Subgroups — #25155496

Order 11008.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145123	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144461	C₂	1	2	yes	maximal
3	#19143190	C₂	1	1	yes	maximal, characteristic
4	#19136109	C₂	1	2	yes	maximal, complemented
5	#19135941	C₂	1	2	yes	maximal, complemented
6	#19135376	C₂	1	1	yes	maximal, characteristic, complemented
7	#19135341	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134819	C₂	1	2	yes	maximal
9	#19134750	C₂	1	1	yes	maximal, characteristic, complemented
10	#19134430	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1815	C₂	1	1	yes	maximal, characteristic
12	#14583530	C₂²	1	2	yes	—
13	#14583382	C₂²	1	2	yes	—
14	#14583306	—	2	2	no	—
15	#14583299	—	2	1	no	—
16	#14583296	—	2	1	no	—
17	#14583208	—	2	2	no	—
18	#14583197	—	2	2	no	—
19	#14583187	—	2	2	no	—
20	#14583182	—	2	2	no	—
21	#14582941	—	2	2	no	—
22	#14582941	C₂²	1	2	yes	—
23	#14582941	C₂²	1	2	yes	—
24	#14582939	C₂²	1	2	yes	—
25	#14582937	—	2	1	no	—
26	#14582937	C₂²	1	1	yes	characteristic
27	C₂×#13063257	—	2	1	no	—
28	C₂×#13063257	C₂²	1	1	yes	characteristic
29	C₂×#13063256	C₂²	1	2	yes	—
30	#14582927	C₂²	1	2	yes	—
31	C₂×#13063252	C₂²	1	1	yes	characteristic
32	#14582895	C₂²	1	1	yes	characteristic
33	#14582854	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
35	#14582768	C₂²	1	2	yes	—
36	#14582667	C₂²	1	2	yes	—
37	#14582663	C₂²	1	1	yes	characteristic
38	#14582660	C₂²	1	2	yes	—
39	#14582660	C₂²	1	1	yes	characteristic
40	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
43	C₄₃×#417	C₂²	1	2	yes	—
44	C₄₃×#343	C₂²	1	2	yes	—
45	C₂×#10621877	C₂³	1	1	yes	characteristic
46	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
47	#13063324	—	2	2	no	—
48	#13063324	—	2	2	no	—
49	#13063316	—	2	2	no	—
50	#13063316	—	2	2	no	—
51	#13063304	—	4	2	no	—
52	Q₈×Dic₄₃	—	2	2	no	—
53	D₄×Dic₄₃	—	2	2	no	—
54	D₄×Dic₄₃	—	4	2	no	—
55	#13063284	—	2	2	no	—
56	#13063284	C₂³	1	1	yes	characteristic
57	#13063284	C₂³	1	2	yes	—
58	C₂×C₄⋊Dic₄₃	—	2	2	no	—
59	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
60	C₂×Dic₄₃⋊C₄	—	4	2	no	—
61	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
62	#13063259	—	2	2	no	—
63	#13063259	—	2	2	no	—
64	#13063259	D₄	1	2	yes	—
65	#13063258	—	2	2	no	—
66	#13063257	—	2	1	no	—
67	#13063257	—	2	1	no	—
68	#13063257	—	2	2	no	—
69	#13063257	—	2	2	no	—
70	#13063257	D₄	1	1	yes	characteristic
71	#13063257	D₄	1	1	yes	characteristic
72	#13063256	—	2	2	no	—
73	#13063256	—	2	2	no	—
74	#13063256	—	2	2	no	—
75	Dic₄₃⋊C₈	—	2	2	no	—
76	Dic₄₃⋊C₈	—	2	2	no	—
77	#13063252	Q₈	1	4	yes	complemented
78	#13063249	D₄	1	2	yes	—
79	#13063246	D₄	1	2	yes	—
80	#13063235	—	2	1	no	—
81	C₈×Dic₄₃	—	2	1	no	—
82	C₈×Dic₄₃	—	2	2	no	—
83	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
84	#131×C₄₃	—	2	2	no	—
85	#131×C₄₃	Q₈	1	4	yes	complemented
86	C₂×M₁₆×C₄₃	—	2	1	no	—
87	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
88	C₂×M₁₆×C₄₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₄₃	—	2	1	no	—
90	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
92	#108×C₄₃	—	2	2	no	—
93	#106×C₄₃	—	2	1	no	—
94	#106×C₄₃	—	2	1	no	—
95	#103×C₄₃	—	2	2	no	—
96	#102×C₄₃	—	2	2	no	—
97	#41×C₄₃	—	2	1	no	—
98	#41×C₄₃	—	2	1	no	—
99	#41×C₄₃	C₂×Q₈	1	2	yes	—
100	#41×C₄₃	C₂×Q₈	1	2	yes	—
101	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₄₃	—	2	1	no	—
103	C₂×D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₄₃	—	2	1	no	—
105	C₂²×C₄×C₄₃	—	2	2	no	—
106	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₂²×Dic₄₃	—	2	1	no	—
108	C₂²×Dic₄₃	—	2	2	no	—
109	C₂²×Dic₄₃	—	4	2	no	—
110	C₂×Dic₈₆	—	4	2	no	—
111	M₁₆×C₄₃	—	2	1	no	—
112	M₁₆×C₄₃	—	2	4	no	—
113	M₁₆×C₄₃	—	2	2	no	—
114	M₁₆×C₄₃	—	2	2	no	—
115	M₁₆×C₄₃	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₄₃	—	2	1	no	—
117	C₂×C₈×C₄₃	—	2	4	no	—
118	C₂×C₈×C₄₃	—	2	2	no	—
119	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
122	C₂×C₈×C₄₃	#41	1	2	yes	—
123	C₄⋊C₄×C₄₃	—	2	2	no	—
124	C₄⋊C₄×C₄₃	—	2	2	no	—
125	C₂²⋊Dic₄₃	—	2	2	no	—
126	C₂²⋊Dic₄₃	—	2	1	no	—
127	C₂²⋊Dic₄₃	—	4	2	no	—
128	C₄⋊Dic₄₃	—	2	2	no	—
129	C₄⋊Dic₄₃	—	2	2	no	—
130	C₄⋊Dic₄₃	—	4	2	no	—
131	C₄⋊Dic₄₃	—	4	2	no	—
132	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
133	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
134	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
135	Dic₄₃⋊C₄	—	4	2	no	—
136	Dic₄₃⋊C₄	—	4	2	no	—
137	Dic₄₃⋊C₄	—	8	4	no	—
138	C₄×Dic₄₃	—	2	1	no	—
139	C₄×Dic₄₃	—	2	2	no	—
140	C₄×Dic₄₃	—	2	1	no	—
141	C₄×Dic₄₃	—	4	2	no	—
142	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
143	#10621877	—	2	1	no	—
144	#10621877	C₂×Q₈	1	2	yes	—
145	C₂×C₄₃⋊C₈	—	2	1	no	—
146	C₂×C₄₃⋊C₈	—	2	1	no	—
147	C₂×C₄₃⋊C₈	—	2	1	no	—
148	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
149	C₂³×C₄₃	—	2	1	no	—
150	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
151	Q₈×C₄₃	—	2	1	no	—
152	Q₈×C₄₃	C₄⋊Q₈	1	2	yes	—
153	D₄×C₄₃	—	2	2	no	—
154	D₄×C₄₃	—	2	2	no	—
155	D₄×C₄₃	—	2	1	no	—
156	D₄×C₄₃	C₄⋊Q₈	1	2	yes	—
157	C₂×C₄×C₄₃	—	2	1	no	—
158	C₂×C₄×C₄₃	—	2	2	no	—
159	C₂×C₄×C₄₃	—	2	2	no	—
160	C₂×C₄×C₄₃	—	4	2	no	—
161	C₂×C₄×C₄₃	—	8	2	no	—
162	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×Dic₄₃	—	2	4	no	—
169	C₂×Dic₄₃	—	2	2	no	—
170	C₂×Dic₄₃	—	2	2	no	—
171	C₂×Dic₄₃	—	2	2	no	—
172	C₂×Dic₄₃	—	2	2	no	—
173	C₂×Dic₄₃	—	4	2	no	—
174	C₂×Dic₄₃	—	8	2	no	—
175	C₂×Dic₄₃	—	16	2	no	—
176	Dic₈₆	—	16	2	no	—
177	C₈×C₄₃	—	2	2	no	—
178	C₈×C₄₃	—	4	2	no	—
179	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
180	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
181	C₄₃⋊C₈	—	2	2	no	—
182	C₄₃⋊C₈	—	2	2	no	—
183	#26081	—	43	1	no	maximal, Sylow
184	C₂²×C₄₃	—	2	2	no	—
185	C₂²×C₄₃	—	2	2	no	—
186	C₂²×C₄₃	—	4	2	no	—
187	C₂²×C₄₃	#544	1	1	yes	characteristic
188	C₂²×C₄₃	D₄²	1	1	yes	characteristic
189	C₂²×C₄₃	#554	1	1	yes	characteristic
190	C₄×C₄₃	—	2	2	no	—
191	C₄×C₄₃	—	8	2	no	—
192	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
193	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
194	C₄×C₄₃	#547	1	1	yes	characteristic
195	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
196	Dic₄₃	—	2	4	no	—
197	Dic₄₃	—	4	2	no	—
198	Dic₄₃	—	4	2	no	—
199	Dic₄₃	—	4	2	no	—
200	Dic₄₃	—	8	2	no	—
201	#3163	—	43	1	no	—
202	#3084	—	43	2	no	—
203	#2912	—	43	1	no	—
204	#2090	—	43	2	no	—
205	#2060	—	43	2	no	—
206	#1951	—	43	1	no	—
207	#1945	—	43	1	no	—
208	#1859	—	43	2	no	—
209	#1844	—	43	1	no	—
210	#1815	D₄₃	1	1	yes	characteristic
211	#1764	—	43	1	no	—
212	C₂×C₄₃	—	2	1	no	—
213	C₂×C₄₃	—	2	1	no	—
214	C₂×C₄₃	—	4	2	no	—
215	C₂×C₄₃	#2440	1	1	yes	characteristic
216	C₂×C₄₃	#3291	1	2	yes	—
217	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
218	#544	—	43	2	no	—
219	#520	—	43	2	no	—
220	#501	—	86	2	no	—
221	#499	—	86	1	no	—
222	#498	—	86	1	no	—
223	#479	—	86	2	no	—
224	#478	—	86	2	no	—
225	#476	—	86	2	no	—
226	#475	—	86	2	no	—
227	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
228	#428	—	43	2	no	—
229	#428	—	43	2	no	—
230	#428	—	43	2	no	—
231	#428	—	86	2	no	—
232	#427	—	43	1	no	—
233	#427	—	86	1	no	—
234	C₂×#106	—	43	1	no	—
235	C₂×#106	—	43	2	no	—
236	C₂×#106	—	86	1	no	—
237	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
238	#424	—	43	2	no	—
239	C₂×#104	—	43	1	no	—
240	#417	C₂×D₄₃	1	2	yes	—
241	#416	—	43	1	no	—
242	#405	—	43	1	no	—
243	C₂×C₄×C₈	—	43	1	no	—
244	#384	—	43	2	no	—
245	#343	—	43	2	no	—
246	#343	C₂×D₄₃	1	2	yes	—
247	#341	—	43	1	no	—
248	#339	—	43	1	no	—
249	#339	—	43	2	no	—
250	C₄₃	#26081	1	1	yes	characteristic, complemented, Sylow
251	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
252	#131	—	2	2	no	—
253	#131	Dic₈₆	1	4	yes	complemented
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	43	1	no	—
256	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
257	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
258	C₂²×C₈	—	2	1	no	—
259	C₂²×C₈	—	43	1	no	—
260	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
261	C₄⋊Q₈	—	86	2	no	—
262	C₄⋊Q₈	—	86	2	no	—
263	#125	—	86	2	no	—
264	#125	—	86	2	no	—
265	C₂²⋊Q₈	—	172	2	no	—
266	C₄×Q₈	—	86	2	no	—
267	C₄×D₄	—	86	2	no	—
268	C₄×D₄	—	172	2	no	—
269	#117	—	43	2	no	—
270	#117	—	43	1	no	—
271	#117	—	86	2	no	—
272	C₂×C₄⋊C₄	—	43	2	no	—
273	C₂×C₄⋊C₄	—	86	2	no	—
274	C₂×C₄⋊C₄	—	172	2	no	—
275	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
276	C₂×C₄²	—	43	1	no	—
277	#108	—	2	2	no	—
278	#107	—	43	2	no	—
279	#107	—	86	2	no	—
280	#107	—	86	2	no	—
281	#107	—	86	2	no	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	43	1	no	—
285	#106	—	43	1	no	—
286	#106	—	86	2	no	—
287	#106	—	86	2	no	—
288	#106	—	86	2	no	—
289	#106	—	86	2	no	—
290	#106	—	86	2	no	—
291	#106	—	86	1	no	—
292	#106	—	86	1	no	—
293	C₄⋊C₈	—	86	2	no	—
294	C₄⋊C₈	—	86	2	no	—
295	#104	—	43	4	no	—
296	#103	—	2	2	no	—
297	#103	—	43	2	no	—
298	#102	—	2	2	no	—
299	#102	—	43	2	no	—
300	#97	—	86	1	no	—
301	C₄×C₈	—	86	1	no	—
302	C₄×C₈	—	86	2	no	—
303	#41	—	2	1	no	—
304	#41	—	2	1	no	—
305	#41	C₂×Dic₈₆	1	2	yes	—
306	#41	Q₈×D₄₃	1	2	yes	—
307	C₂×Q₈	—	172	2	no	—
308	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	D₄×D₄₃	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	86	1	no	—
314	C₂²×C₄	—	86	2	no	—
315	C₂²×C₄	—	172	2	no	—
316	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
317	M₁₆	—	2	1	no	—
318	M₁₆	—	2	4	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	43	2	no	—
322	M₁₆	—	86	1	no	—
323	M₁₆	C₂×Dic₈₆	1	2	yes	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	43	2	no	—
328	C₂×C₈	—	86	1	no	—
329	C₂×C₈	—	86	1	no	—
330	C₂×C₈	—	86	1	no	—
331	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
332	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
334	C₂×C₈	#10621900	1	2	yes	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	43	2	no	—
338	C₄⋊C₄	—	43	2	no	—
339	C₄⋊C₄	—	43	2	no	—
340	C₄⋊C₄	—	86	2	no	—
341	C₄⋊C₄	—	86	2	no	—
342	C₄⋊C₄	—	344	4	no	—
343	C₄⋊C₄	—	172	2	no	—
344	C₄⋊C₄	—	172	2	no	—
345	C₄⋊C₄	—	172	2	no	—
346	C₄⋊C₄	—	172	2	no	—
347	C₂²⋊C₄	—	86	2	no	—
348	C₂²⋊C₄	—	86	1	no	—
349	C₂²⋊C₄	—	172	2	no	—
350	C₄²	—	43	2	no	—
351	C₄²	—	86	1	no	—
352	C₄²	—	86	2	no	—
353	C₄²	—	86	1	no	—
354	C₄²	—	172	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	#13063371	1	1	yes	characteristic
357	Q₈	—	2	1	no	—
358	Q₈	—	688	2	no	—
359	Q₈	#13063324	1	2	yes	—
360	D₄	—	2	2	no	—
361	D₄	—	2	2	no	—
362	D₄	—	2	1	no	—
363	D₄	#13063324	1	2	yes	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	86	4	no	—
370	C₂×C₄	—	86	2	no	—
371	C₂×C₄	—	86	2	no	—
372	C₂×C₄	—	86	2	no	—
373	C₂×C₄	—	86	2	no	—
374	C₂×C₄	—	344	2	no	—
375	C₂×C₄	—	172	2	no	—
376	C₂×C₄	—	688	2	no	—
377	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
379	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
381	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
382	C₂×C₄	#13063372	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	86	2	no	—
386	C₈	—	86	2	no	—
387	C₈	#13063302	1	2	yes	—
388	C₈	#13063302	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	#14583527	1	1	yes	characteristic
393	C₂²	#14583533	1	1	yes	characteristic
394	C₂²	#14583607	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	8	2	no	—
397	C₄	—	86	4	no	—
398	C₄	—	344	2	no	—
399	C₄	—	172	2	no	—
400	C₄	—	172	2	no	—
401	C₄	—	172	2	no	—
402	C₄	C₂×#13063302	1	1	yes	characteristic
403	C₄	C₂×#13063324	1	1	yes	characteristic
404	C₄	#14583549	1	1	yes	characteristic
405	C₄	#14583562	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	4	2	no	—
409	C₂	#19138298	1	1	yes	characteristic
410	C₂	#19146387	1	2	yes	—

	subgroup	G/H
center	C₂²	#14583533
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1815	C₂
socle	C₂²×C₄₃	D₄²
prime radical	C₂×#41×C₄₃	C₂³
hypercenter	#1815	D₄₃
hypocenter	C₄₃	#26081
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1815

Complements

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

N	H ≅ G/N	type
1. #19145123	C₂	semidirect
4. #19136109	C₂	semidirect
5. #19135941	C₂	semidirect
6. #19135376	C₂	semidirect
7. #19135341	C₂	semidirect
9. #19134750	C₂	semidirect
77. #13063252	Q₈	semidirect
85. #131×C₄₃	Q₈	semidirect
250. C₄₃	#26081	semidirect
253. #131	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145123	2
#19144461	2
#19143190	2
#19136109	2
#19135941	2
#19135376	2
#19135341	2
#19134819	2
#19134750	2
#19134430	2
C₄₃×#1815	2
#26081	43