Subgroups — #25012893

Order 11008.

426 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582936	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₄₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#19134621	C₂	1	1	yes	maximal, characteristic
4	#19132687	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×#13063283	—	2	1	no	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14582936	—	2	2	no	—
12	#14582936	C₄	1	4	yes	complemented
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063256	—	2	1	no	—
16	#14582756	—	2	1	no	—
17	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
18	#14582661	—	2	1	no	—
19	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
24	C₂×C₈×D₄₃	—	2	1	no	—
25	C₂×C₈×D₄₃	—	2	1	no	—
26	C₂×C₈×D₄₃	—	2	1	no	—
27	C₂×C₈×D₄₃	—	2	1	no	—
28	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₄₃	D₄	1	2	yes	—
30	#13063283	—	2	4	no	—
31	#13063283	—	4	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×C₄⋊Dic₄₃	—	2	1	no	—
34	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
35	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
36	C₂×C₄×Dic₄₃	—	4	1	no	—
37	C₂×C₄₃⋊C₁₆	—	2	1	no	—
38	C₂×C₄₃⋊C₁₆	—	2	1	no	—
39	C₂×C₄₃⋊C₁₆	—	2	1	no	—
40	#13063259	—	2	1	no	—
41	#13063259	—	2	1	no	—
42	#13063259	—	2	1	no	—
43	#13063259	C₂×C₄	1	2	yes	—
44	#13063256	—	2	4	no	—
45	C₄₃⋊#95	—	4	1	no	—
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
49	C₂×C₁₆×C₄₃	—	2	1	no	—
50	C₂×C₁₆×C₄₃	—	2	1	no	—
51	C₂×C₁₆×C₄₃	—	2	1	no	—
52	#107×C₄₃	—	2	1	no	—
53	#107×C₄₃	—	2	1	no	—
54	#107×C₄₃	—	2	1	no	—
55	#107×C₄₃	C₂×C₄	1	2	yes	—
56	C₂³×D₄₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	4	1	no	—
58	C₂²×C₄×C₄₃	—	4	1	no	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	4	1	no	—
61	C₂²×Dic₄₃	—	4	1	no	—
62	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	—	2	2	no	—
76	C₄⋊C₄×C₄₃	—	4	1	no	—
77	C₄⋊C₄×C₄₃	—	4	1	no	—
78	C₄⋊C₄×C₄₃	—	4	1	no	—
79	(C₂×D₄₃)⋊C₄	—	4	4	no	—
80	(C₂×D₄₃)⋊C₄	—	8	2	no	—
81	C₄⋊Dic₄₃	—	2	2	no	—
82	C₄⋊Dic₄₃	—	4	1	no	—
83	C₄⋊Dic₄₃	—	4	1	no	—
84	C₄⋊Dic₄₃	—	4	1	no	—
85	C₄×Dic₄₃	—	4	4	no	—
86	C₂×C₄₃⋊C₈	—	2	1	no	—
87	C₂×C₄₃⋊C₈	—	2	1	no	—
88	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
89	C₈×D₄₃	—	2	2	no	—
90	C₈×D₄₃	—	2	2	no	—
91	C₈×D₄₃	—	2	2	no	—
92	C₈×D₄₃	—	2	2	no	—
93	C₁₆×C₄₃	—	4	1	no	—
94	C₄₃⋊C₁₆	—	4	1	no	—
95	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂²×D₄₃	—	2	1	no	—
97	C₂²×D₄₃	—	2	1	no	—
98	C₂²×D₄₃	—	2	1	no	—
99	C₂²×D₄₃	—	2	1	no	—
100	C₂²×D₄₃	—	2	1	no	—
101	C₂²×D₄₃	—	2	1	no	—
102	C₂²×D₄₃	#102	1	2	yes	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	8	1	no	—
105	C₂×C₄×C₄₃	—	8	1	no	—
106	C₂×C₄×C₄₃	—	8	1	no	—
107	C₂×C₄×C₄₃	—	8	1	no	—
108	C₂×C₄×C₄₃	—	8	1	no	—
109	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	8	1	no	—
119	C₂×Dic₄₃	—	8	1	no	—
120	C₂×Dic₄₃	—	8	1	no	—
121	C₂×Dic₄₃	—	8	1	no	—
122	C₂×Dic₄₃	—	8	1	no	—
123	C₂×Dic₄₃	#102	1	2	yes	—
124	C₄×D₄₃	—	2	2	no	—
125	C₄×D₄₃	—	2	2	no	—
126	C₄×D₄₃	—	2	2	no	—
127	C₄×D₄₃	—	2	2	no	—
128	C₈×C₄₃	#117	1	1	yes	characteristic
129	C₈×C₄₃	#117	1	1	yes	characteristic
130	C₈×C₄₃	#123	1	1	yes	characteristic
131	C₈×C₄₃	#123	1	1	yes	characteristic
132	C₄₃⋊C₈	—	2	2	no	—
133	#12868	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
135	C₂²×C₄₃	#480	1	1	yes	characteristic
136	C₂²×C₄₃	#480	1	1	yes	characteristic
137	C₂²×C₄₃	#508	1	1	yes	characteristic
138	C₂²×C₄₃	#508	1	1	yes	characteristic
139	C₂²×C₄₃	#509	1	1	yes	characteristic
140	C₂²×C₄₃	#510	1	1	yes	characteristic
141	C₂×D₄₃	—	2	2	no	—
142	C₂×D₄₃	—	2	2	no	—
143	C₂×D₄₃	—	2	2	no	—
144	C₂×D₄₃	—	4	1	no	—
145	C₂×D₄₃	—	4	1	no	—
146	C₂×D₄₃	—	4	1	no	—
147	C₂×D₄₃	—	4	1	no	—
148	C₄×C₄₃	—	8	2	no	—
149	C₄×C₄₃	—	16	1	no	—
150	C₄×C₄₃	#381	1	1	yes	characteristic
151	C₄×C₄₃	#419	1	1	yes	characteristic
152	C₄×C₄₃	#483	1	1	yes	characteristic
153	C₄×C₄₃	#483	1	1	yes	characteristic
154	Dic₄₃	—	2	2	no	—
155	Dic₄₃	—	8	2	no	—
156	Dic₄₃	—	16	1	no	—
157	C₂×#427	—	43	1	no	—
158	C₂×C₂²⋊C₁₆	—	43	1	no	—
159	#1808	—	43	1	no	—
160	#1380	—	43	1	no	—
161	#1380	—	43	1	no	—
162	#1380	—	43	1	no	—
163	#1380	D₄₃	1	1	yes	characteristic, complemented
164	C₂×C₄₃	#1823	1	1	yes	characteristic
165	C₂×C₄₃	#2138	1	1	yes	characteristic
166	C₂×C₄₃	#2139	1	1	yes	characteristic
167	C₂×C₄₃	#2234	1	1	yes	characteristic
168	C₂×C₄₃	#2234	1	1	yes	characteristic
169	C₂×C₄₃	#2237	1	1	yes	characteristic
170	C₂×C₄₃	#2237	1	1	yes	characteristic
171	D₄₃	—	4	2	no	—
172	C₂³×C₈	—	43	1	no	—
173	C₂×#117	—	86	1	no	—
174	C₂²×C₁₆	—	43	1	no	—
175	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
176	#427	—	43	4	no	—
177	#427	—	86	2	no	—
178	C₂×#107	—	43	1	no	—
179	C₂×#107	—	43	1	no	—
180	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
181	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
182	C₂×#106	—	86	1	no	—
183	#380	—	86	1	no	—
184	C₂²⋊C₁₆	—	86	2	no	—
185	#340	—	86	1	no	—
186	C₄₃	#12868	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	—	43	1	no	—
188	C₂²×C₈	—	43	1	no	—
189	C₂²×C₈	—	43	2	no	—
190	C₂²×C₈	—	43	2	no	—
191	C₂²×C₈	—	86	1	no	—
192	C₂²×C₈	—	86	1	no	—
193	C₂²×C₈	—	86	1	no	—
194	C₂²×C₈	—	86	1	no	—
195	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
196	#117	—	86	4	no	—
197	#117	—	172	2	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	86	1	no	—
201	C₂×C₄⋊C₄	—	86	1	no	—
202	C₂×C₂²⋊C₄	—	172	1	no	—
203	C₂×C₂²⋊C₄	—	172	1	no	—
204	C₂×C₄²	—	172	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	86	1	no	—
209	C₂×C₁₆	—	86	1	no	—
210	C₂×C₁₆	—	86	1	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	—	43	2	no	—
215	#107	—	86	1	no	—
216	#107	—	86	1	no	—
217	#107	—	86	1	no	—
218	#107	C₄×D₄₃	1	2	yes	—
219	#106	—	86	4	no	—
220	#95	—	172	1	no	—
221	C₂⁴	—	43	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	43	1	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	—	43	2	no	—
227	C₂²×C₄	—	86	1	no	—
228	C₂²×C₄	—	86	1	no	—
229	C₂²×C₄	—	86	1	no	—
230	C₂²×C₄	—	86	1	no	—
231	C₂²×C₄	—	172	1	no	—
232	C₂²×C₄	—	172	1	no	—
233	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
234	C₂×C₈	—	43	2	no	—
235	C₂×C₈	—	86	1	no	—
236	C₂×C₈	—	86	1	no	—
237	C₂×C₈	—	86	2	no	—
238	C₂×C₈	—	86	2	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
242	C₂×C₈	#10621898	1	1	yes	characteristic
243	C₂×C₈	#10621898	1	1	yes	characteristic
244	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
245	C₂×C₈	#10621900	1	1	yes	characteristic
246	C₂×C₈	#10621902	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	172	1	no	—
253	C₄⋊C₄	—	172	1	no	—
254	C₄⋊C₄	—	172	1	no	—
255	C₂²⋊C₄	—	344	2	no	—
256	C₂²⋊C₄	—	172	4	no	—
257	C₄²	—	172	4	no	—
258	C₁₆	—	4	1	no	—
259	C₁₆	—	172	1	no	—
260	C₂³	—	43	2	no	—
261	C₂³	—	86	1	no	—
262	C₂³	—	86	1	no	—
263	C₂³	—	86	1	no	—
264	C₂³	—	86	1	no	—
265	C₂³	—	86	1	no	—
266	C₂³	—	86	1	no	—
267	C₂³	D₈×D₄₃	1	1	yes	characteristic
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	43	2	no	—
275	C₂×C₄	—	86	1	no	—
276	C₂×C₄	—	86	1	no	—
277	C₂×C₄	—	86	2	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	—	86	2	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	344	1	no	—
282	C₂×C₄	—	344	1	no	—
283	C₂×C₄	—	344	1	no	—
284	C₂×C₄	—	344	1	no	—
285	C₂×C₄	—	344	1	no	—
286	C₂×C₄	—	172	2	no	—
287	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
288	C₂×C₄	#13063307	1	1	yes	characteristic
289	C₂×C₄	#13063310	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
291	C₂×C₄	#13063354	1	1	yes	characteristic
292	C₂×C₄	#13063357	1	1	yes	characteristic
293	C₈	—	86	2	no	—
294	C₈	#13063282	1	1	yes	characteristic
295	C₈	#13063283	1	1	yes	characteristic
296	C₈	#13063307	1	1	yes	characteristic
297	C₈	#13063310	1	1	yes	characteristic
298	C₂²	—	86	2	no	—
299	C₂²	—	86	2	no	—
300	C₂²	—	86	2	no	—
301	C₂²	—	172	1	no	—
302	C₂²	—	172	1	no	—
303	C₂²	—	172	1	no	—
304	C₂²	—	172	1	no	—
305	C₂²	#102×D₄₃	1	1	yes	characteristic
306	C₂²	#14583211	1	1	yes	characteristic
307	C₂²	#14583214	1	1	yes	characteristic
308	C₂²	#14583330	1	1	yes	characteristic
309	C₂²	#14583332	1	1	yes	characteristic
310	C₂²	#14583333	1	1	yes	characteristic
311	C₂²	#14583338	1	1	yes	characteristic
312	C₄	—	8	2	no	—
313	C₄	—	16	1	no	—
314	C₄	—	86	2	no	—
315	C₄	—	344	2	no	—
316	C₄	—	688	1	no	—
317	C₄	#14582761	1	1	yes	characteristic
318	C₄	#14582906	1	1	yes	characteristic
319	C₄	#14583228	1	1	yes	characteristic
320	C₄	#14583231	1	1	yes	characteristic
321	C₂	—	172	2	no	—
322	C₂	#19134680	1	1	yes	characteristic
323	C₂	#19136301	1	1	yes	characteristic
324	C₂	#19136304	1	1	yes	characteristic
325	C₂	#19136785	1	1	yes	characteristic
326	C₂	#19136789	1	1	yes	characteristic
327	C₂	#19136802	1	1	yes	characteristic
328	C₂	#19136806	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂³×D₄₃	D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#12868
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. #19132687	C₂	semidirect
5. #19132686	C₂	semidirect
6. #19132686	C₂	semidirect
7. C₄₃×#1380	C₂	semidirect
12. #14582936	C₄	semidirect
163. #1380	D₄₃	semidirect
186. C₄₃	#12868	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582936	2
C₂×(C₂×D₄₃)⋊C₁₆	2
#19134621	2
#19132687	2
#19132686	2
C₄₃×#1380	2
#12868	43