Subgroups — #25068657

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145163	C₂	1	1	yes	maximal, characteristic
2	#19143756	C₂	1	2	yes	maximal, complemented
3	#19133503	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133436	C₂	1	2	yes	maximal, complemented
5	#19133428	C₂	1	2	yes	maximal, complemented
6	#19133419	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133415	C₂	1	1	yes	maximal, characteristic, complemented
8	#334×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#19133005	C₂	1	2	yes	maximal, complemented
10	#19132881	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1594	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583431	C₂²	1	2	yes	—
13	#14583300	C₄	1	1	yes	characteristic
14	#14583282	C₄	1	1	yes	characteristic
15	#14583252	C₄	1	2	yes	—
16	#14583141	C₄	1	1	yes	characteristic
17	#14583132	C₄	1	1	yes	characteristic
18	#14583127	C₄	1	2	yes	—
19	#14583026	C₂²	1	2	yes	—
20	C₈×Dic₈₆	C₂²	1	2	yes	—
21	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
22	C₄×#10621883	C₂²	1	2	yes	—
23	#14582968	—	2	2	no	—
24	D₄×C₄₃⋊C₈	C₂²	1	2	yes	—
25	(C₂²⋊D₄₃)⋊C₈	—	2	2	no	—
26	#14582957	—	2	2	no	—
27	#14582950	C₂²	1	1	yes	characteristic
28	#14582927	—	2	2	no	—
29	C₄⋊C₈×D₄₃	C₂²	1	2	yes	—
30	C₄×#10621872	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
32	#14582650	C₂²	1	1	yes	characteristic
33	#14582649	C₂²	1	2	yes	—
34	#14582631	C₂²	1	2	yes	—
35	#14582629	C₂²	1	2	yes	—
36	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
37	#14582621	C₂²	1	1	yes	characteristic
38	#14582620	C₂²	1	1	yes	characteristic
39	#14582620	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×#436	C₂²	1	2	yes	—
42	C₄₃×#334	C₂²	1	1	yes	characteristic
43	C₄₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₄₃×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#10621883	C₂×C₄	1	2	yes	—
47	C₂×#10621872	—	2	1	no	—
48	C₂²×C₄₃⋊C₈	—	2	2	no	—
49	C₂×C₈×D₄₃	—	2	2	no	—
50	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
52	#13063318	—	2	2	no	—
53	#13063316	C₂×C₄	1	2	yes	—
54	#13063312	C₂×C₄	1	2	yes	—
55	#13063307	—	2	2	no	—
56	#13063305	C₂×C₄	1	2	yes	—
57	#13063296	C₂×C₄	1	2	yes	—
58	C₄×Dic₈₆	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₄₃	—	2	2	no	—
60	#13063284	—	2	2	no	—
61	C₄²×D₄₃	C₂³	1	1	yes	characteristic
62	#13063259	C₂×C₄	1	1	yes	characteristic
63	#13063256	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
65	Dic₄₃⋊C₈	—	2	2	no	—
66	Dic₄₃⋊C₈	—	2	2	no	—
67	Dic₄₃⋊C₈	—	2	2	no	—
68	Dic₄₃⋊C₈	D₄	1	2	yes	—
69	#13063250	C₂×C₄	1	2	yes	—
70	#13063248	C₂×C₄	1	2	yes	—
71	#13063246	C₂×C₄	1	2	yes	—
72	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
73	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
74	#13063235	—	2	1	no	—
75	#13063235	—	2	2	no	—
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₄₃	—	2	2	no	—
80	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
82	C₂×D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
83	C₂×M₁₆×C₄₃	—	2	2	no	—
84	C₄×D₄×C₄₃	C₂³	1	2	yes	—
85	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
87	#102×C₄₃	C₂×C₄	1	2	yes	—
88	C₂²⋊C₈×C₄₃	—	2	2	no	—
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₄₃	—	2	2	no	—
94	C₂×C₂²⋊D₄₃	—	2	2	no	—
95	C₂×C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
98	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
99	D₈×C₄₃	—	2	2	no	—
100	M₁₆×C₄₃	—	8	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
107	C₂²⋊C₄×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₄₃	—	2	2	no	—
110	C₂²⋊Dic₄₃	—	2	2	no	—
111	C₂²⋊Dic₄₃	—	2	2	no	—
112	#10621883	—	4	2	no	—
113	(C₂×D₄₃)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	—	2	2	no	—
116	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	Dic₄₃⋊C₄	—	2	2	no	—
119	C₄×Dic₄₃	—	2	2	no	—
120	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₃	#41	1	1	yes	characteristic
123	C₂×C₄₃⋊C₈	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	2	1	no	—
125	C₂×C₄₃⋊C₈	—	8	2	no	—
126	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
129	Dic₁₇₂	—	2	2	no	—
130	#10621872	—	4	2	no	—
131	C₈×D₄₃	—	2	2	no	—
132	C₈×D₄₃	—	4	1	no	—
133	C₈×D₄₃	—	8	2	no	—
134	C₂³×C₄₃	—	2	2	no	—
135	C₂²×D₄₃	C₄×D₄	1	1	yes	characteristic
136	D₄×C₄₃	—	2	2	no	—
137	D₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	4	4	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
150	C₄×D₄₃	—	2	2	no	—
151	C₄×D₄₃	—	2	1	no	—
152	C₄×D₄₃	—	2	1	no	—
153	C₄×D₄₃	—	2	2	no	—
154	Dic₈₆	—	2	2	no	—
155	Dic₈₆	—	2	2	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	—	4	1	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₈×C₄₃	#131	1	2	yes	—
161	C₄₃⋊C₈	—	2	1	no	—
162	C₄₃⋊C₈	—	2	1	no	—
163	C₄₃⋊C₈	—	2	2	no	—
164	C₄₃⋊C₈	—	4	1	no	—
165	C₄₃⋊C₈	—	4	2	no	—
166	#18093	—	43	1	no	maximal, Sylow
167	C₂²×C₄₃	—	4	2	no	—
168	C₂²×C₄₃	—	4	2	no	—
169	C₂²×C₄₃	—	4	2	no	—
170	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₄₃	—	2	1	no	—
172	C₂×D₄₃	—	2	1	no	—
173	C₂×D₄₃	#444	1	2	yes	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	4	2	no	—
176	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₄₃	#568	1	1	yes	characteristic
178	C₄×C₄₃	#577	1	2	yes	—
179	Dic₄₃	—	2	1	no	—
180	Dic₄₃	—	2	1	no	—
181	Dic₄₃	—	2	1	no	—
182	Dic₄₃	—	4	2	no	—
183	Dic₄₃	—	4	2	no	—
184	#3168	—	43	1	no	—
185	#2994	—	43	2	no	—
186	#1594	D₄₃	1	1	yes	characteristic, complemented
187	#1593	—	43	1	no	—
188	#1583	—	43	2	no	—
189	#1582	—	43	2	no	—
190	#1580	—	43	1	no	—
191	#1579	—	43	1	no	—
192	C₂×#334	—	43	1	no	—
193	#1483	—	43	2	no	—
194	#1451	—	43	1	no	—
195	C₂×C₄₃	—	8	2	no	—
196	C₂×C₄₃	#2935	1	1	yes	characteristic
197	C₂×C₄₃	#2952	1	1	yes	characteristic
198	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
199	D₄₃	—	2	2	no	—
200	#529	—	43	2	no	—
201	#499	—	43	1	no	—
202	#495	—	43	1	no	—
203	#488	—	43	2	no	—
204	#467	—	43	1	no	—
205	#466	—	43	1	no	—
206	#465	—	43	2	no	—
207	#446	—	43	2	no	—
208	C₈×Q₈	—	43	2	no	—
209	C₄×Q₁₆	—	43	1	no	—
210	C₄×SD₁₆	—	43	2	no	—
211	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
212	#436	C₂×D₄₃	1	2	yes	—
213	#435	—	86	2	no	—
214	C₈×D₄	—	43	2	no	—
215	C₈×D₄	—	86	2	no	—
216	#433	—	86	2	no	—
217	#431	—	43	1	no	—
218	#424	—	86	2	no	—
219	C₂×C₄⋊C₈	—	43	2	no	—
220	C₄×M₁₆	—	43	1	no	—
221	C₂×C₄×C₈	—	43	1	no	—
222	#334	—	43	2	no	—
223	#334	—	43	1	no	—
224	#334	C₂×D₄₃	1	1	yes	characteristic
225	Q₈⋊C₈	—	43	2	no	—
226	Q₈⋊C₈	—	43	2	no	—
227	D₄⋊C₈	—	43	2	no	—
228	D₄⋊C₈	C₂×D₄₃	1	2	yes	—
229	#322	—	43	1	no	—
230	#322	—	43	1	no	—
231	#322	—	43	1	no	—
232	#322	C₂×D₄₃	1	1	yes	characteristic
233	C₄₃	#18093	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	43	1	no	—
235	C₂×SD₁₆	—	43	2	no	—
236	C₂×D₈	C₄×D₄₃	1	1	yes	characteristic
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	86	1	no	—
239	C₂²×C₈	—	43	1	no	—
240	C₂²×C₈	—	43	1	no	—
241	C₂²×C₈	—	86	2	no	—
242	C₂²×C₈	—	86	2	no	—
243	#126	—	86	2	no	—
244	#125	—	43	2	no	—
245	#124	—	43	2	no	—
246	#123	—	86	2	no	—
247	C₂²⋊Q₈	—	43	2	no	—
248	#121	—	43	2	no	—
249	C₄×Q₈	—	43	2	no	—
250	C₄×D₄	—	86	2	no	—
251	C₄×D₄	C₂²×D₄₃	1	2	yes	—
252	#117	—	86	2	no	—
253	C₂×C₄²	—	43	1	no	—
254	#107	—	43	1	no	—
255	#107	C₄×D₄₃	1	1	yes	characteristic
256	#106	—	43	2	no	—
257	C₄⋊C₈	—	43	2	no	—
258	C₄⋊C₈	—	43	2	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	#103	—	43	2	no	—
264	#103	—	43	2	no	—
265	#102	—	43	2	no	—
266	#102	C₄×D₄₃	1	2	yes	—
267	C₂²⋊C₈	—	2	2	no	—
268	C₂²⋊C₈	—	43	2	no	—
269	C₂²⋊C₈	—	86	2	no	—
270	#97	—	86	1	no	—
271	#97	—	86	2	no	—
272	C₄×C₈	—	43	1	no	—
273	C₄×C₈	—	43	1	no	—
274	C₄×C₈	—	43	1	no	—
275	C₄×C₈	—	43	1	no	—
276	C₄×C₈	—	43	1	no	—
277	C₄×C₈	—	86	2	no	—
278	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
281	C₂×Q₈	—	43	2	no	—
282	C₂×D₄	—	86	2	no	—
283	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	43	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	43	1	no	—
288	Q₁₆	—	86	2	no	—
289	SD₁₆	—	172	2	no	—
290	D₈	—	2	2	no	—
291	M₁₆	—	8	2	no	—
292	M₁₆	—	172	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	43	1	no	—
296	C₂×C₈	—	43	1	no	—
297	C₂×C₈	—	43	2	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	—	86	2	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	344	2	no	—
302	C₂×C₈	—	344	2	no	—
303	C₂×C₈	—	172	1	no	—
304	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
306	C₂×C₈	D₄×D₄₃	1	2	yes	—
307	C₄⋊C₄	—	43	2	no	—
308	C₄⋊C₄	—	86	2	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	86	2	no	—
315	C₂²⋊C₄	—	86	2	no	—
316	C₂²⋊C₄	—	86	2	no	—
317	C₂²⋊C₄	—	86	2	no	—
318	C₄²	—	43	1	no	—
319	C₄²	—	43	1	no	—
320	C₄²	—	43	1	no	—
321	C₄²	—	86	2	no	—
322	C₄²	C₂³×D₄₃	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	—	43	1	no	—
325	Q₈	—	86	2	no	—
326	Q₈	—	86	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	172	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	43	1	no	—
334	C₂×C₄	—	43	1	no	—
335	C₂×C₄	—	43	1	no	—
336	C₂×C₄	—	86	2	no	—
337	C₂×C₄	—	86	1	no	—
338	C₂×C₄	—	86	1	no	—
339	C₂×C₄	—	86	2	no	—
340	C₂×C₄	—	86	2	no	—
341	C₂×C₄	—	86	2	no	—
342	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	86	1	no	—
350	C₈	—	86	1	no	—
351	C₈	—	86	2	no	—
352	C₈	—	172	1	no	—
353	C₈	—	172	2	no	—
354	C₈	#13063333	1	2	yes	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	43	2	no	—
359	C₂²	—	86	1	no	—
360	C₂²	—	86	1	no	—
361	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	86	1	no	—
365	C₄	—	86	1	no	—
366	C₄	—	86	1	no	—
367	C₄	—	172	2	no	—
368	C₄	—	172	2	no	—
369	C₄	C₂×#13063333	1	1	yes	characteristic
370	C₄	#14583664	1	1	yes	characteristic
371	C₄	#14583709	1	2	yes	—
372	C₂	—	8	2	no	—
373	C₂	—	86	2	no	—
374	C₂	#19143340	1	1	yes	characteristic
375	C₂	#19143477	1	1	yes	characteristic
376	C₂	#444×D₄₃	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×#1594	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583132	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1594	D₄₃
hypocenter	C₄₃	#18093
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1594

Complements

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

N	H ≅ G/N	type
2. #19143756	C₂	semidirect
3. #19133503	C₂	semidirect
4. #19133436	C₂	semidirect
5. #19133428	C₂	semidirect
6. #19133419	C₂	semidirect
7. #19133415	C₂	semidirect
8. #334×D₄₃	C₂	semidirect
9. #19133005	C₂	semidirect
10. #19132881	C₂	semidirect
11. C₄₃×#1594	C₂	semidirect
186. #1594	D₄₃	semidirect
233. C₄₃	#18093	semidirect

Maximal subgroups

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

subgroup	index
#19145163	2
#19143756	2
#19133503	2
#19133436	2
#19133428	2
#19133419	2
#19133415	2
#334×D₄₃	2
#19133005	2
#19132881	2
C₄₃×#1594	2
#18093	43