Subgroups — #25110253

Order 11008.

658 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137337	C₂	1	2	yes	maximal
2	#19135565	C₂	1	2	yes	maximal
3	#19135189	C₂	1	2	yes	maximal
4	#19135182	C₂	1	2	yes	maximal
5	#19135175	C₂	1	2	yes	maximal
6	#19135168	C₂	1	2	yes	maximal
7	#19134723	C₂	1	1	yes	maximal, characteristic
8	#19134346	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₃×#499	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063236	C₂²	1	1	yes	characteristic
11	#14582798	—	2	2	no	—
12	#14582798	—	2	2	no	—
13	#14582798	C₂²	1	2	yes	—
14	#14582798	C₂²	1	2	yes	—
15	#14582785	—	2	2	no	—
16	#14582785	—	2	2	no	—
17	#14582769	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
19	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
20	#14582745	C₂²	1	1	yes	characteristic
21	#14582662	C₂²	1	2	yes	—
22	#14582662	C₂²	1	1	yes	characteristic
23	#14582662	C₂²	1	1	yes	characteristic
24	#14582661	C₂²	1	2	yes	—
25	#14582661	C₂²	1	2	yes	—
26	#14582661	C₂²	1	2	yes	—
27	#14582661	C₂²	1	2	yes	—
28	#14582653	C₂²	1	2	yes	—
29	#14582653	C₂²	1	2	yes	—
30	C₂×#125×C₄₃	C₂²	1	2	yes	—
31	C₄₃×#499	C₄	1	8	yes	complemented
32	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#106×C₄₃	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₃	—	2	2	no	—
38	C₂×C₄⋊Dic₄₃	—	2	2	no	—
39	C₂×C₄⋊Dic₄₃	—	2	2	no	—
40	C₂×C₄⋊Dic₄₃	—	2	2	no	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	C₂×C₄×Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
46	#13063236	—	2	2	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	2	no	—
49	C₄₃⋊#95	—	2	2	no	—
50	C₄₃⋊#95	—	2	2	no	—
51	C₄₃⋊#95	D₄	1	1	yes	characteristic
52	C₄₃⋊#95	D₄	1	2	yes	—
53	C₄₃⋊#95	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	#125×C₄₃	—	2	4	no	—
57	#125×C₄₃	C₂×C₄	1	8	yes	—
58	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	#107×C₄₃	C₂×C₄	1	4	yes	—
64	#107×C₄₃	C₂×C₄	1	4	yes	—
65	#106×C₄₃	C₂×C₄	1	8	yes	—
66	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	2	2	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	2	2	no	—
73	C₂²×Dic₄₃	—	2	2	no	—
74	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
75	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
80	C₄⋊C₄×C₄₃	—	2	8	no	—
81	C₄⋊C₄×C₄₃	—	2	8	no	—
82	C₄⋊C₄×C₄₃	—	2	2	no	—
83	C₄⋊C₄×C₄₃	—	2	2	no	—
84	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
86	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₃	#41	1	2	yes	—
88	C₄⋊Dic₄₃	—	2	2	no	—
89	C₄⋊Dic₄₃	—	2	2	no	—
90	C₄⋊Dic₄₃	—	2	2	no	—
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄⋊Dic₄₃	—	8	2	no	—
93	C₄⋊Dic₄₃	—	8	2	no	—
94	C₄⋊Dic₄₃	—	8	2	no	—
95	C₄⋊Dic₄₃	—	8	2	no	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	C₄×Dic₄₃	—	8	2	no	—
99	C₄×Dic₄₃	—	8	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	—	4	2	no	—
110	C₂×C₄×C₄₃	—	4	2	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
118	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
119	C₂×Dic₄₃	—	2	2	no	—
120	C₂×Dic₄₃	—	2	2	no	—
121	C₂×Dic₄₃	—	2	2	no	—
122	C₂×Dic₄₃	—	2	2	no	—
123	C₂×Dic₄₃	—	2	2	no	—
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₈×C₄₃	—	2	2	no	—
132	C₈×C₄₃	—	2	2	no	—
133	C₄₃⋊C₈	—	4	2	no	—
134	#22252	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₃	#536	1	2	yes	—
137	C₂²×C₄₃	#576	1	1	yes	characteristic
138	C₂²×C₄₃	#577	1	2	yes	—
139	C₂²×C₄₃	#578	1	1	yes	characteristic
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	4	4	no	—
143	C₄×C₄₃	—	4	4	no	—
144	C₄×C₄₃	#519	1	1	yes	characteristic
145	C₄×C₄₃	#519	1	1	yes	characteristic
146	C₄×C₄₃	#541	1	2	yes	—
147	Dic₄₃	—	4	2	no	—
148	Dic₄₃	—	4	2	no	—
149	Dic₄₃	—	8	2	no	—
150	Dic₄₃	—	8	2	no	—
151	C₂×#499	D₄₃	1	1	yes	characteristic
152	#2330	—	43	2	no	—
153	#1992	—	43	2	no	—
154	#1921	—	43	2	no	—
155	#1920	—	43	2	no	—
156	#1919	—	43	2	no	—
157	#1918	—	43	2	no	—
158	#1834	—	43	1	no	—
159	#1746	—	43	1	no	—
160	C₂×C₄₃	#2327	1	1	yes	characteristic
161	C₂×C₄₃	#2951	1	1	yes	characteristic
162	C₂×C₄₃	#2952	1	1	yes	characteristic
163	C₂×C₄₃	#3241	1	2	yes	—
164	C₂×C₄₃	#3242	1	2	yes	—
165	C₂×#125	C₂×D₄₃	1	2	yes	—
166	#499	Dic₄₃	1	8	yes	complemented
167	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
168	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
169	C₂×#106	C₂×D₄₃	1	2	yes	—
170	C₂×#97	—	43	1	no	—
171	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
172	#389	—	43	2	no	—
173	#389	—	43	2	no	—
174	#389	—	86	2	no	—
175	#389	—	86	2	no	—
176	#387	—	86	2	no	—
177	#387	—	86	2	no	—
178	#384	—	43	2	no	—
179	C₄×C₄⋊C₄	—	43	2	no	—
180	C₄×C₄⋊C₄	—	43	2	no	—
181	#376	—	43	1	no	—
182	#340	—	43	2	no	—
183	#340	—	43	1	no	—
184	#340	—	43	1	no	—
185	#340	—	43	2	no	—
186	#340	—	43	2	no	—
187	#340	—	43	2	no	—
188	#340	—	43	2	no	—
189	#336	—	43	2	no	—
190	#336	—	43	2	no	—
191	C₄₃	#22252	1	1	yes	characteristic, complemented, Sylow
192	C₂²×C₈	—	43	2	no	—
193	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
195	#125	—	2	4	no	—
196	#125	C₂×Dic₄₃	1	8	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	43	2	no	—
200	C₂×C₄⋊C₄	—	43	2	no	—
201	C₂×C₄⋊C₄	—	86	2	no	—
202	C₂×C₄⋊C₄	—	86	2	no	—
203	C₂×C₄⋊C₄	—	86	2	no	—
204	C₂×C₄⋊C₄	—	86	2	no	—
205	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
207	C₂×C₄²	—	43	2	no	—
208	C₂×C₄²	—	86	2	no	—
209	C₂×C₄²	—	86	2	no	—
210	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
211	#107	C₂×Dic₄₃	1	4	yes	—
212	#107	C₂×Dic₄₃	1	4	yes	—
213	#106	C₂×Dic₄₃	1	8	yes	—
214	#97	—	86	2	no	—
215	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
216	#95	—	43	1	no	—
217	#95	—	43	2	no	—
218	#95	—	43	1	no	—
219	#95	—	86	2	no	—
220	#95	—	86	2	no	—
221	#95	—	86	2	no	—
222	#95	—	86	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	—	43	2	no	—
227	C₂²×C₄	—	86	2	no	—
228	C₂²×C₄	—	86	2	no	—
229	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	86	2	no	—
235	C₂×C₈	—	86	2	no	—
236	C₂×C₈	—	86	2	no	—
237	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
238	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	86	2	no	—
244	C₄⋊C₄	—	86	2	no	—
245	C₄⋊C₄	—	86	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	344	2	no	—
248	C₄⋊C₄	—	344	2	no	—
249	C₄⋊C₄	—	344	2	no	—
250	C₄⋊C₄	—	344	2	no	—
251	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
252	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
253	C₄²	—	86	2	no	—
254	C₄²	—	86	2	no	—
255	C₄²	—	344	2	no	—
256	C₄²	—	344	2	no	—
257	C₄²	#10621900	1	2	yes	—
258	C₄²	C₂²×Dic₄₃	1	2	yes	—
259	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	86	2	no	—
270	C₂×C₄	—	86	2	no	—
271	C₂×C₄	—	86	2	no	—
272	C₂×C₄	—	86	2	no	—
273	C₂×C₄	—	86	2	no	—
274	C₂×C₄	—	86	2	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	—	172	2	no	—
277	C₂×C₄	—	172	2	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
282	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
283	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
284	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
285	C₂×C₄	#13063373	1	2	yes	—
286	C₂×C₄	#13063373	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	172	2	no	—
290	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
291	C₂²	#14583474	1	2	yes	—
292	C₂²	#14583701	1	1	yes	characteristic
293	C₂²	#14583706	1	2	yes	—
294	C₂²	#14583713	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	4	no	—
299	C₄	—	344	2	no	—
300	C₄	—	344	2	no	—
301	C₄	—	172	2	no	—
302	C₄	—	172	2	no	—
303	C₄	#14583377	1	1	yes	characteristic
304	C₄	#14583377	1	1	yes	characteristic
305	C₄	#14583510	1	2	yes	—
306	C₂	#19137315	1	1	yes	characteristic
307	C₂	#19143471	1	1	yes	characteristic
308	C₂	#19143479	1	1	yes	characteristic
309	C₂	#19145851	1	2	yes	—
310	C₂	#19145866	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#499	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#499	D₄₃
hypocenter	C₄₃	#22252
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
31. C₄₃×#499	C₄	semidirect
166. #499	Dic₄₃	semidirect
191. C₄₃	#22252	semidirect

Maximal subgroups

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

subgroup	index
#19137337	2
#19135565	2
#19135189	2
#19135182	2
#19135175	2
#19135168	2
#19134723	2
#19134346	2
C₂×C₄₃×#499	2
#22252	43