Subgroups — #25251468

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	#19143195	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136800	C₂	1	2	yes	maximal, complemented
3	#19136790	C₂	1	2	yes	maximal, complemented
4	#19136783	C₂	1	2	yes	maximal, complemented
5	#19136778	C₂	1	2	yes	maximal, complemented
6	#19136381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582726	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136307	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136302	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136209	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3061	C₂	1	1	yes	maximal, characteristic
12	#14583382	—	2	1	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14582941	—	2	1	no	—
16	#14582941	—	2	2	no	—
17	#14582937	C₂²	1	2	yes	—
18	#14582937	C₂²	1	1	yes	characteristic
19	C₂×#13063259	C₂²	1	2	yes	—
20	C₂×#13063259	C₂²	1	1	yes	characteristic
21	#14582726	C₂²	1	2	yes	—
22	#14582726	C₂²	1	2	yes	—
23	#14582726	C₂²	1	2	yes	—
24	#14582724	C₂²	1	2	yes	—
25	#14582703	C₂²	1	1	yes	characteristic
26	#14582703	C₂²	1	2	yes	—
27	#14582703	C₂²	1	1	yes	characteristic
28	#14582700	C₂²	1	1	yes	characteristic
29	#14582700	C₂²	1	2	yes	—
30	#14582700	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
32	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
33	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#537	—	2	1	no	—
36	C₄₃×#468	—	2	2	no	—
37	C₄₃×#467	C₂²	1	2	yes	—
38	C₄₃×#466	C₂²	1	2	yes	—
39	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#417	—	2	1	no	—
41	Q₈×Dic₄₃	—	4	1	no	—
42	D₄×Dic₄₃	—	4	2	no	—
43	D₄×Dic₄₃	—	4	1	no	—
44	#13063284	—	2	2	no	—
45	#13063284	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	2	no	—
47	C₂×C₄⋊Dic₄₃	—	2	1	no	—
48	#13063265	—	2	2	no	—
49	C₂×C₄₃⋊C₁₆	—	2	2	no	—
50	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
53	#13063259	—	2	2	no	—
54	#13063259	—	2	1	no	—
55	#13063259	C₂³	1	1	yes	characteristic
56	#13063259	C₂³	1	2	yes	—
57	#13063259	C₂³	1	1	yes	characteristic
58	#13063257	—	2	2	no	—
59	#13063257	—	2	1	no	—
60	C₂×#41×C₄₃	—	2	1	no	—
61	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
62	#135×C₄₃	—	2	2	no	—
63	#135×C₄₃	—	2	2	no	—
64	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×SD₁₆×C₄₃	—	2	1	no	—
66	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
67	#131×C₄₃	—	2	2	no	—
68	#131×C₄₃	—	2	2	no	—
69	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
70	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
71	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²⋊Q₈×C₄₃	—	2	2	no	—
74	C₂²⋊Q₈×C₄₃	—	4	1	no	—
75	#121×C₄₃	—	2	2	no	—
76	#121×C₄₃	—	4	2	no	—
77	#121×C₄₃	—	4	1	no	—
78	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
79	#107×C₄₃	—	2	1	no	—
80	#107×C₄₃	C₂³	1	2	yes	—
81	#103×C₄₃	—	2	2	no	—
82	#102×C₄₃	—	2	2	no	—
83	#41×C₄₃	—	2	2	no	—
84	#41×C₄₃	—	2	2	no	—
85	#41×C₄₃	—	4	2	no	—
86	#41×C₄₃	—	4	2	no	—
87	C₂×Q₈×C₄₃	—	2	1	no	—
88	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
89	C₂×D₄×C₄₃	—	2	1	no	—
90	C₂×D₄×C₄₃	—	4	1	no	—
91	C₂×D₄×C₄₃	—	4	2	no	—
92	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
93	C₂×D₄×C₄₃	D₈	1	2	yes	—
94	C₂²×C₄×C₄₃	—	4	1	no	—
95	C₂²×C₄×C₄₃	—	4	1	no	—
96	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×Dic₄₃	—	4	2	no	—
99	C₂²×Dic₄₃	—	4	1	no	—
100	Q₁₆×C₄₃	—	2	1	no	—
101	Q₁₆×C₄₃	—	2	1	no	—
102	SD₁₆×C₄₃	—	4	2	no	—
103	D₈×C₄₃	—	2	1	no	—
104	D₈×C₄₃	—	2	1	no	—
105	M₁₆×C₄₃	—	2	2	no	—
106	M₁₆×C₄₃	—	2	4	no	—
107	C₂×C₈×C₄₃	—	2	2	no	—
108	C₂×C₈×C₄₃	—	2	2	no	—
109	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	#41	1	2	yes	—
112	C₂×C₈×C₄₃	#41	1	2	yes	—
113	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₄₃	—	2	2	no	—
115	C₄⋊C₄×C₄₃	—	4	1	no	—
116	C₄⋊C₄×C₄₃	—	4	2	no	—
117	C₂²⋊C₄×C₄₃	—	4	2	no	—
118	C₂²⋊C₄×C₄₃	—	4	2	no	—
119	C₂²⋊Dic₄₃	—	4	2	no	—
120	C₂²⋊Dic₄₃	—	4	1	no	—
121	C₄⋊Dic₄₃	—	2	1	no	—
122	C₄⋊Dic₄₃	—	2	2	no	—
123	C₄⋊Dic₄₃	—	2	1	no	—
124	C₄⋊Dic₄₃	—	4	2	no	—
125	C₄⋊Dic₄₃	—	4	1	no	—
126	C₄×Dic₄₃	—	4	2	no	—
127	C₄×Dic₄₃	—	4	1	no	—
128	C₄₃⋊C₁₆	—	2	1	no	—
129	C₄₃⋊C₁₆	—	2	1	no	—
130	C₄₃⋊C₁₆	—	2	2	no	—
131	C₂³×C₄₃	—	4	1	no	—
132	C₂³×C₄₃	C₂×D₈	1	2	yes	—
133	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
134	Q₈×C₄₃	—	2	2	no	—
135	Q₈×C₄₃	—	4	1	no	—
136	Q₈×C₄₃	—	4	1	no	—
137	D₄×C₄₃	—	2	4	no	—
138	D₄×C₄₃	—	2	2	no	—
139	D₄×C₄₃	—	4	1	no	—
140	D₄×C₄₃	—	4	1	no	—
141	D₄×C₄₃	—	8	2	no	—
142	D₄×C₄₃	—	16	2	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	2	2	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	—	4	1	no	—
147	C₂×C₄×C₄₃	—	8	2	no	—
148	C₂×C₄×C₄₃	—	16	1	no	—
149	C₂×C₄×C₄₃	#123	1	2	yes	—
150	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₄₃	#137	1	2	yes	—
153	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₄₃	—	4	1	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	1	no	—
157	C₂×Dic₄₃	—	16	1	no	—
158	C₂×Dic₄₃	—	16	2	no	—
159	C₈×C₄₃	—	2	2	no	—
160	C₈×C₄₃	#123	1	2	yes	—
161	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
163	#33653	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	2	no	—
167	C₂²×C₄₃	—	2	2	no	—
168	C₂²×C₄₃	—	2	2	no	—
169	C₂²×C₄₃	—	8	1	no	—
170	C₂²×C₄₃	—	8	1	no	—
171	C₂²×C₄₃	—	8	1	no	—
172	C₂²×C₄₃	#480	1	2	yes	—
173	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₄₃	—	2	2	no	—
175	C₄×C₄₃	—	8	1	no	—
176	C₄×C₄₃	—	8	2	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	—	8	1	no	—
181	Dic₄₃	—	8	2	no	—
182	Dic₄₃	—	8	1	no	—
183	#3061	D₄₃	1	1	yes	characteristic
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	—	43	1	no	—
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	—	4	2	no	—
196	C₂×C₄₃	—	16	1	no	—
197	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
198	C₂×C₄₃	#3416	1	1	yes	characteristic
199	C₂×C₄₃	#3416	1	1	yes	characteristic
200	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
201	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
202	#537	—	2	1	no	—
203	#520	—	86	1	no	—
204	C₂×M₃₂	—	43	1	no	—
205	C₂²×C₁₆	—	43	1	no	—
206	#468	—	2	2	no	—
207	#467	C₂×D₄₃	1	2	yes	—
208	#466	C₂×D₄₃	1	2	yes	—
209	#428	—	86	1	no	—
210	#428	—	86	2	no	—
211	#427	—	43	1	no	—
212	#427	—	43	2	no	—
213	C₂×#107	—	43	1	no	—
214	C₂×#107	—	43	2	no	—
215	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
216	#417	—	2	1	no	—
217	#367	—	43	2	no	—
218	#367	—	43	2	no	—
219	#367	—	43	2	no	—
220	#366	—	43	2	no	—
221	#358	—	43	1	no	—
222	#358	—	43	1	no	—
223	#358	—	43	2	no	—
224	#357	—	43	1	no	—
225	#357	—	43	1	no	—
226	#357	—	43	2	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	—	2	1	no	—
231	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
232	#135	—	2	2	no	—
233	#135	—	2	2	no	—
234	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
235	C₂×SD₁₆	—	2	1	no	—
236	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
237	#131	—	2	2	no	—
238	#131	—	2	2	no	—
239	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
240	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
241	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
242	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₂²⋊Q₈	—	2	2	no	—
244	C₂²⋊Q₈	—	4	1	no	—
245	#121	—	2	2	no	—
246	#121	—	4	1	no	—
247	#121	—	4	2	no	—
248	C₄×Q₈	—	172	1	no	—
249	C₄×D₄	—	172	2	no	—
250	C₄×D₄	—	172	1	no	—
251	#117	—	86	2	no	—
252	#117	—	86	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	86	2	no	—
255	C₂×C₄⋊C₄	—	86	1	no	—
256	M₃₂	—	86	2	no	—
257	C₂×C₁₆	—	43	2	no	—
258	C₂×C₁₆	—	43	1	no	—
259	C₂×C₁₆	—	43	1	no	—
260	C₂×C₁₆	—	86	2	no	—
261	#107	—	2	1	no	—
262	#107	—	43	1	no	—
263	#107	—	43	1	no	—
264	#107	—	43	2	no	—
265	#107	—	86	2	no	—
266	#107	—	86	1	no	—
267	#107	C₂²×D₄₃	1	2	yes	—
268	#106	—	86	2	no	—
269	#106	—	86	1	no	—
270	#103	—	2	2	no	—
271	#102	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	4	2	no	—
275	#41	—	4	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	#10621882	1	1	yes	characteristic
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	#10621882	1	2	yes	—
282	C₂×D₄	#10621882	1	1	yes	characteristic
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	172	2	no	—
286	C₂²×C₄	—	172	1	no	—
287	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
288	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
289	Q₁₆	—	2	1	no	—
290	Q₁₆	—	2	1	no	—
291	SD₁₆	—	4	2	no	—
292	D₈	—	2	1	no	—
293	D₈	—	2	1	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	#10621900	1	2	yes	—
299	C₂×C₈	#10621900	1	2	yes	—
300	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
301	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
302	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	86	1	no	—
309	C₄⋊C₄	—	172	2	no	—
310	C₄⋊C₄	—	172	1	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	172	2	no	—
314	C₂²⋊C₄	—	172	1	no	—
315	C₄²	—	172	2	no	—
316	C₄²	—	172	1	no	—
317	C₁₆	—	86	1	no	—
318	C₁₆	—	86	1	no	—
319	C₁₆	—	86	2	no	—
320	C₂³	—	4	1	no	—
321	C₂³	C₂×#10621882	1	2	yes	—
322	C₂³	C₂×#10621882	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	4	1	no	—
326	D₄	—	2	4	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	1	no	—
330	D₄	—	8	2	no	—
331	D₄	—	16	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	16	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	2	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	688	1	no	—
342	C₂×C₄	—	688	2	no	—
343	C₂×C₄	#13063306	1	2	yes	—
344	C₂×C₄	C₂×#10621882	1	2	yes	—
345	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
346	C₂×C₄	#13063355	1	2	yes	—
347	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	#13063306	1	2	yes	—
350	C₈	C₂×#10621900	1	1	yes	characteristic
351	C₈	C₂×#10621900	1	1	yes	characteristic
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	#14583210	1	2	yes	—
361	C₂²	C₂²×#10621882	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	2	no	—
365	C₄	—	344	2	no	—
366	C₄	—	344	1	no	—
367	C₄	—	344	1	no	—
368	C₄	#14583210	1	2	yes	—
369	C₄	C₂×#13063306	1	1	yes	characteristic
370	C₄	C₂×#13063355	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	4	2	no	—
373	C₂	—	16	1	no	—
374	C₂	C₂×#14583210	1	1	yes	characteristic
375	C₂	#19147739	1	1	yes	characteristic
376	C₂	#19147742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3061	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3061	D₄₃
hypocenter	C₄₃	#33653
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3061

Complements

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

N	H ≅ G/N	type
1. #19143195	C₂	semidirect
2. #19136800	C₂	semidirect
3. #19136790	C₂	semidirect
4. #19136783	C₂	semidirect
5. #19136778	C₂	semidirect
6. #19136381	C₂	semidirect
7. C₂×#14582726	C₂	semidirect
8. #19136307	C₂	semidirect
9. #19136302	C₂	semidirect
10. #19136209	C₂	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
#19143195	2
#19136800	2
#19136790	2
#19136783	2
#19136778	2
#19136381	2
C₂×#14582726	2
#19136307	2
#19136302	2
#19136209	2
C₄₃×#3061	2
#33653	43