Subgroups — #25251952

Order 11008.

578 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	#19145106	C₂	1	1	yes	maximal, characteristic
2	#494×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19136839	C₂	1	4	yes	maximal, complemented
4	#19136824	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136823	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136822	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136313	C₂	1	4	yes	maximal
8	C₄×#13063264	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₄₃	—	2	1	no	—
11	C₂×Q₁₆×D₄₃	C₂²	1	2	yes	—
12	#14583454	—	2	1	no	—
13	C₄⋊Q₈×D₄₃	—	2	1	no	—
14	C₂×#13063264	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583300	—	2	1	no	—
17	#14583300	C₂²	1	2	yes	—
18	#14583295	—	2	1	no	—
19	#14583279	C₂²	1	1	yes	characteristic
20	#14583278	—	2	1	no	—
21	#14583277	—	2	1	no	—
22	#14583277	C₂²	1	2	yes	—
23	#14582936	—	2	1	no	—
24	#14582936	C₂²	1	2	yes	—
25	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
26	#14582704	C₂²	1	4	yes	—
27	#14582703	C₂²	1	4	yes	—
28	#14582702	C₂²	1	4	yes	—
29	#14582674	C₂²	1	2	yes	—
30	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₄₃×#500	C₂²	1	1	yes	characteristic
32	C₄₃×#494	C₂²	1	1	yes	characteristic
33	C₄₃×#358	C₂²	1	4	yes	—
34	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₄₃	—	2	2	no	—
36	C₂×Dic₁₇₂	—	2	1	no	—
37	C₂×Dic₁₇₂	C₂³	1	2	yes	—
38	C₂×#10621885	—	2	2	no	—
39	C₂×#10621885	—	2	2	no	—
40	Q₁₆×D₄₃	—	2	8	no	—
41	Q₁₆×D₄₃	—	4	4	no	—
42	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
44	#13063325	—	2	1	no	—
45	#13063325	—	2	1	no	—
46	#13063324	—	4	1	no	—
47	#13063324	—	4	1	no	—
48	#13063323	—	2	2	no	—
49	#13063316	—	2	2	no	—
50	#13063305	—	4	1	no	—
51	#13063301	—	4	1	no	—
52	Q₈×Dic₄₃	—	4	1	no	—
53	#13063290	—	4	1	no	—
54	#13063283	—	2	2	no	—
55	C₄⋊C₄×D₄₃	—	4	1	no	—
56	C₄⋊C₄×D₄₃	—	4	1	no	—
57	C₄²×D₄₃	D₄	1	1	yes	characteristic
58	#13063264	—	2	4	no	—
59	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
60	#13063259	—	2	1	no	—
61	#13063259	C₂³	1	2	yes	—
62	#13063256	—	2	2	no	—
63	#13063256	—	2	2	no	—
64	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₄₃	D₄	1	1	yes	characteristic
66	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₄₃	—	2	1	no	—
68	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₃	—	2	1	no	—
70	C₄⋊Q₈×C₄₃	—	2	1	no	—
71	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
72	#107×C₄₃	—	2	1	no	—
73	#107×C₄₃	C₂³	1	2	yes	—
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	4	1	no	—
77	Q₈×D₄₃	—	4	4	no	—
78	Q₈×D₄₃	—	4	4	no	—
79	C₂×C₄×D₄₃	—	4	2	no	—
80	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
81	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
82	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₈₆	—	2	2	no	—
84	C₂×Dic₈₆	—	4	1	no	—
85	C₂×Dic₈₆	—	4	2	no	—
86	Q₁₆×C₄₃	—	2	4	no	—
87	Q₁₆×C₄₃	—	4	2	no	—
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	4	1	no	—
92	C₄⋊C₄×C₄₃	—	4	1	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	#10621885	—	4	4	no	—
96	#10621885	—	4	4	no	—
97	(C₂×D₄₃)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	—	4	1	no	—
101	C₄⋊Dic₄₃	—	4	1	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	Dic₄₃⋊C₄	—	4	2	no	—
104	Dic₄₃⋊C₄	—	4	2	no	—
105	Dic₄₃⋊C₄	—	4	2	no	—
106	C₄×Dic₄₃	—	4	2	no	—
107	C₄×Dic₄₃	D₈	1	1	yes	characteristic
108	C₄×Dic₄₃	D₈	1	1	yes	characteristic
109	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
112	Dic₁₇₂	—	2	4	no	—
113	Dic₁₇₂	—	4	2	no	—
114	C₈×D₄₃	SD₁₆	1	4	yes	—
115	C₈×D₄₃	#41	1	4	yes	—
116	C₁₆×C₄₃	—	2	2	no	—
117	C₄₃⋊C₁₆	—	2	2	no	—
118	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₄₃	—	4	2	no	—
120	Q₈×C₄₃	—	4	2	no	—
121	Q₈×C₄₃	—	8	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	2	no	—
134	C₄×D₄₃	—	8	4	no	—
135	C₄×D₄₃	#124	1	2	yes	—
136	C₄×D₄₃	C₂×SD₁₆	1	2	yes	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	—	4	2	no	—
139	Dic₈₆	—	8	2	no	—
140	Dic₈₆	—	8	4	no	—
141	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
142	C₈×C₄₃	C₂×#41	1	2	yes	—
143	C₄₃⋊C₈	#124	1	2	yes	—
144	C₄₃⋊C₈	C₂×SD₁₆	1	2	yes	—
145	#33705	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₄₃	—	2	1	no	—
148	C₂×D₄₃	—	2	1	no	—
149	C₂×D₄₃	#486	1	2	yes	—
150	C₄×C₄₃	—	8	2	no	—
151	C₄×C₄₃	—	8	2	no	—
152	C₄×C₄₃	#510	1	2	yes	—
153	C₄×C₄₃	#510	1	2	yes	—
154	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
155	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
156	Dic₄₃	—	2	1	no	—
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	8	2	no	—
159	Dic₄₃	—	8	2	no	—
160	Dic₄₃	#486	1	2	yes	—
161	#3160	—	43	1	no	—
162	C₂×#494	—	43	1	no	—
163	#2245	—	43	4	no	—
164	#2241	—	43	1	no	—
165	#2241	—	43	1	no	—
166	#2241	—	43	1	no	—
167	#2241	D₄₃	1	1	yes	characteristic, complemented
168	#2141	—	43	4	no	—
169	C₄×M₃₂	—	43	1	no	—
170	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
171	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
173	D₄₃	—	2	2	no	—
174	C₂²×Q₁₆	—	43	2	no	—
175	C₂²×Q₁₆	—	86	1	no	—
176	#533	—	86	1	no	—
177	C₂×C₄⋊Q₈	—	86	1	no	—
178	C₂×M₃₂	—	43	2	no	—
179	#500	—	43	1	no	—
180	#500	C₂×D₄₃	1	1	yes	characteristic
181	#499	—	43	2	no	—
182	#499	—	86	1	no	—
183	#498	—	86	1	no	—
184	#494	—	43	2	no	—
185	#494	—	43	1	no	—
186	#494	—	86	1	no	—
187	#494	—	86	1	no	—
188	#494	C₂×D₄₃	1	1	yes	characteristic
189	#427	—	43	2	no	—
190	#427	—	86	1	no	—
191	C₂×C₄×C₈	—	43	1	no	—
192	#358	—	43	4	no	—
193	#358	—	43	4	no	—
194	#358	—	43	4	no	—
195	#358	C₂×D₄₃	1	4	yes	—
196	#346	—	43	2	no	—
197	C₄×C₁₆	—	43	1	no	—
198	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
199	C₄₃	#33705	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	86	2	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	43	2	no	—
203	C₂×Q₁₆	—	86	8	no	—
204	C₂×Q₁₆	—	86	2	no	—
205	C₂×Q₁₆	—	86	2	no	—
206	C₂×Q₁₆	—	86	1	no	—
207	C₂×Q₁₆	—	172	4	no	—
208	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
209	C₂²×C₈	—	43	1	no	—
210	C₂²×C₈	—	43	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	86	2	no	—
214	C₄⋊Q₈	—	86	1	no	—
215	C₄⋊Q₈	—	86	1	no	—
216	C₄⋊Q₈	—	172	1	no	—
217	C₄⋊Q₈	—	172	1	no	—
218	#125	—	86	2	no	—
219	C₂²⋊Q₈	—	172	1	no	—
220	C₂²⋊Q₈	—	172	1	no	—
221	C₄×Q₈	—	172	1	no	—
222	C₄×Q₈	—	172	1	no	—
223	#117	—	86	2	no	—
224	C₂×C₄⋊C₄	—	172	1	no	—
225	C₂×C₄⋊C₄	—	172	1	no	—
226	C₂×C₄²	—	43	1	no	—
227	M₃₂	—	86	4	no	—
228	C₂×C₁₆	—	43	2	no	—
229	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	43	2	no	—
232	#107	—	86	1	no	—
233	#107	C₂²×D₄₃	1	2	yes	—
234	#106	—	86	2	no	—
235	#106	—	86	2	no	—
236	C₄×C₈	—	43	1	no	—
237	C₄×C₈	—	43	1	no	—
238	C₄×C₈	—	43	1	no	—
239	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	86	2	no	—
243	C₂×Q₈	—	172	2	no	—
244	C₂×Q₈	—	172	4	no	—
245	C₂×Q₈	—	172	4	no	—
246	C₂×Q₈	—	172	1	no	—
247	C₂²×C₄	—	43	1	no	—
248	C₂²×C₄	—	43	1	no	—
249	C₂²×C₄	—	43	1	no	—
250	C₂²×C₄	—	172	2	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	86	4	no	—
254	Q₁₆	—	172	4	no	—
255	Q₁₆	—	172	4	no	—
256	Q₁₆	—	172	2	no	—
257	C₂×C₈	—	43	1	no	—
258	C₂×C₈	—	43	1	no	—
259	C₂×C₈	—	43	4	no	—
260	C₂×C₈	—	43	4	no	—
261	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	86	2	no	—
268	C₄⋊C₄	—	172	2	no	—
269	C₄⋊C₄	—	172	2	no	—
270	C₄⋊C₄	—	172	2	no	—
271	C₄⋊C₄	—	172	2	no	—
272	C₄⋊C₄	—	172	1	no	—
273	C₄⋊C₄	—	172	1	no	—
274	C₄⋊C₄	—	172	1	no	—
275	C₂²⋊C₄	—	172	2	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	43	1	no	—
278	C₄²	—	43	1	no	—
279	C₄²	—	172	2	no	—
280	C₄²	D₄×D₄₃	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	86	2	no	—
283	C₂³	—	43	1	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	344	4	no	—
288	Q₈	—	344	2	no	—
289	Q₈	—	172	2	no	—
290	Q₈	—	172	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	43	1	no	—
294	C₂×C₄	—	43	2	no	—
295	C₂×C₄	—	43	2	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	43	1	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	344	4	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
304	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
306	C₈	—	43	2	no	—
307	C₈	—	43	2	no	—
308	C₈	SD₁₆×D₄₃	1	2	yes	—
309	C₈	#41×D₄₃	1	2	yes	—
310	C₂²	—	43	2	no	—
311	C₂²	—	86	1	no	—
312	C₂²	—	86	1	no	—
313	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	43	2	no	—
317	C₄	—	86	1	no	—
318	C₄	—	86	1	no	—
319	C₄	—	344	2	no	—
320	C₄	—	344	2	no	—
321	C₄	#14583338	1	2	yes	—
322	C₄	#14583341	1	2	yes	—
323	C₄	#124×D₄₃	1	1	yes	characteristic
324	C₄	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
325	C₂	—	86	2	no	—
326	C₂	#486×D₄₃	1	1	yes	characteristic
327	C₂	C₂×#14583338	1	1	yes	characteristic
328	C₂	C₂×#14583341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2241	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₄₃	C₂×D₈
hypercenter	#2241	D₄₃
hypocenter	C₄₃	#33705
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #19136839	C₂	semidirect
4. #19136824	C₂	semidirect
5. #19136823	C₂	semidirect
6. #19136822	C₂	semidirect
9. C₄₃×#2241	C₂	semidirect
167. #2241	D₄₃	semidirect
199. C₄₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#19145106	2
#494×D₄₃	2
#19136839	2
#19136824	2
#19136823	2
#19136822	2
#19136313	2
C₄×#13063264	2
C₄₃×#2241	2
#33705	43