Subgroups — #25104939

Order 11008.

658 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137903	C₂	1	1	yes	maximal, characteristic
2	#19135318	C₂	1	1	yes	maximal, characteristic
3	#19135160	C₂	1	2	yes	maximal
4	#19135156	C₂	1	2	yes	maximal
5	#19134714	C₂	1	1	yes	maximal, characteristic
6	#19134460	C₂	1	2	yes	maximal
7	#19134460	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#19134456	C₂	1	2	yes	maximal
10	#19134283	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1940	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×#13063235	C₂²	1	1	yes	characteristic
15	C₂×#13063235	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
21	#14582841	C₄	1	2	yes	—
22	#14582838	C₄	1	2	yes	—
23	#14582832	C₄	1	2	yes	—
24	#14582831	C₄	1	2	yes	—
25	C₄×Dic₄₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
28	#14582745	C₂²	1	1	yes	characteristic
29	#14582654	C₂²	1	2	yes	—
30	#14582653	C₂²	1	2	yes	—
31	#14582653	C₂²	1	2	yes	—
32	#14582653	C₂²	1	2	yes	—
33	#14582653	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#336	C₂²	1	2	yes	—
39	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
45	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
50	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
51	Dic₄₃⋊C₈	—	2	4	no	—
52	#13063235	—	2	2	no	—
53	#13063235	—	2	2	no	—
54	C₄×C₄₃⋊C₈	—	2	4	no	—
55	C₄×C₄₃⋊C₈	Q₈	1	4	yes	—
56	C₈×Dic₄₃	—	2	2	no	—
57	C₈×Dic₄₃	—	2	4	no	—
58	C₈×Dic₄₃	—	2	2	no	—
59	C₄₃⋊#95	C₂×C₄	1	2	yes	—
60	C₄₃⋊#95	C₂×C₄	1	2	yes	—
61	C₄₃⋊#95	C₂×C₄	1	1	yes	characteristic
62	C₄₃⋊#95	C₂×C₄	1	2	yes	—
63	C₄₃⋊#95	C₂×C₄	1	2	yes	—
64	C₄₃⋊#95	C₂×C₄	1	1	yes	characteristic
65	C₄₃⋊#95	C₂×C₄	1	2	yes	—
66	C₄₃⋊#95	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₃	—	2	4	no	—
74	C₄⋊C₈×C₄₃	—	2	2	no	—
75	#97×C₄₃	—	2	2	no	—
76	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
77	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₃	—	2	1	no	—
88	C₂×C₈×C₄₃	—	2	1	no	—
89	C₂×C₈×C₄₃	—	2	1	no	—
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	#41	1	2	yes	—
95	C₂×C₈×C₄₃	#41	1	2	yes	—
96	C₄⋊C₄×C₄₃	—	2	4	no	—
97	C₄²×C₄₃	—	2	2	no	—
98	C₄²×C₄₃	—	2	2	no	—
99	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
100	C₄²×C₄₃	#41	1	2	yes	—
101	C₄⋊Dic₄₃	—	2	4	no	—
102	C₄⋊Dic₄₃	—	2	4	no	—
103	C₄⋊Dic₄₃	—	2	4	no	—
104	Dic₄₃⋊C₄	—	2	4	no	—
105	Dic₄₃⋊C₄	—	2	4	no	—
106	C₄×Dic₄₃	—	2	2	no	—
107	C₄×Dic₄₃	—	2	1	no	—
108	C₄×Dic₄₃	—	2	1	no	—
109	C₄×Dic₄₃	—	2	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	C₄×Dic₄₃	—	2	2	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	1	no	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
118	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₄₃⋊C₈	#41	1	2	yes	—
120	C₂×C₄₃⋊C₈	#41	1	2	yes	—
121	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₄₃	#131	1	4	yes	—
132	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	#131	1	4	yes	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	—	2	2	no	—
152	C₈×C₄₃	—	4	2	no	—
153	C₄₃⋊C₈	—	2	2	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	C₄₃⋊C₈	#125	1	4	yes	—
156	#21676	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₄₃	C₄×#41	1	1	yes	characteristic
159	C₂²×C₄₃	C₄×#41	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#131	1	2	yes	—
161	C₂²×C₄₃	C₂×#131	1	2	yes	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	4	no	—
165	C₄×C₄₃	—	4	2	no	—
166	C₄×C₄₃	C₂×#125	1	1	yes	characteristic
167	C₄×C₄₃	#529	1	1	yes	characteristic
168	C₄×C₄₃	#529	1	1	yes	characteristic
169	C₄×C₄₃	#533	1	1	yes	characteristic
170	Dic₄₃	—	2	4	no	—
171	Dic₄₃	—	4	2	no	—
172	Dic₄₃	—	4	2	no	—
173	Dic₄₃	—	4	2	no	—
174	#2402	—	43	1	no	—
175	#1940	—	43	1	no	—
176	#1940	D₄₃	1	1	yes	characteristic
177	#1917	—	43	2	no	—
178	#1917	—	43	2	no	—
179	#1832	—	43	1	no	—
180	C₈×C₄⋊C₄	—	43	2	no	—
181	C₈×C₄⋊C₄	—	43	1	no	—
182	C₈×C₄⋊C₄	—	43	1	no	—
183	C₈×C₄⋊C₄	—	43	2	no	—
184	#1726	—	43	1	no	—
185	C₂×C₄₃	C₄×#125	1	1	yes	characteristic
186	C₂×C₄₃	#2960	1	1	yes	characteristic
187	C₂×C₄₃	#2960	1	1	yes	characteristic
188	C₂×C₄₃	#2966	1	2	yes	—
189	C₂×C₄₃	#2966	1	2	yes	—
190	C₂×C₄⋊C₈	—	43	2	no	—
191	C₂×C₄⋊C₈	—	43	1	no	—
192	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
194	C₂×#97	—	43	1	no	—
195	C₂×#97	—	43	1	no	—
196	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
197	C₂×C₄×C₈	—	43	1	no	—
198	C₂×C₄×C₈	—	43	1	no	—
199	C₂×C₄×C₈	—	43	1	no	—
200	C₂×C₄×C₈	—	43	2	no	—
201	C₂×C₄×C₈	—	43	2	no	—
202	#400	—	43	2	no	—
203	#400	—	43	2	no	—
204	#398	—	43	2	no	—
205	#398	—	43	2	no	—
206	C₄×C₄⋊C₄	—	43	2	no	—
207	C₄×C₄⋊C₄	—	43	2	no	—
208	C₄×C₄⋊C₄	—	43	1	no	—
209	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
210	#376	—	43	1	no	—
211	#336	—	43	2	no	—
212	#336	—	43	2	no	—
213	#336	—	43	2	no	—
214	#336	—	43	2	no	—
215	#336	—	43	2	no	—
216	#336	C₂×D₄₃	1	2	yes	—
217	C₄₃	#21676	1	1	yes	characteristic, complemented, Sylow
218	C₂²×C₈	—	43	1	no	—
219	C₂²×C₈	—	43	1	no	—
220	C₂²×C₈	—	43	2	no	—
221	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
222	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₂×C₄⋊C₄	—	43	2	no	—
225	C₂×C₄⋊C₄	—	43	2	no	—
226	C₂×C₄⋊C₄	—	43	2	no	—
227	C₂×C₄⋊C₄	—	43	2	no	—
228	C₂×C₄⋊C₄	—	43	2	no	—
229	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
230	C₂×C₄²	—	43	1	no	—
231	C₂×C₄²	—	43	1	no	—
232	C₂×C₄²	—	43	2	no	—
233	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
234	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	4	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	86	4	no	—
238	C₄⋊C₈	—	86	2	no	—
239	#97	—	2	2	no	—
240	#97	—	86	2	no	—
241	#97	—	86	2	no	—
242	C₄×C₈	—	43	4	no	—
243	C₄×C₈	—	86	4	no	—
244	C₄×C₈	—	86	2	no	—
245	C₄×C₈	—	86	2	no	—
246	C₄×C₈	—	86	4	no	—
247	#95	—	43	2	no	—
248	#95	—	43	2	no	—
249	#95	—	43	1	no	—
250	#95	—	43	2	no	—
251	#95	—	43	2	no	—
252	#95	—	43	2	no	—
253	#95	—	43	2	no	—
254	#95	—	43	1	no	—
255	#95	C₄×D₄₃	1	1	yes	characteristic
256	#95	C₄×D₄₃	1	1	yes	characteristic
257	C₂²×C₄	—	43	2	no	—
258	C₂²×C₄	—	43	2	no	—
259	C₂²×C₄	—	43	2	no	—
260	C₂²×C₄	—	43	2	no	—
261	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
262	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
265	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	43	2	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	43	2	no	—
276	C₂×C₈	—	43	2	no	—
277	C₂×C₈	—	86	1	no	—
278	C₂×C₈	—	86	1	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	2	no	—
281	C₂×C₈	—	86	2	no	—
282	C₂×C₈	#10621900	1	2	yes	—
283	C₂×C₈	#10621902	1	2	yes	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	86	4	no	—
286	C₄⋊C₄	—	86	4	no	—
287	C₄⋊C₄	—	86	4	no	—
288	C₄⋊C₄	—	86	4	no	—
289	C₄⋊C₄	—	86	4	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	86	2	no	—
293	C₄²	—	86	1	no	—
294	C₄²	—	86	1	no	—
295	C₄²	—	86	1	no	—
296	C₄²	—	86	1	no	—
297	C₄²	—	86	2	no	—
298	C₄²	#10621900	1	2	yes	—
299	C₄²	Q₈×D₄₃	1	2	yes	—
300	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	43	4	no	—
309	C₂×C₄	—	86	2	no	—
310	C₂×C₄	—	86	2	no	—
311	C₂×C₄	—	86	2	no	—
312	C₂×C₄	—	86	2	no	—
313	C₂×C₄	—	86	2	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	#13063290	1	2	yes	—
321	C₂×C₄	#13063290	1	2	yes	—
322	C₂×C₄	#13063333	1	4	yes	—
323	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
324	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
325	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
326	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
327	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
328	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	43	4	no	—
333	C₈	—	86	2	no	—
334	C₈	—	172	2	no	—
335	C₂²	C₂×#13063290	1	1	yes	characteristic
336	C₂²	C₄×#10621900	1	1	yes	characteristic
337	C₂²	C₄×#10621902	1	1	yes	characteristic
338	C₂²	#131×D₄₃	1	2	yes	—
339	C₂²	C₂×#13063333	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	86	4	no	—
345	C₄	—	172	2	no	—
346	C₄	—	172	2	no	—
347	C₄	—	172	2	no	—
348	C₄	#125×D₄₃	1	1	yes	characteristic
349	C₄	#14583430	1	1	yes	characteristic
350	C₄	#14583435	1	1	yes	characteristic
351	C₄	#14583454	1	1	yes	characteristic
352	C₂	#19137154	1	1	yes	characteristic
353	C₂	#19143525	1	1	yes	characteristic
354	C₂	#19143528	1	1	yes	characteristic
355	C₂	#19143565	1	2	yes	—
356	C₂	#19143569	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#1940	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1940	D₄₃
hypocenter	C₄₃	#21676
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1940

Complements

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

N	H ≅ G/N	type
217. C₄₃	#21676	semidirect

Maximal subgroups

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

subgroup	index
#19137903	2
#19135318	2
#19135160	2
#19135156	2
#19134714	2
#19134460	2
C₄⋊C₄×C₄₃⋊C₈	2
#19134456	2
#19134283	2
C₄₃×#1940	2
#21676	43