Subgroups — #25155485

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 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	#19144450	C₂	1	2	yes	maximal
3	#19143190	C₂	1	1	yes	maximal, characteristic
4	#19136104	C₂	1	2	yes	maximal, complemented
5	#19135946	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	#19134826	C₂	1	2	yes	maximal
9	#19134756	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	#14583204	—	2	2	no	—
18	#14583201	—	2	2	no	—
19	#14583192	—	2	2	no	—
20	#14583177	—	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×#13063258	C₂²	1	2	yes	—
28	C₂×#13063257	—	2	1	no	—
29	C₂×#13063257	C₂²	1	1	yes	characteristic
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	#14582764	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×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
61	#13063259	—	2	2	no	—
62	#13063259	—	2	2	no	—
63	#13063259	D₄	1	2	yes	—
64	#13063258	—	2	2	no	—
65	#13063258	—	2	2	no	—
66	#13063258	—	2	2	no	—
67	#13063257	—	2	1	no	—
68	#13063257	—	2	1	no	—
69	#13063257	—	2	2	no	—
70	#13063257	—	2	2	no	—
71	#13063257	D₄	1	1	yes	characteristic
72	#13063257	D₄	1	1	yes	characteristic
73	#13063256	—	2	2	no	—
74	Dic₄₃⋊C₈	—	2	2	no	—
75	Dic₄₃⋊C₈	—	2	2	no	—
76	#13063252	Q₈	1	4	yes	complemented
77	#13063249	D₄	1	2	yes	—
78	#13063246	D₄	1	2	yes	—
79	#13063235	—	2	1	no	—
80	C₈×Dic₄₃	—	2	1	no	—
81	C₈×Dic₄₃	—	2	2	no	—
82	C₄₃⋊#95	—	4	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	C₄×Dic₄₃	—	2	1	no	—
138	C₄×Dic₄₃	—	2	2	no	—
139	C₄×Dic₄₃	—	2	1	no	—
140	C₄×Dic₄₃	—	4	2	no	—
141	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
142	#10621877	—	2	1	no	—
143	#10621877	C₂×Q₈	1	2	yes	—
144	C₂×C₄₃⋊C₈	—	2	1	no	—
145	C₂×C₄₃⋊C₈	—	2	1	no	—
146	C₂×C₄₃⋊C₈	—	2	1	no	—
147	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
148	C₂³×C₄₃	—	2	1	no	—
149	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
150	Q₈×C₄₃	—	2	1	no	—
151	Q₈×C₄₃	C₄⋊Q₈	1	2	yes	—
152	D₄×C₄₃	—	2	2	no	—
153	D₄×C₄₃	—	2	2	no	—
154	D₄×C₄₃	—	2	1	no	—
155	D₄×C₄₃	C₄⋊Q₈	1	2	yes	—
156	C₂×C₄×C₄₃	—	2	1	no	—
157	C₂×C₄×C₄₃	—	2	2	no	—
158	C₂×C₄×C₄₃	—	2	2	no	—
159	C₂×C₄×C₄₃	—	4	2	no	—
160	C₂×C₄×C₄₃	—	8	2	no	—
161	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×Dic₄₃	—	2	2	no	—
168	C₂×Dic₄₃	—	2	2	no	—
169	C₂×Dic₄₃	—	2	2	no	—
170	C₂×Dic₄₃	—	2	2	no	—
171	C₂×Dic₄₃	—	4	2	no	—
172	C₂×Dic₄₃	—	4	2	no	—
173	C₂×Dic₄₃	—	8	2	no	—
174	C₂×Dic₄₃	—	16	2	no	—
175	Dic₈₆	—	16	2	no	—
176	C₈×C₄₃	—	2	2	no	—
177	C₈×C₄₃	—	4	2	no	—
178	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
179	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
180	C₄₃⋊C₈	—	2	2	no	—
181	C₄₃⋊C₈	—	2	2	no	—
182	#26080	—	43	1	no	maximal, Sylow
183	C₂²×C₄₃	—	2	2	no	—
184	C₂²×C₄₃	—	2	2	no	—
185	C₂²×C₄₃	—	4	2	no	—
186	C₂²×C₄₃	#544	1	1	yes	characteristic
187	C₂²×C₄₃	D₄²	1	1	yes	characteristic
188	C₂²×C₄₃	#554	1	1	yes	characteristic
189	C₄×C₄₃	—	2	2	no	—
190	C₄×C₄₃	—	8	2	no	—
191	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
193	C₄×C₄₃	#547	1	1	yes	characteristic
194	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
195	Dic₄₃	—	4	2	no	—
196	Dic₄₃	—	4	2	no	—
197	Dic₄₃	—	4	2	no	—
198	Dic₄₃	—	4	2	no	—
199	Dic₄₃	—	8	2	no	—
200	#3163	—	43	1	no	—
201	#3083	—	43	2	no	—
202	#2912	—	43	1	no	—
203	#2089	—	43	2	no	—
204	#2061	—	43	2	no	—
205	#1951	—	43	1	no	—
206	#1945	—	43	1	no	—
207	#1860	—	43	2	no	—
208	#1846	—	43	1	no	—
209	#1815	D₄₃	1	1	yes	characteristic
210	#1764	—	43	1	no	—
211	C₂×C₄₃	—	2	1	no	—
212	C₂×C₄₃	—	2	1	no	—
213	C₂×C₄₃	—	4	2	no	—
214	C₂×C₄₃	#2440	1	1	yes	characteristic
215	C₂×C₄₃	#3290	1	2	yes	—
216	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
217	#544	—	43	2	no	—
218	#520	—	43	2	no	—
219	#501	—	86	2	no	—
220	#499	—	86	1	no	—
221	#498	—	86	1	no	—
222	#479	—	86	2	no	—
223	#478	—	86	2	no	—
224	#477	—	86	2	no	—
225	#474	—	86	2	no	—
226	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
227	#428	—	43	2	no	—
228	#428	—	43	2	no	—
229	#428	—	43	2	no	—
230	#428	—	86	2	no	—
231	#427	—	43	1	no	—
232	#427	—	86	1	no	—
233	C₂×#107	—	43	2	no	—
234	C₂×#106	—	43	1	no	—
235	C₂×#106	—	86	1	no	—
236	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
237	#424	—	43	2	no	—
238	C₂×#104	—	43	1	no	—
239	#417	C₂×D₄₃	1	2	yes	—
240	#416	—	43	1	no	—
241	#405	—	43	1	no	—
242	C₂×C₄×C₈	—	43	1	no	—
243	#382	—	43	2	no	—
244	#343	—	43	2	no	—
245	#343	C₂×D₄₃	1	2	yes	—
246	#341	—	43	1	no	—
247	#339	—	43	1	no	—
248	#339	—	43	2	no	—
249	C₄₃	#26080	1	1	yes	characteristic, complemented, Sylow
250	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
251	#131	—	2	2	no	—
252	#131	Dic₈₆	1	4	yes	complemented
253	C₂×M₁₆	—	2	1	no	—
254	C₂×M₁₆	—	43	1	no	—
255	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
256	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	—	43	1	no	—
259	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₄⋊Q₈	—	86	2	no	—
261	C₄⋊Q₈	—	86	2	no	—
262	#125	—	86	2	no	—
263	#125	—	86	2	no	—
264	C₂²⋊Q₈	—	172	2	no	—
265	C₄×Q₈	—	86	2	no	—
266	C₄×D₄	—	86	2	no	—
267	C₄×D₄	—	172	2	no	—
268	#117	—	43	2	no	—
269	#117	—	43	1	no	—
270	#117	—	86	2	no	—
271	C₂×C₄⋊C₄	—	43	2	no	—
272	C₂×C₄⋊C₄	—	86	2	no	—
273	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
274	C₂×C₄²	—	43	1	no	—
275	#108	—	2	2	no	—
276	#107	—	43	2	no	—
277	#107	—	86	2	no	—
278	#107	—	86	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	1	no	—
290	#106	—	86	1	no	—
291	C₄⋊C₈	—	86	2	no	—
292	C₄⋊C₈	—	86	2	no	—
293	#104	—	43	4	no	—
294	#103	—	2	2	no	—
295	#103	—	43	2	no	—
296	#102	—	2	2	no	—
297	#102	—	43	2	no	—
298	#97	—	86	1	no	—
299	C₄×C₈	—	86	1	no	—
300	C₄×C₈	—	86	2	no	—
301	#95	—	172	2	no	—
302	#41	—	2	1	no	—
303	#41	—	2	1	no	—
304	#41	C₂×Dic₈₆	1	2	yes	—
305	#41	Q₈×D₄₃	1	2	yes	—
306	C₂×Q₈	—	172	2	no	—
307	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	D₄×D₄₃	1	1	yes	characteristic
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	86	1	no	—
313	C₂²×C₄	—	86	2	no	—
314	C₂²×C₄	—	172	2	no	—
315	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
316	M₁₆	—	2	1	no	—
317	M₁₆	—	2	4	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	43	2	no	—
321	M₁₆	—	86	1	no	—
322	M₁₆	C₂×Dic₈₆	1	2	yes	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	43	2	no	—
327	C₂×C₈	—	86	1	no	—
328	C₂×C₈	—	86	1	no	—
329	C₂×C₈	—	86	1	no	—
330	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
331	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
332	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₈	#10621900	1	2	yes	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	43	2	no	—
337	C₄⋊C₄	—	43	2	no	—
338	C₄⋊C₄	—	43	2	no	—
339	C₄⋊C₄	—	86	2	no	—
340	C₄⋊C₄	—	86	2	no	—
341	C₄⋊C₄	—	172	2	no	—
342	C₄⋊C₄	—	172	2	no	—
343	C₄⋊C₄	—	172	2	no	—
344	C₄⋊C₄	—	172	2	no	—
345	C₂²⋊C₄	—	86	2	no	—
346	C₂²⋊C₄	—	86	1	no	—
347	C₂²⋊C₄	—	172	2	no	—
348	C₄²	—	43	2	no	—
349	C₄²	—	86	1	no	—
350	C₄²	—	86	2	no	—
351	C₄²	—	86	1	no	—
352	C₄²	—	172	2	no	—
353	C₂³	—	2	1	no	—
354	C₂³	#13063371	1	1	yes	characteristic
355	Q₈	—	2	1	no	—
356	Q₈	—	688	2	no	—
357	Q₈	#13063324	1	2	yes	—
358	D₄	—	2	2	no	—
359	D₄	—	2	2	no	—
360	D₄	—	2	1	no	—
361	D₄	#13063324	1	2	yes	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	86	2	no	—
368	C₂×C₄	—	86	2	no	—
369	C₂×C₄	—	86	2	no	—
370	C₂×C₄	—	86	2	no	—
371	C₂×C₄	—	344	2	no	—
372	C₂×C₄	—	172	2	no	—
373	C₂×C₄	—	172	2	no	—
374	C₂×C₄	—	688	2	no	—
375	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
377	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
378	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
379	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
380	C₂×C₄	#13063372	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	86	2	no	—
384	C₈	—	86	2	no	—
385	C₈	#13063302	1	2	yes	—
386	C₈	#13063302	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	#14583527	1	1	yes	characteristic
391	C₂²	#14583533	1	1	yes	characteristic
392	C₂²	#14583607	1	1	yes	characteristic
393	C₄	—	2	2	no	—
394	C₄	—	8	2	no	—
395	C₄	—	344	2	no	—
396	C₄	—	172	2	no	—
397	C₄	—	172	2	no	—
398	C₄	—	172	2	no	—
399	C₄	—	172	2	no	—
400	C₄	C₂×#13063302	1	1	yes	characteristic
401	C₄	C₂×#13063324	1	1	yes	characteristic
402	C₄	#14583549	1	1	yes	characteristic
403	C₄	#14583562	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	4	2	no	—
407	C₂	#19138298	1	1	yes	characteristic
408	C₂	#19146378	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	#26080
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. #19136104	C₂	semidirect
5. #19135946	C₂	semidirect
6. #19135376	C₂	semidirect
7. #19135341	C₂	semidirect
9. #19134756	C₂	semidirect
76. #13063252	Q₈	semidirect
85. #131×C₄₃	Q₈	semidirect
249. C₄₃	#26080	semidirect
252. #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
#19144450	2
#19143190	2
#19136104	2
#19135946	2
#19135376	2
#19135341	2
#19134826	2
#19134756	2
#19134430	2
C₄₃×#1815	2
#26080	43