Subgroups — #25251650

Order 11008.

542 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145179	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19136791	C₂	1	2	yes	maximal
4	#19136738	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136737	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136642	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136527	C₂	1	2	yes	maximal
8	#19136466	C₂	1	2	yes	maximal, complemented
9	#19136463	C₂	1	2	yes	maximal, complemented
10	#19136397	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₃	C₂²	1	1	yes	characteristic
13	#14583454	—	2	1	no	—
14	C₄×Q₈×D₄₃	—	2	1	no	—
15	C₂×#13063271	C₂²	1	2	yes	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583249	—	2	1	no	—
18	#14583137	—	2	1	no	—
19	#14583137	C₂²	1	2	yes	—
20	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₄₃	—	2	1	no	—
22	Q₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₄₃	C₂²	1	2	yes	—
24	#14582993	C₂²	1	1	yes	characteristic
25	C₄×#10621885	—	2	1	no	—
26	#14582991	—	2	1	no	—
27	#107×D₄₃	C₂²	1	1	yes	characteristic
28	#103×D₄₃	—	2	1	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	#14582727	C₂²	1	2	yes	—
31	#14582726	C₂²	1	2	yes	—
32	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
33	#14582704	C₂²	1	2	yes	—
34	#14582703	C₂²	1	2	yes	—
35	#14582703	C₂²	1	2	yes	—
36	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
37	#14582674	C₂²	1	2	yes	—
38	C₂×Q₃₂×C₄₃	C₂²	1	2	yes	—
39	C₄₃×#494	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₃×#358	C₂²	1	2	yes	—
43	C₂×Q₈×D₄₃	—	2	1	no	—
44	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
45	C₂×#10621885	—	2	1	no	—
46	Q₁₆×D₄₃	—	2	2	no	—
47	Q₁₆×D₄₃	—	2	2	no	—
48	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
49	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
50	#13063325	—	2	1	no	—
51	#13063316	—	4	1	no	—
52	#13063305	—	4	1	no	—
53	#13063301	—	2	2	no	—
54	Q₈×Dic₄₃	—	2	1	no	—
55	Q₈×Dic₄₃	—	2	2	no	—
56	C₄×Dic₈₆	—	2	1	no	—
57	C₄×Dic₈₆	—	4	1	no	—
58	#13063290	—	2	1	no	—
59	#13063290	—	4	1	no	—
60	#13063290	—	4	1	no	—
61	#13063283	—	4	1	no	—
62	C₄⋊C₄×D₄₃	—	2	1	no	—
63	C₄⋊C₄×D₄₃	—	4	1	no	—
64	C₄²×D₄₃	—	4	1	no	—
65	C₄²×D₄₃	D₄	1	1	yes	characteristic
66	#13063271	—	2	4	no	—
67	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
68	#13063259	—	2	1	no	—
69	#13063259	C₂³	1	2	yes	—
70	#13063259	C₂³	1	1	yes	characteristic
71	#13063258	—	2	1	no	—
72	#13063250	—	2	1	no	—
73	#13063249	—	2	2	no	—
74	#13063249	—	2	2	no	—
75	#13063249	—	2	1	no	—
76	#13063248	—	2	1	no	—
77	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
78	C₈×Dic₄₃	D₄	1	1	yes	characteristic
79	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₄₃	—	2	1	no	—
81	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
82	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₄₃	—	2	1	no	—
84	C₄×Q₈×C₄₃	—	2	1	no	—
85	Q₃₂×C₄₃	—	2	4	no	—
86	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
87	#107×C₄₃	C₂³	1	1	yes	characteristic
88	#103×C₄₃	—	2	1	no	—
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₄₃	—	2	1	no	—
91	C₂×Q₈×C₄₃	—	2	2	no	—
92	Q₈×D₄₃	—	2	4	no	—
93	Q₈×D₄₃	—	4	2	no	—
94	C₂×C₄×D₄₃	—	4	1	no	—
95	C₂×C₄×D₄₃	—	4	1	no	—
96	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
97	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
98	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₈₆	—	2	1	no	—
100	C₂×Dic₈₆	—	4	1	no	—
101	C₂×Dic₈₆	—	4	1	no	—
102	Q₁₆×C₄₃	—	2	1	no	—
103	Q₁₆×C₄₃	—	2	2	no	—
104	Q₁₆×C₄₃	—	2	2	no	—
105	Q₁₆×C₄₃	—	4	2	no	—
106	Q₁₆×C₄₃	#41	1	2	yes	—
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₄₃	—	2	1	no	—
110	C₄⋊C₄×C₄₃	—	4	1	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₄⋊C₄×C₄₃	—	4	1	no	—
113	C₄²×C₄₃	—	4	1	no	—
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	#10621885	—	2	4	no	—
116	(C₂×D₄₃)⋊C₄	—	4	2	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	2	1	no	—
119	C₄⋊Dic₄₃	—	4	1	no	—
120	C₄⋊Dic₄₃	—	4	1	no	—
121	C₄⋊Dic₄₃	—	4	2	no	—
122	Dic₄₃⋊C₄	—	4	1	no	—
123	Dic₄₃⋊C₄	—	4	1	no	—
124	Dic₄₃⋊C₄	—	4	1	no	—
125	Dic₄₃⋊C₄	—	4	2	no	—
126	C₄×Dic₄₃	—	4	1	no	—
127	C₄×Dic₄₃	—	4	1	no	—
128	C₄×Dic₄₃	—	4	1	no	—
129	C₄×Dic₄₃	—	4	2	no	—
130	C₄×Dic₄₃	D₈	1	1	yes	characteristic
131	C₄×Dic₄₃	D₈	1	1	yes	characteristic
132	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
135	Dic₁₇₂	—	2	1	no	—
136	Dic₁₇₂	#41	1	2	yes	—
137	C₈×D₄₃	—	2	1	no	—
138	C₈×D₄₃	—	2	1	no	—
139	C₈×D₄₃	—	4	1	no	—
140	C₁₆×C₄₃	—	2	2	no	—
141	C₄₃⋊C₁₆	—	2	2	no	—
142	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
143	Q₈×C₄₃	—	2	2	no	—
144	Q₈×C₄₃	—	4	1	no	—
145	Q₈×C₄₃	—	4	2	no	—
146	Q₈×C₄₃	—	4	2	no	—
147	C₂×C₄×C₄₃	—	4	1	no	—
148	C₂×C₄×C₄₃	—	4	1	no	—
149	C₂×C₄×C₄₃	—	4	2	no	—
150	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	4	1	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
158	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
159	C₄×D₄₃	—	2	2	no	—
160	C₄×D₄₃	—	2	1	no	—
161	C₄×D₄₃	—	2	1	no	—
162	C₄×D₄₃	—	2	2	no	—
163	C₄×D₄₃	—	4	4	no	—
164	C₄×D₄₃	—	8	2	no	—
165	Dic₈₆	—	2	2	no	—
166	Dic₈₆	—	4	4	no	—
167	Dic₈₆	—	4	1	no	—
168	Dic₈₆	—	8	2	no	—
169	C₈×C₄₃	—	2	1	no	—
170	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
171	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
172	C₄₃⋊C₈	—	2	1	no	—
173	C₄₃⋊C₈	—	2	1	no	—
174	#33673	—	43	1	no	maximal, Sylow
175	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₃	—	2	1	no	—
177	C₂×D₄₃	—	2	1	no	—
178	C₂×D₄₃	#510	1	2	yes	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	4	2	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	—	8	1	no	—
183	C₄×C₄₃	#510	1	2	yes	—
184	C₄×C₄₃	#525	1	1	yes	characteristic
185	C₄×C₄₃	#575	1	1	yes	characteristic
186	Dic₄₃	—	2	1	no	—
187	Dic₄₃	—	2	1	no	—
188	Dic₄₃	—	2	1	no	—
189	Dic₄₃	—	4	2	no	—
190	Dic₄₃	—	8	2	no	—
191	Dic₄₃	—	8	1	no	—
192	#3170	—	43	1	no	—
193	C₂×C₄×Q₁₆	—	43	1	no	—
194	#2235	—	43	2	no	—
195	#2225	—	43	1	no	—
196	#2225	—	43	1	no	—
197	#2209	—	43	1	no	—
198	#2209	D₄₃	1	1	yes	characteristic, complemented
199	#2188	—	43	2	no	—
200	#2176	—	43	2	no	—
201	#2176	—	43	2	no	—
202	#2163	—	43	1	no	—
203	C₂×C₄₃	#3100	1	1	yes	characteristic
204	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
205	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
206	D₄₃	—	2	2	no	—
207	C₂²×Q₁₆	—	43	1	no	—
208	#533	—	86	1	no	—
209	C₂×C₄×Q₈	—	86	1	no	—
210	C₂×Q₃₂	—	43	2	no	—
211	C₂×Q₃₂	C₂×D₄₃	1	2	yes	—
212	#500	—	43	1	no	—
213	#494	C₂×D₄₃	1	1	yes	characteristic
214	#487	—	86	1	no	—
215	#467	—	43	2	no	—
216	#467	—	86	1	no	—
217	C₄×Q₁₆	—	43	2	no	—
218	C₄×Q₁₆	—	43	1	no	—
219	C₄×Q₁₆	—	43	1	no	—
220	C₄×Q₁₆	—	43	1	no	—
221	C₄×Q₁₆	—	86	1	no	—
222	C₄×Q₁₆	—	86	1	no	—
223	C₄×Q₁₆	—	86	1	no	—
224	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
225	C₂×#107	—	43	1	no	—
226	C₂×#103	—	86	1	no	—
227	C₂×C₄×C₈	—	43	1	no	—
228	#367	—	43	2	no	—
229	#367	—	43	2	no	—
230	C₄⋊C₁₆	—	43	1	no	—
231	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
232	#358	—	43	2	no	—
233	#358	—	43	2	no	—
234	#358	—	43	2	no	—
235	#358	C₂×D₄₃	1	2	yes	—
236	C₂²⋊C₁₆	—	43	2	no	—
237	#346	—	43	2	no	—
238	C₄₃	#33673	1	1	yes	characteristic, complemented, Sylow
239	C₂²×Q₈	—	86	1	no	—
240	C₂×Q₁₆	—	2	1	no	—
241	C₂×Q₁₆	—	43	1	no	—
242	C₂×Q₁₆	—	86	1	no	—
243	C₂×Q₁₆	—	86	2	no	—
244	C₂×Q₁₆	—	86	2	no	—
245	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
246	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
247	C₂²×C₈	—	43	1	no	—
248	C₂²×C₈	—	43	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	86	1	no	—
251	#125	—	172	1	no	—
252	C₂²⋊Q₈	—	86	2	no	—
253	C₂²⋊Q₈	—	172	1	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	86	1	no	—
256	C₄×Q₈	—	86	1	no	—
257	C₄×Q₈	—	86	2	no	—
258	C₄×Q₈	—	86	1	no	—
259	C₄×Q₈	—	172	1	no	—
260	C₄×Q₈	—	172	1	no	—
261	C₄×Q₈	—	172	1	no	—
262	#117	—	172	1	no	—
263	C₂×C₄⋊C₄	—	86	1	no	—
264	C₂×C₄⋊C₄	—	172	1	no	—
265	C₂×C₄²	—	43	1	no	—
266	C₂×C₄²	—	172	1	no	—
267	Q₃₂	—	2	4	no	—
268	Q₃₂	—	86	4	no	—
269	C₂×C₁₆	—	43	2	no	—
270	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
271	#107	—	43	1	no	—
272	#107	—	43	2	no	—
273	#107	—	86	1	no	—
274	#107	—	86	1	no	—
275	#107	C₂²×D₄₃	1	1	yes	characteristic
276	#103	—	2	1	no	—
277	#103	—	86	1	no	—
278	#103	—	86	1	no	—
279	#103	—	86	2	no	—
280	#103	—	86	2	no	—
281	#103	—	86	1	no	—
282	C₄×C₈	—	43	1	no	—
283	C₄×C₈	—	43	1	no	—
284	C₄×C₈	—	43	1	no	—
285	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	86	4	no	—
289	C₂×Q₈	—	86	1	no	—
290	C₂×Q₈	—	172	1	no	—
291	C₂×Q₈	—	172	2	no	—
292	C₂×Q₈	—	172	1	no	—
293	C₂²×C₄	—	43	1	no	—
294	C₂²×C₄	—	43	1	no	—
295	C₂²×C₄	—	43	1	no	—
296	C₂²×C₄	—	172	1	no	—
297	C₂²×C₄	—	172	1	no	—
298	Q₁₆	—	2	1	no	—
299	Q₁₆	—	2	2	no	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	4	2	no	—
302	Q₁₆	—	43	2	no	—
303	Q₁₆	—	86	4	no	—
304	Q₁₆	—	86	1	no	—
305	Q₁₆	#10621900	1	2	yes	—
306	C₂×C₈	—	43	1	no	—
307	C₂×C₈	—	43	1	no	—
308	C₂×C₈	—	86	1	no	—
309	C₂×C₈	—	86	1	no	—
310	C₂×C₈	—	172	1	no	—
311	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	86	1	no	—
318	C₄⋊C₄	—	86	2	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄⋊C₄	—	172	1	no	—
321	C₄⋊C₄	—	172	1	no	—
322	C₄⋊C₄	—	172	2	no	—
323	C₄⋊C₄	—	172	2	no	—
324	C₄⋊C₄	—	172	1	no	—
325	C₄⋊C₄	—	172	1	no	—
326	C₂²⋊C₄	—	172	2	no	—
327	C₄²	—	4	1	no	—
328	C₄²	—	43	1	no	—
329	C₄²	—	43	1	no	—
330	C₄²	—	43	1	no	—
331	C₄²	—	172	1	no	—
332	C₄²	—	172	1	no	—
333	C₄²	—	172	2	no	—
334	C₄²	—	172	1	no	—
335	C₄²	D₄×D₄₃	1	1	yes	characteristic
336	C₁₆	—	2	2	no	—
337	C₁₆	—	86	2	no	—
338	C₂³	—	43	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	1	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	86	2	no	—
344	Q₈	—	344	2	no	—
345	Q₈	—	172	4	no	—
346	Q₈	—	172	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	43	1	no	—
351	C₂×C₄	—	43	1	no	—
352	C₂×C₄	—	43	1	no	—
353	C₂×C₄	—	86	2	no	—
354	C₂×C₄	—	86	2	no	—
355	C₂×C₄	—	86	1	no	—
356	C₂×C₄	—	86	1	no	—
357	C₂×C₄	—	344	2	no	—
358	C₂×C₄	—	172	4	no	—
359	C₂×C₄	—	172	1	no	—
360	C₂×C₄	—	172	2	no	—
361	C₂×C₄	—	172	1	no	—
362	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
363	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	86	1	no	—
367	C₈	—	86	1	no	—
368	C₈	#41×D₄₃	1	1	yes	characteristic
369	C₈	C₂×#10621900	1	1	yes	characteristic
370	C₂²	—	43	2	no	—
371	C₂²	—	86	1	no	—
372	C₂²	—	86	1	no	—
373	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	86	1	no	—
379	C₄	—	86	1	no	—
380	C₄	—	86	1	no	—
381	C₄	—	344	1	no	—
382	C₄	—	344	2	no	—
383	C₄	—	172	2	no	—
384	C₄	#14583341	1	2	yes	—
385	C₄	#14583409	1	1	yes	characteristic
386	C₄	#14583695	1	1	yes	characteristic
387	C₂	—	86	2	no	—
388	C₂	#19144594	1	1	yes	characteristic
389	C₂	#510×D₄₃	1	1	yes	characteristic
390	C₂	C₂×#14583341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2209	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₄₃	C₂×D₈
hypercenter	#2209	D₄₃
hypocenter	C₄₃	#33673
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
4. #19136738	C₂	semidirect
5. #19136737	C₂	semidirect
6. #19136642	C₂	semidirect
8. #19136466	C₂	semidirect
9. #19136463	C₂	semidirect
11. C₄₃×#2209	C₂	semidirect
198. #2209	D₄₃	semidirect
238. C₄₃	#33673	semidirect

Maximal subgroups

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

subgroup	index
#19145179	2
C₄×Q₁₆×D₄₃	2
#19136791	2
#19136738	2
#19136737	2
#19136642	2
#19136527	2
#19136466	2
#19136463	2
#19136397	2
C₄₃×#2209	2
#33673	43