Subgroups — #25067153

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145178	C₂	1	1	yes	maximal, characteristic
2	#19143526	C₂	1	2	yes	maximal
3	#19133505	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133502	C₂	1	2	yes	maximal, complemented
5	#19133414	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133413	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133411	C₂	1	2	yes	maximal, complemented
8	#19133349	C₂	1	1	yes	maximal, characteristic
9	#19133048	C₂	1	2	yes	maximal
10	#322×D₄₃	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583453	C₂²	1	2	yes	—
13	#14583303	C₄	1	1	yes	characteristic
14	#14583277	C₄	1	1	yes	characteristic
15	#14583248	C₄	1	2	yes	—
16	#14583141	C₄	1	1	yes	characteristic
17	#14583139	C₄	1	2	yes	—
18	#14583137	C₄	1	1	yes	characteristic
19	#14583029	C₂²	1	2	yes	—
20	#14583027	—	2	2	no	—
21	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	—
22	#14583023	—	2	2	no	—
23	#14583023	C₂²	1	2	yes	—
24	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
25	C₄×#10621885	C₂²	1	2	yes	—
26	#14582924	—	2	2	no	—
27	#14582921	C₂²	1	2	yes	—
28	#14582856	—	2	2	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
32	#14582650	C₂²	1	1	yes	characteristic
33	#14582649	C₂²	1	2	yes	—
34	#14582631	C₂²	1	2	yes	—
35	Q₈⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
36	#14582629	C₂²	1	2	yes	—
37	#14582621	C₂²	1	1	yes	characteristic
38	#14582621	C₂²	1	1	yes	characteristic
39	#14582620	C₂²	1	1	yes	characteristic
40	C₄₃×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
42	C₄₃×#334	C₂²	1	1	yes	characteristic
43	C₄₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₄₃×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#10621885	C₂×C₄	1	2	yes	—
47	C₂×#10621872	—	2	2	no	—
48	C₂×C₈×D₄₃	—	2	1	no	—
49	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₄₃	D₄	1	2	yes	—
52	#13063324	C₂×C₄	1	2	yes	—
53	#13063323	C₂×C₄	1	2	yes	—
54	#13063315	—	2	2	no	—
55	#13063305	C₂×C₄	1	2	yes	—
56	#13063303	—	2	2	no	—
57	#13063301	C₂×C₄	1	2	yes	—
58	C₄×Dic₈₆	—	2	2	no	—
59	C₄×Dic₈₆	C₂³	1	2	yes	—
60	#13063282	—	2	2	no	—
61	C₄²×D₄₃	C₂³	1	1	yes	characteristic
62	#13063259	C₂×C₄	1	1	yes	characteristic
63	#13063258	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
65	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
66	Dic₄₃⋊C₈	—	2	2	no	—
67	Dic₄₃⋊C₈	—	2	2	no	—
68	Dic₄₃⋊C₈	—	2	2	no	—
69	#13063250	C₂×C₄	1	2	yes	—
70	#13063249	C₂×C₄	1	2	yes	—
71	#13063248	C₂×C₄	1	2	yes	—
72	#13063235	—	2	2	no	—
73	C₄×C₄₃⋊C₈	—	2	2	no	—
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₃	—	2	1	no	—
78	C₈×Dic₄₃	—	2	2	no	—
79	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₃	D₄	1	2	yes	—
82	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
84	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₄₃	—	2	2	no	—
86	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
87	#103×C₄₃	C₂×C₄	1	2	yes	—
88	#97×C₄₃	—	2	2	no	—
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
96	C₂×Dic₈₆	—	2	2	no	—
97	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	1	no	—
101	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₃	—	2	2	no	—
107	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
108	#10621885	—	2	4	no	—
109	(C₂×D₄₃)⋊C₄	—	2	2	no	—
110	(C₂×D₄₃)⋊C₄	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
113	Dic₄₃⋊C₄	—	2	2	no	—
114	Dic₄₃⋊C₄	—	2	2	no	—
115	Dic₄₃⋊C₄	—	2	2	no	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₃	#41	1	1	yes	characteristic
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	2	1	no	—
124	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₁₇₂	—	2	2	no	—
128	#10621872	—	4	4	no	—
129	C₈×D₄₃	—	2	4	no	—
130	C₈×D₄₃	—	2	2	no	—
131	C₈×D₄₃	—	2	2	no	—
132	C₈×D₄₃	—	4	2	no	—
133	C₂²×D₄₃	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₄₃	—	2	2	no	—
135	Q₈×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
146	C₄×D₄₃	—	2	1	no	—
147	C₄×D₄₃	—	2	1	no	—
148	C₄×D₄₃	—	2	2	no	—
149	C₄×D₄₃	#137	1	4	yes	—
150	Dic₈₆	—	2	2	no	—
151	Dic₈₆	—	2	2	no	—
152	Dic₈₆	—	4	4	no	—
153	C₈×C₄₃	—	2	1	no	—
154	C₈×C₄₃	—	2	2	no	—
155	C₈×C₄₃	—	4	1	no	—
156	C₈×C₄₃	—	4	2	no	—
157	C₈×C₄₃	#131	1	2	yes	—
158	C₄₃⋊C₈	—	2	1	no	—
159	C₄₃⋊C₈	—	2	2	no	—
160	C₄₃⋊C₈	—	4	1	no	—
161	C₄₃⋊C₈	—	4	2	no	—
162	C₄₃⋊C₈	#131	1	2	yes	—
163	#17961	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₄₃	—	2	1	no	—
166	C₂×D₄₃	—	2	1	no	—
167	C₂×D₄₃	#441	1	2	yes	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	4	2	no	—
170	C₄×C₄₃	—	4	2	no	—
171	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
172	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
173	C₄×C₄₃	C₂×#137	1	2	yes	—
174	Dic₄₃	—	2	1	no	—
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	4	2	no	—
177	Dic₄₃	#441	1	2	yes	—
178	Dic₄₃	#444	1	2	yes	—
179	#3170	—	43	1	no	—
180	#2960	—	43	2	no	—
181	#1593	—	43	2	no	—
182	#1593	—	43	1	no	—
183	#1593	D₄₃	1	1	yes	characteristic, complemented
184	#1579	—	43	1	no	—
185	#1579	—	43	1	no	—
186	#1579	—	43	2	no	—
187	#1566	—	43	1	no	—
188	#1493	—	43	2	no	—
189	C₂×#322	—	43	1	no	—
190	C₂×C₄₃	#2931	1	1	yes	characteristic
191	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
192	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
193	D₄₃	—	2	2	no	—
194	#533	—	43	2	no	—
195	#500	—	43	1	no	—
196	#494	—	43	1	no	—
197	#487	—	43	2	no	—
198	#467	—	43	1	no	—
199	#467	—	43	1	no	—
200	#467	—	43	2	no	—
201	#446	—	43	2	no	—
202	#446	—	86	2	no	—
203	#446	C₂×D₄₃	1	2	yes	—
204	C₈×Q₈	—	43	2	no	—
205	C₈×Q₈	—	43	2	no	—
206	C₈×Q₈	—	86	2	no	—
207	C₄×Q₁₆	—	43	2	no	—
208	C₄×Q₁₆	—	43	1	no	—
209	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
210	#424	—	86	2	no	—
211	#423	—	43	2	no	—
212	#405	—	86	2	no	—
213	C₂×C₄×C₈	—	43	1	no	—
214	C₂×C₄×C₈	—	43	1	no	—
215	C₂×C₄×C₈	—	43	1	no	—
216	#334	—	43	2	no	—
217	#334	—	43	1	no	—
218	#334	C₂×D₄₃	1	1	yes	characteristic
219	Q₈⋊C₈	—	43	2	no	—
220	Q₈⋊C₈	—	43	2	no	—
221	Q₈⋊C₈	—	43	2	no	—
222	Q₈⋊C₈	C₂×D₄₃	1	2	yes	—
223	#322	—	43	1	no	—
224	#322	—	43	1	no	—
225	#322	—	43	1	no	—
226	#322	C₂×D₄₃	1	1	yes	characteristic
227	C₄₃	#17961	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₁₆	—	43	2	no	—
229	C₂×Q₁₆	—	43	1	no	—
230	C₂×Q₁₆	C₄×D₄₃	1	1	yes	characteristic
231	C₂×M₁₆	—	86	2	no	—
232	C₂²×C₈	—	43	1	no	—
233	C₂²×C₈	—	43	1	no	—
234	C₂²×C₈	—	43	2	no	—
235	C₂²×C₈	—	86	1	no	—
236	C₄⋊Q₈	—	43	2	no	—
237	C₄⋊Q₈	—	43	2	no	—
238	#125	—	86	2	no	—
239	C₂²⋊Q₈	—	43	2	no	—
240	C₂²⋊Q₈	—	43	2	no	—
241	C₂²⋊Q₈	—	86	2	no	—
242	C₄×Q₈	—	43	2	no	—
243	C₄×Q₈	—	86	2	no	—
244	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
245	#117	—	86	2	no	—
246	C₂×C₄²	—	43	1	no	—
247	#107	—	43	1	no	—
248	#107	—	43	2	no	—
249	#107	C₄×D₄₃	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	43	2	no	—
252	C₄⋊C₈	—	86	2	no	—
253	C₄⋊C₈	—	86	2	no	—
254	C₄⋊C₈	—	86	2	no	—
255	C₄⋊C₈	—	86	2	no	—
256	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
257	#103	—	43	2	no	—
258	#103	—	43	2	no	—
259	#103	—	43	2	no	—
260	#103	C₄×D₄₃	1	2	yes	—
261	#97	—	2	2	no	—
262	#97	—	86	2	no	—
263	C₄×C₈	—	43	1	no	—
264	C₄×C₈	—	43	1	no	—
265	C₄×C₈	—	43	2	no	—
266	C₄×C₈	—	43	1	no	—
267	C₄×C₈	—	43	1	no	—
268	C₄×C₈	—	43	1	no	—
269	C₄×C₈	—	86	2	no	—
270	C₄×C₈	—	86	1	no	—
271	C₄×C₈	—	86	2	no	—
272	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
275	C₂×Q₈	—	43	2	no	—
276	C₂×Q₈	—	86	2	no	—
277	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	43	1	no	—
280	C₂²×C₄	—	43	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	86	4	no	—
283	Q₁₆	—	86	2	no	—
284	M₁₆	—	172	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	43	1	no	—
288	C₂×C₈	—	43	1	no	—
289	C₂×C₈	—	43	2	no	—
290	C₂×C₈	—	86	2	no	—
291	C₂×C₈	—	86	4	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	86	1	no	—
295	C₂×C₈	—	172	2	no	—
296	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₈	D₄×D₄₃	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	43	2	no	—
301	C₄⋊C₄	—	86	2	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	—	86	2	no	—
306	C₄⋊C₄	—	86	2	no	—
307	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
308	C₂²⋊C₄	—	86	2	no	—
309	C₂²⋊C₄	—	86	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	43	1	no	—
312	C₄²	—	43	1	no	—
313	C₄²	—	43	1	no	—
314	C₄²	—	86	2	no	—
315	C₄²	C₂³×D₄₃	1	1	yes	characteristic
316	C₂³	—	43	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	86	2	no	—
320	Q₈	—	86	2	no	—
321	Q₈	—	172	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	43	1	no	—
325	C₂×C₄	—	43	1	no	—
326	C₂×C₄	—	43	4	no	—
327	C₂×C₄	—	43	1	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	86	1	no	—
330	C₂×C₄	—	86	1	no	—
331	C₂×C₄	—	86	2	no	—
332	C₂×C₄	—	86	2	no	—
333	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
335	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	43	2	no	—
341	C₈	—	86	1	no	—
342	C₈	—	86	2	no	—
343	C₈	—	172	1	no	—
344	C₈	—	172	2	no	—
345	C₈	#13063333	1	2	yes	—
346	C₂²	—	43	2	no	—
347	C₂²	—	86	1	no	—
348	C₂²	—	86	1	no	—
349	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	43	2	no	—
354	C₄	—	43	2	no	—
355	C₄	—	86	1	no	—
356	C₄	—	172	2	no	—
357	C₄	—	172	2	no	—
358	C₄	#131×D₄₃	1	1	yes	characteristic
359	C₄	C₂×#13063333	1	1	yes	characteristic
360	C₄	#137×D₄₃	1	2	yes	—
361	C₂	—	86	2	no	—
362	C₂	#19143304	1	1	yes	characteristic
363	C₂	#441×D₄₃	1	1	yes	characteristic
364	C₂	#444×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1593	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂²×D₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1593	D₄₃
hypocenter	C₄₃	#17961
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #19133505	C₂	semidirect
4. #19133502	C₂	semidirect
5. #19133414	C₂	semidirect
6. #19133413	C₂	semidirect
7. #19133411	C₂	semidirect
11. C₄₃×#1593	C₂	semidirect
183. #1593	D₄₃	semidirect
227. C₄₃	#17961	semidirect

Maximal subgroups

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

subgroup	index
#19145178	2
#19143526	2
#19133505	2
#19133502	2
#19133414	2
#19133413	2
#19133411	2
#19133349	2
#19133048	2
#322×D₄₃	2
C₄₃×#1593	2
#17961	43