Subgroups — #25079713

Order 11008.

650 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145725	C₂	1	1	yes	maximal, characteristic
2	#19145723	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583302	C₂	1	1	yes	maximal, characteristic, complemented
4	#19143306	C₂	1	1	yes	maximal, characteristic
5	#19134117	C₂	1	4	yes	maximal, complemented
6	#19133993	C₂	1	4	yes	maximal, complemented
7	#19133341	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133277	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063333	—	2	1	no	—
11	#14583449	C₂²	1	1	yes	characteristic
12	C₂×#13063325	C₂²	1	1	yes	characteristic
13	#14583302	—	2	1	no	—
14	#14583302	—	2	1	no	—
15	#14583302	C₂²	1	2	yes	—
16	#14583302	C₂²	1	2	yes	—
17	#14583297	—	2	1	no	—
18	#14583295	—	2	1	no	—
19	#14583247	—	2	1	no	—
20	#14583246	—	2	1	no	—
21	#14583218	—	2	2	no	—
22	#14583217	—	2	2	no	—
23	#14583190	C₂²	1	4	yes	—
24	#14583186	C₂²	1	4	yes	—
25	#14582976	—	2	1	no	—
26	#14582976	C₂²	1	2	yes	—
27	#14582964	—	2	1	no	—
28	#14582964	C₂²	1	2	yes	—
29	#14582952	C₂²	1	1	yes	characteristic
30	C₂×#13063258	—	2	1	no	—
31	C₂×#13063258	—	2	1	no	—
32	#14582867	—	2	1	no	—
33	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
34	#14582651	C₂²	1	2	yes	—
35	#14582649	C₂²	1	2	yes	—
36	#14582647	C₂²	1	4	yes	—
37	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#431	C₂²	1	1	yes	characteristic
39	C₄₃×#423	C₂²	1	1	yes	characteristic
40	C₄₃×#333	C₂²	1	4	yes	—
41	C₂×#10621898	—	2	1	no	—
42	C₂²×Dic₈₆	—	2	1	no	—
43	#13063333	—	4	4	no	—
44	C₂×#10621872	—	2	2	no	—
45	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
46	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₈×D₄₃	—	2	2	no	—
48	#13063325	—	2	1	no	—
49	#13063325	—	2	1	no	—
50	#13063325	—	2	1	no	—
51	#13063325	D₄	1	1	yes	characteristic
52	#13063325	C₂³	1	2	yes	—
53	#13063322	D₄	1	1	yes	characteristic
54	#13063313	—	2	1	no	—
55	#13063299	—	2	1	no	—
56	#13063299	D₄	1	2	yes	—
57	C₄×D₁₇₂	—	2	1	no	—
58	C₄×D₁₇₂	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₄₃	—	2	1	no	—
60	C₂×C₄⋊Dic₄₃	—	2	1	no	—
61	#13063258	—	2	2	no	—
62	#13063258	—	2	2	no	—
63	#13063258	—	2	2	no	—
64	#13063258	—	2	2	no	—
65	#13063257	—	2	4	no	—
66	#13063256	—	2	4	no	—
67	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
68	#13063247	—	2	4	no	—
69	#13063245	—	2	4	no	—
70	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
71	(C₂×D₄₃)⋊C₈	—	2	2	no	—
72	(C₂×D₄₃)⋊C₈	—	2	2	no	—
73	C₄×C₄₃⋊C₈	—	2	1	no	—
74	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₄₃	—	2	1	no	—
79	C₂×M₁₆×C₄₃	—	2	1	no	—
80	C₄⋊Q₈×C₄₃	—	2	1	no	—
81	C₄⋊Q₈×C₄₃	—	2	1	no	—
82	C₄⋊Q₈×C₄₃	—	2	1	no	—
83	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
86	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₃	—	2	1	no	—
88	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
89	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
90	C₂²⋊C₈×C₄₃	—	2	1	no	—
91	C₄×C₈×C₄₃	—	2	1	no	—
92	C₂×Q₈×C₄₃	—	2	2	no	—
93	C₂×Q₈×C₄₃	—	4	1	no	—
94	C₂×Q₈×C₄₃	—	8	2	no	—
95	C₂²×C₄×C₄₃	—	4	1	no	—
96	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
99	C₂×C₂²⋊D₄₃	—	2	2	no	—
100	C₂²×Dic₄₃	—	4	1	no	—
101	#10621898	—	4	2	no	—
102	#10621898	—	4	2	no	—
103	C₂×D₁₇₂	—	2	1	no	—
104	C₂×D₁₇₂	—	2	1	no	—
105	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
106	C₂×C₄×D₄₃	—	2	2	no	—
107	C₂×Dic₈₆	—	2	1	no	—
108	C₂×Dic₈₆	—	2	2	no	—
109	C₂×Dic₈₆	—	4	1	no	—
110	C₂×Dic₈₆	—	8	2	no	—
111	M₁₆×C₄₃	—	4	2	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	—	2	2	no	—
114	C₄⋊C₄×C₄₃	—	2	2	no	—
115	C₄⋊C₄×C₄₃	—	2	2	no	—
116	C₄⋊C₄×C₄₃	—	4	1	no	—
117	C₄⋊C₄×C₄₃	—	4	1	no	—
118	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₃	#41	1	1	yes	characteristic
121	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
122	(C₂×D₄₃)⋊C₄	—	2	2	no	—
123	(C₂×D₄₃)⋊C₄	—	2	2	no	—
124	C₄⋊Dic₄₃	—	2	1	no	—
125	C₄⋊Dic₄₃	—	2	2	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	—	4	1	no	—
128	C₄⋊Dic₄₃	—	4	1	no	—
129	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
130	C₂×C₄₃⋊C₈	—	2	2	no	—
131	C₂×C₄₃⋊C₈	—	2	2	no	—
132	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	—
133	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
137	#10621872	—	4	4	no	—
138	C₈×D₄₃	—	4	4	no	—
139	C₂³×C₄₃	#121	1	1	yes	characteristic
140	C₂²×D₄₃	—	2	2	no	—
141	Q₈×C₄₃	—	8	2	no	—
142	Q₈×C₄₃	—	8	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	—	8	2	no	—
147	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₄₃	#137	1	2	yes	—
152	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₂²⋊D₄₃	—	4	4	no	—
156	C₂×Dic₄₃	—	2	2	no	—
157	C₂×Dic₄₃	—	4	2	no	—
158	C₂×Dic₄₃	—	8	2	no	—
159	D₁₇₂	—	2	4	no	—
160	D₁₇₂	—	4	2	no	—
161	D₁₇₂	—	4	2	no	—
162	C₄×D₄₃	—	4	2	no	—
163	C₄×D₄₃	—	4	2	no	—
164	Dic₈₆	—	4	2	no	—
165	Dic₈₆	—	8	2	no	—
166	Dic₈₆	—	8	2	no	—
167	C₈×C₄₃	—	4	2	no	—
168	C₈×C₄₃	—	4	2	no	—
169	C₄₃⋊C₈	—	2	1	no	—
170	C₄₃⋊C₈	—	2	1	no	—
171	C₄₃⋊C₈	—	4	2	no	—
172	C₄₃⋊C₈	C₂×SD₁₆	1	2	yes	—
173	C₄₃⋊C₈	C₂×SD₁₆	1	2	yes	—
174	#19193	—	43	1	no	maximal, Sylow
175	C₂²×C₄₃	—	2	1	no	—
176	C₂²×C₄₃	—	2	1	no	—
177	C₂²×C₄₃	#461	1	2	yes	—
178	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
179	C₂×D₄₃	—	4	2	no	—
180	C₂×D₄₃	—	4	2	no	—
181	C₂×D₄₃	—	4	2	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	8	2	no	—
185	C₄×C₄₃	#461	1	2	yes	—
186	C₄×C₄₃	#472	1	2	yes	—
187	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
188	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
189	C₄×C₄₃	#578	1	2	yes	—
190	Dic₄₃	—	4	2	no	—
191	Dic₄₃	—	8	2	no	—
192	#3227	—	43	1	no	—
193	#3227	—	43	1	no	—
194	C₂×#500	—	43	1	no	—
195	#2931	—	43	1	no	—
196	#1695	—	43	4	no	—
197	#1675	—	43	4	no	—
198	#1564	—	43	1	no	—
199	#1548	—	43	1	no	—
200	#1548	D₄₃	1	1	yes	characteristic, complemented
201	C₂×C₄₃	—	2	2	no	—
202	C₂×C₄₃	C₂×#461	1	1	yes	characteristic
203	C₂×C₄₃	#3059	1	1	yes	characteristic
204	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
205	D₄₃	—	8	2	no	—
206	C₂×#131	—	86	1	no	—
207	#532	—	43	1	no	—
208	C₂×C₄⋊Q₈	—	43	1	no	—
209	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
210	#500	—	43	2	no	—
211	#500	—	43	2	no	—
212	#500	—	86	1	no	—
213	#500	—	86	1	no	—
214	#498	—	86	1	no	—
215	#498	—	86	1	no	—
216	#486	—	86	1	no	—
217	#486	—	86	1	no	—
218	#481	—	86	2	no	—
219	#481	—	86	2	no	—
220	#476	—	43	4	no	—
221	#476	—	43	4	no	—
222	#436	—	43	2	no	—
223	#436	—	86	1	no	—
224	C₈×D₄	—	43	2	no	—
225	C₈×D₄	—	86	1	no	—
226	#431	—	43	1	no	—
227	#431	C₂×D₄₃	1	1	yes	characteristic
228	C₂×#107	—	86	1	no	—
229	C₂×#107	—	86	1	no	—
230	#423	C₂×D₄₃	1	1	yes	characteristic
231	#408	—	86	1	no	—
232	C₂×C₄×C₈	—	43	1	no	—
233	#335	—	43	2	no	—
234	#334	—	43	2	no	—
235	#333	—	43	4	no	—
236	#333	C₂×D₄₃	1	4	yes	—
237	C₄₃	#19193	1	1	yes	characteristic, complemented, Sylow
238	C₂×#41	—	86	1	no	—
239	C₂²×Q₈	—	2	1	no	—
240	C₂²×Q₈	—	86	1	no	—
241	#131	—	172	4	no	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	86	2	no	—
244	C₂²×C₈	—	43	1	no	—
245	C₂²×C₈	—	43	1	no	—
246	C₂²×C₈	—	86	2	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	43	1	no	—
251	C₄⋊Q₈	—	43	2	no	—
252	C₄⋊Q₈	—	86	1	no	—
253	C₄⋊Q₈	—	86	1	no	—
254	C₄⋊Q₈	—	86	1	no	—
255	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
256	#127	—	43	1	no	—
257	#124	—	86	1	no	—
258	#121	—	43	2	no	—
259	#121	—	86	1	no	—
260	C₄×D₄	—	43	2	no	—
261	C₄×D₄	—	86	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	86	1	no	—
265	C₂×C₄⋊C₄	—	86	1	no	—
266	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
267	#107	—	86	2	no	—
268	#107	—	86	2	no	—
269	#107	—	86	2	no	—
270	#107	—	86	2	no	—
271	#106	—	86	4	no	—
272	#106	—	86	4	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	43	2	no	—
275	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
276	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
277	#102	—	86	4	no	—
278	#102	—	86	4	no	—
279	C₂²⋊C₈	—	2	1	no	—
280	C₂²⋊C₈	—	86	2	no	—
281	C₂²⋊C₈	—	86	2	no	—
282	C₂²⋊C₈	—	86	1	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	43	1	no	—
285	C₄×C₈	—	43	1	no	—
286	C₄×C₈	—	43	1	no	—
287	C₄×C₈	—	43	1	no	—
288	C₄×C₈	—	86	1	no	—
289	#41	—	172	2	no	—
290	#41	—	172	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×Q₈	—	8	2	no	—
294	C₂×Q₈	—	86	1	no	—
295	C₂×Q₈	—	86	2	no	—
296	C₂×Q₈	—	344	2	no	—
297	C₂×Q₈	—	172	1	no	—
298	C₂×D₄	—	43	2	no	—
299	C₂×D₄	—	86	2	no	—
300	C₂×D₄	—	86	1	no	—
301	C₂×D₄	—	86	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	86	2	no	—
304	C₂²×C₄	—	172	1	no	—
305	C₂²×C₄	#10621898	1	1	yes	characteristic
306	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
307	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
308	M₁₆	—	4	2	no	—
309	M₁₆	—	172	4	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	43	4	no	—
313	C₂×C₈	—	43	1	no	—
314	C₂×C₈	—	43	1	no	—
315	C₂×C₈	—	43	1	no	—
316	C₂×C₈	—	43	1	no	—
317	C₂×C₈	—	86	2	no	—
318	C₂×C₈	—	86	2	no	—
319	C₂×C₈	—	172	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	43	2	no	—
325	C₄⋊C₄	—	86	1	no	—
326	C₄⋊C₄	—	86	2	no	—
327	C₄⋊C₄	—	86	2	no	—
328	C₄⋊C₄	—	172	1	no	—
329	C₄⋊C₄	—	172	1	no	—
330	C₂²⋊C₄	—	86	2	no	—
331	C₂²⋊C₄	—	86	2	no	—
332	C₄²	#10621898	1	1	yes	characteristic
333	C₄²	D₄×D₄₃	1	1	yes	characteristic
334	C₄²	D₄×D₄₃	1	1	yes	characteristic
335	C₄²	C₂³×D₄₃	1	1	yes	characteristic
336	C₂³	—	86	2	no	—
337	C₂³	#13063298	1	1	yes	characteristic
338	Q₈	—	8	2	no	—
339	Q₈	—	8	2	no	—
340	Q₈	—	344	2	no	—
341	Q₈	—	344	2	no	—
342	Q₈	—	172	2	no	—
343	D₄	—	86	4	no	—
344	D₄	—	172	4	no	—
345	D₄	—	172	2	no	—
346	D₄	—	172	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	86	2	no	—
352	C₂×C₄	—	344	2	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	—	172	2	no	—
356	C₂×C₄	#13063298	1	1	yes	characteristic
357	C₂×C₄	#13063298	1	1	yes	characteristic
358	C₂×C₄	#13063298	1	1	yes	characteristic
359	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
360	C₂×C₄	#13063357	1	2	yes	—
361	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
362	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
363	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
364	C₈	—	4	2	no	—
365	C₈	—	4	2	no	—
366	C₈	—	43	2	no	—
367	C₈	—	43	2	no	—
368	C₈	—	86	1	no	—
369	C₈	—	86	1	no	—
370	C₈	—	172	2	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	172	2	no	—
374	C₂²	—	172	2	no	—
375	C₂²	—	172	2	no	—
376	C₂²	#14583103	1	2	yes	—
377	C₂²	C₂×#13063298	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	8	2	no	—
381	C₄	—	344	2	no	—
382	C₄	—	172	2	no	—
383	C₄	#14583103	1	2	yes	—
384	C₄	#14583168	1	2	yes	—
385	C₄	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
386	C₄	C₂×#13063357	1	1	yes	characteristic
387	C₄	#14583715	1	2	yes	—
388	C₂	—	2	2	no	—
389	C₂	—	344	2	no	—
390	C₂	C₂×#14583103	1	1	yes	characteristic
391	C₂	#19144249	1	1	yes	characteristic
392	C₂	C₂×#14583168	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063298
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1548	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	#1548	D₄₃
hypocenter	C₄₃	#19193
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1548

Complements

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

N	H ≅ G/N	type
3. C₂×#14583302	C₂	semidirect
5. #19134117	C₂	semidirect
6. #19133993	C₂	semidirect
7. #19133341	C₂	semidirect
8. #19133277	C₂	semidirect
9. C₄₃×#1548	C₂	semidirect
200. #1548	D₄₃	semidirect
237. C₄₃	#19193	semidirect

Maximal subgroups

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

subgroup	index
#19145725	2
#19145723	2
C₂×#14583302	2
#19143306	2
#19134117	2
#19133993	2
#19133341	2
#19133277	2
C₄₃×#1548	2
#19193	43