Subgroups — #24992238

Order 11008.

588 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582952	C₂	1	1	yes	maximal, characteristic
2	#19134575	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582648	C₂	1	2	yes	maximal, complemented
4	#19132478	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2857	C₂	1	1	yes	maximal, characteristic
6	#14582952	—	2	4	no	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
10	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
11	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
12	#14582654	—	2	1	no	—
13	#14582648	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₄₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#391	C₄	1	2	yes	—
18	C₄₃×#389	—	2	1	no	—
19	C₄₃×#389	—	2	1	no	—
20	C₄₃×#384	C₂²	1	1	yes	characteristic
21	C₄₃×#380	—	2	1	no	—
22	C₄₃×#380	—	2	1	no	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
29	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	4	2	no	—
31	C₄×C₄₃⋊C₈	—	2	4	no	—
32	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	—	2	2	no	—
38	C₂²⋊Q₈×C₄₃	—	4	4	no	—
39	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄²×C₄₃	—	2	2	no	—
47	C₂×C₄²×C₄₃	—	2	2	no	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	#95×C₄₃	—	2	2	no	—
53	#95×C₄₃	—	2	2	no	—
54	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₃	—	2	4	no	—
56	C₂×Q₈×C₄₃	—	4	2	no	—
57	C₂×Q₈×C₄₃	—	4	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂²×C₄×C₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	—	4	2	no	—
78	C₄⋊C₄×C₄₃	—	4	4	no	—
79	C₄⋊C₄×C₄₃	—	4	1	no	—
80	C₄⋊C₄×C₄₃	—	4	2	no	—
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	2	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊C₄×C₄₃	—	4	4	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	—	2	1	no	—
89	C₄²×C₄₃	C₂×D₄	1	2	yes	—
90	C₄²×C₄₃	#41	1	2	yes	—
91	C₂×C₄₃⋊C₈	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂×C₄₃⋊C₈	—	4	2	no	—
95	C₂×C₄₃⋊C₈	—	4	2	no	—
96	C₂×C₄₃⋊C₈	—	4	2	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	—	2	1	no	—
99	C₂³×C₄₃	—	2	1	no	—
100	C₂³×C₄₃	—	2	1	no	—
101	C₂³×C₄₃	—	2	1	no	—
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	#103	1	2	yes	—
104	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	Q₈×C₄₃	—	8	4	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	#100	1	2	yes	—
127	C₂×C₄×C₄₃	#103	1	2	yes	—
128	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#121	1	2	yes	—
131	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#135	1	2	yes	—
135	C₂×C₄×C₄₃	#135	1	2	yes	—
136	C₂×C₄×C₄₃	#137	1	2	yes	—
137	C₄₃⋊C₈	—	4	4	no	—
138	C₄₃⋊C₈	—	8	2	no	—
139	#9128	—	43	1	no	maximal, Sylow
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	4	1	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	#386	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
149	C₂²×C₄₃	#413	1	1	yes	characteristic
150	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
151	C₂²×C₄₃	#416	1	1	yes	characteristic
152	C₂²×C₄₃	#416	1	1	yes	characteristic
153	C₂²×C₄₃	#419	1	1	yes	characteristic
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	—	2	2	no	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	4	1	no	—
158	C₄×C₄₃	—	8	2	no	—
159	C₄×C₄₃	—	8	2	no	—
160	C₄×C₄₃	#452	1	2	yes	—
161	C₄×C₄₃	#465	1	2	yes	—
162	C₄×C₄₃	#467	1	2	yes	—
163	C₄×C₄₃	#485	1	2	yes	—
164	C₂×#431	—	43	1	no	—
165	#2857	D₄₃	1	1	yes	characteristic
166	#1797	—	43	1	no	—
167	C₂×#333	—	43	2	no	—
168	#1296	—	43	2	no	—
169	C₂×C₄₃	—	4	2	no	—
170	C₂×C₄₃	#1542	1	1	yes	characteristic
171	C₂×C₄₃	#1543	1	1	yes	characteristic
172	C₂×C₄₃	#1548	1	1	yes	characteristic
173	C₂×C₄₃	#1554	1	1	yes	characteristic
174	C₂×C₄₃	#1873	1	1	yes	characteristic
175	C₂×C₄₃	#1892	1	1	yes	characteristic
176	C₂×C₄₃	#1900	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
178	C₂×C₂²⋊Q₈	Dic₄₃	1	2	yes	—
179	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
180	#431	—	86	4	no	—
181	C₂×C₄⋊C₈	—	43	2	no	—
182	C₂×C₄⋊C₈	—	43	2	no	—
183	C₂×C₂²⋊C₈	—	86	1	no	—
184	C₂×C₂²⋊C₈	—	86	1	no	—
185	C₂×C₄×C₈	—	86	1	no	—
186	#391	Dic₄₃	1	2	yes	—
187	#389	—	2	1	no	—
188	#389	—	2	1	no	—
189	#384	C₂×D₄₃	1	1	yes	characteristic
190	#380	—	2	1	no	—
191	#380	—	2	1	no	—
192	#336	—	86	1	no	—
193	#333	—	86	4	no	—
194	C₄₃	#9128	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	C₂×Dic₄₃	1	2	yes	—
196	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
198	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
199	C₂²×C₈	—	86	2	no	—
200	C₂²×C₈	—	86	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	4	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊C₈	—	43	4	no	—
218	C₄⋊C₈	—	86	2	no	—
219	C₄⋊C₈	—	86	2	no	—
220	C₄⋊C₈	—	86	2	no	—
221	C₂²⋊C₈	—	86	4	no	—
222	C₂²⋊C₈	—	172	2	no	—
223	C₄×C₈	—	86	4	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	#10621884	1	2	yes	—
242	C₂²×C₄	#10621885	1	2	yes	—
243	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
249	C₂×C₈	—	86	4	no	—
250	C₂×C₈	—	172	2	no	—
251	C₂×C₈	—	172	2	no	—
252	C₂×C₈	—	172	2	no	—
253	C₂×C₈	—	172	2	no	—
254	C₂×C₈	—	172	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	D₄×D₄₃	1	2	yes	—
268	C₄²	#10621900	1	2	yes	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	#13063249	1	2	yes	—
276	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
277	Q₈	—	8	4	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	#13063242	1	2	yes	—
299	C₂×C₄	#13063249	1	2	yes	—
300	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
301	C₂×C₄	#13063293	1	1	yes	characteristic
302	C₂×C₄	#13063297	1	2	yes	—
303	C₂×C₄	#13063306	1	1	yes	characteristic
304	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
305	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
306	C₂×C₄	#13063345	1	2	yes	—
307	C₂×C₄	#13063345	1	2	yes	—
308	C₂×C₄	#13063356	1	2	yes	—
309	C₈	—	344	2	no	—
310	C₈	—	172	4	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	#14582779	1	1	yes	characteristic
319	C₂²	C₂×#13063242	1	1	yes	characteristic
320	C₂²	#14582883	1	1	yes	characteristic
321	C₂²	C₂×#13063249	1	1	yes	characteristic
322	C₂²	#14582895	1	1	yes	characteristic
323	C₂²	#14582895	1	1	yes	characteristic
324	C₂²	#14582908	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	#14583059	1	2	yes	—
332	C₄	#14583130	1	2	yes	—
333	C₄	#14583138	1	2	yes	—
334	C₄	#14583237	1	2	yes	—
335	C₂	—	4	2	no	—
336	C₂	#19133252	1	1	yes	characteristic
337	C₂	#19133257	1	1	yes	characteristic
338	C₂	#19133276	1	1	yes	characteristic
339	C₂	#19133305	1	1	yes	characteristic
340	C₂	#19134908	1	1	yes	characteristic
341	C₂	#19135019	1	1	yes	characteristic
342	C₂	#19135059	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2857	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2857	D₄₃
hypocenter	C₄₃	#9128
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. C₂×#14582648	C₂	semidirect
4. #19132478	C₂	semidirect
194. C₄₃	#9128	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582952	2
#19134575	2
C₂×#14582648	2
#19132478	2
C₄₃×#2857	2
#9128	43