	—	2	2	no	—
65	#13063259	—	2	2	no	—
66	#13063259	—	2	4	no	—
67	#13063258	—	2	4	no	—
68	#13063258	—	2	4	no	—
69	Dic₄₃⋊C₈	D₄	1	2	yes	—
70	#13063246	—	2	4	no	—
71	#13063246	—	2	4	no	—
72	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
73	#13063235	D₄	1	2	yes	—
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₃	D₄	1	2	yes	—
78	C₂×SD₁₆×C₄₃	—	2	4	no	—
79	C₂×D₈×C₄₃	—	2	2	no	—
80	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
81	#127×C₄₃	C₂³	1	2	yes	—
82	#107×C₄₃	—	2	2	no	—
83	#102×C₄₃	—	2	4	no	—
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	C₂×D₄×C₄₃	—	2	2	no	—
89	C₂×D₄×C₄₃	—	2	2	no	—
90	C₂×D₄×C₄₃	—	2	2	no	—
91	C₂×C₂²⋊D₄₃	—	4	2	no	—
92	C₂²×Dic₄₃	—	4	2	no	—
93	Q₈×D₄₃	—	4	8	no	—
94	#10621900	—	4	8	no	—
95	C₂×C₄×D₄₃	—	4	2	no	—
96	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₈₆	—	2	2	no	—
100	C₂×Dic₈₆	—	2	2	no	—
101	C₂×Dic₈₆	—	4	2	no	—
102	SD₁₆×C₄₃	—	4	8	no	—
103	D₈×C₄₃	—	4	4	no	—
104	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	2	no	—
109	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₄₃	—	4	2	no	—
111	C₂²⋊Dic₄₃	—	4	2	no	—
112	#10621885	—	4	8	no	—
113	#10621883	—	4	8	no	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	—	2	2	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	Dic₄₃⋊C₄	—	4	2	no	—
119	Dic₄₃⋊C₄	—	4	2	no	—
120	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
124	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
125	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
126	Dic₁₇₂	—	4	8	no	—
127	#10621872	D₈	1	8	yes	complemented
128	C₈×D₄₃	—	2	4	no	—
129	C₂³×C₄₃	—	4	2	no	—
130	C₂²×D₄₃	#127	1	1	yes	characteristic
131	Q₈×C₄₃	—	4	4	no	—
132	D₄×C₄₃	—	4	4	no	—
133	D₄×C₄₃	—	4	4	no	—
134	D₄×C₄₃	—	4	4	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₃	—	8	4	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	8	4	no	—
143	C₂×Dic₄₃	#127	1	1	yes	characteristic
144	C₂×Dic₄₃	#127	1	1	yes	characteristic
145	C₂×Dic₄₃	#127	1	1	yes	characteristic
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	8	4	no	—
149	C₄×D₄₃	C₂×D₈	1	4	yes	—
150	Dic₈₆	—	4	4	no	—
151	Dic₈₆	—	4	4	no	—
152	Dic₈₆	—	8	4	no	—
153	C₈×C₄₃	—	2	2	no	—
154	C₈×C₄₃	C₂×D₈	1	4	yes	—
155	C₈×C₄₃	#137	1	4	yes	complemented
156	C₄₃⋊C₈	—	2	2	no	—
157	C₄₃⋊C₈	—	2	2	no	—
158	C₄₃⋊C₈	C₂×D₈	1	4	yes	—
159	#19630	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	—	8	2	no	—
161	C₂²×C₄₃	—	8	2	no	—
162	C₂²×C₄₃	—	8	2	no	—
163	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
164	C₂×D₄₃	—	2	1	no	—
165	C₂×D₄₃	—	2	1	no	—
166	C₂×D₄₃	#493	1	2	yes	—
167	C₄×C₄₃	—	8	2	no	—
168	C₄×C₄₃	C₂²×D₈	1	2	yes	—
169	C₄×C₄₃	C₂×#137	1	2	yes	—
170	C₄×C₄₃	#578	1	2	yes	—
171	Dic₄₃	—	2	1	no	—
172	Dic₄₃	—	2	1	no	—
173	Dic₄₃	—	8	2	no	—
174	Dic₄₃	—	8	2	no	—
175	Dic₄₃	#493	1	2	yes	—
176	#3226	—	43	2	no	—
177	#3169	—	43	1	no	—
178	C₂×#500	—	43	1	no	—
179	#3152	—	43	2	no	—
180	#1722	—	43	2	no	—
181	#1722	—	43	2	no	—
182	#1713	—	43	1	no	—
183	#1713	—	43	1	no	—
184	#1706	—	43	1	no	—
185	#1706	D₄₃	1	1	yes	characteristic, complemented
186	C₈×M₁₆	—	43	1	no	—
187	C₂×C₄₃	—	16	2	no	—
188	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
189	C₂×C₄₃	#3152	1	1	yes	characteristic
190	C₂×C₄₃	#3156	1	1	yes	characteristic
191	D₄₃	—	2	2	no	—
192	C₂×#137	—	86	4	no	—
193	#532	—	43	2	no	—
194	C₂×C₄⋊Q₈	—	43	2	no	—
195	#500	—	43	2	no	—
196	#500	—	43	2	no	—
197	#500	—	43	1	no	—
198	#500	—	43	1	no	—
199	#500	—	43	2	no	—
200	#500	—	86	2	no	—
201	#500	C₂×D₄₃	1	1	yes	characteristic
202	#497	—	86	4	no	—
203	#494	—	43	2	no	—
204	#494	—	43	2	no	—
205	#493	C₂×D₄₃	1	1	yes	characteristic
206	#492	—	43	2	no	—
207	#492	C₂×D₄₃	1	2	yes	—
208	#486	—	43	2	no	—
209	#486	—	43	2	no	—
210	#486	C₂×D₄₃	1	2	yes	—
211	#480	—	86	4	no	—
212	#474	—	86	4	no	—
213	#466	—	86	2	no	—
214	C₄×D₈	—	86	2	no	—
215	#431	—	43	1	no	—
216	C₂×#107	—	86	2	no	—
217	C₄×M₁₆	—	43	1	no	—
218	C₂×C₄×C₈	—	43	1	no	—
219	#322	—	43	1	no	—
220	#322	—	43	1	no	—
221	C₈²	—	43	1	no	—
222	C₈²	C₂×D₄₃	1	1	yes	characteristic
223	C₄₃	#19630	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	86	2	no	—
225	C₂²×Q₈	—	86	2	no	—
226	#137	—	172	16	no	—
227	C₂×Q₁₆	—	86	4	no	—
228	C₂×Q₁₆	—	86	4	no	—
229	C₂×SD₁₆	—	2	4	no	—
230	C₂×SD₁₆	—	86	4	no	—
231	C₂×D₈	—	2	2	no	—
232	C₂×M₁₆	—	43	2	no	—
233	C₂²×C₈	—	43	2	no	—
234	C₄⋊Q₈	—	43	2	no	—
235	C₄⋊Q₈	—	43	2	no	—
236	C₄⋊Q₈	—	86	2	no	—
237	C₄⋊Q₈	—	86	2	no	—
238	C₄⋊Q₈	—	86	2	no	—
239	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
240	#127	C₂²×D₄₃	1	2	yes	—
241	#124	—	86	2	no	—
242	#121	—	86	2	no	—
243	#121	—	86	2	no	—
244	C₄×D₄	—	86	2	no	—
245	C₄×D₄	—	86	2	no	—
246	C₂×C₄⋊C₄	—	86	2	no	—
247	C₂×C₄⋊C₄	—	86	2	no	—
248	C₂×C₄²	—	43	1	no	—
249	#107	—	2	2	no	—
250	#107	—	86	4	no	—
251	#107	—	86	4	no	—
252	#107	—	86	2	no	—
253	#107	—	86	2	no	—
254	#107	—	86	4	no	—
255	C₄⋊C₈	—	43	2	no	—
256	#102	—	2	4	no	—
257	#102	—	86	4	no	—
258	#102	—	86	4	no	—
259	C₂²⋊C₈	—	43	2	no	—
260	#97	—	43	2	no	—
261	C₄×C₈	—	43	2	no	—
262	C₄×C₈	—	43	1	no	—
263	C₄×C₈	—	43	1	no	—
264	C₄×C₈	—	43	1	no	—
265	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
268	#41	—	172	8	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	86	2	no	—
271	C₂×Q₈	—	86	2	no	—
272	C₂×Q₈	—	172	2	no	—
273	C₂×Q₈	—	172	8	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	172	2	no	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	43	1	no	—
280	C₂²×C₄	—	43	1	no	—
281	C₂²×C₄	—	172	2	no	—
282	C₂²×C₄	—	172	2	no	—
283	Q₁₆	—	172	8	no	—
284	Q₁₆	—	172	8	no	—
285	SD₁₆	—	4	8	no	—
286	SD₁₆	—	172	8	no	—
287	D₈	—	4	4	no	—
288	M₁₆	—	43	8	no	—
289	C₂×C₈	—	43	2	no	—
290	C₂×C₈	—	43	2	no	—
291	C₂×C₈	—	43	2	no	—
292	C₂×C₈	—	86	4	no	—
293	C₂×C₈	D₄×D₄₃	1	2	yes	—
294	C₂×C₈	D₄×D₄₃	1	2	yes	—
295	C₂×C₈	D₄×D₄₃	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	2	no	—
302	C₄⋊C₄	—	172	2	no	—
303	C₄⋊C₄	—	172	2	no	—
304	C₂²⋊C₄	—	172	2	no	—
305	C₂²⋊C₄	—	172	2	no	—
306	C₄²	—	43	1	no	—
307	C₄²	—	43	1	no	—
308	C₄²	—	43	1	no	—
309	C₄²	C₂³×D₄₃	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	—	43	1	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	344	4	no	—
314	Q₈	—	172	4	no	—
315	Q₈	—	172	4	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	4	4	no	—
319	D₄	—	344	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	43	1	no	—
322	C₂×C₄	—	43	4	no	—
323	C₂×C₄	—	43	1	no	—
324	C₂×C₄	—	43	1	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	86	2	no	—
327	C₂×C₄	—	344	4	no	—
328	C₂×C₄	—	344	4	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	43	4	no	—
336	C₈	—	86	2	no	—
337	C₈	—	86	2	no	—
338	C₈	D₈×D₄₃	1	4	yes	—
339	C₈	#13063354	1	4	yes	complemented
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	43	2	no	—
344	C₂²	—	86	1	no	—
345	C₂²	—	86	1	no	—
346	C₂²	#127×D₄₃	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	43	2	no	—
349	C₄	—	86	1	no	—
350	C₄	—	86	1	no	—
351	C₄	—	344	2	no	—
352	C₄	—	344	2	no	—
353	C₄	C₂×D₈×D₄₃	1	2	yes	—
354	C₄	C₂×#13063354	1	2	yes	—
355	C₄	#14583713	1	2	yes	—
356	C₂	—	16	2	no	—
357	C₂	—	86	2	no	—
358	C₂	#493×D₄₃	1	1	yes	characteristic
359	C₂	#19145049	1	1	yes	characteristic
360	C₂	#19145085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1706	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	#19145172	C₂
hypercenter	#1706	D₄₃
hypocenter	C₄₃	#19630
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #19145712	C₂	semidirect
3. #500×D₄₃	C₂	semidirect
4. #19145049	C₂	semidirect
5. #19134268	C₂	semidirect
6. #19134264	C₂	semidirect
7. #19134213	C₂	semidirect
8. #19134212	C₂	semidirect
9. #19134184	C₂	semidirect
10. C₈×#10621872	C₂	semidirect
11. C₄₃×#1706	C₂	semidirect
127. #10621872	D₈	semidirect
155. C₈×C₄₃	#137	semidirect
185. #1706	D₄₃	semidirect
223. C₄₃	#19630	semidirect
339. C₈	#13063354	semidirect

Maximal subgroups

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

subgroup	index
#19145712	2
#19145172	2
#500×D₄₃	2
#19145049	2
#19134268	2
#19134264	2
#19134213	2
#19134212	2
#19134184	2
C₈×#10621872	2
C₄₃×#1706	2
#19630	43