Subgroups — #25105454

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143536	C₂	1	2	yes	maximal
2	#19135577	C₂	1	2	yes	maximal
3	#19135349	C₂	1	2	yes	maximal
4	#19135344	C₂	1	2	yes	maximal
5	#19134767	C₂	1	1	yes	maximal, characteristic
6	#19134481	C₂	1	2	yes	maximal
7	#19134476	C₂	1	2	yes	maximal
8	#19134281	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#3160	C₂	1	1	yes	maximal, characteristic
10	#14583030	—	2	8	no	—
11	C₂×#13063261	C₂²	1	2	yes	—
12	C₂×#13063261	C₂²	1	2	yes	—
13	#14582926	C₂²	1	4	yes	—
14	#14582923	C₂²	1	2	yes	—
15	#14582855	C₂²	1	4	yes	—
16	C₂×#13063236	C₂²	1	1	yes	characteristic
17	C₂×#13063235	C₂²	1	1	yes	characteristic
18	C₂×#13063235	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
21	#14582745	C₂²	1	1	yes	characteristic
22	#14582658	C₂²	1	4	yes	—
23	#14582658	C₂²	1	4	yes	—
24	C₄₃×#533	C₂²	1	2	yes	—
25	C₄₃×#500	C₄	1	2	yes	complemented
26	C₄₃×#499	C₄	1	2	yes	complemented
27	C₄₃×#499	C₄	1	2	yes	complemented
28	C₄₃×#498	C₄	1	2	yes	complemented
29	C₄₃×#427	C₂²	1	2	yes	—
30	C₄₃×#427	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×#10621877	C₂³	1	4	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
37	#13063261	—	2	4	no	—
38	#13063261	—	2	4	no	—
39	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
41	#13063236	—	2	4	no	—
42	#13063236	D₄	1	4	yes	—
43	#13063235	—	2	2	no	—
44	#13063235	Q₈	1	4	yes	complemented
45	C₄×C₄₃⋊C₈	—	2	4	no	—
46	C₈×Dic₄₃	—	2	2	no	—
47	C₈×Dic₄₃	Q₈	1	4	yes	complemented
48	C₄₃⋊#95	—	2	2	no	—
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
52	#125×C₄₃	C₂×C₄	1	4	yes	—
53	C₂²⋊Q₈×C₄₃	—	2	4	no	—
54	C₄×Q₈×C₄₃	—	2	4	no	—
55	#117×C₄₃	C₂³	1	4	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	#107×C₄₃	C₂×C₄	1	4	yes	—
58	#107×C₄₃	C₂×C₄	1	4	yes	—
59	#106×C₄₃	C₂×C₄	1	4	yes	—
60	#106×C₄₃	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂×Q₈×C₄₃	—	2	4	no	—
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	1	no	—
69	C₂²×Dic₄₃	—	2	1	no	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₄₃	#41	1	2	yes	—
77	C₂×C₈×C₄₃	#41	1	2	yes	—
78	C₄⋊C₄×C₄₃	—	2	8	no	—
79	C₄⋊C₄×C₄₃	—	2	4	no	—
80	C₄⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
81	C₂²⋊C₄×C₄₃	—	2	4	no	—
82	C₄²×C₄₃	—	2	4	no	—
83	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂×D₄	1	2	yes	—
86	C₄×Dic₄₃	—	2	1	no	—
87	C₄×Dic₄₃	—	2	1	no	—
88	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
89	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
90	#10621877	—	2	4	no	—
91	#10621877	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	2	1	no	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
97	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
99	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
100	Q₈×C₄₃	—	8	4	no	—
101	C₂×C₄×C₄₃	—	2	8	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₄₃	—	2	2	no	—
112	C₂×Dic₄₃	—	2	2	no	—
113	C₂×Dic₄₃	—	4	2	no	—
114	C₂×Dic₄₃	—	4	2	no	—
115	C₈×C₄₃	C₄×Q₈	1	2	yes	—
116	C₈×C₄₃	C₄×Q₈	1	2	yes	—
117	C₈×C₄₃	C₄×Q₈	1	2	yes	—
118	C₈×C₄₃	C₄×Q₈	1	2	yes	—
119	C₄₃⋊C₈	—	2	2	no	—
120	C₄₃⋊C₈	—	4	4	no	—
121	C₄₃⋊C₈	C₄⋊Q₈	1	4	yes	—
122	#21733	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₄₃	#533	1	1	yes	characteristic
126	C₂²×C₄₃	#533	1	1	yes	characteristic
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	—	2	2	no	—
129	C₄×C₄₃	—	4	8	no	—
130	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₄×C₄₃	#532	1	1	yes	characteristic
134	Dic₄₃	—	4	2	no	—
135	Dic₄₃	—	4	2	no	—
136	#3160	D₄₃	1	1	yes	characteristic
137	#2961	—	43	2	no	—
138	#1995	—	43	2	no	—
139	#1946	—	43	2	no	—
140	#1945	—	43	2	no	—
141	#1848	—	43	1	no	—
142	#1773	—	43	2	no	—
143	#1772	—	43	2	no	—
144	#1726	—	43	1	no	—
145	C₂×C₄₃	—	2	1	no	—
146	C₂×C₄₃	—	2	1	no	—
147	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₄₃	#2957	1	2	yes	—
149	#533	C₂×D₄₃	1	2	yes	—
150	#500	Dic₄₃	1	2	yes	complemented
151	#499	Dic₄₃	1	2	yes	complemented
152	#499	Dic₄₃	1	2	yes	complemented
153	#498	Dic₄₃	1	2	yes	complemented
154	#446	—	86	8	no	—
155	C₂×#108	—	43	2	no	—
156	C₂×#108	—	43	2	no	—
157	#427	C₂×D₄₃	1	2	yes	—
158	#427	C₂×D₄₃	1	2	yes	—
159	#424	—	43	4	no	—
160	#423	—	43	2	no	—
161	#405	—	43	4	no	—
162	C₂×#97	—	43	1	no	—
163	C₂×#97	—	43	1	no	—
164	C₂×#97	—	43	1	no	—
165	C₂×C₄×C₈	—	43	1	no	—
166	C₂×C₄×C₈	—	43	1	no	—
167	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
168	#376	—	43	1	no	—
169	#339	—	43	4	no	—
170	#339	—	43	4	no	—
171	C₄₃	#21733	1	1	yes	characteristic, complemented, Sylow
172	C₂×M₁₆	—	43	4	no	—
173	C₂²×C₈	—	43	1	no	—
174	C₂²×C₈	—	43	1	no	—
175	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
177	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
178	#125	C₂×Dic₄₃	1	4	yes	—
179	C₂²⋊Q₈	—	2	4	no	—
180	C₄×Q₈	—	2	4	no	—
181	#117	C₂²×D₄₃	1	4	yes	—
182	C₂×C₄²	—	43	1	no	—
183	C₂×C₄²	—	43	1	no	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	#108	—	86	4	no	—
186	#108	—	86	4	no	—
187	#107	C₂×Dic₄₃	1	4	yes	—
188	#107	C₂×Dic₄₃	1	4	yes	—
189	#106	C₂×Dic₄₃	1	4	yes	—
190	#106	C₂×Dic₄₃	1	4	yes	—
191	C₄⋊C₈	—	86	8	no	—
192	C₄⋊C₈	—	86	4	no	—
193	#97	—	43	4	no	—
194	#97	—	43	4	no	—
195	#97	—	86	4	no	—
196	#97	—	86	2	no	—
197	C₄×C₈	—	43	4	no	—
198	C₄×C₈	—	86	4	no	—
199	C₄×C₈	—	86	2	no	—
200	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
201	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
202	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
203	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
204	#95	—	86	2	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂²×C₄	—	86	1	no	—
207	C₂²×C₄	—	86	1	no	—
208	C₂²×C₄	#10621900	1	2	yes	—
209	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
210	M₁₆	—	86	4	no	—
211	M₁₆	—	86	4	no	—
212	C₂×C₈	—	43	2	no	—
213	C₂×C₈	—	43	2	no	—
214	C₂×C₈	—	43	2	no	—
215	C₂×C₈	—	43	2	no	—
216	C₂×C₈	—	86	1	no	—
217	C₂×C₈	—	86	1	no	—
218	C₂×C₈	—	86	4	no	—
219	C₂×C₈	Q₈×D₄₃	1	2	yes	—
220	C₂×C₈	Q₈×D₄₃	1	2	yes	—
221	C₂×C₈	#10621902	1	2	yes	—
222	C₂×C₈	#10621902	1	2	yes	—
223	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂²×Dic₄₃	1	8	yes	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	43	2	no	—
233	C₄²	—	43	2	no	—
234	C₄²	—	86	1	no	—
235	C₄²	—	86	1	no	—
236	C₄²	D₄×D₄₃	1	2	yes	—
237	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
238	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
239	C₂³	C₂×#10621900	1	1	yes	characteristic
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	86	2	no	—
244	C₂×C₄	—	86	2	no	—
245	C₂×C₄	—	172	2	no	—
246	C₂×C₄	—	172	2	no	—
247	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
248	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
249	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
254	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
255	C₈	—	43	4	no	—
256	C₈	—	86	2	no	—
257	C₈	—	172	4	no	—
258	C₈	Q₈×Dic₄₃	1	2	yes	—
259	C₈	Q₈×Dic₄₃	1	2	yes	—
260	C₈	Q₈×Dic₄₃	1	2	yes	—
261	C₈	Q₈×Dic₄₃	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
264	C₂²	#14583454	1	1	yes	characteristic
265	C₂²	#14583454	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	8	no	—
269	C₄	—	172	2	no	—
270	C₄	—	172	2	no	—
271	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
272	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
273	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
274	C₄	#14583450	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₄⋊Q₈×Dic₄₃	1	1	yes	characteristic
278	C₂	#19143512	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#3160	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#3160	D₄₃
hypocenter	C₄₃	#21733
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
25. C₄₃×#500	C₄	semidirect
26. C₄₃×#499	C₄	semidirect
27. C₄₃×#499	C₄	semidirect
28. C₄₃×#498	C₄	semidirect
44. #13063235	Q₈	semidirect
47. C₈×Dic₄₃	Q₈	semidirect
150. #500	Dic₄₃	semidirect
151. #499	Dic₄₃	semidirect
152. #499	Dic₄₃	semidirect
153. #498	Dic₄₃	semidirect
171. C₄₃	#21733	semidirect

Maximal subgroups

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

subgroup	index
#19143536	2
#19135577	2
#19135349	2
#19135344	2
#19134767	2
#19134481	2
#19134476	2
#19134281	2
C₄₃×#3160	2
#21733	43