Subgroups — #25251945

Order 11008.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145089	C₂	1	1	yes	maximal, characteristic
2	#19136842	C₂	1	2	yes	maximal, complemented
3	#19136834	C₂	1	2	yes	maximal, complemented
4	#19136829	C₂	1	2	yes	maximal, complemented
5	#19136824	C₂	1	2	yes	maximal, complemented
6	#19136314	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136305	C₂	1	1	yes	maximal, characteristic
8	#19136300	C₂	1	1	yes	maximal, characteristic
9	C₂×#14582702	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136192	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063358	—	2	1	no	—
13	C₂×#13063347	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
15	#14583475	—	2	1	no	—
16	C₂×#13063265	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	#14583293	—	2	2	no	—
19	#14583283	C₂²	1	2	yes	—
20	#14583279	C₂²	1	2	yes	—
21	#14582702	C₂²	1	2	yes	—
22	#14582702	C₂²	1	2	yes	—
23	#14582702	C₂²	1	1	yes	characteristic
24	#14582702	C₂²	1	1	yes	characteristic
25	#14582702	C₂²	1	2	yes	—
26	#14582702	C₂²	1	2	yes	—
27	#14582702	C₂²	1	2	yes	—
28	#14582699	C₂²	1	1	yes	characteristic
29	#14582699	C₂²	1	1	yes	characteristic
30	#14582699	C₂²	1	2	yes	—
31	#14582675	C₂²	1	2	yes	—
32	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
33	C₄₃×#544	—	2	1	no	—
34	C₄₃×#501	—	2	2	no	—
35	C₄₃×#500	C₂²	1	2	yes	—
36	C₄₃×#499	C₂²	1	2	yes	—
37	C₄₃×#428	—	2	1	no	—
38	C₄₃×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#405	C₂²	1	1	yes	characteristic
41	C₂×#10621898	—	2	1	no	—
42	C₂²×Dic₈₆	—	2	1	no	—
43	#13063358	—	4	4	no	—
44	#13063347	—	2	2	no	—
45	#13063347	—	2	2	no	—
46	C₂×Dic₁₇₂	—	2	2	no	—
47	C₂×Dic₁₇₂	—	2	2	no	—
48	C₂×Dic₁₇₂	—	2	1	no	—
49	C₂×Dic₁₇₂	—	2	2	no	—
50	C₂×Dic₁₇₂	C₂³	1	2	yes	—
51	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
52	C₂×#10621873	—	2	1	no	—
53	C₂×#10621873	—	2	2	no	—
54	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
55	#13063325	—	2	2	no	—
56	#13063313	—	2	2	no	—
57	#13063309	—	4	2	no	—
58	#13063305	—	4	1	no	—
59	#13063305	—	4	1	no	—
60	#13063265	—	2	2	no	—
61	C₂×C₄₃⋊C₁₆	—	2	2	no	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
65	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₄₃	—	2	2	no	—
68	C₄⋊Q₈×C₄₃	—	4	1	no	—
69	#125×C₄₃	—	2	2	no	—
70	#125×C₄₃	—	4	1	no	—
71	C₂²⋊Q₈×C₄₃	—	4	2	no	—
72	#117×C₄₃	—	2	1	no	—
73	#117×C₄₃	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
75	#107×C₄₃	—	2	2	no	—
76	#107×C₄₃	—	2	1	no	—
77	#107×C₄₃	C₂³	1	1	yes	characteristic
78	#107×C₄₃	C₂³	1	2	yes	—
79	#107×C₄₃	C₂³	1	1	yes	characteristic
80	#106×C₄₃	—	2	2	no	—
81	#106×C₄₃	—	2	1	no	—
82	#97×C₄₃	D₄	1	2	yes	—
83	C₄×C₈×C₄₃	C₂³	1	2	yes	—
84	C₂×Q₈×C₄₃	—	4	2	no	—
85	C₂²×C₄×C₄₃	—	4	1	no	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₃	—	4	1	no	—
88	C₂²×Dic₄₃	—	4	1	no	—
89	#10621898	—	4	2	no	—
90	#10621898	—	4	2	no	—
91	C₂×D₁₇₂	—	2	1	no	—
92	C₂×C₄×D₄₃	—	4	1	no	—
93	C₂×Dic₈₆	—	2	2	no	—
94	C₂×Dic₈₆	—	2	1	no	—
95	C₂×Dic₈₆	—	4	2	no	—
96	C₂×Dic₈₆	—	4	2	no	—
97	C₂×Dic₈₆	—	4	1	no	—
98	M₁₆×C₄₃	SD₁₆	1	4	yes	—
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	#41	1	2	yes	—
102	C₂×C₈×C₄₃	#41	1	2	yes	—
103	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₃	—	2	1	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	1	no	—
107	C₄⋊C₄×C₄₃	—	4	1	no	—
108	C₄⋊C₄×C₄₃	—	4	2	no	—
109	C₄⋊C₄×C₄₃	—	4	2	no	—
110	C₄⋊C₄×C₄₃	—	4	2	no	—
111	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
112	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
113	C₂²⋊C₄×C₄₃	—	4	1	no	—
114	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
115	C₄²×C₄₃	—	4	1	no	—
116	C₄²×C₄₃	C₂×D₄	1	2	yes	—
117	(C₂×D₄₃)⋊C₄	—	4	2	no	—
118	(C₂×D₄₃)⋊C₄	—	4	2	no	—
119	C₄⋊Dic₄₃	—	4	2	no	—
120	C₄⋊Dic₄₃	—	4	2	no	—
121	Dic₁₇₂	—	2	2	no	—
122	Dic₁₇₂	—	2	2	no	—
123	Dic₁₇₂	—	2	1	no	—
124	Dic₁₇₂	—	2	1	no	—
125	Dic₁₇₂	—	4	2	no	—
126	Dic₁₇₂	—	4	4	no	—
127	D₃₄₄	—	2	1	no	—
128	D₃₄₄	—	2	1	no	—
129	#10621873	—	4	2	no	—
130	#10621873	—	4	4	no	—
131	C₄₃⋊C₁₆	—	2	1	no	—
132	C₄₃⋊C₁₆	—	2	1	no	—
133	C₄₃⋊C₁₆	—	2	2	no	—
134	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₄₃	—	4	1	no	—
136	Q₈×C₄₃	—	16	2	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	16	1	no	—
141	C₂×C₄×C₄₃	#124	1	2	yes	—
142	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₄₃	—	8	2	no	—
148	C₂×Dic₄₃	—	4	1	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	8	2	no	—
151	D₁₇₂	—	4	1	no	—
152	D₁₇₂	—	4	1	no	—
153	C₄×D₄₃	—	8	2	no	—
154	Dic₈₆	—	4	1	no	—
155	Dic₈₆	—	4	2	no	—
156	Dic₈₆	—	4	2	no	—
157	Dic₈₆	—	4	1	no	—
158	Dic₈₆	—	8	2	no	—
159	C₈×C₄₃	#124	1	2	yes	—
160	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
161	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
163	#33703	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	#486	1	2	yes	—
167	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₄₃	—	8	1	no	—
169	C₂×D₄₃	—	8	1	no	—
170	C₂×D₄₃	—	8	1	no	—
171	C₄×C₄₃	—	2	2	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	—	8	1	no	—
175	C₄×C₄₃	—	8	1	no	—
176	C₄×C₄₃	#486	1	2	yes	—
177	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
178	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
179	Dic₄₃	—	8	2	no	—
180	Dic₄₃	—	8	1	no	—
181	#3164	D₄₃	1	1	yes	characteristic, complemented
182	#3156	—	43	1	no	—
183	#2245	—	43	2	no	—
184	#2243	—	43	2	no	—
185	#2242	—	43	2	no	—
186	#2241	—	43	2	no	—
187	#2141	—	43	1	no	—
188	#2139	—	43	1	no	—
189	#2138	—	43	1	no	—
190	C₂×#358	—	43	1	no	—
191	#2108	—	43	1	no	—
192	C₂×C₄₃	—	2	2	no	—
193	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
194	C₂×C₄₃	#3416	1	1	yes	characteristic
195	C₂×C₄₃	#3417	1	1	yes	characteristic
196	D₄₃	—	16	1	no	—
197	C₂×#137	—	86	1	no	—
198	C₂×#135	—	43	1	no	—
199	C₂²×Q₁₆	—	43	1	no	—
200	#544	—	2	1	no	—
201	#536	—	86	1	no	—
202	C₂×M₃₂	—	43	1	no	—
203	C₂²×C₁₆	—	43	1	no	—
204	#501	—	2	2	no	—
205	#500	C₂×D₄₃	1	2	yes	—
206	#499	C₂×D₄₃	1	2	yes	—
207	#497	—	86	2	no	—
208	#495	—	43	2	no	—
209	#494	—	43	2	no	—
210	#428	—	2	1	no	—
211	#427	C₂×D₄₃	1	1	yes	characteristic
212	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
213	#405	C₂×D₄₃	1	1	yes	characteristic
214	#358	—	43	2	no	—
215	#358	—	43	2	no	—
216	#358	—	43	1	no	—
217	#358	—	43	1	no	—
218	#358	—	43	2	no	—
219	#358	—	43	2	no	—
220	#358	—	43	2	no	—
221	#357	—	43	1	no	—
222	#357	—	43	1	no	—
223	#357	—	43	2	no	—
224	#346	—	43	2	no	—
225	C₄×C₁₆	—	43	2	no	—
226	C₄₃	#33703	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	—	86	1	no	—
228	C₂²×Q₈	—	86	1	no	—
229	#137	—	172	4	no	—
230	#135	—	86	2	no	—
231	#135	—	86	2	no	—
232	C₂×Q₁₆	—	43	2	no	—
233	C₂×Q₁₆	—	43	1	no	—
234	C₂×Q₁₆	—	86	2	no	—
235	C₂×Q₁₆	—	86	2	no	—
236	C₂×Q₁₆	—	86	1	no	—
237	C₂×Q₁₆	—	86	2	no	—
238	C₂×SD₁₆	—	86	1	no	—
239	C₂×SD₁₆	—	86	2	no	—
240	C₂×D₈	—	43	1	no	—
241	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
242	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₄⋊Q₈	—	2	2	no	—
244	C₄⋊Q₈	—	4	1	no	—
245	C₄⋊Q₈	—	86	2	no	—
246	#125	—	2	2	no	—
247	#125	—	4	1	no	—
248	#124	—	86	2	no	—
249	#123	—	172	2	no	—
250	C₂²⋊Q₈	—	4	2	no	—
251	C₂²⋊Q₈	—	172	1	no	—
252	C₂²⋊Q₈	—	172	1	no	—
253	#117	—	2	1	no	—
254	#117	C₂²⋊D₄₃	1	1	yes	characteristic
255	C₂×C₄⋊C₄	—	2	1	no	—
256	M₃₂	—	86	2	no	—
257	C₂×C₁₆	—	43	1	no	—
258	C₂×C₁₆	—	43	2	no	—
259	C₂×C₁₆	—	43	1	no	—
260	C₂×C₁₆	—	86	2	no	—
261	#107	—	2	1	no	—
262	#107	—	2	2	no	—
263	#107	C₂²×D₄₃	1	2	yes	—
264	#107	C₂²×D₄₃	1	1	yes	characteristic
265	#107	C₂²×D₄₃	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	2	2	no	—
268	#97	C₂²⋊D₄₃	1	2	yes	—
269	C₄×C₈	C₂²×D₄₃	1	2	yes	—
270	#41	—	172	2	no	—
271	#41	—	172	2	no	—
272	C₂×Q₈	—	4	2	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	86	1	no	—
275	C₂×Q₈	—	172	2	no	—
276	C₂×Q₈	—	172	2	no	—
277	C₂×Q₈	—	172	1	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂×D₄	—	172	1	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	172	1	no	—
282	C₂²×C₄	—	172	1	no	—
283	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
284	Q₁₆	—	86	2	no	—
285	Q₁₆	—	86	2	no	—
286	Q₁₆	—	86	1	no	—
287	Q₁₆	—	86	1	no	—
288	Q₁₆	—	172	2	no	—
289	Q₁₆	—	172	4	no	—
290	SD₁₆	—	172	2	no	—
291	SD₁₆	—	172	4	no	—
292	D₈	—	86	1	no	—
293	D₈	—	86	1	no	—
294	M₁₆	#10621884	1	4	yes	—
295	C₂×C₈	#10621902	1	2	yes	—
296	C₂×C₈	#10621902	1	2	yes	—
297	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
299	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	172	2	no	—
308	C₄⋊C₄	—	172	2	no	—
309	C₄⋊C₄	#10621882	1	1	yes	characteristic
310	C₄⋊C₄	#10621882	1	1	yes	characteristic
311	C₂²⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	172	2	no	—
313	C₂²⋊C₄	—	172	2	no	—
314	C₂²⋊C₄	#10621882	1	2	yes	—
315	C₄²	—	4	1	no	—
316	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
317	C₁₆	—	86	1	no	—
318	C₁₆	—	86	1	no	—
319	C₁₆	—	86	2	no	—
320	C₂³	—	172	1	no	—
321	C₂³	C₂×#10621882	1	1	yes	characteristic
322	Q₈	—	16	2	no	—
323	Q₈	—	344	2	no	—
324	Q₈	—	172	2	no	—
325	Q₈	—	172	1	no	—
326	Q₈	—	172	1	no	—
327	Q₈	—	172	2	no	—
328	D₄	—	344	2	no	—
329	D₄	—	172	1	no	—
330	D₄	—	172	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	16	1	no	—
335	C₂×C₄	—	344	2	no	—
336	C₂×C₄	—	344	2	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	#13063314	1	2	yes	—
340	C₂×C₄	C₂×#10621882	1	2	yes	—
341	C₂×C₄	C₂×#10621882	1	2	yes	—
342	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
343	C₂×C₄	C₂×#10621884	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
345	C₈	#13063314	1	2	yes	—
346	C₈	C₂×#10621884	1	4	yes	—
347	C₈	C₂×#10621902	1	1	yes	characteristic
348	C₈	C₂×#10621902	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	344	1	no	—
352	C₂²	—	344	1	no	—
353	C₂²	—	344	1	no	—
354	C₂²	#14583247	1	2	yes	—
355	C₂²	C₂²×#10621882	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	344	1	no	—
362	C₄	—	344	2	no	—
363	C₄	#14583247	1	2	yes	—
364	C₄	C₂×#13063314	1	1	yes	characteristic
365	C₄	C₂²×#10621884	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	688	1	no	—
368	C₂	C₂×#14583247	1	1	yes	characteristic
369	C₂	#19147742	1	1	yes	characteristic
370	C₂	#19147746	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3164	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#13063347	C₂²
hypercenter	#3164	D₄₃
hypocenter	C₄₃	#33703
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #19136842	C₂	semidirect
3. #19136834	C₂	semidirect
4. #19136829	C₂	semidirect
5. #19136824	C₂	semidirect
6. #19136314	C₂	semidirect
9. C₂×#14582702	C₂	semidirect
10. #19136192	C₂	semidirect
11. C₄₃×#3164	C₂	semidirect
181. #3164	D₄₃	semidirect
226. C₄₃	#33703	semidirect

Maximal subgroups

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

subgroup	index
#19145089	2
#19136842	2
#19136834	2
#19136829	2
#19136824	2
#19136314	2
#19136305	2
#19136300	2
C₂×#14582702	2
#19136192	2
C₄₃×#3164	2
#33703	43