Subgroups — #25068430

Order 11008.

592 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	#19143600	C₂	1	2	yes	maximal, complemented
2	#19133498	C₂	1	2	yes	maximal, complemented
3	#19133467	C₂	1	2	yes	maximal, complemented
4	#19133460	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#19133343	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133003	C₂	1	2	yes	maximal, complemented
8	#19132884	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3168	C₂	1	1	yes	maximal, characteristic
10	#14583030	C₂²	1	2	yes	—
11	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	—
12	#14582977	—	2	2	no	—
13	#14582977	C₂²	1	2	yes	—
14	#14582971	—	2	2	no	—
15	D₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	#14582960	—	2	2	no	—
17	#14582952	C₂²	1	2	yes	—
18	#14582952	C₂²	1	1	yes	characteristic
19	#14582855	—	2	2	no	—
20	C₄×#10621877	C₂²	1	1	yes	characteristic
21	#14582652	C₂²	1	1	yes	characteristic
22	#14582652	C₂²	1	1	yes	characteristic
23	#14582650	C₂²	1	2	yes	—
24	Q₈⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
25	Q₈⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
26	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
27	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
28	#14582621	C₂²	1	2	yes	—
29	#14582620	C₂²	1	2	yes	—
30	C₄₃×#529	C₂²	1	2	yes	—
31	C₄₃×#499	C₄	1	1	yes	characteristic
32	C₄₃×#495	C₄	1	1	yes	characteristic
33	C₄₃×#488	C₄	1	2	yes	—
34	C₄₃×#467	C₄	1	1	yes	characteristic
35	C₄₃×#466	C₄	1	1	yes	characteristic
36	C₄₃×#465	C₄	1	2	yes	—
37	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₄₃	C₂²	1	2	yes	—
39	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#10621877	—	2	2	no	—
42	C₂×#10621877	D₄	1	1	yes	characteristic
43	C₂×#10621877	D₄	1	1	yes	characteristic
44	C₂²×C₄₃⋊C₈	—	2	2	no	—
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
48	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
49	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
50	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
52	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
53	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
54	#13063236	—	2	2	no	—
55	#13063236	D₄	1	2	yes	—
56	C₄×C₄₃⋊C₈	—	2	2	no	—
57	C₄×C₄₃⋊C₈	—	2	2	no	—
58	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
59	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂×SD₁₆×C₄₃	C₂×C₄	1	2	yes	—
62	C₂×D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	#126×C₄₃	—	2	2	no	—
66	#125×C₄₃	C₂×C₄	1	2	yes	—
67	#124×C₄₃	C₂×C₄	1	2	yes	—
68	#123×C₄₃	—	2	2	no	—
69	C₂²⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
70	#121×C₄₃	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
72	C₄×D₄×C₄₃	—	2	2	no	—
73	C₄×D₄×C₄₃	C₂³	1	2	yes	—
74	#117×C₄₃	—	2	2	no	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
77	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
78	#106×C₄₃	C₂×C₄	1	2	yes	—
79	#103×C₄₃	C₂×C₄	1	2	yes	—
80	#103×C₄₃	C₂×C₄	1	2	yes	—
81	#102×C₄₃	C₂×C₄	1	2	yes	—
82	#102×C₄₃	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₄₃	—	2	2	no	—
89	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₄₃	—	2	2	no	—
91	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
94	Q₁₆×C₄₃	—	4	1	no	—
95	SD₁₆×C₄₃	—	2	4	no	—
96	D₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	4	1	no	—
99	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
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	—	2	2	no	—
107	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
109	C₂²⋊C₄×C₄₃	—	2	2	no	—
110	C₂²⋊C₄×C₄₃	—	2	2	no	—
111	C₂²⋊C₄×C₄₃	—	2	2	no	—
112	C₂²⋊C₄×C₄₃	—	2	2	no	—
113	C₂²⋊C₄×C₄₃	—	2	2	no	—
114	C₄²×C₄₃	—	2	2	no	—
115	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₃	#41	1	1	yes	characteristic
118	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
119	#10621877	—	4	1	no	—
120	#10621877	—	4	1	no	—
121	#10621877	—	4	4	no	—
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	—	4	4	no	—
126	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
127	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
128	C₂³×C₄₃	—	2	2	no	—
129	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₄₃	—	2	2	no	—
131	Q₈×C₄₃	—	2	2	no	—
132	D₄×C₄₃	—	2	2	no	—
133	D₄×C₄₃	—	2	2	no	—
134	D₄×C₄₃	—	8	2	no	—
135	C₂×C₄×C₄₃	—	2	1	no	—
136	C₂×C₄×C₄₃	—	2	1	no	—
137	C₂×C₄×C₄₃	—	2	1	no	—
138	C₂×C₄×C₄₃	—	2	1	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	2	2	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	4	1	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
145	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	—	2	1	no	—
153	C₈×C₄₃	—	2	1	no	—
154	C₈×C₄₃	#131	1	2	yes	—
155	C₄₃⋊C₈	—	2	2	no	—
156	C₄₃⋊C₈	—	2	2	no	—
157	C₄₃⋊C₈	—	4	2	no	—
158	C₄₃⋊C₈	—	4	2	no	—
159	C₄₃⋊C₈	—	4	2	no	—
160	#18074	—	43	1	no	maximal, Sylow
161	C₂²×C₄₃	—	2	1	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	4	2	no	—
165	C₂²×C₄₃	—	4	2	no	—
166	C₂²×C₄₃	—	4	2	no	—
167	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	4	2	no	—
173	C₄×C₄₃	—	4	2	no	—
174	C₄×C₄₃	—	4	2	no	—
175	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
176	C₄×C₄₃	#568	1	1	yes	characteristic
177	C₄×C₄₃	#575	1	1	yes	characteristic
178	C₄×C₄₃	#575	1	1	yes	characteristic
179	#3168	D₄₃	1	1	yes	characteristic
180	#2969	—	43	2	no	—
181	#1592	—	43	2	no	—
182	#1587	—	43	2	no	—
183	#1586	—	43	2	no	—
184	SD₁₆⋊C₈	—	43	2	no	—
185	#1564	—	43	1	no	—
186	#1482	—	43	2	no	—
187	#1451	—	43	1	no	—
188	C₂×C₄₃	—	4	1	no	—
189	C₂×C₄₃	—	8	2	no	—
190	C₂×C₄₃	#2935	1	1	yes	characteristic
191	C₂×C₄₃	#2943	1	1	yes	characteristic
192	C₂×C₄₃	#2957	1	1	yes	characteristic
193	#529	C₂×D₄₃	1	2	yes	—
194	#499	Dic₄₃	1	1	yes	characteristic
195	#495	Dic₄₃	1	1	yes	characteristic
196	#488	Dic₄₃	1	2	yes	—
197	#467	Dic₄₃	1	1	yes	characteristic
198	#466	Dic₄₃	1	1	yes	characteristic
199	#465	Dic₄₃	1	2	yes	—
200	#446	—	43	2	no	—
201	C₈×Q₈	—	43	2	no	—
202	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
203	C₄×SD₁₆	C₂×D₄₃	1	2	yes	—
204	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
205	#436	—	43	2	no	—
206	#436	—	86	2	no	—
207	#435	—	86	2	no	—
208	C₈×D₄	—	43	2	no	—
209	#433	—	86	2	no	—
210	#431	—	43	1	no	—
211	#431	—	43	2	no	—
212	#405	—	86	2	no	—
213	C₄×M₁₆	—	43	1	no	—
214	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
215	#335	—	43	1	no	—
216	#335	—	43	1	no	—
217	#334	—	43	2	no	—
218	Q₈⋊C₈	—	43	2	no	—
219	Q₈⋊C₈	—	43	2	no	—
220	D₄⋊C₈	—	43	2	no	—
221	D₄⋊C₈	—	43	2	no	—
222	#322	—	43	2	no	—
223	#322	—	43	2	no	—
224	C₄₃	#18074	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₁₆	C₂×Dic₄₃	1	1	yes	characteristic
226	C₂×SD₁₆	C₂×Dic₄₃	1	2	yes	—
227	C₂×D₈	C₂×Dic₄₃	1	1	yes	characteristic
228	C₂×M₁₆	—	43	1	no	—
229	C₂×M₁₆	—	43	1	no	—
230	C₂×M₁₆	—	86	2	no	—
231	C₂²×C₈	—	86	2	no	—
232	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
233	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
234	#126	—	2	2	no	—
235	#125	C₂×Dic₄₃	1	2	yes	—
236	#124	C₂×Dic₄₃	1	2	yes	—
237	#123	—	2	2	no	—
238	C₂²⋊Q₈	C₂×Dic₄₃	1	2	yes	—
239	#121	C₂×Dic₄₃	1	2	yes	—
240	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	C₂²×D₄₃	1	2	yes	—
243	#117	—	2	2	no	—
244	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
245	#107	C₂×Dic₄₃	1	1	yes	characteristic
246	#107	C₂×Dic₄₃	1	1	yes	characteristic
247	#106	C₂×Dic₄₃	1	2	yes	—
248	C₄⋊C₈	—	43	2	no	—
249	C₄⋊C₈	—	43	2	no	—
250	C₄⋊C₈	—	86	2	no	—
251	C₄⋊C₈	—	86	2	no	—
252	C₄⋊C₈	—	86	1	no	—
253	#103	C₂×Dic₄₃	1	2	yes	—
254	#103	C₂×Dic₄₃	1	2	yes	—
255	#102	C₂×Dic₄₃	1	2	yes	—
256	#102	C₂×Dic₄₃	1	2	yes	—
257	C₂²⋊C₈	—	86	2	no	—
258	C₂²⋊C₈	—	86	2	no	—
259	C₂²⋊C₈	—	86	2	no	—
260	C₂²⋊C₈	—	86	1	no	—
261	#97	—	43	2	no	—
262	#97	—	86	2	no	—
263	C₄×C₈	—	43	2	no	—
264	C₄×C₈	—	43	2	no	—
265	C₄×C₈	—	86	2	no	—
266	C₄×C₈	—	86	2	no	—
267	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
268	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₂×Q₈	C₂²×Dic₄₃	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₄₃	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
276	C₂²×C₄	#10621900	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
278	Q₁₆	—	4	1	no	—
279	SD₁₆	—	2	4	no	—
280	D₈	—	4	1	no	—
281	M₁₆	—	172	1	no	—
282	M₁₆	—	172	1	no	—
283	M₁₆	—	172	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	43	2	no	—
287	C₂×C₈	—	43	2	no	—
288	C₂×C₈	—	86	2	no	—
289	C₂×C₈	—	86	2	no	—
290	C₂×C₈	—	86	2	no	—
291	C₂×C₈	—	172	4	no	—
292	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂²×Dic₄₃	1	2	yes	—
301	C₄⋊C₄	C₂²×Dic₄₃	1	2	yes	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	D₄×D₄₃	1	1	yes	characteristic
309	C₄²	#10621900	1	1	yes	characteristic
310	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
311	C₄²	C₂³×D₄₃	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	8	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
330	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	86	2	no	—
339	C₈	—	86	2	no	—
340	C₈	—	172	2	no	—
341	C₈	—	172	2	no	—
342	C₈	—	172	2	no	—
343	C₈	#13063332	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	C₂×#13063332	1	1	yes	characteristic
359	C₄	#14583666	1	1	yes	characteristic
360	C₄	#14583696	1	1	yes	characteristic
361	C₄	#14583696	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	8	2	no	—
364	C₂	#19143339	1	1	yes	characteristic
365	C₂	#19143396	1	1	yes	characteristic
366	C₂	#19143512	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3168	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₄₃×#466	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#3168	D₄₃
hypocenter	C₄₃	#18074
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #19143600	C₂	semidirect
2. #19133498	C₂	semidirect
3. #19133467	C₂	semidirect
4. #19133460	C₂	semidirect
5. SD₁₆⋊(C₄₃⋊C₈)	C₂	semidirect
6. #19133343	C₂	semidirect
7. #19133003	C₂	semidirect
8. #19132884	C₂	semidirect
224. C₄₃	#18074	semidirect

Maximal subgroups

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

subgroup	index
#19143600	2
#19133498	2
#19133467	2
#19133460	2
SD₁₆⋊(C₄₃⋊C₈)	2
#19133343	2
#19133003	2
#19132884	2
C₄₃×#3168	2
#18074	43