Subgroups — #25105372

Order 11008.

750 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	C₂×#14583302	C₂	1	1	yes	maximal, characteristic
2	#19137344	C₂	1	1	yes	maximal, characteristic
3	#19137341	C₂	1	1	yes	maximal, characteristic
4	#19135568	C₂	1	1	yes	maximal, characteristic
5	#19135332	C₂	1	2	yes	maximal
6	#19135327	C₂	1	2	yes	maximal
7	#19134757	C₂	1	1	yes	maximal, characteristic
8	#19134491	C₂	1	2	yes	maximal
9	#19134487	C₂	1	2	yes	maximal
10	#19134283	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1993	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063325	C₂²	1	1	yes	characteristic
13	#14583302	C₄	1	8	yes	complemented
14	C₂×#13063258	C₂²	1	2	yes	—
15	C₂×#13063258	C₂²	1	2	yes	—
16	C₂×#13063258	C₂²	1	2	yes	—
17	C₂×#13063256	C₂²	1	2	yes	—
18	C₂×#13063235	C₂²	1	1	yes	characteristic
19	C₂×#13063235	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
23	#14582806	—	2	2	no	—
24	#14582803	—	2	2	no	—
25	#14582798	C₂²	1	2	yes	—
26	#14582795	C₂²	1	2	yes	—
27	#14582785	—	2	2	no	—
28	#14582784	—	2	2	no	—
29	#14582765	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
31	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
32	#14582745	C₂²	1	1	yes	characteristic
33	#14582662	C₂²	1	2	yes	—
34	#14582662	C₂²	1	2	yes	—
35	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×#382	C₂²	1	1	yes	characteristic
38	C₄₃×#340	C₂²	1	2	yes	—
39	C₄₃×#340	C₂²	1	2	yes	—
40	C₂²×Dic₈₆	—	2	1	no	—
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	#13063325	—	2	2	no	—
44	#13063325	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	2	no	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
49	C₂×Dic₄₃⋊C₄	—	2	2	no	—
50	C₂×C₄×Dic₄₃	—	2	2	no	—
51	C₂×C₄×Dic₄₃	—	2	2	no	—
52	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
54	#13063258	—	2	4	no	—
55	#13063258	C₂×C₄	1	8	yes	—
56	#13063258	C₂×C₄	1	8	yes	—
57	#13063256	—	2	4	no	—
58	#13063235	—	2	2	no	—
59	#13063235	Q₈	1	4	yes	complemented
60	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
61	C₈×Dic₄₃	—	2	2	no	—
62	C₈×Dic₄₃	Q₈	1	4	yes	complemented
63	C₄₃⋊#95	—	2	1	no	—
64	C₄₃⋊#95	—	2	1	no	—
65	C₄₃⋊#95	—	2	2	no	—
66	C₄₃⋊#95	—	2	2	no	—
67	C₄₃⋊#95	—	2	2	no	—
68	C₄₃⋊#95	—	2	2	no	—
69	C₄₃⋊#95	D₄	1	2	yes	—
70	C₄₃⋊#95	D₄	1	2	yes	—
71	C₂²×Q₈×C₄₃	—	2	1	no	—
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₄₃	—	2	2	no	—
75	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	#97×C₄₃	—	2	2	no	—
81	#95×C₄₃	—	2	1	no	—
82	#95×C₄₃	—	2	1	no	—
83	C₂×Q₈×C₄₃	—	2	4	no	—
84	C₂×Q₈×C₄₃	—	8	2	no	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₄₃	—	2	2	no	—
91	C₂²×Dic₄₃	—	2	2	no	—
92	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
93	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₈₆	—	2	4	no	—
95	C₂×Dic₈₆	—	8	2	no	—
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	1	no	—
98	C₂×C₈×C₄₃	—	2	1	no	—
99	C₂×C₈×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
102	C₄⋊C₄×C₄₃	—	2	4	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
107	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₄₃	#41	1	2	yes	—
109	C₄⋊Dic₄₃	—	2	4	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	8	2	no	—
114	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
115	Dic₄₃⋊C₄	—	8	2	no	—
116	C₄×Dic₄₃	—	2	1	no	—
117	C₄×Dic₄₃	—	2	1	no	—
118	C₄×Dic₄₃	—	8	2	no	—
119	C₄×Dic₄₃	—	8	2	no	—
120	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
121	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
125	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
126	C₂×C₄₃⋊C₈	#41	1	2	yes	—
127	C₂×C₄₃⋊C₈	#41	1	2	yes	—
128	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₄₃	—	8	4	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	4	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	4	no	—
149	C₂×Dic₄₃	—	2	2	no	—
150	C₂×Dic₄₃	—	2	2	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	C₄⋊Q₈	1	4	yes	—
158	Dic₈₆	—	8	4	no	—
159	C₈×C₄₃	—	2	2	no	—
160	C₈×C₄₃	—	2	2	no	—
161	C₄₃⋊C₈	C₄×Q₈	1	4	yes	—
162	C₄₃⋊C₈	C₄×Q₈	1	4	yes	—
163	#21725	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄₃	#532	1	1	yes	characteristic
167	C₂²×C₄₃	#576	1	2	yes	—
168	C₂²×C₄₃	#578	1	2	yes	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	4	4	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₄×C₄₃	#533	1	1	yes	characteristic
176	C₄×C₄₃	#533	1	1	yes	characteristic
177	Dic₄₃	—	2	4	no	—
178	Dic₄₃	—	4	2	no	—
179	Dic₄₃	—	4	4	no	—
180	Dic₄₃	—	8	2	no	—
181	C₂×#500	—	43	1	no	—
182	#2331	—	43	1	no	—
183	#2331	—	43	1	no	—
184	#1993	—	43	1	no	—
185	#1993	D₄₃	1	1	yes	characteristic
186	#1943	—	43	2	no	—
187	#1942	—	43	2	no	—
188	#1846	—	43	1	no	—
189	#1776	—	43	2	no	—
190	C₄×#107	—	43	2	no	—
191	#1726	—	43	1	no	—
192	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
193	C₂×C₄₃	#2948	1	1	yes	characteristic
194	C₂×C₄₃	#2950	1	1	yes	characteristic
195	C₂×C₄₃	#3164	1	2	yes	—
196	C₂×C₄₃	#3175	1	2	yes	—
197	C₂×C₄⋊Q₈	—	43	1	no	—
198	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
199	#500	—	43	8	no	—
200	C₂×#107	—	43	2	no	—
201	C₂×#107	—	43	2	no	—
202	C₂×#107	—	43	2	no	—
203	C₂×#106	—	43	2	no	—
204	C₂×#97	—	43	1	no	—
205	C₂×#97	—	43	1	no	—
206	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
207	C₂×C₄×C₈	—	43	1	no	—
208	C₂×C₄×C₈	—	43	1	no	—
209	C₂×C₄×C₈	—	43	1	no	—
210	#391	—	86	2	no	—
211	#391	—	86	2	no	—
212	#389	—	43	2	no	—
213	#389	—	43	2	no	—
214	#387	—	86	2	no	—
215	#387	—	86	2	no	—
216	#382	—	43	1	no	—
217	#382	C₂×D₄₃	1	1	yes	characteristic
218	C₄×C₄⋊C₄	—	43	2	no	—
219	C₄×C₄⋊C₄	—	43	2	no	—
220	#376	—	43	1	no	—
221	#340	—	43	2	no	—
222	#340	—	43	2	no	—
223	#340	C₂×D₄₃	1	2	yes	—
224	#340	C₂×D₄₃	1	2	yes	—
225	C₄₃	#21725	1	1	yes	characteristic, complemented, Sylow
226	C₂²×Q₈	—	2	1	no	—
227	C₂²×Q₈	—	86	1	no	—
228	C₂²×C₈	—	43	1	no	—
229	C₂²×C₈	—	43	1	no	—
230	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊Q₈	—	2	2	no	—
233	C₄⋊Q₈	—	43	4	no	—
234	C₄⋊Q₈	—	86	2	no	—
235	C₄⋊Q₈	C₄×D₄₃	1	4	yes	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	43	2	no	—
238	C₂×C₄⋊C₄	—	43	2	no	—
239	C₂×C₄⋊C₄	—	86	1	no	—
240	C₂×C₄⋊C₄	—	86	2	no	—
241	C₂×C₄⋊C₄	—	86	2	no	—
242	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
244	C₂×C₄²	—	43	1	no	—
245	C₂×C₄²	—	43	1	no	—
246	C₂×C₄²	—	86	2	no	—
247	C₂×C₄²	—	86	2	no	—
248	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
249	#107	—	43	8	no	—
250	#107	—	43	8	no	—
251	#107	—	86	4	no	—
252	#106	—	86	4	no	—
253	#97	—	2	2	no	—
254	#97	—	43	4	no	—
255	#97	—	86	2	no	—
256	C₄×C₈	—	43	4	no	—
257	C₄×C₈	—	43	4	no	—
258	C₄×C₈	—	86	2	no	—
259	#95	—	2	1	no	—
260	#95	—	2	1	no	—
261	#95	—	43	2	no	—
262	#95	—	43	2	no	—
263	#95	—	86	2	no	—
264	#95	—	86	2	no	—
265	#95	—	86	2	no	—
266	#95	—	86	2	no	—
267	#95	—	86	1	no	—
268	#95	—	86	1	no	—
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	8	2	no	—
271	C₂×Q₈	—	86	4	no	—
272	C₂×Q₈	—	344	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	43	2	no	—
276	C₂²×C₄	—	43	2	no	—
277	C₂²×C₄	—	86	2	no	—
278	C₂²×C₄	—	86	2	no	—
279	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
281	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	43	2	no	—
287	C₂×C₈	—	43	2	no	—
288	C₂×C₈	—	43	2	no	—
289	C₂×C₈	—	43	2	no	—
290	C₂×C₈	—	43	2	no	—
291	C₂×C₈	—	43	2	no	—
292	C₂×C₈	Q₈×D₄₃	1	2	yes	—
293	C₂×C₈	Q₈×D₄₃	1	2	yes	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	43	4	no	—
299	C₄⋊C₄	—	86	4	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	2	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	344	2	no	—
304	C₄⋊C₄	—	344	2	no	—
305	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
306	C₄²	—	43	2	no	—
307	C₄²	—	43	2	no	—
308	C₄²	—	86	1	no	—
309	C₄²	—	86	1	no	—
310	C₄²	—	344	2	no	—
311	C₄²	—	344	2	no	—
312	C₄²	C₂×C₄×D₄₃	1	2	yes	—
313	C₄²	#10621900	1	2	yes	—
314	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
315	Q₈	—	8	4	no	—
316	Q₈	—	344	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	43	4	no	—
326	C₂×C₄	—	86	2	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	4	no	—
329	C₂×C₄	—	86	2	no	—
330	C₂×C₄	—	86	2	no	—
331	C₂×C₄	—	86	2	no	—
332	C₂×C₄	—	172	2	no	—
333	C₂×C₄	—	172	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
338	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
339	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
340	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
341	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
343	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	2	2	no	—
347	C₈	—	43	4	no	—
348	C₈	—	43	4	no	—
349	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
350	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
351	C₂²	#14583448	1	1	yes	characteristic
352	C₂²	#14583703	1	2	yes	—
353	C₂²	#14583715	1	2	yes	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	4	no	—
357	C₄	—	8	2	no	—
358	C₄	—	86	4	no	—
359	C₄	—	344	2	no	—
360	C₄	—	172	2	no	—
361	C₄	—	172	4	no	—
362	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
363	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
364	C₄	#14583451	1	1	yes	characteristic
365	C₄	#14583451	1	1	yes	characteristic
366	C₂	#19137164	1	1	yes	characteristic
367	C₂	#19143443	1	1	yes	characteristic
368	C₂	#19143459	1	1	yes	characteristic
369	C₂	#19145130	1	2	yes	—
370	C₂	#19145233	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1993	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1993	D₄₃
hypocenter	C₄₃	#21725
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1993

Complements

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

N	H ≅ G/N	type
13. #14583302	C₄	semidirect
59. #13063235	Q₈	semidirect
62. C₈×Dic₄₃	Q₈	semidirect
225. C₄₃	#21725	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583302	2
#19137344	2
#19137341	2
#19135568	2
#19135332	2
#19135327	2
#19134757	2
#19134491	2
#19134487	2
#19134283	2
C₄₃×#1993	2
#21725	43