Subgroups — #25251894

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 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, complemented
2	#19145022	C₂	1	1	yes	maximal, characteristic
3	#19136825	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	#19136818	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136299	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063344	—	2	1	no	—
11	C₂×#13063344	C₂²	1	2	yes	—
12	#14583454	—	2	1	no	—
13	#14583447	—	2	1	no	—
14	C₂×C₁₆×D₄₃	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	#14583282	—	2	1	no	—
20	#14583282	C₂²	1	2	yes	—
21	#14583279	C₂²	1	1	yes	characteristic
22	#14583271	—	2	1	no	—
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	#14582702	C₂²	1	4	yes	—
28	#14582700	C₂²	1	4	yes	—
29	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₄₃	C₂²	1	2	yes	—
31	C₄₃×#500	C₂²	1	1	yes	characteristic
32	C₄₃×#493	C₂²	1	1	yes	characteristic
33	C₄₃×#357	C₂²	1	4	yes	—
34	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×#10621900	—	2	2	no	—
36	#13063344	—	2	8	no	—
37	#13063344	—	4	4	no	—
38	C₂×Dic₁₇₂	—	2	1	no	—
39	C₂×Dic₁₇₂	C₂³	1	2	yes	—
40	C₂×#10621883	—	2	2	no	—
41	C₂×#10621883	—	2	2	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	#13063316	—	2	2	no	—
47	#13063312	—	2	2	no	—
48	#13063305	—	4	1	no	—
49	#13063301	—	4	1	no	—
50	#13063296	—	4	1	no	—
51	#13063296	—	4	1	no	—
52	Q₈×Dic₄₃	—	4	1	no	—
53	#13063290	—	4	1	no	—
54	D₄×Dic₄₃	—	4	1	no	—
55	D₄×Dic₄₃	—	4	1	no	—
56	#13063283	—	2	2	no	—
57	C₄²×D₄₃	D₄	1	1	yes	characteristic
58	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
59	C₁₆×D₄₃	—	2	4	no	—
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×D₈×C₄₃	—	2	1	no	—
68	C₂×D₈×C₄₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₃	—	2	1	no	—
70	#127×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	—	4	1	no	—
76	C₂×D₄×C₄₃	—	2	2	no	—
77	C₂×D₄×C₄₃	—	4	1	no	—
78	C₂×D₄×C₄₃	—	4	1	no	—
79	C₂×C₂²⋊D₄₃	—	4	2	no	—
80	C₂²×Dic₄₃	—	4	2	no	—
81	#10621900	—	4	4	no	—
82	#10621900	—	4	4	no	—
83	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
84	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₈₆	—	2	2	no	—
87	C₂×Dic₈₆	—	4	1	no	—
88	D₈×C₄₃	—	2	4	no	—
89	D₈×C₄₃	—	4	2	no	—
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	—	4	2	no	—
96	C₂²⋊Dic₄₃	—	4	2	no	—
97	#10621883	—	4	4	no	—
98	#10621883	—	4	4	no	—
99	(C₂×D₄₃)⋊C₄	—	4	2	no	—
100	C₄⋊Dic₄₃	—	2	2	no	—
101	C₄⋊Dic₄₃	—	4	1	no	—
102	C₄⋊Dic₄₃	—	4	1	no	—
103	C₄⋊Dic₄₃	—	4	1	no	—
104	C₄⋊Dic₄₃	—	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	C₂³×C₄₃	—	4	2	no	—
119	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₄₃	—	8	2	no	—
121	D₄×C₄₃	—	4	2	no	—
122	D₄×C₄₃	—	4	2	no	—
123	D₄×C₄₃	—	8	2	no	—
124	D₄×C₄₃	—	8	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₃	—	8	4	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	—	8	4	no	—
133	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
136	C₄×D₄₃	—	2	2	no	—
137	C₄×D₄₃	—	2	2	no	—
138	C₄×D₄₃	#124	1	2	yes	—
139	C₄×D₄₃	C₂×SD₁₆	1	2	yes	—
140	Dic₈₆	—	4	2	no	—
141	Dic₈₆	—	4	2	no	—
142	Dic₈₆	—	8	2	no	—
143	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
144	C₈×C₄₃	C₂×#41	1	2	yes	—
145	C₄₃⋊C₈	#124	1	2	yes	—
146	C₄₃⋊C₈	C₂×SD₁₆	1	2	yes	—
147	#33697	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	—	8	2	no	—
150	C₂²×C₄₃	—	8	2	no	—
151	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	#486	1	2	yes	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	#508	1	2	yes	—
157	C₄×C₄₃	#508	1	2	yes	—
158	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
159	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	2	1	no	—
162	Dic₄₃	—	8	2	no	—
163	Dic₄₃	—	8	2	no	—
164	Dic₄₃	#486	1	2	yes	—
165	#3160	—	43	1	no	—
166	#3147	—	43	1	no	—
167	#2242	—	43	4	no	—
168	#2241	—	43	1	no	—
169	#2241	—	43	1	no	—
170	#2240	—	43	1	no	—
171	#2240	D₄₃	1	1	yes	characteristic, complemented
172	#2138	—	43	4	no	—
173	C₂×C₄×C₁₆	—	43	1	no	—
174	C₂×C₄₃	—	16	2	no	—
175	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
176	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
177	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
178	D₄₃	—	2	2	no	—
179	C₂×#135	—	43	2	no	—
180	C₂×#135	—	86	1	no	—
181	#533	—	86	1	no	—
182	#532	—	86	1	no	—
183	C₂²×C₁₆	—	43	2	no	—
184	#500	—	43	1	no	—
185	#500	C₂×D₄₃	1	1	yes	characteristic
186	#499	—	43	2	no	—
187	#499	—	86	1	no	—
188	#498	—	86	1	no	—
189	#495	—	43	2	no	—
190	#495	—	86	1	no	—
191	#494	—	43	1	no	—
192	#493	C₂×D₄₃	1	1	yes	characteristic
193	#492	—	86	1	no	—
194	#427	—	43	2	no	—
195	#427	—	86	1	no	—
196	C₂×C₄×C₈	—	43	1	no	—
197	#358	—	43	4	no	—
198	#358	—	43	4	no	—
199	#357	—	43	4	no	—
200	#357	C₂×D₄₃	1	4	yes	—
201	C₄×C₁₆	—	43	2	no	—
202	C₄×C₁₆	—	43	1	no	—
203	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₄₃	#33697	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	86	2	no	—
206	#135	—	86	8	no	—
207	#135	—	172	4	no	—
208	C₂×Q₁₆	—	43	2	no	—
209	C₂×Q₁₆	—	86	1	no	—
210	C₂×SD₁₆	—	86	2	no	—
211	C₂×SD₁₆	—	86	2	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	C₂²×D₄₃	1	2	yes	—
214	C₂²×C₈	—	43	1	no	—
215	C₂²×C₈	—	43	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	86	1	no	—
218	C₄⋊Q₈	—	86	1	no	—
219	#127	—	2	1	no	—
220	#125	—	86	2	no	—
221	#124	—	86	2	no	—
222	C₂²⋊Q₈	—	172	1	no	—
223	C₂²⋊Q₈	—	172	1	no	—
224	#121	—	172	1	no	—
225	#121	—	172	1	no	—
226	C₄×Q₈	—	172	1	no	—
227	C₄×Q₈	—	172	1	no	—
228	C₄×D₄	—	172	1	no	—
229	C₄×D₄	—	172	1	no	—
230	#117	—	86	2	no	—
231	C₂×C₄²	—	43	1	no	—
232	C₂×C₁₆	—	43	2	no	—
233	C₂×C₁₆	—	86	4	no	—
234	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	43	2	no	—
237	#107	—	86	1	no	—
238	#107	C₂²×D₄₃	1	2	yes	—
239	#106	—	86	2	no	—
240	#106	—	86	2	no	—
241	C₄×C₈	—	43	1	no	—
242	C₄×C₈	—	43	1	no	—
243	C₄×C₈	—	43	1	no	—
244	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
245	#41	—	172	4	no	—
246	#41	—	172	4	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	86	2	no	—
249	C₂×Q₈	—	172	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	172	2	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	43	1	no	—
257	C₂²×C₄	—	172	2	no	—
258	Q₁₆	—	86	4	no	—
259	Q₁₆	—	172	2	no	—
260	SD₁₆	—	172	4	no	—
261	SD₁₆	—	172	4	no	—
262	D₈	—	2	4	no	—
263	D₈	—	4	2	no	—
264	C₂×C₈	—	43	1	no	—
265	C₂×C₈	—	43	1	no	—
266	C₂×C₈	—	43	4	no	—
267	C₂×C₈	—	43	4	no	—
268	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	86	2	no	—
273	C₄⋊C₄	—	172	2	no	—
274	C₄⋊C₄	—	172	2	no	—
275	C₄⋊C₄	—	172	1	no	—
276	C₄⋊C₄	—	172	1	no	—
277	C₄⋊C₄	—	172	1	no	—
278	C₂²⋊C₄	—	172	2	no	—
279	C₂²⋊C₄	—	172	2	no	—
280	C₂²⋊C₄	—	172	2	no	—
281	C₄²	—	43	1	no	—
282	C₄²	—	43	1	no	—
283	C₄²	—	43	1	no	—
284	C₄²	—	172	2	no	—
285	C₄²	D₄×D₄₃	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	86	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	43	1	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	344	2	no	—
292	Q₈	—	172	2	no	—
293	Q₈	—	172	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	344	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	43	2	no	—
302	C₂×C₄	—	43	2	no	—
303	C₂×C₄	—	43	1	no	—
304	C₂×C₄	—	43	1	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	—	86	2	no	—
307	C₂×C₄	—	344	4	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
313	C₈	—	43	2	no	—
314	C₈	—	43	2	no	—
315	C₈	SD₁₆×D₄₃	1	2	yes	—
316	C₈	#41×D₄₃	1	2	yes	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	43	2	no	—
321	C₂²	—	86	1	no	—
322	C₂²	—	86	1	no	—
323	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	43	2	no	—
326	C₄	—	86	1	no	—
327	C₄	—	86	1	no	—
328	C₄	—	344	2	no	—
329	C₄	—	344	2	no	—
330	C₄	#14583328	1	2	yes	—
331	C₄	#14583330	1	2	yes	—
332	C₄	#124×D₄₃	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
334	C₂	—	16	2	no	—
335	C₂	—	86	2	no	—
336	C₂	#486×D₄₃	1	1	yes	characteristic
337	C₂	C₂×#14583328	1	1	yes	characteristic
338	C₂	C₂×#14583330	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×#2240	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19145022	C₂
hypercenter	#2240	D₄₃
hypocenter	C₄₃	#33697
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #19145106	C₂	semidirect
3. #19136825	C₂	semidirect
4. #19136824	C₂	semidirect
5. #19136823	C₂	semidirect
6. #19136818	C₂	semidirect
7. #19136299	C₂	semidirect
8. C₄×C₁₆×D₄₃	C₂	semidirect
9. C₄₃×#2240	C₂	semidirect
171. #2240	D₄₃	semidirect
204. C₄₃	#33697	semidirect

Maximal subgroups

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

subgroup	index
#19145106	2
#19145022	2
#19136825	2
#19136824	2
#19136823	2
#19136818	2
#19136299	2
C₄×C₁₆×D₄₃	2
C₄₃×#2240	2
#33697	43