Subgroups — #25251476

Order 11008.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144267	C₂	1	1	yes	maximal, characteristic
2	#19143192	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136805	C₂	1	2	yes	maximal, complemented
4	#19136794	C₂	1	2	yes	maximal, complemented
5	#19136789	C₂	1	2	yes	maximal, complemented
6	#19136782	C₂	1	2	yes	maximal, complemented
7	C₂×#14582726	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136308	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136300	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136210	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	C₂×#13063331	C₂²	1	1	yes	characteristic
14	#14583480	—	2	1	no	—
15	#14583379	—	2	1	no	—
16	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	#14583146	—	2	2	no	—
18	#14583141	C₂²	1	2	yes	—
19	#14583136	C₂²	1	2	yes	—
20	#14582942	—	2	1	no	—
21	#14582938	—	2	2	no	—
22	#14582936	C₂²	1	2	yes	—
23	C₂×#13063259	C₂²	1	1	yes	characteristic
24	#107×D₄₃	C₂²	1	2	yes	—
25	#14582898	—	2	1	no	—
26	#14582726	C₂²	1	2	yes	—
27	#14582726	C₂²	1	2	yes	—
28	#14582704	C₂²	1	2	yes	—
29	#14582702	C₂²	1	1	yes	characteristic
30	#14582702	C₂²	1	1	yes	characteristic
31	#14582701	C₂²	1	2	yes	—
32	#14582699	C₂²	1	1	yes	characteristic
33	#14582699	C₂²	1	1	yes	characteristic
34	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
35	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
36	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×#427	C₂²	1	1	yes	characteristic
38	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#367	C₂²	1	2	yes	—
40	C₄₃×#366	C₂²	1	2	yes	—
41	C₂×#10621898	—	2	1	no	—
42	C₂×#10621898	D₄	1	1	yes	characteristic
43	#13063347	—	2	2	no	—
44	#13063347	—	2	2	no	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	#13063331	—	2	2	no	—
49	#13063331	—	2	2	no	—
50	C₂×#10621877	D₄	1	1	yes	characteristic
51	C₂×#10621872	D₄	1	2	yes	—
52	C₂×C₈×D₄₃	C₂³	1	2	yes	—
53	#13063305	—	2	2	no	—
54	#13063305	—	4	1	no	—
55	#13063300	—	2	2	no	—
56	#13063300	—	4	1	no	—
57	#13063299	—	4	2	no	—
58	C₄×Dic₈₆	—	4	1	no	—
59	C₄×D₁₇₂	—	4	1	no	—
60	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
61	#13063283	—	2	2	no	—
62	C₂×C₄⋊Dic₄₃	—	2	1	no	—
63	C₄⋊C₄×D₄₃	—	2	2	no	—
64	C₂×C₄₃⋊C₁₆	—	2	2	no	—
65	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
67	#13063259	—	2	1	no	—
68	#13063259	C₂³	1	2	yes	—
69	#13063258	—	2	2	no	—
70	#13063256	—	2	2	no	—
71	#13063248	—	2	2	no	—
72	#13063245	—	2	2	no	—
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	#117×C₄₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
76	M₃₂×C₄₃	—	2	2	no	—
77	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
78	#107×C₄₃	—	2	1	no	—
79	#107×C₄₃	C₂³	1	2	yes	—
80	#107×C₄₃	C₂³	1	1	yes	characteristic
81	#107×C₄₃	C₂³	1	1	yes	characteristic
82	#106×C₄₃	—	2	1	no	—
83	C₂²×C₄×C₄₃	—	4	1	no	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₃	—	4	1	no	—
86	C₂×C₂²⋊D₄₃	D₈	1	2	yes	—
87	C₂²×Dic₄₃	—	4	1	no	—
88	#10621898	—	2	2	no	—
89	#10621898	—	2	2	no	—
90	#10621898	—	4	2	no	—
91	#10621898	—	4	2	no	—
92	C₂×D₁₇₂	—	2	1	no	—
93	C₂×D₁₇₂	—	4	2	no	—
94	C₂×D₁₇₂	D₈	1	1	yes	characteristic
95	C₂×C₄×D₄₃	—	4	1	no	—
96	C₂×C₄×D₄₃	—	4	2	no	—
97	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
98	C₂×Dic₈₆	—	2	1	no	—
99	C₂×Dic₈₆	D₈	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	2	yes	—
102	C₂×C₈×C₄₃	#41	1	2	yes	—
103	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₃	—	2	1	no	—
105	C₄⋊C₄×C₄₃	—	2	1	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	1	no	—
108	C₂²⋊C₄×C₄₃	—	4	1	no	—
109	C₄²×C₄₃	—	4	1	no	—
110	(C₂×D₄₃)⋊C₄	—	4	2	no	—
111	(C₂×D₄₃)⋊C₄	—	4	2	no	—
112	(C₂×D₄₃)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₄₃	—	2	2	no	—
114	C₄⋊Dic₄₃	—	4	1	no	—
115	C₄⋊Dic₄₃	—	4	2	no	—
116	Dic₄₃⋊C₄	—	4	2	no	—
117	C₄×Dic₄₃	—	4	2	no	—
118	#10621877	—	2	2	no	—
119	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
120	Dic₁₇₂	—	2	1	no	—
121	Dic₁₇₂	—	2	1	no	—
122	D₃₄₄	—	2	1	no	—
123	D₃₄₄	—	2	1	no	—
124	#10621873	—	4	2	no	—
125	#10621872	—	2	4	no	—
126	C₈×D₄₃	—	2	2	no	—
127	C₈×D₄₃	—	2	2	no	—
128	C₁₆×C₄₃	—	2	2	no	—
129	C₄₃⋊C₁₆	—	2	1	no	—
130	C₄₃⋊C₁₆	—	2	1	no	—
131	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
132	C₂²×D₄₃	—	4	1	no	—
133	C₂²×D₄₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	16	1	no	—
138	C₂×C₄×C₄₃	#123	1	2	yes	—
139	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#137	1	2	yes	—
141	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₄₃	—	2	4	no	—
143	C₂²⋊D₄₃	—	8	2	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	4	1	no	—
146	C₂×Dic₄₃	—	16	1	no	—
147	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
148	D₁₇₂	—	2	2	no	—
149	D₁₇₂	—	4	1	no	—
150	D₁₇₂	—	4	1	no	—
151	D₁₇₂	—	16	2	no	—
152	C₄×D₄₃	—	2	2	no	—
153	C₄×D₄₃	—	2	2	no	—
154	C₄×D₄₃	—	8	2	no	—
155	C₄×D₄₃	—	16	2	no	—
156	Dic₈₆	—	2	2	no	—
157	Dic₈₆	—	4	1	no	—
158	Dic₈₆	—	4	1	no	—
159	C₈×C₄₃	#123	1	2	yes	—
160	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₄₃⋊C₈	—	2	2	no	—
163	#33653	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	#480	1	2	yes	—
167	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₄₃	—	2	2	no	—
169	C₂×D₄₃	—	2	2	no	—
170	C₂×D₄₃	—	2	2	no	—
171	C₂×D₄₃	—	8	1	no	—
172	C₂×D₄₃	—	8	1	no	—
173	C₂×D₄₃	—	8	1	no	—
174	C₄×C₄₃	—	8	1	no	—
175	C₄×C₄₃	—	8	2	no	—
176	C₄×C₄₃	—	8	1	no	—
177	C₄×C₄₃	#480	1	2	yes	—
178	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
179	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
180	Dic₄₃	—	2	2	no	—
181	Dic₄₃	—	8	1	no	—
182	Dic₄₃	—	8	2	no	—
183	#3061	—	43	1	no	—
184	#2913	—	43	1	no	—
185	#2237	—	43	2	no	—
186	#2235	—	43	2	no	—
187	#2234	—	43	2	no	—
188	#2233	—	43	2	no	—
189	#2159	D₄₃	1	1	yes	characteristic, complemented
190	C₂×#367	—	43	1	no	—
191	#2139	—	43	1	no	—
192	#2138	—	43	1	no	—
193	#2113	—	43	1	no	—
194	C₂×C₄₃	—	2	2	no	—
195	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
196	C₂×C₄₃	#3416	1	1	yes	characteristic
197	C₂×C₄₃	#3416	1	1	yes	characteristic
198	D₄₃	—	4	2	no	—
199	D₄₃	—	16	1	no	—
200	C₂×#135	—	43	1	no	—
201	C₂×#131	—	43	1	no	—
202	#537	—	86	1	no	—
203	#520	—	86	1	no	—
204	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
205	C₂²×C₁₆	—	43	1	no	—
206	#468	—	86	2	no	—
207	#467	—	43	2	no	—
208	#466	—	43	2	no	—
209	#428	—	86	2	no	—
210	#428	—	86	1	no	—
211	#427	—	43	2	no	—
212	#427	C₂×D₄₃	1	1	yes	characteristic
213	C₂×#107	—	43	1	no	—
214	C₂×#107	—	43	2	no	—
215	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
216	#417	—	86	1	no	—
217	#367	—	43	2	no	—
218	#367	—	43	2	no	—
219	#367	C₂×D₄₃	1	2	yes	—
220	#366	C₂×D₄₃	1	2	yes	—
221	#358	—	43	2	no	—
222	#358	—	43	1	no	—
223	#358	—	43	1	no	—
224	#357	—	43	2	no	—
225	#357	—	43	1	no	—
226	#357	—	43	1	no	—
227	C₂²⋊C₁₆	—	43	2	no	—
228	C₂²⋊C₁₆	—	43	2	no	—
229	C₄₃	#33653	1	1	yes	characteristic, complemented, Sylow
230	C₂×#41	—	43	1	no	—
231	C₂×#41	—	86	1	no	—
232	#135	—	86	2	no	—
233	#135	—	86	2	no	—
234	C₂×Q₁₆	—	43	1	no	—
235	C₂×SD₁₆	—	86	1	no	—
236	C₂×D₈	—	43	1	no	—
237	#131	—	86	2	no	—
238	#131	—	86	2	no	—
239	C₂×M₁₆	—	43	1	no	—
240	C₂×M₁₆	—	43	2	no	—
241	C₂²×C₈	—	43	2	no	—
242	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₂²⋊Q₈	—	86	2	no	—
244	C₂²⋊Q₈	—	172	1	no	—
245	#121	—	86	2	no	—
246	#121	—	172	1	no	—
247	#121	—	172	2	no	—
248	C₄×Q₈	—	172	1	no	—
249	C₄×D₄	—	172	1	no	—
250	C₄×D₄	—	172	2	no	—
251	#117	—	2	1	no	—
252	#117	—	86	2	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	86	2	no	—
255	C₂×C₄⋊C₄	—	86	1	no	—
256	M₃₂	—	2	2	no	—
257	C₂×C₁₆	—	43	1	no	—
258	C₂×C₁₆	—	43	1	no	—
259	C₂×C₁₆	—	86	2	no	—
260	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
261	#107	—	2	1	no	—
262	#107	—	43	2	no	—
263	#107	—	86	2	no	—
264	#107	—	86	1	no	—
265	#107	C₂²×D₄₃	1	1	yes	characteristic
266	#107	C₂²×D₄₃	1	1	yes	characteristic
267	#107	C₂²×D₄₃	1	2	yes	—
268	#106	—	2	1	no	—
269	#106	—	86	2	no	—
270	#103	—	86	2	no	—
271	#102	—	86	2	no	—
272	#41	—	86	2	no	—
273	#41	—	86	2	no	—
274	#41	—	172	2	no	—
275	#41	—	172	2	no	—
276	C₂×Q₈	—	43	1	no	—
277	C₂×Q₈	—	86	1	no	—
278	C₂×D₄	—	43	2	no	—
279	C₂×D₄	—	43	1	no	—
280	C₂×D₄	—	86	1	no	—
281	C₂×D₄	—	172	2	no	—
282	C₂×D₄	—	172	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	43	2	no	—
285	C₂²×C₄	—	172	2	no	—
286	C₂²×C₄	—	172	1	no	—
287	C₂²×C₄	—	172	1	no	—
288	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
289	Q₁₆	—	86	1	no	—
290	Q₁₆	—	86	1	no	—
291	SD₁₆	—	172	2	no	—
292	D₈	—	86	1	no	—
293	D₈	—	86	1	no	—
294	M₁₆	—	86	2	no	—
295	M₁₆	—	86	4	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	86	2	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	#10621898	1	2	yes	—
300	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
301	C₂×C₈	#10621902	1	2	yes	—
302	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	172	2	no	—
309	C₄⋊C₄	—	172	2	no	—
310	C₄⋊C₄	—	172	1	no	—
311	C₂²⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	172	2	no	—
313	C₂²⋊C₄	—	172	2	no	—
314	C₂²⋊C₄	—	172	2	no	—
315	C₄²	—	4	1	no	—
316	C₄²	—	172	2	no	—
317	C₁₆	—	2	2	no	—
318	C₁₆	—	86	1	no	—
319	C₁₆	—	86	1	no	—
320	C₂³	—	43	2	no	—
321	C₂³	—	172	1	no	—
322	C₂³	D₈×D₄₃	1	1	yes	characteristic
323	Q₈	—	86	2	no	—
324	Q₈	—	172	1	no	—
325	Q₈	—	172	1	no	—
326	D₄	—	86	4	no	—
327	D₄	—	86	2	no	—
328	D₄	—	344	2	no	—
329	D₄	—	172	1	no	—
330	D₄	—	172	1	no	—
331	D₄	—	688	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	16	1	no	—
336	C₂×C₄	—	43	2	no	—
337	C₂×C₄	—	86	2	no	—
338	C₂×C₄	—	86	2	no	—
339	C₂×C₄	—	344	2	no	—
340	C₂×C₄	—	172	2	no	—
341	C₂×C₄	—	172	1	no	—
342	C₂×C₄	—	688	2	no	—
343	C₂×C₄	—	688	1	no	—
344	C₂×C₄	#13063310	1	2	yes	—
345	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
346	C₂×C₄	#13063357	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
348	C₈	—	86	2	no	—
349	C₈	#13063310	1	2	yes	—
350	C₈	C₂×#10621898	1	1	yes	characteristic
351	C₈	C₂×#10621902	1	1	yes	characteristic
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	86	2	no	—
355	C₂²	—	86	2	no	—
356	C₂²	—	86	2	no	—
357	C₂²	—	344	1	no	—
358	C₂²	—	344	1	no	—
359	C₂²	—	344	1	no	—
360	C₂²	#14583214	1	2	yes	—
361	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	86	2	no	—
366	C₄	—	344	1	no	—
367	C₄	—	344	2	no	—
368	C₄	#14583214	1	2	yes	—
369	C₄	C₂×#13063310	1	1	yes	characteristic
370	C₄	C₂×#13063357	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	172	2	no	—
373	C₂	—	688	1	no	—
374	C₂	C₂×#14583214	1	1	yes	characteristic
375	C₂	#19147740	1	1	yes	characteristic
376	C₂	#19147741	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×#2159	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19144267	C₂
hypercenter	#2159	D₄₃
hypocenter	C₄₃	#33653
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
2. #19143192	C₂	semidirect
3. #19136805	C₂	semidirect
4. #19136794	C₂	semidirect
5. #19136789	C₂	semidirect
6. #19136782	C₂	semidirect
7. C₂×#14582726	C₂	semidirect
8. #19136308	C₂	semidirect
9. #19136300	C₂	semidirect
10. #19136210	C₂	semidirect
11. C₄₃×#2159	C₂	semidirect
189. #2159	D₄₃	semidirect
229. C₄₃	#33653	semidirect

Maximal subgroups

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

subgroup	index
#19144267	2
#19143192	2
#19136805	2
#19136794	2
#19136789	2
#19136782	2
C₂×#14582726	2
#19136308	2
#19136300	2
#19136210	2
C₄₃×#2159	2
#33653	43