Subgroups — #25105390

Order 11008.

854 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583296	C₂	1	1	yes	maximal, characteristic
2	#19137166	C₂	1	2	yes	maximal
3	#19135554	C₂	1	2	yes	maximal
4	#19135338	C₂	1	1	yes	maximal, characteristic
5	#19135335	C₂	1	2	yes	maximal
6	#19134753	C₂	1	1	yes	maximal, characteristic
7	#19134495	C₂	1	1	yes	maximal, characteristic
8	#19134494	C₂	1	1	yes	maximal, characteristic
9	#19134491	C₂	1	2	yes	maximal
10	#97×Dic₄₃	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1944	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063324	C₂²	1	2	yes	—
13	C₂×#13063290	—	2	2	no	—
14	#14583296	C₄	1	8	yes	complemented
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×#13063258	C₂²	1	2	yes	—
17	C₂×#13063257	C₂²	1	1	yes	characteristic
18	C₂×#13063256	C₂²	1	2	yes	—
19	C₂×#13063236	C₂²	1	1	yes	characteristic
20	C₂×#13063235	C₂²	1	1	yes	characteristic
21	C₂×#13063235	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
24	#14582803	—	2	2	no	—
25	#14582797	C₂²	1	2	yes	—
26	#14582794	—	2	2	no	—
27	#14582768	C₂²	1	2	yes	—
28	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
29	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
30	#14582750	—	2	2	no	—
31	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
32	#14582662	C₂²	1	2	yes	—
33	#14582661	C₂²	1	2	yes	—
34	#14582661	C₂²	1	2	yes	—
35	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#389	C₂²	1	2	yes	—
39	C₄₃×#340	C₂²	1	2	yes	—
40	C₂²×Dic₈₆	—	2	2	no	—
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	#13063324	—	2	4	no	—
44	#13063324	C₂×C₄	1	8	yes	—
45	#13063290	—	4	8	no	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×Dic₄₃⋊C₄	—	2	2	no	—
49	C₂×Dic₄₃⋊C₄	—	2	2	no	—
50	C₂×Dic₄₃⋊C₄	—	2	2	no	—
51	C₂×Dic₄₃⋊C₄	—	2	2	no	—
52	C₂×C₄×Dic₄₃	—	2	2	no	—
53	C₂×C₄×Dic₄₃	—	2	2	no	—
54	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
55	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
56	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
58	#13063259	—	2	2	no	—
59	#13063258	—	2	4	no	—
60	#13063257	C₂×C₄	1	4	yes	—
61	#13063256	C₂×C₄	1	8	yes	—
62	#13063236	—	2	2	no	—
63	#13063235	Q₈	1	4	yes	complemented
64	#13063235	Q₈	1	4	yes	complemented
65	C₈×Dic₄₃	—	2	2	no	—
66	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
67	C₄₃⋊#95	—	2	2	no	—
68	C₄₃⋊#95	—	2	2	no	—
69	C₄₃⋊#95	—	2	2	no	—
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	#107×C₄₃	—	2	2	no	—
77	#106×C₄₃	C₂×C₄	1	4	yes	—
78	#97×C₄₃	—	2	2	no	—
79	#95×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₃	—	2	2	no	—
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
88	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
89	C₂×Dic₈₆	—	2	8	no	—
90	C₂×Dic₈₆	—	4	4	no	—
91	C₂×Dic₈₆	—	4	4	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	2	1	no	—
94	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₄₃	#41	1	2	yes	—
98	C₄⋊C₄×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	4	4	no	—
102	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₄₃	—	2	1	no	—
104	C₄²×C₄₃	—	2	1	no	—
105	C₄⋊Dic₄₃	—	2	2	no	—
106	C₄⋊Dic₄₃	—	2	2	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
109	Dic₄₃⋊C₄	—	2	8	no	—
110	Dic₄₃⋊C₄	—	4	4	no	—
111	Dic₄₃⋊C₄	—	4	4	no	—
112	Dic₄₃⋊C₄	—	4	4	no	—
113	C₄×Dic₄₃	—	2	4	no	—
114	C₄×Dic₄₃	—	2	4	no	—
115	C₄×Dic₄₃	—	4	4	no	—
116	C₄×Dic₄₃	—	4	4	no	—
117	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
119	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
120	C₄×Dic₄₃	#41	1	2	yes	—
121	C₂×C₄₃⋊C₈	—	2	1	no	—
122	C₂×C₄₃⋊C₈	—	2	1	no	—
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	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	—	2	4	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	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	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	4	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	C₄×D₄	1	4	yes	—
153	C₂×Dic₄₃	C₄⋊Q₈	1	4	yes	—
154	C₂×Dic₄₃	AGL(1,ℤ/8)	1	4	yes	—
155	C₂×Dic₄₃	#137	1	4	yes	—
156	Dic₈₆	—	4	16	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	C₄×Q₈	1	4	yes	complemented
159	C₄₃⋊C₈	—	2	2	no	—
160	C₄₃⋊C₈	C₄×Q₈	1	4	yes	complemented
161	#21727	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
164	C₂²×C₄₃	#532	1	1	yes	characteristic
165	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
166	C₂²×C₄₃	C₂×#137	1	2	yes	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	—	4	1	no	—
169	C₄×C₄₃	—	4	4	no	—
170	C₄×C₄₃	—	8	2	no	—
171	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
172	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
173	C₄×C₄₃	#533	1	1	yes	characteristic
174	C₄×C₄₃	#533	1	1	yes	characteristic
175	Dic₄₃	—	2	4	no	—
176	Dic₄₃	—	4	4	no	—
177	Dic₄₃	—	8	2	no	—
178	Dic₄₃	#440	1	8	yes	—
179	C₂×#498	—	43	1	no	—
180	C₄×C₄⋊Q₈	—	43	2	no	—
181	#1990	—	43	2	no	—
182	#1944	—	43	1	no	—
183	#1944	—	43	2	no	—
184	#1944	D₄₃	1	1	yes	characteristic
185	#1845	—	43	1	no	—
186	#1776	—	43	1	no	—
187	#1776	—	43	1	no	—
188	#1776	—	43	2	no	—
189	C₄×#97	—	43	1	no	—
190	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
191	C₂×C₄₃	C₂×#440	1	2	yes	—
192	C₂×C₄₃	#3165	1	2	yes	—
193	C₂×C₄₃	#3184	1	1	yes	characteristic
194	C₂×C₄₃	#3185	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	—	43	2	no	—
196	C₂×C₄×Q₈	—	86	2	no	—
197	#498	—	43	8	no	—
198	C₂×#107	—	43	1	no	—
199	C₂×#107	—	43	2	no	—
200	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
201	C₂×#106	—	43	1	no	—
202	C₂×#106	—	43	2	no	—
203	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
204	C₂×#97	—	43	1	no	—
205	C₂×#97	—	43	1	no	—
206	C₂×#97	—	43	1	no	—
207	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₄×C₈	—	43	1	no	—
209	C₂×C₄×C₈	—	43	1	no	—
210	#391	—	86	2	no	—
211	#389	—	43	2	no	—
212	#389	—	86	2	no	—
213	#389	C₂×D₄₃	1	2	yes	—
214	#384	—	43	2	no	—
215	C₄×C₄⋊C₄	—	43	2	no	—
216	C₄×C₄⋊C₄	—	43	2	no	—
217	C₄×C₄⋊C₄	—	86	2	no	—
218	C₄³	—	43	1	no	—
219	#340	—	43	2	no	—
220	#340	—	43	2	no	—
221	#340	—	43	2	no	—
222	#340	C₂×D₄₃	1	2	yes	—
223	C₄₃	#21727	1	1	yes	characteristic, complemented, Sylow
224	C₂²×Q₈	—	86	2	no	—
225	C₂²×C₈	—	43	1	no	—
226	C₂²×C₈	—	43	1	no	—
227	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
229	C₄⋊Q₈	—	43	8	no	—
230	C₄⋊Q₈	—	86	4	no	—
231	C₄×Q₈	—	172	8	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	43	2	no	—
234	C₂×C₄⋊C₄	—	43	2	no	—
235	C₂×C₄⋊C₄	—	86	2	no	—
236	C₂×C₄⋊C₄	—	86	2	no	—
237	C₂×C₄⋊C₄	—	86	2	no	—
238	C₂×C₄⋊C₄	—	86	2	no	—
239	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
240	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
241	C₂×C₄²	—	43	1	no	—
242	C₂×C₄²	—	43	1	no	—
243	C₂×C₄²	—	43	2	no	—
244	C₂×C₄²	—	43	2	no	—
245	C₂×C₄²	—	86	2	no	—
246	C₂×C₄²	—	86	2	no	—
247	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
248	#107	—	2	2	no	—
249	#107	—	86	4	no	—
250	#107	—	86	2	no	—
251	#106	—	43	4	no	—
252	#106	—	43	8	no	—
253	#106	C₄×D₄₃	1	4	yes	—
254	#97	—	2	2	no	—
255	#97	—	43	4	no	—
256	#97	—	43	4	no	—
257	#97	—	86	2	no	—
258	C₄×C₈	—	43	4	no	—
259	C₄×C₈	—	86	2	no	—
260	#95	—	2	2	no	—
261	#95	—	86	2	no	—
262	#95	—	86	2	no	—
263	#95	—	86	2	no	—
264	C₂×Q₈	—	86	8	no	—
265	C₂×Q₈	—	172	4	no	—
266	C₂×Q₈	—	172	4	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	43	2	no	—
270	C₂²×C₄	—	43	2	no	—
271	C₂²×C₄	—	86	2	no	—
272	C₂²×C₄	—	86	2	no	—
273	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
275	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	43	2	no	—
279	C₂×C₈	—	43	2	no	—
280	C₂×C₈	—	43	2	no	—
281	C₂×C₈	—	43	2	no	—
282	C₂×C₈	—	86	1	no	—
283	C₂×C₈	—	86	1	no	—
284	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
285	C₂×C₈	#10621900	1	2	yes	—
286	C₂×C₈	Q₈×D₄₃	1	2	yes	—
287	C₂×C₈	Q₈×D₄₃	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	4	no	—
292	C₄⋊C₄	—	43	4	no	—
293	C₄⋊C₄	—	86	8	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₄⋊C₄	—	86	2	no	—
296	C₄⋊C₄	—	86	2	no	—
297	C₄⋊C₄	—	172	4	no	—
298	C₄⋊C₄	—	172	4	no	—
299	C₄⋊C₄	—	172	4	no	—
300	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	43	2	no	—
304	C₄²	—	43	2	no	—
305	C₄²	—	43	2	no	—
306	C₄²	—	43	2	no	—
307	C₄²	—	86	4	no	—
308	C₄²	—	86	4	no	—
309	C₄²	—	172	4	no	—
310	C₄²	—	172	4	no	—
311	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
312	Q₈	—	172	16	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	43	4	no	—
324	C₂×C₄	—	43	4	no	—
325	C₂×C₄	—	43	4	no	—
326	C₂×C₄	—	43	4	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	86	4	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	—	172	2	no	—
333	C₂×C₄	—	172	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
336	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
337	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
338	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
339	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
340	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	43	4	no	—
343	C₈	—	86	2	no	—
344	C₈	#13063290	1	4	yes	complemented
345	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
346	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
347	C₂²	#14583444	1	1	yes	characteristic
348	C₂²	AGL(1,ℤ/8)×D₄₃	1	2	yes	—
349	C₂²	#137×D₄₃	1	2	yes	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	2	no	—
354	C₄	—	43	8	no	—
355	C₄	—	86	4	no	—
356	C₄	—	344	2	no	—
357	C₄	—	172	4	no	—
358	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
359	C₄	C₂×#13063290	1	1	yes	characteristic
360	C₄	#14583451	1	1	yes	characteristic
361	C₄	#14583455	1	1	yes	characteristic
362	C₂	#19137166	1	1	yes	characteristic
363	C₂	#440×D₄₃	1	2	yes	—
364	C₂	#19145140	1	2	yes	—
365	C₂	#19145328	1	1	yes	characteristic
366	C₂	#19145335	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1944	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1944	D₄₃
hypocenter	C₄₃	#21727
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1944

Complements

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

N	H ≅ G/N	type
14. #14583296	C₄	semidirect
63. #13063235	Q₈	semidirect
64. #13063235	Q₈	semidirect
158. C₈×C₄₃	C₄×Q₈	semidirect
160. C₄₃⋊C₈	C₄×Q₈	semidirect
223. C₄₃	#21727	semidirect
344. C₈	#13063290	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583296	2
#19137166	2
#19135554	2
#19135338	2
#19135335	2
#19134753	2
#19134495	2
#19134494	2
#19134491	2
#97×Dic₄₃	2
C₄₃×#1944	2
#21727	43