Subgroups — #25251958

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145108	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19136833	C₂	1	2	yes	maximal, complemented
4	#19136830	C₂	1	2	yes	maximal, complemented
5	#19136829	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136827	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136826	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136313	C₂	1	2	yes	maximal, complemented
9	#19136309	C₂	1	2	yes	maximal
10	C₄×#13063264	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2242	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×D₄₃	—	2	1	no	—
14	C₂×D₈×D₄₃	C₂²	1	1	yes	characteristic
15	#14583455	—	2	1	no	—
16	#124×D₄₃	—	2	1	no	—
17	C₂×#13063264	C₂²	1	2	yes	—
18	#14583303	C₂²	1	2	yes	complemented
19	#14583301	C₂²	1	1	yes	characteristic, complemented
20	#14583298	—	2	1	no	—
21	#14583296	—	2	1	no	—
22	#14583284	C₂²	1	1	yes	characteristic
23	#14583283	C₂²	1	1	yes	characteristic
24	#14583282	C₂²	1	1	yes	characteristic
25	#14583281	—	2	1	no	—
26	#14583280	—	2	1	no	—
27	#14582936	C₂²	1	2	yes	—
28	#14582934	—	2	1	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	#14582704	C₂²	1	2	yes	complemented
31	#14582703	C₂²	1	2	yes	complemented
32	#14582702	C₂²	1	2	yes	complemented
33	#14582701	C₂²	1	2	yes	—
34	#14582700	C₂²	1	2	yes	—
35	#14582699	C₂²	1	2	yes	—
36	#14582674	C₂²	1	2	yes	complemented
37	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
38	C₄₃×#499	C₂²	1	1	yes	characteristic, complemented
39	C₄₃×#495	C₂²	1	1	yes	characteristic
40	C₄₃×#358	C₂²	1	2	yes	complemented
41	C₄₃×#357	C₂²	1	2	yes	—
42	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₄₃	—	2	1	no	—
44	C₂×D₄×D₄₃	—	2	1	no	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621885	—	2	1	no	—
47	Q₁₆×D₄₃	—	2	4	no	—
48	C₂×#10621873	—	2	1	no	—
49	C₂×#10621883	—	2	1	no	—
50	C₂×#10621884	—	2	1	no	—
51	SD₁₆×D₄₃	—	4	4	no	—
52	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
53	C₂×#10621882	—	2	1	no	—
54	D₈×D₄₃	—	2	4	no	—
55	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
57	#13063324	—	2	2	no	—
58	#13063317	—	2	1	no	—
59	#13063314	—	2	1	no	—
60	#13063313	—	2	1	no	—
61	#13063312	—	2	1	no	—
62	#13063311	—	4	2	no	—
63	#13063303	—	4	2	no	—
64	#13063290	—	4	2	no	—
65	#13063283	—	2	2	no	—
66	C₂²⋊C₄×D₄₃	—	4	2	no	—
67	C₄²×D₄₃	D₄	1	1	yes	characteristic
68	#13063264	—	2	4	no	—
69	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
70	#13063259	C₂³	1	2	yes	—
71	#13063258	—	2	2	no	—
72	#13063257	—	2	1	no	—
73	#13063256	—	2	2	no	—
74	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
75	C₈×Dic₄₃	D₄	1	1	yes	characteristic
76	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₄₃	—	2	1	no	—
79	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
80	#125×C₄₃	—	2	1	no	—
81	#124×C₄₃	—	2	1	no	—
82	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
83	#107×C₄₃	C₂³	1	2	yes	—
84	#106×C₄₃	—	2	1	no	—
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₂³×D₄₃	—	4	1	no	—
87	C₂×Q₈×C₄₃	—	2	1	no	—
88	C₂×D₄×C₄₃	—	2	1	no	—
89	C₂×C₂²⋊D₄₃	—	4	1	no	—
90	Q₈×D₄₃	—	4	2	no	—
91	Q₈×D₄₃	—	4	2	no	—
92	D₄×D₄₃	—	4	2	no	—
93	D₄×D₄₃	—	4	2	no	—
94	C₂×D₁₇₂	—	2	1	no	—
95	C₂×C₄×D₄₃	—	4	1	no	—
96	C₂×C₄×D₄₃	D₈	1	2	yes	—
97	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×Dic₈₆	—	2	1	no	—
99	C₂×Dic₈₆	—	4	2	no	—
100	C₂×Dic₈₆	—	4	1	no	—
101	Q₁₆×C₄₃	—	2	2	no	—
102	SD₁₆×C₄₃	—	4	2	no	—
103	D₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	2	no	—
108	C₂²⋊C₄×C₄₃	—	4	2	no	—
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂²⋊Dic₄₃	—	4	2	no	—
111	#10621885	—	4	2	no	—
112	#10621884	—	4	2	no	—
113	#10621883	—	4	2	no	—
114	#10621882	—	4	2	no	—
115	(C₂×D₄₃)⋊C₄	—	4	2	no	—
116	(C₂×D₄₃)⋊C₄	—	4	2	no	—
117	(C₂×D₄₃)⋊C₄	—	4	2	no	—
118	C₄⋊Dic₄₃	—	2	2	no	—
119	Dic₄₃⋊C₄	—	4	2	no	—
120	Dic₄₃⋊C₄	—	4	2	no	—
121	C₄×Dic₄₃	—	4	2	no	—
122	C₄×Dic₄₃	D₈	1	2	yes	—
123	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
126	Dic₁₇₂	—	2	2	no	—
127	D₃₄₄	—	2	2	no	—
128	#10621873	—	4	2	no	—
129	C₈×D₄₃	SD₁₆	1	4	yes	—
130	C₈×D₄₃	#41	1	4	yes	—
131	C₁₆×C₄₃	—	2	2	no	—
132	C₄₃⋊C₁₆	—	2	2	no	—
133	C₂³×C₄₃	—	4	1	no	—
134	C₂²×D₄₃	—	4	1	no	—
135	C₂²×D₄₃	—	8	4	no	—
136	C₂²×D₄₃	—	8	2	no	—
137	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₄₃	—	4	1	no	—
139	Q₈×C₄₃	—	4	1	no	—
140	D₄×C₄₃	—	4	1	no	—
141	D₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	1	no	—
144	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₄₃	—	8	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	1	no	—
149	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
150	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
151	D₁₇₂	—	4	1	no	—
152	D₁₇₂	—	4	1	no	—
153	C₄×D₄₃	—	4	2	no	—
154	C₄×D₄₃	—	8	2	no	—
155	C₄×D₄₃	#124	1	2	yes	—
156	C₄×D₄₃	C₂×SD₁₆	1	2	yes	—
157	Dic₈₆	—	4	1	no	—
158	Dic₈₆	—	4	1	no	—
159	Dic₈₆	—	8	4	no	—
160	Dic₈₆	—	8	2	no	—
161	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
162	C₈×C₄₃	C₂×#41	1	2	yes	—
163	C₄₃⋊C₈	#124	1	2	yes	—
164	C₄₃⋊C₈	C₂×SD₁₆	1	2	yes	—
165	#33706	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	8	1	no	—
167	C₂²×C₄₃	—	8	2	no	—
168	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₄₃	—	2	2	no	—
170	C₂×D₄₃	—	8	1	no	—
171	C₂×D₄₃	—	8	2	no	—
172	C₂×D₄₃	—	16	4	no	—
173	C₂×D₄₃	#486	1	2	yes	—
174	C₄×C₄₃	—	2	2	no	—
175	C₄×C₄₃	—	8	1	no	—
176	C₄×C₄₃	—	8	2	no	—
177	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
178	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
179	Dic₄₃	—	2	2	no	—
180	Dic₄₃	—	8	1	no	—
181	Dic₄₃	—	8	2	no	—
182	Dic₄₃	#486	1	2	yes	—
183	#3160	—	43	1	no	—
184	C₂×#495	—	43	1	no	—
185	#2243	—	43	2	no	—
186	#2243	—	43	2	no	—
187	#2242	—	43	1	no	—
188	#2242	—	43	1	no	—
189	#2242	—	43	1	no	—
190	#2242	D₄₃	1	1	yes	characteristic, complemented
191	#2141	—	43	2	no	—
192	#2140	—	43	2	no	—
193	C₄×M₃₂	—	43	1	no	—
194	C₂×C₄₃	—	16	1	no	—
195	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
196	C₂×C₄₃	#3414	1	2	yes	—
197	D₄₃	—	2	2	no	—
198	D₄₃	—	16	1	no	—
199	C₂²×Q₁₆	—	43	1	no	—
200	C₂²×SD₁₆	—	86	1	no	—
201	C₂²×D₈	—	43	1	no	—
202	#533	—	86	1	no	—
203	C₂×#124	—	86	1	no	—
204	C₂×M₃₂	—	43	2	no	—
205	#500	—	43	2	no	—
206	#499	—	43	1	no	—
207	#499	—	86	1	no	—
208	#499	C₂×D₄₃	1	1	yes	characteristic, complemented
209	#498	—	86	1	no	—
210	#495	—	43	1	no	—
211	#495	—	43	1	no	—
212	#495	—	43	1	no	—
213	#495	—	86	1	no	—
214	#495	—	86	1	no	—
215	#495	C₂×D₄₃	1	1	yes	characteristic
216	#427	—	43	2	no	—
217	#427	—	86	1	no	—
218	C₂×C₄×C₈	—	43	1	no	—
219	#358	—	43	2	no	—
220	#358	—	43	2	no	—
221	#358	—	43	2	no	—
222	#358	C₂×D₄₃	1	2	yes	complemented
223	#357	—	43	2	no	—
224	#357	—	43	2	no	—
225	#357	—	43	2	no	—
226	#357	C₂×D₄₃	1	2	yes	—
227	#346	—	43	2	no	—
228	C₄×C₁₆	—	43	1	no	—
229	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
230	C₄₃	#33706	1	1	yes	characteristic, complemented, Sylow
231	C₂²×Q₈	—	86	1	no	—
232	C₂²×D₄	—	86	1	no	—
233	C₂×Q₁₆	—	43	1	no	—
234	C₂×Q₁₆	—	86	4	no	—
235	C₂×Q₁₆	—	86	1	no	—
236	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×SD₁₆	—	86	1	no	—
239	C₂×SD₁₆	—	86	1	no	—
240	C₂×SD₁₆	—	86	1	no	—
241	C₂×SD₁₆	—	172	4	no	—
242	C₂×D₈	—	43	1	no	—
243	C₂×D₈	—	86	4	no	—
244	C₂×D₈	—	86	1	no	—
245	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂²×C₈	—	43	1	no	—
247	C₂²×C₈	—	43	1	no	—
248	C₄⋊Q₈	—	86	2	no	—
249	#125	—	2	1	no	—
250	#125	—	86	1	no	—
251	#124	—	2	1	no	—
252	#124	—	86	1	no	—
253	#124	—	86	1	no	—
254	#124	—	86	1	no	—
255	#124	—	172	2	no	—
256	C₂²⋊Q₈	—	172	2	no	—
257	C₄×Q₈	—	172	2	no	—
258	#117	—	86	2	no	—
259	C₂×C₂²⋊C₄	—	172	2	no	—
260	C₂×C₄²	—	43	1	no	—
261	M₃₂	—	86	4	no	—
262	C₂×C₁₆	—	43	2	no	—
263	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
264	#107	—	43	2	no	—
265	#107	—	86	2	no	—
266	#107	C₂²×D₄₃	1	2	yes	—
267	#106	—	2	1	no	—
268	#106	—	86	2	no	—
269	#106	—	86	1	no	—
270	C₄×C₈	—	43	1	no	—
271	C₄×C₈	—	43	1	no	—
272	C₄×C₈	—	43	1	no	—
273	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
274	C₂⁴	—	172	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	86	1	no	—
277	C₂×Q₈	—	172	2	no	—
278	C₂×Q₈	—	172	1	no	—
279	C₂×Q₈	—	172	2	no	—
280	C₂×Q₈	—	172	2	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	86	1	no	—
283	C₂×D₄	—	172	2	no	—
284	C₂×D₄	—	172	2	no	—
285	C₂×D₄	—	172	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	43	2	no	—
288	C₂²×C₄	—	172	1	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	86	2	no	—
291	Q₁₆	—	172	2	no	—
292	SD₁₆	—	4	2	no	—
293	SD₁₆	—	172	2	no	—
294	SD₁₆	—	172	2	no	—
295	SD₁₆	—	172	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	86	2	no	—
298	D₈	—	172	2	no	—
299	C₂×C₈	—	43	1	no	—
300	C₂×C₈	—	43	1	no	—
301	C₂×C₈	—	43	4	no	—
302	C₂×C₈	—	43	4	no	—
303	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	172	2	no	—
309	C₄⋊C₄	—	172	2	no	—
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	—	172	2	no	—
312	C₂²⋊C₄	—	172	2	no	—
313	C₂²⋊C₄	—	172	2	no	—
314	C₂²⋊C₄	—	172	2	no	—
315	C₄²	—	43	2	no	—
316	C₄²	—	43	1	no	—
317	C₄²	—	172	2	no	—
318	C₄²	D₄×D₄₃	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	86	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	43	1	no	—
323	C₂³	—	344	2	no	—
324	C₂³	—	344	4	no	—
325	C₂³	—	172	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	344	2	no	—
329	Q₈	—	344	4	no	—
330	Q₈	—	172	1	no	—
331	Q₈	—	172	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	4	1	no	—
334	D₄	—	344	2	no	—
335	D₄	—	172	1	no	—
336	D₄	—	172	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	43	1	no	—
340	C₂×C₄	—	43	2	no	—
341	C₂×C₄	—	43	2	no	—
342	C₂×C₄	—	43	2	no	—
343	C₂×C₄	—	344	2	no	—
344	C₂×C₄	—	172	2	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	—	172	1	no	—
347	C₂×C₄	D₈×D₄₃	1	2	yes	—
348	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
349	C₈	—	43	2	no	—
350	C₈	—	43	2	no	—
351	C₈	SD₁₆×D₄₃	1	2	yes	—
352	C₈	#41×D₄₃	1	2	yes	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	43	2	no	—
356	C₂²	—	86	2	no	—
357	C₂²	—	344	1	no	—
358	C₂²	—	344	2	no	—
359	C₂²	—	688	4	no	—
360	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	43	2	no	—
365	C₄	—	86	2	no	—
366	C₄	—	344	1	no	—
367	C₄	—	344	2	no	—
368	C₄	#124×D₄₃	1	1	yes	characteristic
369	C₄	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
370	C₂	—	16	1	no	—
371	C₂	—	86	2	no	—
372	C₂	—	688	1	no	—
373	C₂	#486×D₄₃	1	1	yes	characteristic
374	C₂	#19147728	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2242	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×D₈×D₄₃	C₂²
hypercenter	#2242	D₄₃
hypocenter	C₄₃	#33706
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

Complements

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

N	H ≅ G/N	type
1. #19145108	C₂	semidirect
3. #19136833	C₂	semidirect
4. #19136830	C₂	semidirect
5. #19136829	C₂	semidirect
6. #19136827	C₂	semidirect
7. #19136826	C₂	semidirect
8. #19136313	C₂	semidirect
10. C₄×#13063264	C₂	semidirect
11. C₄₃×#2242	C₂	semidirect
18. #14583303	C₂²	semidirect
19. #14583301	C₂²	semidirect
30. #14582704	C₂²	semidirect
31. #14582703	C₂²	semidirect
32. #14582702	C₂²	semidirect
36. #14582674	C₂²	semidirect
37. C₄×C₄₃⋊C₁₆	C₂²	semidirect
38. C₄₃×#499	C₂²	semidirect
40. C₄₃×#358	C₂²	semidirect
42. C₄×C₁₆×C₄₃	C₂²	semidirect
190. #2242	D₄₃	semidirect
208. #499	C₂×D₄₃	semidirect
222. #358	C₂×D₄₃	semidirect
229. C₄×C₁₆	C₂×D₄₃	semidirect
230. C₄₃	#33706	semidirect

Maximal subgroups

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

subgroup	index
#19145108	2
#495×D₄₃	2
#19136833	2
#19136830	2
#19136829	2
#19136827	2
#19136826	2
#19136313	2
#19136309	2
C₄×#13063264	2
C₄₃×#2242	2
#33706	43