Subgroups — #25105232

Order 11008.

1006 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145040	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583014	C₂	1	4	yes	maximal, complemented
3	#19143308	C₂	1	1	yes	maximal, characteristic, complemented
4	#19135487	C₂	1	4	yes	maximal, complemented
5	#19135428	C₂	1	1	yes	maximal, characteristic, complemented
6	#19135426	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134769	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063345	C₂²	1	4	yes	—
11	C₂×#13063331	C₂²	1	2	yes	—
12	#14583449	C₂²	1	1	yes	characteristic
13	#14583294	C₄	1	2	yes	complemented
14	#14583291	C₄	1	2	yes	complemented
15	#14583289	C₄	1	2	yes	complemented
16	#14583286	C₄	1	2	yes	complemented
17	#14583014	—	2	16	no	—
18	#14582978	—	2	2	no	—
19	C₈×D₁₇₂	—	2	2	no	—
20	C₂×#13063260	C₂²	1	4	yes	—
21	#14582923	C₂²	1	1	yes	characteristic
22	C₂×#13063252	C₂²	1	4	yes	—
23	C₂×#13063251	C₂²	1	4	yes	—
24	#14582865	C₂²	1	2	yes	—
25	C₂×#13063236	C₂²	1	1	yes	characteristic
26	C₂×#13063235	C₂²	1	2	yes	—
27	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
28	#14582745	C₂²	1	1	yes	characteristic
29	C₂×#131×C₄₃	C₂²	1	2	yes	—
30	C₄₃×#532	C₂²	1	1	yes	characteristic
31	C₄₃×#436	—	2	2	no	—
32	C₈×D₄×C₄₃	—	2	2	no	—
33	C₄₃×#423	C₂²	1	1	yes	characteristic
34	C₄₃×#408	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#10621898	C₂³	1	2	yes	—
37	#13063345	—	2	8	no	—
38	#13063345	—	2	8	no	—
39	C₂×#10621885	C₂×C₄	1	4	yes	—
40	C₂×#10621883	C₂×C₄	1	4	yes	—
41	C₂×#10621884	C₂×C₄	1	4	yes	—
42	C₂×#10621882	C₂×C₄	1	4	yes	—
43	#13063331	—	2	8	no	—
44	C₂×#10621877	C₂³	1	2	yes	—
45	C₂×#10621872	—	2	2	no	—
46	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
47	C₂×C₈×D₄₃	—	2	2	no	—
48	#13063325	C₂×C₄	1	1	yes	characteristic
49	#13063322	C₂×C₄	1	1	yes	characteristic
50	#13063313	C₂×C₄	1	2	yes	—
51	#13063299	—	2	2	no	—
52	C₄×D₁₇₂	—	2	2	no	—
53	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
54	#13063260	—	2	8	no	—
55	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
56	#13063252	—	2	8	no	—
57	#13063251	—	2	8	no	—
58	(C₂×D₄₃)⋊C₈	—	2	4	no	—
59	#13063236	C₂×C₄	1	2	yes	—
60	#13063236	C₂×C₄	1	2	yes	—
61	#13063235	—	2	4	no	—
62	C₈×Dic₄₃	D₄	1	8	yes	complemented
63	C₄₃⋊#95	—	2	2	no	—
64	C₂×#41×C₄₃	C₂³	1	2	yes	—
65	#131×C₄₃	—	2	8	no	—
66	C₂×M₁₆×C₄₃	—	2	2	no	—
67	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₃	—	2	2	no	—
69	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
72	#127×C₄₃	C₂×C₄	1	1	yes	characteristic
73	#124×C₄₃	C₂×C₄	1	2	yes	—
74	#121×C₄₃	—	2	2	no	—
75	C₄×D₄×C₄₃	—	2	2	no	—
76	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₄₃	—	2	2	no	—
78	C₂²⋊C₈×C₄₃	—	2	4	no	—
79	C₄×C₈×C₄₃	D₄	1	2	yes	—
80	C₄×C₈×C₄₃	D₄	1	2	yes	—
81	#41×C₄₃	—	2	4	no	—
82	#41×C₄₃	—	2	4	no	—
83	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×D₄×C₄₃	—	2	2	no	—
85	C₂×D₄×C₄₃	—	2	2	no	—
86	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	#41	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₃	—	2	2	no	—
91	C₂²×Dic₄₃	—	2	2	no	—
92	#10621898	—	2	4	no	—
93	#10621898	—	2	4	no	—
94	C₂×D₁₇₂	—	2	2	no	—
95	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₄₃	—	2	2	no	—
97	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
98	M₁₆×C₄₃	—	2	4	no	—
99	M₁₆×C₄₃	—	4	4	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	4	4	no	—
103	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
104	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₂²⋊C₄×C₄₃	—	2	4	no	—
109	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₄₃	C₂×D₄	1	2	yes	—
112	#10621885	—	2	8	no	—
113	#10621884	—	2	8	no	—
114	#10621883	—	2	8	no	—
115	#10621882	—	2	8	no	—
116	(C₂×D₄₃)⋊C₄	—	2	4	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
120	#10621877	—	2	4	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
123	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
124	C₂×C₄₃⋊C₈	#41	1	4	yes	—
125	#10621872	—	4	4	no	—
126	C₈×D₄₃	—	4	4	no	—
127	C₂³×C₄₃	—	2	2	no	—
128	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
129	C₂²×D₄₃	—	2	2	no	—
130	Q₈×C₄₃	—	2	4	no	—
131	D₄×C₄₃	—	2	4	no	—
132	D₄×C₄₃	—	4	4	no	—
133	D₄×C₄₃	—	4	4	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
144	C₂²⋊D₄₃	—	4	4	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	4	no	—
147	C₂×Dic₄₃	—	4	4	no	—
148	D₁₇₂	—	2	4	no	—
149	D₁₇₂	—	4	4	no	—
150	C₄×D₄₃	—	4	2	no	—
151	C₄×D₄₃	—	4	2	no	—
152	Dic₈₆	—	2	4	no	—
153	C₈×C₄₃	—	2	2	no	—
154	C₈×C₄₃	—	4	2	no	—
155	C₈×C₄₃	#127	1	2	yes	—
156	C₈×C₄₃	#127	1	2	yes	—
157	C₄₃⋊C₈	—	4	2	no	—
158	C₄₃⋊C₈	C₄×D₄	1	8	yes	—
159	#21708	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	—	4	2	no	—
162	C₂²×C₄₃	—	4	4	no	—
163	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂²×C₄₃	#532	1	2	yes	—
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	4	4	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	4	2	no	—
170	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
171	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
172	C₄×C₄₃	#532	1	1	yes	characteristic
173	Dic₄₃	—	4	4	no	—
174	Dic₄₃	—	4	2	no	—
175	#3150	—	43	1	no	—
176	C₂×#443	—	43	4	no	—
177	#2931	—	43	1	no	—
178	#2931	D₄₃	1	1	yes	characteristic, complemented
179	#1975	—	43	4	no	—
180	#1963	—	43	1	no	—
181	#1963	—	43	1	no	—
182	#1849	—	43	1	no	—
183	#1726	—	43	1	no	—
184	C₂×C₄₃	—	2	2	no	—
185	C₂×C₄₃	—	8	2	no	—
186	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
187	C₂×C₄₃	#2955	1	2	yes	—
188	D₄₃	—	8	2	no	—
189	C₂×#135	—	43	4	no	—
190	C₂×#131	—	43	2	no	—
191	C₂×#131	C₂×D₄₃	1	2	yes	—
192	#532	—	43	1	no	—
193	#532	C₂×D₄₃	1	1	yes	characteristic
194	#497	—	43	2	no	—
195	#497	—	43	2	no	—
196	#496	—	43	2	no	—
197	#496	—	43	2	no	—
198	#443	—	86	16	no	—
199	#436	—	2	2	no	—
200	#436	—	86	2	no	—
201	C₈×D₄	—	2	2	no	—
202	C₈×D₄	—	86	2	no	—
203	C₂×#108	—	43	4	no	—
204	#423	—	43	1	no	—
205	#423	C₂×D₄₃	1	1	yes	characteristic
206	C₂×#104	—	43	4	no	—
207	C₂×#104	—	43	4	no	—
208	#408	—	43	2	no	—
209	#408	C₂×D₄₃	1	2	yes	—
210	C₂×#97	—	43	1	no	—
211	C₂×#97	—	43	2	no	—
212	C₂×C₄×C₈	—	43	2	no	—
213	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
214	#376	—	43	1	no	—
215	C₄₃	#21708	1	1	yes	characteristic, complemented, Sylow
216	C₂×#41	—	43	2	no	—
217	C₂×#41	C₂²×D₄₃	1	2	yes	—
218	#135	—	86	8	no	—
219	#135	—	86	8	no	—
220	C₂×Q₁₆	—	43	4	no	—
221	C₂×SD₁₆	—	43	4	no	—
222	C₂×SD₁₆	—	43	4	no	—
223	C₂×D₈	—	43	4	no	—
224	#131	—	2	8	no	—
225	#131	—	86	8	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	43	2	no	—
228	C₂×M₁₆	—	86	2	no	—
229	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	—	43	2	no	—
232	C₂²×C₈	—	86	2	no	—
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊Q₈	—	43	1	no	—
236	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
237	#127	—	43	1	no	—
238	#127	C₄×D₄₃	1	1	yes	characteristic
239	#124	—	43	2	no	—
240	#124	C₄×D₄₃	1	2	yes	—
241	#121	—	2	2	no	—
242	#121	—	86	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	86	2	no	—
245	C₂×C₄²	—	43	2	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	#108	—	86	8	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	86	2	no	—
250	#104	—	86	8	no	—
251	#104	—	86	8	no	—
252	C₂²⋊C₈	—	2	4	no	—
253	C₂²⋊C₈	—	86	4	no	—
254	#97	—	43	2	no	—
255	#97	—	43	2	no	—
256	#97	—	86	4	no	—
257	C₄×C₈	—	43	8	no	—
258	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
259	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
260	#95	—	86	2	no	—
261	#41	—	2	4	no	—
262	#41	—	2	4	no	—
263	#41	—	86	4	no	—
264	#41	—	86	4	no	—
265	C₂×Q₈	—	43	2	no	—
266	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	43	2	no	—
270	C₂×D₄	—	86	2	no	—
271	C₂×D₄	—	86	2	no	—
272	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	86	2	no	—
275	C₂²×C₄	—	86	2	no	—
276	C₂²×C₄	#10621898	1	2	yes	—
277	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
278	Q₁₆	—	86	8	no	—
279	SD₁₆	—	86	8	no	—
280	SD₁₆	—	86	8	no	—
281	D₈	—	86	8	no	—
282	M₁₆	—	2	4	no	—
283	M₁₆	—	4	4	no	—
284	M₁₆	—	86	4	no	—
285	M₁₆	—	172	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	—	43	4	no	—
290	C₂×C₈	—	43	4	no	—
291	C₂×C₈	—	43	4	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	—	172	4	no	—
294	C₂×C₈	D₄×D₄₃	1	4	yes	—
295	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₂²⋊C₄	—	86	4	no	—
302	C₄²	—	43	4	no	—
303	C₄²	—	86	2	no	—
304	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
306	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	86	2	no	—
309	C₂³	C₂×#10621898	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	86	4	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	86	4	no	—
316	D₄	—	172	4	no	—
317	D₄	—	172	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	86	4	no	—
324	C₂×C₄	—	172	4	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
328	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
329	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
331	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
332	C₂×C₄	#41×D₄₃	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	43	8	no	—
336	C₈	—	172	2	no	—
337	C₈	#13063321	1	2	yes	—
338	C₈	#13063321	1	2	yes	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	172	4	no	—
343	C₂²	—	172	2	no	—
344	C₂²	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
345	C₂²	#14583446	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	172	4	no	—
350	C₄	—	172	2	no	—
351	C₄	C₄×D₄×D₄₃	1	2	yes	—
352	C₄	C₂×#13063321	1	1	yes	characteristic
353	C₄	#14583445	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	8	2	no	—
356	C₂	—	344	2	no	—
357	C₂	C₄×#13063321	1	1	yes	characteristic
358	C₂	#19143496	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2931	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#19145040	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2931	D₄₃
hypocenter	C₄₃	#21708
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#14583014	C₂	semidirect
3. #19143308	C₂	semidirect
4. #19135487	C₂	semidirect
5. #19135428	C₂	semidirect
6. #19135426	C₂	semidirect
7. #19134769	C₂	semidirect
8. #19134281	C₂	semidirect
9. C₄₃×#2931	C₂	semidirect
13. #14583294	C₄	semidirect
14. #14583291	C₄	semidirect
15. #14583289	C₄	semidirect
16. #14583286	C₄	semidirect
62. C₈×Dic₄₃	D₄	semidirect
178. #2931	D₄₃	semidirect
215. C₄₃	#21708	semidirect

Maximal subgroups

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

subgroup	index
#19145040	2
C₂×#14583014	2
#19143308	2
#19135487	2
#19135428	2
#19135426	2
#19134769	2
#19134281	2
C₄₃×#2931	2
#21708	43