Subgroups — #25249904

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583328	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063344	C₂	1	2	yes	maximal, complemented
3	#19136471	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136467	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136465	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136459	C₂	1	2	yes	maximal, complemented
7	#19136456	C₂	1	2	yes	maximal, complemented
8	#19136378	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136299	C₂	1	2	yes	maximal, complemented
10	#346×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063344	C₂²	1	2	yes	—
13	C₄×#10621900	—	2	2	no	—
14	#14583328	C₄	1	8	yes	complemented
15	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
16	C₂×#13063269	C₂²	1	2	yes	—
17	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
19	C₄×Dic₁₇₂	C₂²	1	2	yes	—
20	#14582993	C₂²	1	2	yes	—
21	#14582990	—	2	2	no	—
22	C₄×#10621883	—	2	2	no	—
23	D₈×Dic₄₃	C₂²	1	2	yes	—
24	#14582936	C₂²	1	2	yes	—
25	#14582892	—	2	2	no	—
26	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
27	#14582727	C₂²	1	1	yes	characteristic
28	#14582724	C₂²	1	2	yes	—
29	#14582704	C₂²	1	2	yes	—
30	#14582702	C₂²	1	2	yes	—
31	#14582700	C₂²	1	2	yes	—
32	#14582675	C₂²	1	1	yes	characteristic
33	#14582674	C₂²	1	1	yes	characteristic
34	#14582674	C₂²	1	1	yes	characteristic
35	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₃	C₂²	1	2	yes	—
37	C₄₃×#367	C₂²	1	1	yes	characteristic
38	C₄₃×#357	C₂²	1	2	yes	—
39	C₄₃×#346	C₂²	1	1	yes	characteristic
40	C₂×#10621900	—	2	2	no	—
41	#13063344	—	2	4	no	—
42	#13063344	C₂×C₄	1	8	yes	complemented
43	C₂×Dic₁₇₂	C₂³	1	2	yes	—
44	C₂×#10621883	—	2	2	no	—
45	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
47	C₄×Dic₈₆	—	2	2	no	—
48	#13063290	—	2	2	no	—
49	D₄×Dic₄₃	—	2	2	no	—
50	C₄×C₂²⋊D₄₃	—	4	2	no	—
51	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
52	#13063283	—	2	2	no	—
53	#13063281	—	4	2	no	—
54	C₂×C₄×Dic₄₃	—	4	2	no	—
55	C₄²×D₄₃	D₄	1	1	yes	characteristic
56	Dic₃₄₄	C₂×C₄	1	4	yes	complemented
57	#13063269	C₂×C₄	1	8	yes	complemented
58	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₄₃	—	2	2	no	—
61	C₁₆×D₄₃	C₂×C₄	1	4	yes	complemented
62	#13063259	C₂³	1	2	yes	—
63	#13063256	—	2	2	no	—
64	#13063250	—	2	2	no	—
65	#13063248	—	2	2	no	—
66	#13063246	—	2	2	no	—
67	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
68	C₈×Dic₄₃	D₄	1	1	yes	characteristic
69	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₄₃	C₂³	1	2	yes	—
71	C₄×D₄×C₄₃	—	2	2	no	—
72	D₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
75	#107×C₄₃	C₂³	1	2	yes	—
76	#102×C₄₃	—	2	2	no	—
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	—	4	2	no	—
80	C₂×C₂²⋊D₄₃	—	4	2	no	—
81	C₂²×Dic₄₃	—	4	2	no	—
82	C₂²×Dic₄₃	—	4	2	no	—
83	#10621900	—	2	8	no	—
84	#10621900	—	4	4	no	—
85	C₂×C₄×D₄₃	D₈	1	2	yes	—
86	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₈₆	—	2	2	no	—
88	D₈×C₄₃	—	2	2	no	—
89	D₈×C₄₃	C₂²×C₄	1	4	yes	—
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	C₂²⋊C₄×C₄₃	—	4	2	no	—
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	—	4	2	no	—
96	#10621883	—	2	8	no	—
97	(C₂×D₄₃)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	Dic₄₃⋊C₄	—	4	2	no	—
100	Dic₄₃⋊C₄	—	4	2	no	—
101	C₄×Dic₄₃	—	4	2	no	—
102	C₄×Dic₄₃	—	4	2	no	—
103	C₄×Dic₄₃	—	4	2	no	—
104	C₄×Dic₄₃	D₈	1	2	yes	—
105	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
108	Dic₁₇₂	—	2	2	no	—
109	Dic₁₇₂	C₂²×C₄	1	4	yes	—
110	C₈×D₄₃	—	2	2	no	—
111	C₈×D₄₃	C₂²×C₄	1	2	yes	—
112	C₈×D₄₃	#41	1	2	yes	—
113	C₁₆×C₄₃	—	2	1	no	—
114	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
115	C₄₃⋊C₁₆	—	2	1	no	—
116	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
117	C₂³×C₄₃	—	4	2	no	—
118	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
119	D₄×C₄₃	—	2	4	no	—
120	D₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	8	4	no	—
123	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	4	8	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	8	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	8	4	no	—
130	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
131	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₄₃	—	4	2	no	—
133	C₄×D₄₃	C₄×D₄	1	2	yes	—
134	C₄×D₄₃	#135	1	2	yes	—
135	Dic₈₆	—	2	4	no	—
136	Dic₈₆	—	4	2	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₄₃⋊C₈	—	2	1	no	—
141	C₄₃⋊C₈	C₄×D₄	1	2	yes	—
142	#33451	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	4	4	no	—
144	C₂²×C₄₃	—	8	2	no	—
145	C₂²×C₄₃	—	8	2	no	—
146	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₄₃	—	2	2	no	—
148	C₂×D₄₃	C₄×D₈	1	2	yes	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	8	2	no	—
151	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
153	Dic₄₃	—	2	2	no	—
154	Dic₄₃	—	4	4	no	—
155	Dic₄₃	—	8	2	no	—
156	Dic₄₃	C₄×D₈	1	2	yes	—
157	C₂×#508	—	43	1	no	—
158	C₄×#135	—	43	2	no	—
159	#2177	—	43	1	no	—
160	#2177	D₄₃	1	1	yes	characteristic, complemented
161	#2176	—	43	1	no	—
162	#2176	—	43	1	no	—
163	#2175	—	43	2	no	—
164	#2175	—	43	2	no	—
165	#2158	—	43	1	no	—
166	#2138	—	43	2	no	—
167	C₂×#346	—	43	1	no	—
168	C₂×C₄₃	—	8	4	no	—
169	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
170	C₂×C₄₃	#3414	1	2	yes	—
171	D₄₃	—	2	2	no	—
172	C₂×#135	—	43	2	no	—
173	C₄×#41	—	86	2	no	—
174	#508	—	43	8	no	—
175	C₂×Q₃₂	—	43	1	no	—
176	C₂×SD₃₂	—	43	2	no	—
177	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
178	C₂²×C₁₆	—	43	1	no	—
179	C₂²×C₁₆	—	43	1	no	—
180	C₄×Q₁₆	—	43	2	no	—
181	C₄×Q₁₆	—	43	2	no	—
182	C₄×SD₁₆	—	86	2	no	—
183	C₄×SD₁₆	—	86	2	no	—
184	C₄×D₈	—	43	2	no	—
185	C₄×D₈	C₂×D₄₃	1	2	yes	—
186	#427	—	43	2	no	—
187	#416	—	86	2	no	—
188	C₂×C₄×C₈	—	43	1	no	—
189	#367	—	43	1	no	—
190	#367	C₂×D₄₃	1	1	yes	characteristic
191	#366	—	43	2	no	—
192	#358	—	43	2	no	—
193	#358	—	43	2	no	—
194	#357	—	43	2	no	—
195	#357	C₂×D₄₃	1	2	yes	—
196	#346	—	43	1	no	—
197	#346	—	43	1	no	—
198	#346	—	43	1	no	—
199	#346	C₂×D₄₃	1	1	yes	characteristic
200	C₄₃	#33451	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	86	2	no	—
202	#135	—	43	8	no	—
203	#135	—	86	4	no	—
204	C₂×Q₁₆	—	43	2	no	—
205	C₂×SD₁₆	—	86	2	no	—
206	C₂×D₈	C₂²×D₄₃	1	2	yes	—
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	43	1	no	—
209	C₄×Q₈	—	86	2	no	—
210	C₄×Q₈	—	86	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	86	2	no	—
213	C₄×D₄	—	172	2	no	—
214	C₄×D₄	—	172	2	no	—
215	#117	—	86	2	no	—
216	#117	—	172	2	no	—
217	C₂×C₄²	—	43	1	no	—
218	C₂×C₄²	—	172	2	no	—
219	Q₃₂	—	43	4	no	—
220	SD₃₂	—	43	8	no	—
221	D₁₆	C₄×D₄₃	1	4	yes	complemented
222	C₂×C₁₆	—	43	1	no	—
223	C₂×C₁₆	—	43	4	no	—
224	C₂×C₁₆	—	43	1	no	—
225	C₂×C₁₆	—	86	2	no	—
226	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
228	#107	—	43	2	no	—
229	#107	C₂²×D₄₃	1	2	yes	—
230	#106	—	86	2	no	—
231	#103	—	86	2	no	—
232	#103	—	86	2	no	—
233	#102	—	2	2	no	—
234	#102	—	86	2	no	—
235	C₄×C₈	—	43	1	no	—
236	C₄×C₈	—	43	1	no	—
237	C₄×C₈	—	43	1	no	—
238	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
239	#41	—	86	8	no	—
240	#41	—	172	4	no	—
241	C₂×Q₈	—	86	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	172	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	43	2	no	—
247	C₂²×C₄	—	172	2	no	—
248	C₂²×C₄	—	172	2	no	—
249	Q₁₆	—	43	4	no	—
250	Q₁₆	—	86	2	no	—
251	SD₁₆	—	86	8	no	—
252	D₈	—	2	2	no	—
253	D₈	C₂×C₄×D₄₃	1	4	yes	—
254	C₂×C₈	—	43	1	no	—
255	C₂×C₈	—	43	2	no	—
256	C₂×C₈	—	43	2	no	—
257	C₂×C₈	—	43	1	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	86	2	no	—
263	C₄⋊C₄	—	172	2	no	—
264	C₄⋊C₄	—	172	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	172	2	no	—
267	C₂²⋊C₄	—	172	2	no	—
268	C₄²	—	43	2	no	—
269	C₄²	—	43	1	no	—
270	C₄²	—	172	2	no	—
271	C₄²	—	172	2	no	—
272	C₄²	—	172	2	no	—
273	C₄²	D₄×D₄₃	1	1	yes	characteristic
274	C₁₆	—	2	1	no	—
275	C₁₆	—	43	2	no	—
276	C₁₆	—	86	1	no	—
277	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	—	43	1	no	—
280	Q₈	—	86	4	no	—
281	Q₈	—	172	2	no	—
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	172	8	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	43	1	no	—
288	C₂×C₄	—	43	2	no	—
289	C₂×C₄	—	43	2	no	—
290	C₂×C₄	—	43	2	no	—
291	C₂×C₄	—	344	4	no	—
292	C₂×C₄	—	172	2	no	—
293	C₂×C₄	—	172	8	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	D₈×D₄₃	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	43	2	no	—
300	C₈	—	86	1	no	—
301	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
302	C₈	#41×D₄₃	1	1	yes	characteristic
303	C₂²	—	4	4	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	43	2	no	—
307	C₂²	—	86	2	no	—
308	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	43	2	no	—
312	C₄	—	86	2	no	—
313	C₄	—	344	2	no	—
314	C₄	—	172	4	no	—
315	C₄	C₄×D₄×D₄₃	1	1	yes	characteristic
316	C₄	#135×D₄₃	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	—	86	2	no	—
319	C₂	C₄×D₈×D₄₃	1	1	yes	characteristic
320	C₂	#19147727	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2177	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#14583328	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2177	D₄₃
hypocenter	C₄₃	#33451
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. C₄×#13063344	C₂	semidirect
3. #19136471	C₂	semidirect
4. #19136467	C₂	semidirect
5. #19136465	C₂	semidirect
6. #19136459	C₂	semidirect
7. #19136456	C₂	semidirect
8. #19136378	C₂	semidirect
9. #19136299	C₂	semidirect
10. #346×D₄₃	C₂	semidirect
11. C₄₃×#2177	C₂	semidirect
14. #14583328	C₄	semidirect
42. #13063344	C₂×C₄	semidirect
56. Dic₃₄₄	C₂×C₄	semidirect
57. #13063269	C₂×C₄	semidirect
61. C₁₆×D₄₃	C₂×C₄	semidirect
72. D₁₆×C₄₃	C₂×C₄	semidirect
160. #2177	D₄₃	semidirect
200. C₄₃	#33451	semidirect
221. D₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583328	2
C₄×#13063344	2
#19136471	2
#19136467	2
#19136465	2
#19136459	2
#19136456	2
#19136378	2
#19136299	2
#346×D₄₃	2
C₄₃×#2177	2
#33451	43