Subgroups — #25066528

Order 11008.

610 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144630	C₂	1	1	yes	maximal, characteristic
2	#19143530	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583023	C₂	1	1	yes	maximal, characteristic
4	#19133469	C₂	1	4	yes	maximal, complemented
5	#19133411	C₂	1	4	yes	maximal, complemented
6	#19133365	C₂	1	1	yes	maximal, characteristic
7	#19133025	C₂	1	1	yes	maximal, characteristic
8	#19132888	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1481	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	#14583232	C₄	1	2	yes	—
12	#14583191	C₄	1	2	yes	—
13	#14583106	C₄	1	2	yes	—
14	#14583054	C₄	1	2	yes	—
15	#14583030	—	2	1	no	—
16	#14583030	C₂²	1	2	yes	—
17	Q₈×C₄₃⋊C₈	—	2	1	no	—
18	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	—
19	#14583023	—	2	1	no	—
20	#14583023	—	2	1	no	—
21	#14583023	C₂²	1	2	yes	—
22	#14583023	C₂²	1	2	yes	—
23	C₄×#10621885	C₂²	1	4	yes	—
24	#14582952	C₂²	1	1	yes	characteristic
25	#14582926	—	2	1	no	—
26	#14582923	C₂²	1	1	yes	characteristic
27	C₂×Dic₄₃⋊C₈	—	2	1	no	—
28	#14582855	—	2	1	no	—
29	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₃	—	2	1	no	—
31	#14582649	C₂²	1	4	yes	—
32	#14582631	C₂²	1	4	yes	—
33	#14582621	C₂²	1	2	yes	—
34	#14582620	C₂²	1	2	yes	—
35	C₄₃×#533	C₂²	1	1	yes	characteristic
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×Q₈⋊C₈	C₂²	1	4	yes	—
39	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
40	C₂×#10621885	C₂×C₄	1	4	yes	—
41	C₂×#10621877	—	2	1	no	—
42	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
43	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
44	C₄×Dic₈₆	—	2	1	no	—
45	C₄×Dic₈₆	—	2	1	no	—
46	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₈₆	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×Dic₄₃⋊C₄	—	2	1	no	—
50	C₂×C₄×Dic₄₃	—	2	1	no	—
51	#13063258	C₂×C₄	1	4	yes	—
52	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
54	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
55	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
56	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
57	Dic₄₃⋊C₈	—	2	2	no	—
58	Dic₄₃⋊C₈	—	2	2	no	—
59	#13063249	C₂×C₄	1	4	yes	—
60	#13063248	C₂×C₄	1	4	yes	—
61	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
62	#13063236	—	2	2	no	—
63	C₄×C₄₃⋊C₈	—	2	2	no	—
64	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
65	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
66	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
67	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₈×Dic₄₃	—	2	2	no	—
70	C₈×Dic₄₃	—	2	2	no	—
71	C₂²×C₈×C₄₃	—	2	1	no	—
72	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
74	#125×C₄₃	—	2	1	no	—
75	C₂²⋊Q₈×C₄₃	—	2	1	no	—
76	C₂²⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₄₃	—	2	1	no	—
78	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
79	#117×C₄₃	—	2	1	no	—
80	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₄₃	—	2	1	no	—
82	C₄⋊C₈×C₄₃	—	2	1	no	—
83	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₄₃	—	2	1	no	—
85	C₄×C₈×C₄₃	C₂³	1	2	yes	—
86	C₂×Q₈×C₄₃	—	2	1	no	—
87	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
91	C₂²×Dic₄₃	—	2	1	no	—
92	C₂²×Dic₄₃	—	2	1	no	—
93	C₂×Dic₈₆	—	2	1	no	—
94	C₂×Dic₈₆	—	2	4	no	—
95	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	4	2	no	—
99	C₄⋊C₄×C₄₃	—	2	1	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	1	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
105	C₂²⋊C₄×C₄₃	—	2	2	no	—
106	C₄²×C₄₃	—	2	2	no	—
107	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₃	#41	1	1	yes	characteristic
110	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
111	#10621885	—	2	8	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
114	Dic₄₃⋊C₄	—	2	2	no	—
115	Dic₄₃⋊C₄	—	2	2	no	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	#10621877	—	4	2	no	—
119	C₂×C₄₃⋊C₈	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
123	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
128	Q₈×C₄₃	—	2	4	no	—
129	Q₈×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#137	1	2	yes	—
138	C₂×C₄×C₄₃	#137	1	2	yes	—
139	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₄₃	—	2	4	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	4	2	no	—
146	Dic₈₆	—	2	4	no	—
147	Dic₈₆	—	2	2	no	—
148	Dic₈₆	#131	1	4	yes	—
149	C₈×C₄₃	—	4	2	no	—
150	C₈×C₄₃	—	4	2	no	—
151	C₄₃⋊C₈	—	2	4	no	—
152	C₄₃⋊C₈	—	2	1	no	—
153	C₄₃⋊C₈	—	2	1	no	—
154	C₄₃⋊C₈	—	2	1	no	—
155	C₄₃⋊C₈	—	2	1	no	—
156	C₄₃⋊C₈	—	4	2	no	—
157	#17906	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#441	1	2	yes	—
161	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	4	2	no	—
166	C₄×C₄₃	—	4	2	no	—
167	C₄×C₄₃	#441	1	2	yes	—
168	C₄×C₄₃	C₂×#131	1	2	yes	—
169	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
170	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
171	Dic₄₃	—	2	4	no	—
172	Dic₄₃	—	4	2	no	—
173	#3104	—	43	1	no	—
174	#2960	—	43	1	no	—
175	C₂×C₈×Q₈	—	43	1	no	—
176	#1587	—	43	4	no	—
177	#1579	—	43	4	no	—
178	#1570	—	43	1	no	—
179	#1487	—	43	1	no	—
180	#1481	D₄₃	1	1	yes	characteristic
181	#1452	—	43	1	no	—
182	C₂×C₄₃	—	2	2	no	—
183	C₂×C₄₃	#2929	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
185	C₂×C₄₃	#2955	1	1	yes	characteristic
186	#533	C₂×D₄₃	1	1	yes	characteristic
187	C₂×C₄×Q₈	—	43	1	no	—
188	#484	—	43	2	no	—
189	#477	—	43	2	no	—
190	#462	—	43	2	no	—
191	#451	—	43	2	no	—
192	#446	—	43	2	no	—
193	#446	—	86	1	no	—
194	C₈×Q₈	—	43	2	no	—
195	C₈×Q₈	—	43	2	no	—
196	C₈×Q₈	—	43	2	no	—
197	C₈×Q₈	—	86	1	no	—
198	C₈×Q₈	—	86	1	no	—
199	C₈×Q₈	—	86	1	no	—
200	C₄×Q₁₆	—	43	4	no	—
201	#431	—	43	1	no	—
202	#431	C₂×D₄₃	1	1	yes	characteristic
203	#424	—	86	1	no	—
204	#423	—	43	1	no	—
205	C₂×C₄⋊C₈	—	86	1	no	—
206	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
207	#405	—	86	1	no	—
208	C₂×C₄×C₈	—	43	1	no	—
209	C₂×C₄×C₈	—	86	1	no	—
210	#334	—	43	4	no	—
211	Q₈⋊C₈	—	43	4	no	—
212	Q₈⋊C₈	C₂×D₄₃	1	4	yes	—
213	#322	—	43	2	no	—
214	#322	—	43	2	no	—
215	C₄₃	#17906	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	43	1	no	—
217	C₂×Q₁₆	—	43	4	no	—
218	C₂×M₁₆	—	86	1	no	—
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	—	43	1	no	—
221	C₂²×C₈	—	43	1	no	—
222	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
223	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
224	#125	—	2	1	no	—
225	C₂²⋊Q₈	—	2	1	no	—
226	C₂²⋊Q₈	C₄×D₄₃	1	2	yes	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	43	2	no	—
229	C₄×Q₈	—	43	2	no	—
230	C₄×Q₈	—	86	1	no	—
231	C₄×Q₈	—	86	1	no	—
232	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
233	#117	—	2	1	no	—
234	C₂×C₄⋊C₄	—	43	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₄²	—	86	1	no	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	#107	—	43	4	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	43	2	no	—
242	C₄⋊C₈	—	43	2	no	—
243	C₄⋊C₈	—	86	2	no	—
244	C₄⋊C₈	—	86	2	no	—
245	C₄⋊C₈	—	86	2	no	—
246	C₄⋊C₈	—	86	2	no	—
247	C₄⋊C₈	—	86	1	no	—
248	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
249	#103	—	43	4	no	—
250	#103	—	43	4	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	C₂²⋊C₈	—	43	2	no	—
253	#97	—	86	2	no	—
254	C₄×C₈	—	43	1	no	—
255	C₄×C₈	—	43	1	no	—
256	C₄×C₈	—	43	2	no	—
257	C₄×C₈	—	43	1	no	—
258	C₄×C₈	—	43	1	no	—
259	C₄×C₈	—	86	2	no	—
260	C₄×C₈	—	86	2	no	—
261	C₄×C₈	—	86	2	no	—
262	C₄×C₈	C₂²×D₄₃	1	2	yes	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	43	2	no	—
265	C₂×Q₈	—	86	4	no	—
266	C₂×Q₈	—	86	1	no	—
267	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
268	C₂²×C₄	—	86	1	no	—
269	C₂²×C₄	—	86	1	no	—
270	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	#10621900	1	1	yes	characteristic
273	Q₁₆	—	86	8	no	—
274	M₁₆	—	172	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	43	4	no	—
279	C₂×C₈	—	43	1	no	—
280	C₂×C₈	—	43	1	no	—
281	C₂×C₈	—	43	1	no	—
282	C₂×C₈	—	43	1	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	—	86	2	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	43	2	no	—
292	C₄⋊C₄	—	86	1	no	—
293	C₄⋊C₄	—	86	2	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	86	2	no	—
299	C₄²	—	86	2	no	—
300	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
301	C₄²	D₄×D₄₃	1	1	yes	characteristic
302	C₄²	#10621900	1	1	yes	characteristic
303	C₄²	C₂³×D₄₃	1	1	yes	characteristic
304	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
305	Q₈	—	2	4	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	43	4	no	—
308	Q₈	—	86	4	no	—
309	Q₈	—	86	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	86	4	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
319	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
321	C₂×C₄	#13063354	1	2	yes	—
322	C₂×C₄	#13063357	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	86	4	no	—
329	C₈	—	86	1	no	—
330	C₈	—	86	1	no	—
331	C₈	—	86	1	no	—
332	C₈	—	86	1	no	—
333	C₈	—	172	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	#14583003	1	2	yes	—
337	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	86	4	no	—
344	C₄	—	172	2	no	—
345	C₄	#14583003	1	2	yes	—
346	C₄	#131×D₄₃	1	2	yes	—
347	C₄	C₂×#13063354	1	1	yes	characteristic
348	C₄	C₂×#13063357	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	#19143292	1	1	yes	characteristic
351	C₂	C₂×#14583003	1	1	yes	characteristic
352	C₂	#19143494	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×#1481	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1481	D₄₃
hypocenter	C₄₃	#17906
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

Complements

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

N	H ≅ G/N	type
4. #19133469	C₂	semidirect
5. #19133411	C₂	semidirect
215. C₄₃	#17906	semidirect

Maximal subgroups

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

subgroup	index
#19144630	2
#19143530	2
C₂×#14583023	2
#19133469	2
#19133411	2
#19133365	2
#19133025	2
#19132888	2
C₄₃×#1481	2
#17906	43