Subgroups — #24990457

Order 11008.

482 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143618	C₂	1	1	yes	maximal, characteristic
2	#19135388	C₂	1	1	yes	maximal, characteristic
3	C₄×#13063241	C₂	1	1	yes	maximal, characteristic
4	#19133345	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132433	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
9	C₈×D₁₇₂	—	2	1	no	—
10	C₈×D₁₇₂	—	2	1	no	—
11	#14582964	—	2	1	no	—
12	#14582964	—	2	1	no	—
13	#14582952	C₂²	1	1	yes	characteristic
14	C₂×#13063241	C₄	1	1	yes	characteristic
15	C₂×#13063241	C₄	1	1	yes	characteristic
16	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
17	C₂²⋊C₈×D₄₃	—	2	2	no	—
18	C₄×#10621877	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582667	C₄	1	1	yes	characteristic
21	#14582667	C₄	1	1	yes	characteristic
22	#14582651	—	2	1	no	—
23	#14582649	—	2	1	no	—
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#335	—	2	1	no	—
28	C₄₃×#334	—	2	1	no	—
29	C₂²×D₁₇₂	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₄₃	D₄	1	2	yes	—
31	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
32	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	—	2	1	no	—
34	C₂×C₈×D₄₃	—	2	2	no	—
35	C₂×C₈×D₄₃	—	2	2	no	—
36	C₄×D₁₇₂	—	2	1	no	—
37	C₄×D₁₇₂	—	2	1	no	—
38	C₄×D₁₇₂	—	2	1	no	—
39	C₄×D₁₇₂	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
42	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
43	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
44	#13063241	—	2	2	no	—
45	#13063241	C₈	1	4	yes	complemented
46	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
47	(C₂×D₄₃)⋊C₈	—	2	2	no	—
48	(C₂×D₄₃)⋊C₈	—	2	2	no	—
49	(C₂×D₄₃)⋊C₈	—	4	4	no	—
50	#13063236	—	2	1	no	—
51	C₄×C₄₃⋊C₈	—	2	1	no	—
52	C₄×C₄₃⋊C₈	—	2	1	no	—
53	C₄×C₄₃⋊C₈	—	2	1	no	—
54	C₄×C₄₃⋊C₈	—	2	1	no	—
55	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	—	2	1	no	—
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
62	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₄₃	—	2	1	no	—
64	#97×C₄₃	—	2	1	no	—
65	C₄×C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₂³×D₄₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×Dic₄₃	C₄⋊C₄	1	2	yes	—
72	C₂×D₁₇₂	—	2	1	no	—
73	C₂×D₁₇₂	—	2	1	no	—
74	C₂×D₁₇₂	—	2	1	no	—
75	C₂×D₁₇₂	—	2	1	no	—
76	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
77	C₂×D₁₇₂	M₁₆	1	2	yes	—
78	C₂×C₄×D₄₃	—	2	2	no	—
79	C₂×C₄×D₄₃	—	2	2	no	—
80	C₂×C₄×D₄₃	—	4	2	no	—
81	C₂×C₄×D₄₃	—	4	2	no	—
82	M₁₆×C₄₃	—	2	1	no	—
83	M₁₆×C₄₃	—	2	1	no	—
84	M₁₆×C₄₃	—	2	1	no	—
85	M₁₆×C₄₃	C₂×C₈	1	2	yes	complemented
86	C₂×C₈×C₄₃	—	2	1	no	—
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	—	4	2	no	—
91	C₂×C₈×C₄₃	—	4	1	no	—
92	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	#41	1	1	yes	characteristic
95	C₄²×C₄₃	#41	1	1	yes	characteristic
96	(C₂×D₄₃)⋊C₄	—	2	2	no	—
97	(C₂×D₄₃)⋊C₄	—	2	2	no	—
98	C₄⋊Dic₄₃	—	2	1	no	—
99	C₄⋊Dic₄₃	—	2	1	no	—
100	#10621877	—	2	1	no	—
101	#10621877	—	2	1	no	—
102	#10621877	—	2	1	no	—
103	#10621877	C₂×C₈	1	2	yes	complemented
104	C₂×C₄₃⋊C₈	—	2	1	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	—	2	1	no	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	1	no	—
111	C₈×D₄₃	—	8	2	no	—
112	C₈×D₄₃	—	8	2	no	—
113	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₄₃	—	2	2	no	—
115	C₂²×D₄₃	—	2	2	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	4	2	no	—
118	C₂²×D₄₃	C₄⋊C₈	1	4	yes	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	4	1	no	—
122	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
132	C₂×Dic₄₃	—	2	2	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	D₁₇₂	—	2	2	no	—
135	D₁₇₂	—	2	2	no	—
136	D₁₇₂	—	2	2	no	—
137	D₁₇₂	—	2	2	no	—
138	C₄×D₄₃	—	4	2	no	—
139	C₄×D₄₃	—	4	2	no	—
140	C₄×D₄₃	—	8	2	no	—
141	C₈×C₄₃	—	2	2	no	—
142	C₈×C₄₃	—	4	1	no	—
143	C₈×C₄₃	—	4	1	no	—
144	C₈×C₄₃	—	4	1	no	—
145	C₈×C₄₃	—	8	1	no	—
146	C₄₃⋊C₈	—	2	2	no	—
147	C₄₃⋊C₈	—	4	2	no	—
148	C₄₃⋊C₈	—	4	1	no	—
149	C₄₃⋊C₈	—	4	1	no	—
150	C₄₃⋊C₈	—	4	1	no	—
151	#8835	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	#385	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
156	C₂²×C₄₃	#424	1	1	yes	characteristic
157	C₂×D₄₃	—	2	4	no	—
158	C₂×D₄₃	—	4	2	no	—
159	C₂×D₄₃	—	4	2	no	—
160	C₂×D₄₃	—	4	4	no	—
161	C₂×D₄₃	—	8	2	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
166	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
167	C₄×C₄₃	#435	1	1	yes	characteristic
168	C₄×C₄₃	#435	1	1	yes	characteristic
169	C₄×C₄₃	#453	1	1	yes	characteristic
170	C₄×C₄₃	#455	1	1	yes	characteristic
171	Dic₄₃	—	4	2	no	—
172	#2972	—	43	1	no	—
173	#1954	—	43	1	no	—
174	C₄×#100	—	43	1	no	—
175	#1565	—	43	1	no	—
176	#1565	D₄₃	1	1	yes	characteristic, complemented
177	#1278	—	43	1	no	—
178	#1278	—	43	1	no	—
179	C₂×C₄₃	—	2	1	no	—
180	C₂×C₄₃	—	2	1	no	—
181	C₂×C₄₃	#1850	1	1	yes	characteristic
182	C₂×C₄₃	#1856	1	1	yes	characteristic
183	C₂×C₄₃	#1858	1	1	yes	characteristic
184	D₄₃	—	4	4	no	—
185	C₂×C₄×D₄	—	43	1	no	—
186	C₈×D₄	—	86	1	no	—
187	C₈×D₄	—	86	1	no	—
188	C₈×D₄	—	86	1	no	—
189	C₈×D₄	—	86	1	no	—
190	#431	—	43	1	no	—
191	#431	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
193	C₂×#100	—	43	1	no	—
194	C₂×#100	—	43	1	no	—
195	C₂×C₂²⋊C₈	—	86	2	no	—
196	C₂×C₂²⋊C₈	—	86	2	no	—
197	C₄×M₁₆	—	43	1	no	—
198	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
199	C₂×C₄×C₈	—	43	1	no	—
200	#343	—	43	1	no	—
201	#343	—	43	1	no	—
202	#335	—	2	1	no	—
203	#335	—	86	1	no	—
204	#334	—	2	1	no	—
205	#334	—	86	1	no	—
206	C₄₃	#8835	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₄	—	43	1	no	—
208	C₂³×C₄	—	43	2	no	—
209	C₂×M₁₆	—	43	1	no	—
210	C₂×M₁₆	—	43	1	no	—
211	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
212	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	86	2	no	—
215	C₂²×C₈	—	86	2	no	—
216	C₂²×C₈	—	86	1	no	—
217	C₄×D₄	—	86	1	no	—
218	C₄×D₄	—	86	1	no	—
219	C₄×D₄	—	86	1	no	—
220	C₄×D₄	—	86	1	no	—
221	C₂×C₄⋊C₄	—	43	1	no	—
222	C₂×C₂²⋊C₄	—	43	2	no	—
223	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	43	1	no	—
227	C₄⋊C₈	—	43	1	no	—
228	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
229	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
230	#100	—	43	4	no	—
231	#100	—	86	2	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	86	2	no	—
234	C₂²⋊C₈	—	86	2	no	—
235	C₂²⋊C₈	—	86	1	no	—
236	C₂²⋊C₈	—	172	4	no	—
237	#97	—	2	1	no	—
238	#97	—	86	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	86	1	no	—
242	C₄×C₈	—	86	1	no	—
243	C₄×C₈	—	86	1	no	—
244	C₄×C₈	—	86	1	no	—
245	C₂⁴	—	43	2	no	—
246	C₂×D₄	—	43	2	no	—
247	C₂×D₄	—	43	2	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂×D₄	—	86	1	no	—
250	C₂×D₄	—	86	1	no	—
251	C₂×D₄	—	86	1	no	—
252	C₂²×C₄	—	43	2	no	—
253	C₂²×C₄	—	86	2	no	—
254	C₂²×C₄	—	86	2	no	—
255	C₂²×C₄	—	172	2	no	—
256	C₂²×C₄	—	172	2	no	—
257	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	43	2	no	—
264	M₁₆	—	86	1	no	—
265	M₁₆	—	86	1	no	—
266	M₁₆	—	86	1	no	—
267	M₁₆	C₈×D₄₃	1	2	yes	complemented
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	86	1	no	—
275	C₂×C₈	—	86	2	no	—
276	C₂×C₈	—	86	1	no	—
277	C₂×C₈	—	86	1	no	—
278	C₂×C₈	—	86	1	no	—
279	C₂×C₈	—	344	2	no	—
280	C₂×C₈	—	344	2	no	—
281	C₂×C₈	—	172	1	no	—
282	C₂×C₈	—	172	1	no	—
283	C₄⋊C₄	—	86	1	no	—
284	C₄⋊C₄	—	86	1	no	—
285	C₂²⋊C₄	—	86	2	no	—
286	C₂²⋊C₄	—	86	2	no	—
287	C₄²	#10621898	1	1	yes	characteristic
288	C₄²	D₄×D₄₃	1	1	yes	characteristic
289	C₄²	#10621902	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
291	C₂³	—	43	4	no	—
292	C₂³	—	86	2	no	—
293	C₂³	—	86	2	no	—
294	C₂³	—	172	2	no	—
295	C₂³	—	172	2	no	—
296	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
297	D₄	—	86	2	no	—
298	D₄	—	86	2	no	—
299	D₄	—	86	2	no	—
300	D₄	—	86	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	86	2	no	—
305	C₂×C₄	—	344	2	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
310	C₂×C₄	#13063287	1	1	yes	characteristic
311	C₂×C₄	#13063293	1	1	yes	characteristic
312	C₂×C₄	#13063301	1	1	yes	characteristic
313	C₂×C₄	#13063303	1	1	yes	characteristic
314	C₂×C₄	#13063310	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
316	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
317	C₂×C₄	#13063331	1	1	yes	characteristic
318	C₂×C₄	#13063333	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₈	—	86	2	no	—
325	C₈	—	172	2	no	—
326	C₈	—	172	1	no	—
327	C₈	—	172	1	no	—
328	C₈	—	172	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	86	4	no	—
332	C₂²	—	344	2	no	—
333	C₂²	—	172	4	no	—
334	C₂²	—	172	2	no	—
335	C₂²	—	172	2	no	—
336	C₂²	#14582770	1	1	yes	characteristic
337	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
338	C₂²	#14582924	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	172	2	no	—
343	C₄	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
344	C₄	#14582964	1	1	yes	characteristic
345	C₄	#14582968	1	1	yes	characteristic
346	C₄	#14582969	1	1	yes	characteristic
347	C₄	#14583065	1	1	yes	characteristic
348	C₄	#14583075	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	172	4	no	—
352	C₂	#19134771	1	1	yes	characteristic
353	C₂	#19134803	1	1	yes	characteristic
354	C₂	#19134813	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1565	C₂
socle	C₂²×C₄₃	#385
prime radical	C₂²×D₁₇₂	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1565	D₄₃
hypocenter	C₄₃	#8835
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #19133345	C₂	semidirect
5. #19132433	C₂	semidirect
6. #19132431	C₂	semidirect
7. C₄₃×#1565	C₂	semidirect
45. #13063241	C₈	semidirect
85. M₁₆×C₄₃	C₂×C₈	semidirect
103. #10621877	C₂×C₈	semidirect
176. #1565	D₄₃	semidirect
206. C₄₃	#8835	semidirect
267. M₁₆	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143618	2
#19135388	2
C₄×#13063241	2
#19133345	2
#19132433	2
#19132431	2
C₄₃×#1565	2
#8835	43