Subgroups — #25097007

Order 11008.

1126 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582936	C₂	1	2	yes	maximal
2	C₂×#14582934	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₄₃	C₂	1	1	yes	maximal, characteristic
4	#19137001	C₂	1	2	yes	maximal
5	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134757	C₂	1	2	yes	maximal, complemented
7	#19134753	C₂	1	2	yes	maximal, complemented
8	#19134750	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
12	C₂×#13063283	C₂²	1	4	yes	—
13	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
14	#14582936	—	2	8	no	—
15	#14582934	—	2	8	no	—
16	C₂×#13063259	C₂²	1	2	yes	—
17	C₂×#13063258	C₂²	1	4	yes	—
18	C₂×#13063257	C₂²	1	2	yes	—
19	C₂×#13063256	C₂²	1	4	yes	—
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	C₄×C₈×D₄₃	C₄	1	8	yes	complemented
24	#14582768	C₂²	1	4	yes	—
25	#14582765	C₂²	1	2	yes	—
26	#14582758	—	2	4	no	—
27	#14582750	—	2	4	no	—
28	C₂×#107×C₄₃	C₂²	1	2	yes	—
29	C₂×#106×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#382	C₂²	1	2	yes	—
32	C₂²×C₄×D₄₃	D₄	1	2	yes	—
33	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₄₃	D₄	1	4	yes	—
39	C₂×C₈×D₄₃	D₄	1	4	yes	—
40	C₂×C₈×D₄₃	Q₈	1	4	yes	—
41	C₂×C₈×D₄₃	Q₈	1	4	yes	—
42	#13063283	—	2	8	no	—
43	#13063283	—	2	8	no	—
44	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
45	C₂×Dic₄₃⋊C₄	—	2	8	no	—
46	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
47	C₂×C₄×Dic₄₃	—	2	4	no	—
48	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
49	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
50	C₄²×D₄₃	C₂×C₄	1	4	yes	—
51	C₄²×D₄₃	Q₈	1	4	yes	—
52	#13063259	—	2	4	no	—
53	#13063258	—	2	8	no	—
54	#13063257	—	2	4	no	—
55	#13063256	—	2	8	no	—
56	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
57	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
58	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
59	C₄₃⋊#95	—	2	4	no	—
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	#107×C₄₃	—	2	4	no	—
65	#106×C₄₃	—	2	4	no	—
66	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
67	#95×C₄₃	—	2	4	no	—
68	C₂³×D₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	—	2	4	no	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	2	4	no	—
73	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₄₃	—	2	8	no	—
76	C₂×C₄×D₄₃	C₄⋊C₄	1	8	yes	—
77	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
81	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₄⋊C₄×C₄₃	—	2	4	no	—
91	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
93	(C₂×D₄₃)⋊C₄	—	4	8	no	—
94	C₄⋊Dic₄₃	—	2	4	no	—
95	C₄⋊Dic₄₃	—	2	4	no	—
96	Dic₄₃⋊C₄	—	4	16	no	—
97	C₄×Dic₄₃	—	4	8	no	—
98	C₄×Dic₄₃	C₄⋊C₄	1	8	yes	—
99	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
101	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
105	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
106	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
107	C₈×D₄₃	C₄⋊C₄	1	16	yes	complemented
108	C₈×D₄₃	C₄⋊C₄	1	16	yes	complemented
109	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	4	no	—
111	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂²×D₄₃	#127	1	2	yes	—
113	C₂²×D₄₃	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	4	4	no	—
116	C₂×C₄×C₄₃	—	4	8	no	—
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×Dic₄₃	—	4	4	no	—
125	C₂×Dic₄₃	—	4	8	no	—
126	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×Dic₄₃	#127	1	2	yes	—
129	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
130	C₂×Dic₄₃	#135	1	8	yes	—
131	C₄×D₄₃	—	2	16	no	—
132	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
133	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₄×D₄₃	C₄⋊Q₈	1	4	yes	—
135	C₄×D₄₃	C₄⋊Q₈	1	4	yes	—
136	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
139	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
140	#20797	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
143	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂²×C₄₃	C₂×#135	1	4	yes	—
145	C₂×D₄₃	—	2	4	no	—
146	C₂×D₄₃	—	2	4	no	—
147	C₂×D₄₃	—	2	4	no	—
148	C₂×D₄₃	#384	1	4	yes	—
149	C₄×C₄₃	—	2	4	no	—
150	C₄×C₄₃	—	8	4	no	—
151	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	Dic₄₃	—	8	4	no	—
156	Dic₄₃	#384	1	4	yes	—
157	Dic₄₃	#427	1	8	yes	complemented
158	C₂×#427	—	43	2	no	—
159	C₂×#427	—	43	2	no	—
160	C₂²×C₄×C₈	—	43	1	no	—
161	#2285	—	43	2	no	—
162	#1846	—	43	1	no	—
163	#1846	—	43	2	no	—
164	#1846	D₄₃	1	1	yes	characteristic, complemented
165	#1845	—	43	2	no	—
166	#1844	—	43	2	no	—
167	C₂×C₄₃	C₂×#384	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#427	1	2	yes	—
169	C₂×C₄₃	#3147	1	2	yes	—
170	C₂×C₄₃	#3160	1	2	yes	—
171	D₄₃	—	2	4	no	—
172	C₂³×C₈	—	43	1	no	—
173	C₂³×C₈	—	43	1	no	—
174	C₂×#117	—	43	4	no	—
175	C₂²×C₄²	—	43	1	no	—
176	#427	—	86	8	no	—
177	#427	—	86	8	no	—
178	C₂×#107	—	43	2	no	—
179	C₂×#107	—	43	4	no	—
180	C₂×#107	C₂×D₄₃	1	2	yes	—
181	C₂×#106	—	43	2	no	—
182	C₂×#106	—	43	4	no	—
183	C₂×#106	C₂×D₄₃	1	2	yes	—
184	C₂×C₄×C₈	—	43	1	no	—
185	C₂×C₄×C₈	—	43	1	no	—
186	C₂×C₄×C₈	—	43	1	no	—
187	C₂×C₄×C₈	—	43	8	no	—
188	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
189	#384	—	43	4	no	—
190	#382	—	43	2	no	—
191	#382	C₂×D₄₃	1	2	yes	—
192	#380	—	86	4	no	—
193	C₄×C₄⋊C₄	—	86	4	no	—
194	C₄₃	#20797	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	43	1	no	—
196	C₂³×C₄	—	43	2	no	—
197	C₂²×C₈	—	43	1	no	—
198	C₂²×C₈	—	43	1	no	—
199	C₂²×C₈	—	43	4	no	—
200	C₂²×C₈	—	43	4	no	—
201	C₂²×C₈	—	43	4	no	—
202	C₂²×C₈	—	43	4	no	—
203	C₂²×C₈	—	43	4	no	—
204	C₂²×C₈	—	43	4	no	—
205	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	#117	—	86	8	no	—
208	#117	—	86	8	no	—
209	C₂×C₄⋊C₄	—	43	4	no	—
210	C₂×C₄⋊C₄	—	86	8	no	—
211	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
212	C₂×C₂²⋊C₄	—	86	4	no	—
213	C₂×C₄²	—	43	2	no	—
214	C₂×C₄²	—	43	1	no	—
215	C₂×C₄²	—	43	4	no	—
216	C₂×C₄²	—	43	4	no	—
217	C₂×C₄²	—	86	4	no	—
218	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
219	#107	—	2	4	no	—
220	#107	—	86	8	no	—
221	#107	—	86	4	no	—
222	#106	—	2	4	no	—
223	#106	—	86	8	no	—
224	#106	—	86	4	no	—
225	C₄×C₈	—	43	4	no	—
226	C₄×C₈	—	43	4	no	—
227	C₄×C₈	—	43	4	no	—
228	C₄×C₈	C₄×D₄₃	1	4	yes	—
229	#95	—	2	4	no	—
230	#95	—	86	4	no	—
231	C₂⁴	—	43	1	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	43	1	no	—
234	C₂²×C₄	—	43	2	no	—
235	C₂²×C₄	—	43	2	no	—
236	C₂²×C₄	—	43	2	no	—
237	C₂²×C₄	—	43	2	no	—
238	C₂²×C₄	—	43	2	no	—
239	C₂²×C₄	—	43	2	no	—
240	C₂²×C₄	—	43	2	no	—
241	C₂²×C₄	—	43	8	no	—
242	C₂²×C₄	—	86	8	no	—
243	C₂²×C₄	—	86	4	no	—
244	C₂²×C₄	D₄×D₄₃	1	2	yes	—
245	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
246	C₂×C₈	—	43	2	no	—
247	C₂×C₈	—	43	2	no	—
248	C₂×C₈	—	43	2	no	—
249	C₂×C₈	—	43	2	no	—
250	C₂×C₈	—	43	2	no	—
251	C₂×C₈	—	43	2	no	—
252	C₂×C₈	—	43	16	no	—
253	C₂×C₈	—	43	16	no	—
254	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
256	C₂×C₈	D₄×D₄₃	1	2	yes	—
257	C₂×C₈	D₄×D₄₃	1	2	yes	—
258	C₂×C₈	Q₈×D₄₃	1	2	yes	—
259	C₂×C₈	Q₈×D₄₃	1	2	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	86	4	no	—
263	C₄⋊C₄	—	86	4	no	—
264	C₄⋊C₄	—	172	16	no	—
265	C₂²⋊C₄	—	172	8	no	—
266	C₄²	—	43	8	no	—
267	C₄²	—	43	2	no	—
268	C₄²	—	43	2	no	—
269	C₄²	—	172	8	no	—
270	C₄²	C₂×C₄×D₄₃	1	2	yes	—
271	C₄²	Q₈×D₄₃	1	2	yes	—
272	C₂³	—	43	2	no	—
273	C₂³	—	43	2	no	—
274	C₂³	—	43	2	no	—
275	C₂³	—	86	4	no	—
276	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	43	4	no	—
281	C₂×C₄	—	43	2	no	—
282	C₂×C₄	—	43	8	no	—
283	C₂×C₄	—	43	2	no	—
284	C₂×C₄	—	43	2	no	—
285	C₂×C₄	—	43	4	no	—
286	C₂×C₄	—	43	4	no	—
287	C₂×C₄	—	43	4	no	—
288	C₂×C₄	—	43	4	no	—
289	C₂×C₄	—	86	16	no	—
290	C₂×C₄	—	172	8	no	—
291	C₂×C₄	—	172	4	no	—
292	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
293	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
296	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
297	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
299	C₈	—	43	4	no	—
300	C₈	—	43	4	no	—
301	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
302	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
303	C₂²	—	43	4	no	—
304	C₂²	—	86	4	no	—
305	C₂²	—	86	4	no	—
306	C₂²	—	86	4	no	—
307	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
308	C₂²	#127×D₄₃	1	1	yes	characteristic
309	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
310	C₂²	#135×D₄₃	1	4	yes	—
311	C₄	—	2	4	no	—
312	C₄	—	8	4	no	—
313	C₄	—	43	8	no	—
314	C₄	—	43	4	no	—
315	C₄	—	344	4	no	—
316	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
317	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
318	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
319	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
320	C₂	—	86	4	no	—
321	C₂	#384×D₄₃	1	1	yes	characteristic
322	C₂	#427×D₄₃	1	2	yes	—
323	C₂	#19145020	1	2	yes	—
324	C₂	#19145108	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1846	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂³×D₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#20797
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
5. #19134759	C₂	semidirect
6. #19134757	C₂	semidirect
7. #19134753	C₂	semidirect
8. #19134750	C₂	semidirect
9. C₄₃×#1846	C₂	semidirect
23. C₄×C₈×D₄₃	C₄	semidirect
107. C₈×D₄₃	C₄⋊C₄	semidirect
108. C₈×D₄₃	C₄⋊C₄	semidirect
157. Dic₄₃	#427	semidirect
164. #1846	D₄₃	semidirect
194. C₄₃	#20797	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582936	2
C₂×#14582934	2
C₂×C₄×C₈×D₄₃	2
#19137001	2
#19134759	2
#19134757	2
#19134753	2
#19134750	2
C₄₃×#1846	2
#20797	43