Subgroups — #25081964

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145731	C₂	1	1	yes	maximal, characteristic
2	#19144717	C₂	1	1	yes	maximal, characteristic
3	#19143676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134041	C₂	1	2	yes	maximal, complemented
5	#19134013	C₂	1	2	yes	maximal, complemented
6	#19133997	C₂	1	2	yes	maximal, complemented
7	#19133975	C₂	1	2	yes	maximal, complemented
8	#19133373	C₂	1	1	yes	maximal, characteristic, complemented
9	#19133305	C₂	1	1	yes	maximal, characteristic, complemented
10	#19133255	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583449	C₂²	1	1	yes	characteristic
13	#14583302	C₂²	1	1	yes	characteristic
14	#14583295	C₂²	1	1	yes	characteristic
15	#14583247	C₂²	1	1	yes	characteristic
16	#14583247	C₂²	1	1	yes	characteristic
17	#14583223	—	2	2	no	—
18	#14583212	—	2	1	no	—
19	#14583196	—	2	2	no	—
20	#14583194	C₂²	1	2	yes	—
21	#14583099	C₂²	1	2	yes	—
22	#14583038	—	2	1	no	—
23	Q₈×C₄₃⋊C₈	—	2	2	no	—
24	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	—
25	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	—
26	#14582997	—	2	2	no	—
27	#14582956	C₂²	1	1	yes	characteristic
28	#14582952	—	2	2	no	—
29	#14582952	C₂²	1	1	yes	characteristic
30	#14582952	C₂²	1	1	yes	characteristic
31	#14582651	C₂²	1	2	yes	—
32	#14582649	C₂²	1	2	yes	—
33	#14582648	C₂²	1	2	yes	—
34	#14582647	C₂²	1	2	yes	—
35	#14582645	C₂²	1	2	yes	—
36	#14582642	C₂²	1	2	yes	—
37	C₄₃×#533	C₂²	1	1	yes	characteristic
38	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#484	—	2	1	no	—
40	C₄₃×#477	C₂²	1	2	yes	—
41	C₄₃×#462	C₂²	1	2	yes	—
42	C₄₃×#451	—	2	1	no	—
43	C₄×Q₁₆×C₄₃	—	2	2	no	—
44	C₄₃×#431	C₂²	1	1	yes	characteristic
45	C₂×#10621898	—	2	1	no	—
46	C₂×#10621884	—	2	2	no	—
47	#13063325	C₂³	1	1	yes	characteristic
48	#13063322	C₂³	1	1	yes	characteristic
49	#13063313	—	2	1	no	—
50	#13063299	—	2	1	no	—
51	#13063299	—	2	1	no	—
52	C₄×D₁₇₂	—	2	1	no	—
53	C₄×D₁₇₂	—	2	1	no	—
54	#13063258	—	2	2	no	—
55	#13063258	—	2	2	no	—
56	#13063256	—	2	2	no	—
57	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
58	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
59	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
60	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
61	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
62	#13063249	—	2	2	no	—
63	#13063245	—	2	2	no	—
64	#13063245	—	2	2	no	—
65	#13063245	—	2	2	no	—
66	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
68	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
69	#13063236	—	2	1	no	—
70	C₄×C₄₃⋊C₈	—	2	1	no	—
71	C₄×C₄₃⋊C₈	—	2	2	no	—
72	C₄×C₄₃⋊C₈	—	2	2	no	—
73	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
76	C₂×Q₁₆×C₄₃	—	2	2	no	—
77	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
79	#125×C₄₃	—	2	1	no	—
80	C₂²⋊Q₈×C₄₃	—	2	1	no	—
81	C₂²⋊Q₈×C₄₃	—	2	1	no	—
82	C₄×Q₈×C₄₃	—	2	1	no	—
83	C₄×Q₈×C₄₃	—	2	1	no	—
84	C₄×Q₈×C₄₃	—	2	1	no	—
85	C₄×Q₈×C₄₃	—	2	1	no	—
86	C₄×Q₈×C₄₃	D₄	1	2	yes	—
87	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
88	#117×C₄₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₄₃	—	2	1	no	—
92	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
93	#107×C₄₃	—	2	2	no	—
94	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
95	#103×C₄₃	—	2	2	no	—
96	#103×C₄₃	—	2	2	no	—
97	C₂²⋊C₈×C₄₃	—	2	1	no	—
98	C₄×C₈×C₄₃	—	2	1	no	—
99	C₂×Q₈×C₄₃	—	2	4	no	—
100	C₂×Q₈×C₄₃	—	2	1	no	—
101	C₂×Q₈×C₄₃	—	2	1	no	—
102	C₂×Q₈×C₄₃	—	2	1	no	—
103	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₄₃	—	2	1	no	—
105	C₂²×C₄×C₄₃	—	2	1	no	—
106	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
109	C₂×C₂²⋊D₄₃	—	4	1	no	—
110	#10621898	—	8	2	no	—
111	C₂×D₁₇₂	—	2	1	no	—
112	C₂×D₁₇₂	—	2	1	no	—
113	C₂×D₁₇₂	—	2	1	no	—
114	C₂×C₄×D₄₃	—	4	1	no	—
115	C₂×Dic₈₆	—	2	1	no	—
116	Q₁₆×C₄₃	—	4	4	no	—
117	C₂×C₈×C₄₃	—	2	2	no	—
118	C₄⋊C₄×C₄₃	—	2	2	no	—
119	C₄⋊C₄×C₄₃	—	2	1	no	—
120	C₄⋊C₄×C₄₃	—	2	2	no	—
121	C₄⋊C₄×C₄₃	—	2	1	no	—
122	C₄⋊C₄×C₄₃	—	2	1	no	—
123	C₄⋊C₄×C₄₃	—	2	1	no	—
124	C₄⋊C₄×C₄₃	—	2	1	no	—
125	C₄⋊C₄×C₄₃	—	4	1	no	—
126	C₄⋊C₄×C₄₃	—	4	1	no	—
127	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
128	C₂²⋊C₄×C₄₃	—	4	1	no	—
129	C₄²×C₄₃	—	2	2	no	—
130	C₄²×C₄₃	—	2	2	no	—
131	C₄²×C₄₃	—	4	1	no	—
132	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₄₃	#41	1	1	yes	characteristic
135	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
136	#10621884	—	4	4	no	—
137	(C₂×D₄₃)⋊C₄	—	4	1	no	—
138	(C₂×D₄₃)⋊C₄	—	4	1	no	—
139	C₄⋊Dic₄₃	—	2	1	no	—
140	C₄⋊Dic₄₃	—	2	1	no	—
141	C₂×C₄₃⋊C₈	—	2	2	no	—
142	C₂×C₄₃⋊C₈	—	2	2	no	—
143	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
144	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
145	C₂³×C₄₃	#121	1	1	yes	characteristic
146	C₂²×D₄₃	—	4	1	no	—
147	Q₈×C₄₃	—	2	4	no	—
148	Q₈×C₄₃	—	2	2	no	—
149	Q₈×C₄₃	—	4	2	no	—
150	Q₈×C₄₃	—	8	1	no	—
151	Q₈×C₄₃	#135	1	4	yes	—
152	C₂×C₄×C₄₃	—	2	4	no	—
153	C₂×C₄×C₄₃	—	2	2	no	—
154	C₂×C₄×C₄₃	—	2	2	no	—
155	C₂×C₄×C₄₃	—	2	2	no	—
156	C₂×C₄×C₄₃	—	2	2	no	—
157	C₂×C₄×C₄₃	—	4	1	no	—
158	C₂×C₄×C₄₃	—	4	1	no	—
159	C₂×C₄×C₄₃	—	4	1	no	—
160	C₂×C₄×C₄₃	—	8	1	no	—
161	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
167	C₂²⋊D₄₃	—	8	2	no	—
168	C₂×Dic₄₃	—	4	1	no	—
169	D₁₇₂	—	4	2	no	—
170	D₁₇₂	—	4	2	no	—
171	D₁₇₂	—	8	1	no	—
172	C₄×D₄₃	—	8	2	no	—
173	Dic₈₆	—	8	1	no	—
174	C₈×C₄₃	—	4	2	no	—
175	C₄₃⋊C₈	—	2	2	no	—
176	C₄₃⋊C₈	—	2	2	no	—
177	C₄₃⋊C₈	—	4	2	no	—
178	C₄₃⋊C₈	—	4	2	no	—
179	#19406	—	43	1	no	maximal, Sylow
180	C₂²×C₄₃	—	2	1	no	—
181	C₂²×C₄₃	—	2	1	no	—
182	C₂²×C₄₃	—	2	1	no	—
183	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
184	C₂×D₄₃	—	8	1	no	—
185	C₂×D₄₃	—	8	1	no	—
186	C₂×D₄₃	—	8	1	no	—
187	C₄×C₄₃	—	2	1	no	—
188	C₄×C₄₃	—	2	4	no	—
189	C₄×C₄₃	—	2	1	no	—
190	C₄×C₄₃	—	2	1	no	—
191	C₄×C₄₃	—	2	1	no	—
192	C₄×C₄₃	—	4	2	no	—
193	C₄×C₄₃	—	8	1	no	—
194	C₄×C₄₃	—	8	1	no	—
195	C₄×C₄₃	C₂×#135	1	2	yes	—
196	C₄×C₄₃	#578	1	2	yes	—
197	Dic₄₃	—	8	1	no	—
198	#3228	—	43	1	no	—
199	#3112	—	43	1	no	—
200	#3104	D₄₃	1	1	yes	characteristic, complemented
201	#2982	—	43	1	no	—
202	#1682	—	43	2	no	—
203	#1678	—	43	2	no	—
204	#1676	—	43	2	no	—
205	#1672	—	43	2	no	—
206	#1572	—	43	1	no	—
207	#1554	—	43	1	no	—
208	#1543	—	43	1	no	—
209	C₂×C₄₃	—	4	1	no	—
210	C₂×C₄₃	#3044	1	1	yes	characteristic
211	C₂×C₄₃	#3050	1	1	yes	characteristic
212	C₂×C₄₃	#3068	1	1	yes	characteristic
213	D₄₃	—	16	1	no	—
214	#533	C₂×D₄₃	1	1	yes	characteristic
215	#532	—	43	1	no	—
216	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
217	#500	—	43	1	no	—
218	#498	—	43	1	no	—
219	#486	—	43	1	no	—
220	#486	—	43	1	no	—
221	#484	—	2	1	no	—
222	#482	—	86	2	no	—
223	#480	—	86	1	no	—
224	#478	—	86	2	no	—
225	#477	—	43	2	no	—
226	#477	C₂×D₄₃	1	2	yes	—
227	#462	C₂×D₄₃	1	2	yes	—
228	#460	—	43	2	no	—
229	#451	—	2	1	no	—
230	#448	—	86	1	no	—
231	C₈×Q₈	—	43	2	no	—
232	C₈×Q₈	—	43	2	no	—
233	C₈×Q₈	—	86	2	no	—
234	#440	—	86	2	no	—
235	C₄×Q₁₆	—	2	2	no	—
236	#432	—	43	1	no	—
237	#431	—	43	1	no	—
238	#431	—	43	1	no	—
239	#431	—	86	2	no	—
240	#431	C₂×D₄₃	1	1	yes	characteristic
241	#335	—	43	2	no	—
242	#334	—	43	2	no	—
243	#333	—	43	2	no	—
244	#333	—	43	2	no	—
245	#332	—	43	2	no	—
246	#331	—	43	2	no	—
247	C₄₃	#19406	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	86	1	no	—
249	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×Q₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	86	2	no	—
252	C₄⋊Q₈	—	43	1	no	—
253	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
254	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
255	#127	—	43	1	no	—
256	#125	—	2	1	no	—
257	#124	—	86	1	no	—
258	C₂²⋊Q₈	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	#121	—	86	1	no	—
261	#121	—	86	1	no	—
262	C₄×Q₈	—	2	1	no	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	—	2	1	no	—
265	C₄×Q₈	—	2	1	no	—
266	C₄×Q₈	C₂²⋊D₄₃	1	2	yes	—
267	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
268	C₄×D₄	—	86	1	no	—
269	C₄×D₄	—	86	1	no	—
270	#117	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
273	C₂×C₄²	—	2	1	no	—
274	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
275	#107	—	2	2	no	—
276	#107	—	86	2	no	—
277	#107	—	86	2	no	—
278	#106	—	86	2	no	—
279	C₄⋊C₈	—	43	2	no	—
280	C₄⋊C₈	—	43	2	no	—
281	C₄⋊C₈	—	43	2	no	—
282	C₄⋊C₈	—	86	2	no	—
283	C₄⋊C₈	—	86	2	no	—
284	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
285	#103	—	2	2	no	—
286	#103	—	2	2	no	—
287	#103	—	86	2	no	—
288	#102	—	86	2	no	—
289	#102	—	86	2	no	—
290	#102	—	86	2	no	—
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	43	2	no	—
293	C₂²⋊C₈	—	86	1	no	—
294	C₂²⋊C₈	—	86	1	no	—
295	#97	—	86	1	no	—
296	C₄×C₈	—	2	1	no	—
297	C₄×C₈	—	43	1	no	—
298	C₄×C₈	—	43	1	no	—
299	C₄×C₈	—	86	2	no	—
300	C₄×C₈	—	86	2	no	—
301	C₄×C₈	—	86	1	no	—
302	#41	—	344	2	no	—
303	C₂×Q₈	—	2	4	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	—	2	1	no	—
307	C₂×Q₈	—	86	1	no	—
308	C₂×Q₈	C₂×C₂²⋊D₄₃	1	2	yes	—
309	C₂×D₄	—	86	1	no	—
310	C₂×D₄	—	86	1	no	—
311	C₂×D₄	—	86	1	no	—
312	C₂×D₄	—	172	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	172	1	no	—
316	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
317	C₂²×C₄	#10621900	1	1	yes	characteristic
318	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
319	Q₁₆	—	4	4	no	—
320	SD₁₆	—	172	4	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	43	2	no	—
323	C₂×C₈	—	43	2	no	—
324	C₂×C₈	—	86	2	no	—
325	C₂×C₈	—	86	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	86	1	no	—
336	C₄⋊C₄	—	86	1	no	—
337	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
338	C₂²⋊C₄	—	4	1	no	—
339	C₂²⋊C₄	—	172	1	no	—
340	C₂²⋊C₄	—	172	1	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	4	1	no	—
344	C₄²	D₄×D₄₃	1	1	yes	characteristic
345	C₄²	#10621900	1	1	yes	characteristic
346	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
347	C₄²	C₂³×D₄₃	1	1	yes	characteristic
348	C₂³	—	172	1	no	—
349	C₂³	#13063297	1	1	yes	characteristic
350	Q₈	—	2	4	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	4	2	no	—
353	Q₈	—	8	1	no	—
354	Q₈	—	344	1	no	—
355	Q₈	#13063345	1	4	yes	—
356	D₄	—	344	1	no	—
357	D₄	—	344	2	no	—
358	D₄	—	172	2	no	—
359	D₄	—	172	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	8	1	no	—
369	C₂×C₄	—	344	2	no	—
370	C₂×C₄	—	172	1	no	—
371	C₂×C₄	#13063297	1	1	yes	characteristic
372	C₂×C₄	#13063297	1	1	yes	characteristic
373	C₂×C₄	#13063297	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
375	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
376	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
377	C₈	—	4	2	no	—
378	C₈	—	86	2	no	—
379	C₈	—	86	2	no	—
380	C₈	—	172	2	no	—
381	C₈	—	172	2	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	344	1	no	—
386	C₂²	—	344	1	no	—
387	C₂²	—	344	1	no	—
388	C₂²	C₂×#13063297	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	4	no	—
393	C₄	—	2	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	8	1	no	—
396	C₄	—	8	1	no	—
397	C₄	—	344	1	no	—
398	C₄	C₂×#13063345	1	2	yes	—
399	C₄	#14583712	1	2	yes	—
400	C₂	—	4	1	no	—
401	C₂	—	688	1	no	—
402	C₂	#19144122	1	1	yes	characteristic
403	C₂	#19144173	1	1	yes	characteristic
404	C₂	#19144325	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063297
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3104	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	#3104	D₄₃
hypocenter	C₄₃	#19406
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3104

Complements

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

N	H ≅ G/N	type
3. #19143676	C₂	semidirect
4. #19134041	C₂	semidirect
5. #19134013	C₂	semidirect
6. #19133997	C₂	semidirect
7. #19133975	C₂	semidirect
8. #19133373	C₂	semidirect
9. #19133305	C₂	semidirect
11. C₄₃×#3104	C₂	semidirect
200. #3104	D₄₃	semidirect
247. C₄₃	#19406	semidirect

Maximal subgroups

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

subgroup	index
#19145731	2
#19144717	2
#19143676	2
#19134041	2
#19134013	2
#19133997	2
#19133975	2
#19133373	2
#19133305	2
#19133255	2
C₄₃×#3104	2
#19406	43