Subgroups — #25067077

Order 11008.

558 conjugacy classes of subgroups.

The subgroup lattice has 364 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	#19143690	C₂	1	1	yes	maximal, characteristic
3	#19143678	C₂	1	1	yes	maximal, characteristic
4	#19133501	C₂	1	2	yes	maximal, complemented
5	#19133489	C₂	1	2	yes	maximal, complemented
6	#19133463	C₂	1	2	yes	maximal, complemented
7	#19133442	C₂	1	2	yes	maximal, complemented
8	#19133373	C₂	1	1	yes	maximal, characteristic
9	#19133043	C₂	1	1	yes	maximal, characteristic
10	#19132892	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1485	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
13	#14583232	C₄	1	2	yes	—
14	#14583191	C₄	1	2	yes	—
15	#14583106	C₄	1	2	yes	—
16	#14583054	C₄	1	2	yes	—
17	#14583030	—	2	2	no	—
18	#14583030	C₂²	1	2	yes	—
19	#14583029	—	2	2	no	—
20	#14583026	C₂²	1	2	yes	—
21	#14583026	C₂²	1	2	yes	—
22	Q₈×C₄₃⋊C₈	C₂²	1	2	yes	—
23	C₄×#10621885	C₂²	1	2	yes	—
24	C₄×#10621885	C₂²	1	2	yes	—
25	#14582960	—	2	2	no	—
26	#14582956	C₂²	1	1	yes	characteristic
27	#14582953	—	2	2	no	—
28	#14582952	C₂²	1	1	yes	characteristic
29	#14582952	C₂²	1	1	yes	characteristic
30	#14582651	C₂²	1	2	yes	—
31	#14582649	C₂²	1	2	yes	—
32	#14582631	C₂²	1	2	yes	—
33	#14582631	C₂²	1	2	yes	—
34	#14582621	C₂²	1	1	yes	characteristic
35	#14582621	C₂²	1	1	yes	characteristic
36	#14582620	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#533	C₂²	1	1	yes	characteristic
39	C₄₃×#431	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₄₃×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
44	C₂×#10621885	C₂×C₄	1	2	yes	—
45	C₂×#10621885	C₂×C₄	1	2	yes	—
46	C₄×Dic₈₆	—	2	1	no	—
47	C₄×Dic₈₆	—	2	1	no	—
48	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
49	C₄×Dic₈₆	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×Dic₄₃⋊C₄	—	2	1	no	—
52	C₂×C₄×Dic₄₃	—	2	1	no	—
53	#13063258	C₂×C₄	1	2	yes	—
54	#13063258	C₂×C₄	1	2	yes	—
55	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
56	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
57	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
58	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
59	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
60	Dic₄₃⋊C₈	—	2	2	no	—
61	Dic₄₃⋊C₈	—	2	2	no	—
62	#13063249	C₂×C₄	1	2	yes	—
63	#13063249	C₂×C₄	1	2	yes	—
64	#13063248	C₂×C₄	1	2	yes	—
65	#13063248	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
68	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
69	#13063236	—	2	1	no	—
70	#13063236	—	2	2	no	—
71	#13063235	—	2	2	no	—
72	#13063235	—	2	2	no	—
73	C₄×C₄₃⋊C₈	—	2	2	no	—
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₄₃	—	2	1	no	—
79	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
81	#125×C₄₃	—	2	1	no	—
82	C₂²⋊Q₈×C₄₃	—	2	1	no	—
83	C₂²⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
84	C₄×Q₈×C₄₃	—	2	1	no	—
85	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
86	#117×C₄₃	—	2	1	no	—
87	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
89	C₂²⋊C₈×C₄₃	—	2	1	no	—
90	#97×C₄₃	—	2	1	no	—
91	C₄×C₈×C₄₃	—	2	1	no	—
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₃	—	2	1	no	—
95	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
99	C₂²×Dic₄₃	—	2	1	no	—
100	C₂²×Dic₄₃	—	2	1	no	—
101	C₂×Dic₈₆	—	2	1	no	—
102	C₂×Dic₈₆	—	4	2	no	—
103	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
104	M₁₆×C₄₃	—	4	2	no	—
105	C₂×C₈×C₄₃	—	2	1	no	—
106	C₂×C₈×C₄₃	—	2	2	no	—
107	C₂×C₈×C₄₃	—	2	1	no	—
108	C₄⋊C₄×C₄₃	—	2	1	no	—
109	C₄⋊C₄×C₄₃	—	2	1	no	—
110	C₄⋊C₄×C₄₃	—	2	1	no	—
111	C₄⋊C₄×C₄₃	—	2	2	no	—
112	C₄⋊C₄×C₄₃	—	2	2	no	—
113	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₄₃	—	2	2	no	—
115	C₄²×C₄₃	—	2	2	no	—
116	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₃	#41	1	1	yes	characteristic
119	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
120	#10621885	—	4	2	no	—
121	#10621885	—	4	2	no	—
122	C₄⋊Dic₄₃	—	2	1	no	—
123	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
124	Dic₄₃⋊C₄	—	2	2	no	—
125	Dic₄₃⋊C₄	—	2	2	no	—
126	C₄×Dic₄₃	—	2	2	no	—
127	C₄×Dic₄₃	—	2	2	no	—
128	C₂×C₄₃⋊C₈	—	2	2	no	—
129	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
132	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
133	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₄₃	—	2	2	no	—
135	Q₈×C₄₃	—	2	2	no	—
136	Q₈×C₄₃	—	8	1	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	2	2	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	2	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	Dic₈₆	—	2	2	no	—
153	Dic₈₆	—	2	2	no	—
154	Dic₈₆	—	4	2	no	—
155	C₈×C₄₃	—	2	2	no	—
156	C₈×C₄₃	—	4	1	no	—
157	C₈×C₄₃	—	4	2	no	—
158	C₄₃⋊C₈	—	2	2	no	—
159	C₄₃⋊C₈	—	2	2	no	—
160	C₄₃⋊C₈	—	2	2	no	—
161	C₄₃⋊C₈	—	2	2	no	—
162	C₄₃⋊C₈	—	4	2	no	—
163	#17953	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	1	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	#568	1	2	yes	—
175	C₄×C₄₃	#578	1	2	yes	—
176	Dic₄₃	—	4	2	no	—
177	Dic₄₃	—	4	2	no	—
178	#3104	—	43	1	no	—
179	#2985	—	43	1	no	—
180	#2983	—	43	1	no	—
181	#1593	—	43	2	no	—
182	#1591	—	43	2	no	—
183	#1587	—	43	2	no	—
184	#1584	—	43	2	no	—
185	#1572	—	43	1	no	—
186	#1491	—	43	1	no	—
187	#1485	D₄₃	1	1	yes	characteristic
188	#1453	—	43	1	no	—
189	C₂×C₄₃	—	4	1	no	—
190	C₂×C₄₃	#2930	1	1	yes	characteristic
191	C₂×C₄₃	#2950	1	1	yes	characteristic
192	C₂×C₄₃	#2956	1	1	yes	characteristic
193	#533	C₂×D₄₃	1	1	yes	characteristic
194	C₂×C₄×Q₈	—	43	1	no	—
195	#484	—	43	2	no	—
196	#477	—	43	2	no	—
197	#462	—	43	2	no	—
198	#451	—	43	2	no	—
199	#446	—	43	2	no	—
200	#446	—	43	2	no	—
201	#446	—	43	2	no	—
202	#446	—	86	2	no	—
203	#446	—	86	2	no	—
204	C₈×Q₈	—	43	2	no	—
205	C₄×Q₁₆	—	43	2	no	—
206	C₄×Q₁₆	—	43	2	no	—
207	#433	—	86	2	no	—
208	#432	—	43	1	no	—
209	#432	—	86	2	no	—
210	#431	—	43	1	no	—
211	#431	—	43	1	no	—
212	#431	C₂×D₄₃	1	1	yes	characteristic
213	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
214	#335	—	43	2	no	—
215	#334	—	43	2	no	—
216	Q₈⋊C₈	—	43	2	no	—
217	Q₈⋊C₈	—	43	2	no	—
218	Q₈⋊C₈	C₂×D₄₃	1	2	yes	—
219	Q₈⋊C₈	C₂×D₄₃	1	2	yes	—
220	#322	—	43	1	no	—
221	#322	—	43	1	no	—
222	#322	—	43	1	no	—
223	C₈²	—	43	1	no	—
224	C₄₃	#17953	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	43	1	no	—
226	C₂×Q₁₆	—	43	2	no	—
227	C₂×Q₁₆	—	43	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
230	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
231	#125	—	2	1	no	—
232	C₂²⋊Q₈	—	2	1	no	—
233	C₂²⋊Q₈	C₄×D₄₃	1	2	yes	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×Q₈	—	43	2	no	—
236	C₄×Q₈	—	43	2	no	—
237	C₄×Q₈	—	86	1	no	—
238	C₄×Q₈	—	86	1	no	—
239	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
240	#117	—	2	1	no	—
241	C₂×C₄⋊C₄	—	43	1	no	—
242	C₂×C₄⋊C₄	—	86	1	no	—
243	C₂×C₄²	—	86	1	no	—
244	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
245	#107	—	43	2	no	—
246	#107	—	43	2	no	—
247	C₄⋊C₈	—	43	2	no	—
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	2	no	—
253	C₄⋊C₈	—	86	2	no	—
254	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
255	#103	—	43	2	no	—
256	#103	—	43	2	no	—
257	#103	—	43	2	no	—
258	#103	—	43	2	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	43	2	no	—
261	C₂²⋊C₈	—	43	2	no	—
262	C₂²⋊C₈	—	86	1	no	—
263	#97	—	2	1	no	—
264	#97	—	86	2	no	—
265	#97	—	86	2	no	—
266	#97	—	86	2	no	—
267	#97	—	86	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	43	1	no	—
270	C₄×C₈	—	43	1	no	—
271	C₄×C₈	—	43	1	no	—
272	C₄×C₈	—	43	1	no	—
273	C₄×C₈	—	86	2	no	—
274	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	43	2	no	—
278	C₂×Q₈	—	86	1	no	—
279	C₂×Q₈	—	172	2	no	—
280	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
285	C₂²×C₄	#10621900	1	1	yes	characteristic
286	Q₁₆	—	172	2	no	—
287	Q₁₆	—	172	2	no	—
288	M₁₆	—	4	2	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	43	2	no	—
293	C₂×C₈	—	43	2	no	—
294	C₂×C₈	—	43	2	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	—	86	2	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	43	2	no	—
303	C₄⋊C₄	—	86	1	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	—	86	2	no	—
306	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	86	2	no	—
310	C₄²	—	86	2	no	—
311	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
312	C₄²	D₄×D₄₃	1	1	yes	characteristic
313	C₄²	#10621900	1	1	yes	characteristic
314	C₄²	C₂³×D₄₃	1	1	yes	characteristic
315	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
316	Q₈	—	2	2	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	8	1	no	—
319	Q₈	—	86	2	no	—
320	Q₈	—	86	2	no	—
321	Q₈	—	172	2	no	—
322	C₂×C₄	—	2	2	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	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	344	1	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
332	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	86	2	no	—
341	C₈	—	86	2	no	—
342	C₈	—	86	2	no	—
343	C₈	—	86	2	no	—
344	C₈	—	172	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	172	2	no	—
356	C₄	—	172	2	no	—
357	C₄	#14583664	1	2	yes	—
358	C₄	#14583715	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#19143301	1	1	yes	characteristic
361	C₂	#19143459	1	1	yes	characteristic
362	C₂	#19143501	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×#1485	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1485	D₄₃
hypocenter	C₄₃	#17953
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1485

Complements

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

N	H ≅ G/N	type
4. #19133501	C₂	semidirect
5. #19133489	C₂	semidirect
6. #19133463	C₂	semidirect
7. #19133442	C₂	semidirect
224. C₄₃	#17953	semidirect

Maximal subgroups

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

subgroup	index
#19144630	2
#19143690	2
#19143678	2
#19133501	2
#19133489	2
#19133463	2
#19133442	2
#19133373	2
#19133043	2
#19132892	2
C₄₃×#1485	2
#17953	43