Subgroups — #25003909

Order 11008.

852 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14582916	C₂	1	2	yes	maximal
3	C₂×#14582670	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₃⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2879	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	2	yes	—
8	#14582941	—	2	4	no	—
9	#14582937	—	2	2	no	—
10	#14582916	—	2	4	no	—
11	#14582916	—	2	4	no	—
12	C₂×#13063252	—	2	4	no	—
13	#14582670	—	2	4	no	—
14	C₄₃⋊#338	—	2	2	no	—
15	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#408	—	2	2	no	—
19	C₄₃×#408	C₄	1	4	yes	complemented
20	C₄₃×#407	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	D₄	1	4	yes	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063284	—	2	2	no	—
30	#13063284	—	2	2	no	—
31	#13063284	D₄	1	4	yes	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄×Dic₄₃	—	2	2	no	—
35	#13063259	—	2	4	no	—
36	#13063257	—	2	4	no	—
37	#13063252	—	2	16	no	—
38	C₂×#41×C₄₃	—	2	1	no	—
39	C₂×#41×C₄₃	—	2	1	no	—
40	C₂×#41×C₄₃	—	2	2	no	—
41	C₂×#41×C₄₃	—	2	2	no	—
42	C₂×#41×C₄₃	—	2	2	no	—
43	C₂×#41×C₄₃	—	2	2	no	—
44	C₂×#41×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×#41×C₄₃	D₄	1	2	yes	—
46	C₂²×Q₈×C₄₃	Q₈	1	1	yes	characteristic
47	C₂²×D₄×C₄₃	—	2	1	no	—
48	C₂²×D₄×C₄₃	Q₈	1	1	yes	characteristic
49	C₂³×C₄×C₄₃	—	2	1	no	—
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₄₃	—	2	2	no	—
52	C₂×M₁₆×C₄₃	—	2	2	no	—
53	C₂×M₁₆×C₄₃	—	2	2	no	—
54	C₂×M₁₆×C₄₃	—	2	1	no	—
55	C₂×M₁₆×C₄₃	—	2	1	no	—
56	C₂×M₁₆×C₄₃	—	2	2	no	—
57	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
59	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₄₃	D₄	1	2	yes	—
61	C₂²⋊C₈×C₄₃	—	2	8	no	—
62	C₂⁴×C₄₃	—	2	1	no	—
63	C₂⁴×C₄₃	#41	1	1	yes	characteristic
64	#41×C₄₃	—	2	4	no	—
65	#41×C₄₃	—	2	4	no	—
66	#41×C₄₃	—	2	4	no	—
67	#41×C₄₃	—	2	4	no	—
68	#41×C₄₃	—	4	1	no	—
69	#41×C₄₃	—	4	1	no	—
70	#41×C₄₃	—	4	2	no	—
71	#41×C₄₃	—	4	2	no	—
72	#41×C₄₃	—	4	1	no	—
73	#41×C₄₃	—	4	1	no	—
74	C₂×Q₈×C₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂×Q₈×C₄₃	C₄⋊C₄	1	2	yes	—
78	C₂×D₄×C₄₃	—	2	1	no	—
79	C₂×D₄×C₄₃	—	2	2	no	—
80	C₂×D₄×C₄₃	—	2	2	no	—
81	C₂×D₄×C₄₃	—	2	2	no	—
82	C₂×D₄×C₄₃	—	2	2	no	—
83	C₂×D₄×C₄₃	—	4	2	no	—
84	C₂×D₄×C₄₃	—	4	2	no	—
85	C₂×D₄×C₄₃	C₄⋊C₄	1	2	yes	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	1	no	—
89	C₂²×C₄×C₄₃	—	4	2	no	—
90	C₂²×C₄×C₄₃	—	4	2	no	—
91	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
99	C₂²×C₄×C₄₃	#41	1	2	yes	—
100	C₂²×C₄×C₄₃	#41	1	2	yes	—
101	C₂²×Dic₄₃	—	4	2	no	—
102	M₁₆×C₄₃	—	2	4	no	—
103	M₁₆×C₄₃	—	2	4	no	—
104	M₁₆×C₄₃	—	4	2	no	—
105	M₁₆×C₄₃	—	4	1	no	—
106	M₁₆×C₄₃	—	4	1	no	—
107	C₂×C₈×C₄₃	—	2	2	no	—
108	C₂×C₈×C₄₃	—	2	2	no	—
109	C₂×C₈×C₄₃	—	2	2	no	—
110	C₂×C₈×C₄₃	—	2	4	no	—
111	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
112	C₂²⋊Dic₄₃	—	2	4	no	—
113	C₂²⋊Dic₄₃	—	4	2	no	—
114	C₄⋊Dic₄₃	—	2	4	no	—
115	C₄⋊Dic₄₃	—	4	2	no	—
116	C₄×Dic₄₃	—	2	4	no	—
117	C₄×Dic₄₃	—	4	2	no	—
118	#10621877	—	2	4	no	—
119	#10621877	—	2	4	no	—
120	#10621877	—	4	2	no	—
121	#10621877	—	4	2	no	—
122	C₂×C₄₃⋊C₈	—	2	4	no	—
123	C₂×C₄₃⋊C₈	—	4	2	no	—
124	C₂×C₄₃⋊C₈	—	4	2	no	—
125	C₂³×C₄₃	—	2	2	no	—
126	C₂³×C₄₃	—	4	2	no	—
127	C₂³×C₄₃	—	8	1	no	—
128	C₂³×C₄₃	—	8	2	no	—
129	C₂³×C₄₃	#117	1	1	yes	characteristic
130	C₂³×C₄₃	#121	1	2	yes	—
131	C₂³×C₄₃	#121	1	1	yes	characteristic
132	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂³×C₄₃	#126	1	2	yes	—
134	Q₈×C₄₃	—	2	4	no	—
135	Q₈×C₄₃	—	4	1	no	—
136	Q₈×C₄₃	—	4	1	no	—
137	D₄×C₄₃	—	2	4	no	—
138	D₄×C₄₃	—	4	4	no	—
139	D₄×C₄₃	—	4	2	no	—
140	D₄×C₄₃	—	4	2	no	—
141	D₄×C₄₃	—	4	1	no	—
142	D₄×C₄₃	—	4	1	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	—	4	2	no	—
147	C₂×C₄×C₄₃	—	4	4	no	—
148	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
153	C₂×C₄×C₄₃	#120	1	2	yes	—
154	C₂×C₄×C₄₃	#121	1	2	yes	—
155	C₂×C₄×C₄₃	#121	1	2	yes	—
156	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
159	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
161	C₂×C₄×C₄₃	#123	1	2	yes	—
162	C₂×C₄×C₄₃	#123	1	2	yes	—
163	C₂×C₄×C₄₃	#124	1	2	yes	—
164	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
165	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
166	C₂×Dic₄₃	—	4	4	no	—
167	C₂×Dic₄₃	—	8	4	no	—
168	C₈×C₄₃	—	2	4	no	—
169	C₈×C₄₃	—	4	2	no	—
170	C₄₃⋊C₈	—	4	4	no	—
171	#11109	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	4	1	no	—
173	C₂²×C₄₃	—	4	1	no	—
174	C₂²×C₄₃	—	4	1	no	—
175	C₂²×C₄₃	—	4	2	no	—
176	C₂²×C₄₃	—	4	2	no	—
177	C₂²×C₄₃	—	4	4	no	—
178	C₂²×C₄₃	—	4	2	no	—
179	C₂²×C₄₃	—	8	4	no	—
180	C₂²×C₄₃	#380	1	1	yes	characteristic
181	C₂²×C₄₃	#388	1	2	yes	—
182	C₂²×C₄₃	#392	1	1	yes	characteristic
183	C₂²×C₄₃	#396	1	2	yes	—
184	C₂²×C₄₃	#397	1	1	yes	characteristic
185	C₄×C₄₃	—	4	2	no	—
186	C₄×C₄₃	#385	1	2	yes	—
187	C₄×C₄₃	#386	1	1	yes	characteristic
188	C₄×C₄₃	#390	1	1	yes	characteristic
189	C₄×C₄₃	#393	1	1	yes	characteristic
190	C₄×C₄₃	#394	1	2	yes	—
191	C₄×C₄₃	#399	1	1	yes	characteristic
192	Dic₄₃	—	8	4	no	—
193	#2911	—	43	1	no	—
194	C₂×#421	—	43	2	no	—
195	#2879	D₄₃	1	1	yes	characteristic
196	C₂×#344	—	43	2	no	—
197	C₂×#338	—	43	1	no	—
198	C₂×C₄₃	—	2	1	no	—
199	C₂×C₄₃	—	2	1	no	—
200	C₂×C₄₃	—	2	2	no	—
201	C₂×C₄₃	—	2	2	no	—
202	C₂×C₄₃	—	2	1	no	—
203	C₂×C₄₃	—	8	2	no	—
204	C₂×C₄₃	#1439	1	1	yes	characteristic
205	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
206	C₂²×M₁₆	—	43	2	no	—
207	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
208	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#117	—	43	2	no	—
210	#428	—	86	4	no	—
211	#427	—	86	2	no	—
212	#421	—	86	4	no	—
213	#421	—	86	4	no	—
214	C₂×#104	—	86	4	no	—
215	#408	—	2	2	no	—
216	#408	Dic₄₃	1	4	yes	complemented
217	#407	—	2	2	no	—
218	#344	—	86	4	no	—
219	#338	—	86	2	no	—
220	C₄₃	#11109	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	2	1	no	—
222	C₂×#41	—	2	1	no	—
223	C₂×#41	—	2	2	no	—
224	C₂×#41	—	2	2	no	—
225	C₂×#41	—	2	2	no	—
226	C₂×#41	—	2	2	no	—
227	C₂×#41	D₁₇₂	1	2	yes	—
228	C₂×#41	C₂×Dic₄₃	1	2	yes	—
229	C₂²×Q₈	Dic₈₆	1	1	yes	characteristic
230	C₂²×D₄	—	2	1	no	—
231	C₂²×D₄	Dic₈₆	1	1	yes	characteristic
232	C₂³×C₄	—	2	1	no	—
233	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	43	4	no	—
241	C₂×M₁₆	—	86	2	no	—
242	C₂×M₁₆	—	86	2	no	—
243	C₂×M₁₆	—	86	2	no	—
244	C₂×M₁₆	—	86	2	no	—
245	C₂×M₁₆	—	86	2	no	—
246	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
247	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
248	C₂²×C₈	—	86	2	no	—
249	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
250	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
251	#117	—	43	4	no	—
252	#117	—	86	2	no	—
253	#117	—	86	2	no	—
254	#117	—	86	2	no	—
255	C₂×C₄⋊C₄	—	86	2	no	—
256	C₂×C₂²⋊C₄	—	86	2	no	—
257	C₂×C₄²	—	86	2	no	—
258	#107	—	86	4	no	—
259	#106	—	86	4	no	—
260	#104	—	86	16	no	—
261	C₂²⋊C₈	—	2	8	no	—
262	C₂⁴	—	2	1	no	—
263	C₂⁴	#10621898	1	1	yes	characteristic
264	#41	—	2	4	no	—
265	#41	—	2	4	no	—
266	#41	—	2	4	no	—
267	#41	—	2	4	no	—
268	#41	—	4	1	no	—
269	#41	—	4	1	no	—
270	#41	—	4	2	no	—
271	#41	—	4	2	no	—
272	#41	—	4	1	no	—
273	#41	—	4	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	C₄⋊Dic₄₃	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	2	no	—
285	C₂×D₄	C₄⋊Dic₄₃	1	2	yes	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	4	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	172	2	no	—
292	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
294	C₂²×C₄	#10621898	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₄₃	1	2	yes	—
296	C₂²×C₄	D₄×D₄₃	1	2	yes	—
297	C₂²×C₄	#10621900	1	2	yes	—
298	C₂²×C₄	#10621900	1	2	yes	—
299	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
300	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
302	M₁₆	—	2	4	no	—
303	M₁₆	—	2	4	no	—
304	M₁₆	—	4	2	no	—
305	M₁₆	—	4	1	no	—
306	M₁₆	—	4	1	no	—
307	M₁₆	—	86	4	no	—
308	M₁₆	—	86	4	no	—
309	M₁₆	—	172	2	no	—
310	M₁₆	—	172	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	86	4	no	—
316	C₂×C₈	—	172	2	no	—
317	C₂×C₈	—	172	2	no	—
318	C₂×C₈	C₂²⋊Dic₄₃	1	4	yes	—
319	C₄⋊C₄	—	86	4	no	—
320	C₄⋊C₄	—	172	2	no	—
321	C₂²⋊C₄	—	86	4	no	—
322	C₂²⋊C₄	—	172	2	no	—
323	C₄²	—	86	4	no	—
324	C₄²	—	172	2	no	—
325	C₂³	—	2	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	8	1	no	—
328	C₂³	—	8	2	no	—
329	C₂³	#13063284	1	1	yes	characteristic
330	C₂³	#13063298	1	2	yes	—
331	C₂³	#13063299	1	1	yes	characteristic
332	C₂³	#13063304	1	1	yes	characteristic
333	C₂³	#13063318	1	2	yes	—
334	Q₈	—	2	4	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	4	1	no	—
337	D₄	—	2	4	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	4	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	344	4	no	—
349	C₂×C₄	—	172	4	no	—
350	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
352	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
353	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
354	C₂×C₄	#13063293	1	1	yes	characteristic
355	C₂×C₄	#13063295	1	2	yes	—
356	C₂×C₄	#13063296	1	2	yes	—
357	C₂×C₄	#13063297	1	2	yes	—
358	C₂×C₄	#13063299	1	1	yes	characteristic
359	C₂×C₄	#13063302	1	2	yes	—
360	C₂×C₄	#13063302	1	2	yes	—
361	C₂×C₄	#13063304	1	1	yes	characteristic
362	C₂×C₄	#13063306	1	1	yes	characteristic
363	C₂×C₄	#13063307	1	2	yes	—
364	C₂×C₄	#13063309	1	2	yes	—
365	C₂×C₄	#13063311	1	2	yes	—
366	C₂×C₄	#13063312	1	1	yes	characteristic
367	C₂×C₄	#13063321	1	1	yes	characteristic
368	C₈	—	2	4	no	—
369	C₈	—	4	2	no	—
370	C₈	—	172	4	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	1	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	8	4	no	—
379	C₂²	#14582759	1	1	yes	characteristic
380	C₂²	#14582792	1	2	yes	—
381	C₂²	#14582808	1	1	yes	characteristic
382	C₂²	#14582822	1	2	yes	—
383	C₂²	#14582824	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	344	4	no	—
386	C₄	#14582774	1	2	yes	—
387	C₄	#14582779	1	1	yes	characteristic
388	C₄	#14582802	1	1	yes	characteristic
389	C₄	#14582811	1	1	yes	characteristic
390	C₄	#14582815	1	2	yes	—
391	C₄	#14582835	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	2	2	no	—
395	C₂	—	2	2	no	—
396	C₂	—	2	1	no	—
397	C₂	—	8	2	no	—
398	C₂	#19132840	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14582802
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#2879	C₂
socle	C₂×C₄₃	#1439
prime radical	C₂²×#41×C₄₃	C₂²
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#2879	D₄₃
hypocenter	C₄₃	#11109
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2879

Complements

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

N	H ≅ G/N	type
1. #19143185	C₂	semidirect
3. C₂×#14582670	C₂	semidirect
4. C₂×C₄₃⋊#338	C₂	semidirect
19. C₄₃×#408	C₄	semidirect
216. #408	Dic₄₃	semidirect
220. C₄₃	#11109	semidirect

Maximal subgroups

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

subgroup	index
#19143185	2
C₂×#14582916	2
C₂×#14582670	2
C₂×C₄₃⋊#338	2
C₄₃×#2879	2
#11109	43