Subgroups — #25068636

Order 11008.

644 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145171	C₂	1	1	yes	maximal, characteristic
2	#19143660	C₂	1	2	yes	maximal, complemented
3	#19133508	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133422	C₂	1	2	yes	maximal, complemented
5	#19133421	C₂	1	2	yes	maximal, complemented
6	#19133420	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133419	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133340	C₂	1	1	yes	maximal, characteristic, complemented
9	#19133010	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	#14583446	C₂²	1	2	yes	—
13	#14583303	C₄	1	1	yes	characteristic
14	#14583274	C₄	1	1	yes	characteristic
15	#14583244	C₄	1	2	yes	—
16	#14583136	C₄	1	1	yes	characteristic
17	#14583134	C₄	1	2	yes	—
18	#14583132	C₄	1	1	yes	characteristic
19	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
20	C₄×#10621882	C₂²	1	2	yes	—
21	#14582976	C₂²	1	2	yes	—
22	#14582975	—	2	2	no	—
23	C₈×D₁₇₂	C₂²	1	2	yes	—
24	D₄×C₄₃⋊C₈	C₂²	1	2	yes	—
25	(C₂²⋊D₄₃)⋊C₈	—	2	2	no	—
26	#14582951	C₂²	1	2	yes	—
27	#14582950	C₂²	1	1	yes	characteristic
28	#14582867	—	2	2	no	—
29	#14582864	—	2	2	no	—
30	C₄×#10621872	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
32	#14582651	C₂²	1	2	yes	—
33	#14582650	C₂²	1	1	yes	characteristic
34	#14582628	C₂²	1	2	yes	—
35	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
36	#14582626	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×#10621898	—	2	2	no	—
46	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#10621882	C₂×C₄	1	2	yes	—
48	C₂×#10621872	—	2	1	no	—
49	C₂×#10621872	—	2	2	no	—
50	C₂²×C₄₃⋊C₈	—	2	2	no	—
51	C₂×C₈×D₄₃	—	2	2	no	—
52	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
54	#13063324	C₂×C₄	1	2	yes	—
55	#13063321	C₂×C₄	1	2	yes	—
56	#13063311	—	2	2	no	—
57	#13063300	C₂×C₄	1	2	yes	—
58	#13063298	—	2	2	no	—
59	#13063296	C₂×C₄	1	2	yes	—
60	C₄×D₁₇₂	C₂³	1	2	yes	—
61	C₄×C₂²⋊D₄₃	—	2	2	no	—
62	C₄²×D₄₃	C₂³	1	1	yes	characteristic
63	#13063259	C₂×C₄	1	1	yes	characteristic
64	#13063258	C₂×C₄	1	2	yes	—
65	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
66	Dic₄₃⋊C₈	D₄	1	2	yes	—
67	#13063247	C₂×C₄	1	2	yes	—
68	#13063246	C₂×C₄	1	2	yes	—
69	#13063245	C₂×C₄	1	2	yes	—
70	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
71	(C₂×D₄₃)⋊C₈	—	2	2	no	—
72	(C₂×D₄₃)⋊C₈	—	2	2	no	—
73	(C₂×D₄₃)⋊C₈	—	2	2	no	—
74	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
75	#13063235	—	2	1	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×C₂²⋊D₄₃	—	2	2	no	—
96	C₂×C₂²⋊D₄₃	—	2	2	no	—
97	#10621898	—	8	2	no	—
98	#10621898	—	8	2	no	—
99	C₂×D₁₇₂	—	2	2	no	—
100	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₄₃	—	2	2	no	—
102	C₂×C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
105	C₂×Dic₈₆	—	2	2	no	—
106	D₈×C₄₃	—	2	2	no	—
107	M₁₆×C₄₃	—	8	2	no	—
108	C₂×C₈×C₄₃	—	2	2	no	—
109	C₂×C₈×C₄₃	—	2	1	no	—
110	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₄₃	—	2	2	no	—
115	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂²⋊Dic₄₃	—	2	2	no	—
117	#10621882	—	4	2	no	—
118	(C₂×D₄₃)⋊C₄	—	2	2	no	—
119	(C₂×D₄₃)⋊C₄	—	2	2	no	—
120	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
121	Dic₄₃⋊C₄	—	2	2	no	—
122	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₄₃	#41	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	—	2	2	no	—
126	C₂×C₄₃⋊C₈	—	2	1	no	—
127	C₂×C₄₃⋊C₈	—	8	2	no	—
128	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
131	D₃₄₄	—	2	2	no	—
132	#10621872	—	4	2	no	—
133	#10621872	—	8	2	no	—
134	C₈×D₄₃	—	2	2	no	—
135	C₈×D₄₃	—	4	1	no	—
136	C₈×D₄₃	—	8	2	no	—
137	C₂³×C₄₃	—	2	2	no	—
138	C₂²×D₄₃	—	2	2	no	—
139	C₂²×D₄₃	C₄×D₄	1	1	yes	characteristic
140	D₄×C₄₃	—	2	2	no	—
141	D₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
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	C₂²⋊D₄₃	—	4	4	no	—
149	C₂²⋊D₄₃	—	8	2	no	—
150	C₂²⋊D₄₃	—	8	2	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
155	D₁₇₂	—	2	2	no	—
156	D₁₇₂	—	2	2	no	—
157	D₁₇₂	—	8	2	no	—
158	C₄×D₄₃	—	2	1	no	—
159	C₄×D₄₃	—	2	1	no	—
160	C₄×D₄₃	—	2	2	no	—
161	C₄×D₄₃	—	2	2	no	—
162	C₄×D₄₃	—	4	2	no	—
163	C₄×D₄₃	—	4	2	no	—
164	Dic₈₆	—	8	2	no	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	2	2	no	—
167	C₈×C₄₃	—	4	1	no	—
168	C₈×C₄₃	—	4	2	no	—
169	C₈×C₄₃	#131	1	2	yes	—
170	C₄₃⋊C₈	—	2	1	no	—
171	C₄₃⋊C₈	—	2	1	no	—
172	C₄₃⋊C₈	—	2	2	no	—
173	C₄₃⋊C₈	—	4	1	no	—
174	C₄₃⋊C₈	—	4	2	no	—
175	#18091	—	43	1	no	maximal, Sylow
176	C₂²×C₄₃	—	4	2	no	—
177	C₂²×C₄₃	—	4	2	no	—
178	C₂²×C₄₃	—	4	2	no	—
179	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×D₄₃	—	2	1	no	—
181	C₂×D₄₃	—	2	1	no	—
182	C₂×D₄₃	—	2	1	no	—
183	C₂×D₄₃	—	4	2	no	—
184	C₂×D₄₃	—	4	2	no	—
185	C₂×D₄₃	—	4	2	no	—
186	C₄×C₄₃	—	2	1	no	—
187	C₄×C₄₃	—	4	2	no	—
188	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
189	C₄×C₄₃	#568	1	1	yes	characteristic
190	C₄×C₄₃	#576	1	2	yes	—
191	Dic₄₃	—	2	1	no	—
192	Dic₄₃	—	2	1	no	—
193	Dic₄₃	—	4	2	no	—
194	Dic₄₃	#444	1	2	yes	—
195	#3169	—	43	1	no	—
196	#2979	—	43	2	no	—
197	#1594	—	43	1	no	—
198	#1594	D₄₃	1	1	yes	characteristic, complemented
199	#1581	—	43	2	no	—
200	#1581	—	43	2	no	—
201	#1580	—	43	1	no	—
202	#1580	—	43	1	no	—
203	#1564	—	43	1	no	—
204	#1484	—	43	2	no	—
205	#1451	—	43	1	no	—
206	C₂×C₄₃	—	8	2	no	—
207	C₂×C₄₃	#2935	1	1	yes	characteristic
208	C₂×C₄₃	#2948	1	1	yes	characteristic
209	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
210	D₄₃	—	4	1	no	—
211	D₄₃	—	8	2	no	—
212	#532	—	43	2	no	—
213	#500	—	43	1	no	—
214	#493	—	43	1	no	—
215	#486	—	43	2	no	—
216	#466	—	43	1	no	—
217	#466	—	43	1	no	—
218	#466	—	43	2	no	—
219	C₄×D₈	—	43	2	no	—
220	C₄×D₈	—	43	1	no	—
221	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
222	#436	—	43	2	no	—
223	#436	—	86	2	no	—
224	#436	C₂×D₄₃	1	2	yes	—
225	C₈×D₄	—	43	2	no	—
226	C₈×D₄	—	43	2	no	—
227	C₈×D₄	—	86	2	no	—
228	#431	—	43	1	no	—
229	#431	—	43	2	no	—
230	#408	—	86	2	no	—
231	#408	—	86	2	no	—
232	C₄×M₁₆	—	43	1	no	—
233	C₂×C₄×C₈	—	43	1	no	—
234	#335	—	43	2	no	—
235	#334	—	43	1	no	—
236	#334	C₂×D₄₃	1	1	yes	characteristic
237	D₄⋊C₈	—	43	2	no	—
238	D₄⋊C₈	—	43	2	no	—
239	D₄⋊C₈	—	43	2	no	—
240	D₄⋊C₈	C₂×D₄₃	1	2	yes	—
241	#322	—	43	1	no	—
242	#322	—	43	1	no	—
243	#322	—	43	1	no	—
244	#322	C₂×D₄₃	1	1	yes	characteristic
245	C₄₃	#18091	1	1	yes	characteristic, complemented, Sylow
246	C₂×#41	—	86	2	no	—
247	C₂×D₈	—	43	2	no	—
248	C₂×D₈	—	43	1	no	—
249	C₂×D₈	C₄×D₄₃	1	1	yes	characteristic
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	86	2	no	—
252	C₂×M₁₆	—	86	1	no	—
253	C₂²×C₈	—	43	1	no	—
254	C₂²×C₈	—	43	1	no	—
255	C₂²×C₈	—	86	2	no	—
256	C₂²×C₈	—	86	2	no	—
257	C₄⋊Q₈	—	43	2	no	—
258	#127	—	43	2	no	—
259	#124	—	86	2	no	—
260	#121	—	43	2	no	—
261	#121	—	43	2	no	—
262	#121	—	86	2	no	—
263	C₄×D₄	—	43	2	no	—
264	C₄×D₄	—	86	2	no	—
265	C₄×D₄	C₂²×D₄₃	1	2	yes	—
266	C₂×C₄²	—	43	1	no	—
267	#107	—	43	1	no	—
268	#107	—	43	2	no	—
269	#107	C₄×D₄₃	1	1	yes	characteristic
270	C₄⋊C₈	—	43	2	no	—
271	C₄⋊C₈	—	43	2	no	—
272	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
273	#102	—	43	2	no	—
274	#102	—	43	2	no	—
275	#102	—	43	2	no	—
276	#102	C₄×D₄₃	1	2	yes	—
277	C₂²⋊C₈	—	2	2	no	—
278	C₂²⋊C₈	—	43	2	no	—
279	C₂²⋊C₈	—	86	2	no	—
280	C₂²⋊C₈	—	86	2	no	—
281	C₂²⋊C₈	—	86	2	no	—
282	C₂²⋊C₈	—	86	2	no	—
283	#97	—	86	1	no	—
284	C₄×C₈	—	43	1	no	—
285	C₄×C₈	—	43	1	no	—
286	C₄×C₈	—	43	1	no	—
287	C₄×C₈	—	43	1	no	—
288	C₄×C₈	—	43	1	no	—
289	C₄×C₈	—	86	2	no	—
290	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
293	#41	—	344	2	no	—
294	#41	—	344	2	no	—
295	C₂×Q₈	—	86	2	no	—
296	C₂×D₄	—	43	2	no	—
297	C₂×D₄	—	86	2	no	—
298	C₂×D₄	—	86	2	no	—
299	C₂×D₄	—	86	2	no	—
300	C₂×D₄	—	86	2	no	—
301	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	43	1	no	—
304	C₂²×C₄	—	43	1	no	—
305	C₂²×C₄	—	43	1	no	—
306	C₂²×C₄	—	86	2	no	—
307	D₈	—	2	2	no	—
308	D₈	—	86	2	no	—
309	D₈	—	172	2	no	—
310	M₁₆	—	8	2	no	—
311	M₁₆	—	344	2	no	—
312	M₁₆	—	172	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	43	1	no	—
316	C₂×C₈	—	43	1	no	—
317	C₂×C₈	—	43	2	no	—
318	C₂×C₈	—	86	2	no	—
319	C₂×C₈	—	86	2	no	—
320	C₂×C₈	—	86	1	no	—
321	C₂×C₈	—	344	2	no	—
322	C₂×C₈	—	344	2	no	—
323	C₂×C₈	—	172	1	no	—
324	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
326	C₂×C₈	D₄×D₄₃	1	2	yes	—
327	C₄⋊C₄	—	43	2	no	—
328	C₄⋊C₄	—	86	2	no	—
329	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	86	2	no	—
332	C₂²⋊C₄	—	86	2	no	—
333	C₂²⋊C₄	—	86	2	no	—
334	C₄²	—	43	1	no	—
335	C₄²	—	43	1	no	—
336	C₄²	—	43	1	no	—
337	C₄²	C₂³×D₄₃	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	—	43	1	no	—
340	C₂³	—	86	2	no	—
341	Q₈	—	344	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	86	2	no	—
345	D₄	—	86	2	no	—
346	D₄	—	344	2	no	—
347	D₄	—	344	2	no	—
348	D₄	—	344	2	no	—
349	D₄	—	172	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	43	1	no	—
354	C₂×C₄	—	43	1	no	—
355	C₂×C₄	—	43	1	no	—
356	C₂×C₄	—	86	2	no	—
357	C₂×C₄	—	86	1	no	—
358	C₂×C₄	—	86	1	no	—
359	C₂×C₄	—	86	2	no	—
360	C₂×C₄	—	86	2	no	—
361	C₂×C₄	—	172	2	no	—
362	C₂×C₄	—	172	2	no	—
363	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
365	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	2	no	—
368	C₈	—	4	1	no	—
369	C₈	—	4	2	no	—
370	C₈	—	86	1	no	—
371	C₈	—	86	1	no	—
372	C₈	—	86	2	no	—
373	C₈	—	172	1	no	—
374	C₈	—	172	2	no	—
375	C₈	#13063333	1	2	yes	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	86	1	no	—
380	C₂²	—	86	1	no	—
381	C₂²	—	86	1	no	—
382	C₂²	—	172	2	no	—
383	C₂²	—	172	2	no	—
384	C₂²	—	172	2	no	—
385	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	2	no	—
388	C₄	—	43	2	no	—
389	C₄	—	86	1	no	—
390	C₄	—	86	1	no	—
391	C₄	—	172	2	no	—
392	C₄	C₂×#13063333	1	1	yes	characteristic
393	C₄	#14583664	1	1	yes	characteristic
394	C₄	#14583703	1	2	yes	—
395	C₂	—	8	2	no	—
396	C₂	—	344	2	no	—
397	C₂	—	172	1	no	—
398	C₂	#19143340	1	1	yes	characteristic
399	C₂	#19143444	1	1	yes	characteristic
400	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	#19145171	C₂
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1594	D₄₃
hypocenter	C₄₃	#18091
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. #19143660	C₂	semidirect
3. #19133508	C₂	semidirect
4. #19133422	C₂	semidirect
5. #19133421	C₂	semidirect
6. #19133420	C₂	semidirect
7. #19133419	C₂	semidirect
8. #19133340	C₂	semidirect
9. #19133010	C₂	semidirect
10. #19132881	C₂	semidirect
11. C₄₃×#1594	C₂	semidirect
198. #1594	D₄₃	semidirect
245. C₄₃	#18091	semidirect

Maximal subgroups

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

subgroup	index
#19145171	2
#19143660	2
#19133508	2
#19133422	2
#19133421	2
#19133420	2
#19133419	2
#19133340	2
#19133010	2
#19132881	2
C₄₃×#1594	2
#18091	43