Subgroups — #25110285

Order 11008.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19143525	C₂	1	2	yes	maximal
3	#19135571	C₂	1	2	yes	maximal
4	#19135193	C₂	1	8	yes	maximal
5	#19134340	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1837	C₂	1	1	yes	maximal, characteristic
7	#14583456	C₂²	1	2	yes	—
8	#14583304	C₄	1	1	yes	characteristic, complemented
9	#14583304	C₄	1	1	yes	characteristic, complemented
10	#14583301	C₄	1	4	yes	complemented
11	#14583297	C₄	1	2	yes	complemented
12	#14583026	—	2	2	no	—
13	#14583026	—	2	2	no	—
14	#14583023	—	2	2	no	—
15	#14583023	—	2	2	no	—
16	#14582937	C₂²	1	4	yes	—
17	#14582927	C₂²	1	4	yes	—
18	#14582857	C₂²	1	4	yes	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
21	#14582660	C₂²	1	8	yes	—
22	#14582653	C₂²	1	4	yes	—
23	C₄₃×#423	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#341	C₂²	1	4	yes	—
26	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
27	#13063325	C₂×C₄	1	2	yes	—
28	#13063325	C₂×C₄	1	2	yes	—
29	#13063317	C₂×C₄	1	4	yes	—
30	#13063304	—	2	2	no	—
31	#13063304	—	2	2	no	—
32	C₄×Dic₈₆	—	2	2	no	—
33	C₄×Dic₈₆	—	2	2	no	—
34	#13063284	C₂³	1	4	yes	—
35	C₂×C₄×Dic₄₃	—	2	1	no	—
36	C₂×C₄×Dic₄₃	—	2	1	no	—
37	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
38	#13063259	C₂×C₄	1	4	yes	—
39	#13063259	C₂×C₄	1	4	yes	—
40	#13063257	C₂×C₄	1	8	yes	—
41	Dic₄₃⋊C₈	—	2	4	no	—
42	Dic₄₃⋊C₈	—	2	4	no	—
43	#13063235	—	2	4	no	—
44	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
45	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
46	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
47	C₈×Dic₄₃	—	2	4	no	—
48	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₃	—	2	2	no	—
52	C₄⋊C₈×C₄₃	—	2	2	no	—
53	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₃	—	2	2	no	—
60	C₂×Dic₈₆	—	2	2	no	—
61	C₂×Dic₈₆	—	2	2	no	—
62	M₁₆×C₄₃	—	2	8	no	—
63	C₂×C₈×C₄₃	—	2	4	no	—
64	C₂×C₈×C₄₃	—	2	4	no	—
65	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²⋊Dic₄₃	—	2	4	no	—
72	C₄⋊Dic₄₃	—	2	2	no	—
73	C₄⋊Dic₄₃	—	2	2	no	—
74	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
75	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
76	Dic₄₃⋊C₄	—	2	4	no	—
77	Dic₄₃⋊C₄	—	2	4	no	—
78	C₄×Dic₄₃	—	2	4	no	—
79	C₄×Dic₄₃	—	2	4	no	—
80	C₄×Dic₄₃	—	2	4	no	—
81	C₄×Dic₄₃	—	4	2	no	—
82	C₂×C₄₃⋊C₈	—	2	4	no	—
83	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
84	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
85	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
95	C₂×Dic₄₃	—	2	4	no	—
96	C₂×Dic₄₃	—	2	4	no	—
97	C₂×Dic₄₃	—	2	4	no	—
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	Dic₈₆	—	4	4	no	—
101	Dic₈₆	—	4	4	no	—
102	C₈×C₄₃	—	2	2	no	—
103	C₈×C₄₃	—	2	2	no	—
104	C₈×C₄₃	—	4	4	no	—
105	C₄₃⋊C₈	—	2	2	no	—
106	C₄₃⋊C₈	—	2	2	no	—
107	#22256	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	—	2	1	no	—
110	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₄₃	#541	1	2	yes	—
112	C₄×C₄₃	—	2	1	no	—
113	C₄×C₄₃	—	2	1	no	—
114	C₄×C₄₃	#443	1	2	yes	—
115	C₄×C₄₃	#444	1	2	yes	—
116	C₄×C₄₃	#519	1	2	yes	—
117	C₄×C₄₃	#536	1	2	yes	—
118	Dic₄₃	—	4	4	no	—
119	Dic₄₃	—	4	4	no	—
120	Dic₄₃	—	4	4	no	—
121	#3160	—	43	1	no	—
122	#2960	—	43	2	no	—
123	#1994	—	43	2	no	—
124	#1922	—	43	8	no	—
125	#1837	D₄₃	1	1	yes	characteristic
126	#1744	—	43	1	no	—
127	C₂×C₄₃	—	2	2	no	—
128	C₂×C₄₃	#2327	1	1	yes	characteristic
129	C₂×C₄₃	C₂×#443	1	1	yes	characteristic
130	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
131	#533	—	43	2	no	—
132	#500	—	43	1	no	—
133	#500	—	43	1	no	—
134	#499	—	43	4	no	—
135	#498	—	43	2	no	—
136	#446	—	86	2	no	—
137	#446	—	86	2	no	—
138	C₈×Q₈	—	86	2	no	—
139	C₈×Q₈	—	86	2	no	—
140	#427	—	43	4	no	—
141	#424	—	43	4	no	—
142	#423	C₂×D₄₃	1	2	yes	—
143	#405	—	43	4	no	—
144	C₂×C₄×C₈	—	43	1	no	—
145	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
146	C₄³	—	43	1	no	—
147	#341	C₂×D₄₃	1	4	yes	—
148	#339	—	43	8	no	—
149	#336	—	43	4	no	—
150	C₄₃	#22256	1	1	yes	characteristic, complemented, Sylow
151	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
152	C₂²×C₈	—	43	2	no	—
153	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
154	C₄⋊Q₈	—	43	2	no	—
155	C₄⋊Q₈	—	43	2	no	—
156	#125	—	43	4	no	—
157	C₂²⋊Q₈	—	86	2	no	—
158	C₂²⋊Q₈	—	86	2	no	—
159	C₄×Q₈	—	86	2	no	—
160	C₄×Q₈	—	86	2	no	—
161	#117	—	43	4	no	—
162	C₂×C₄²	—	43	2	no	—
163	C₂×C₄²	—	86	1	no	—
164	C₂×C₄²	—	86	1	no	—
165	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
166	#107	—	43	4	no	—
167	#107	—	43	4	no	—
168	#106	—	43	8	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	86	4	no	—
172	C₄⋊C₈	—	86	4	no	—
173	#97	—	86	4	no	—
174	C₄×C₈	—	43	1	no	—
175	C₄×C₈	—	43	2	no	—
176	C₄×C₈	—	43	1	no	—
177	C₄×C₈	—	86	4	no	—
178	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
179	C₄×C₈	C₄×D₄₃	1	2	yes	—
180	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
181	C₂×Q₈	—	86	2	no	—
182	C₂×Q₈	—	86	2	no	—
183	C₂²×C₄	—	86	2	no	—
184	C₂²×C₄	#10621902	1	1	yes	characteristic
185	C₂²×C₄	#10621902	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	43	2	no	—
191	C₂×C₈	—	43	2	no	—
192	C₂×C₈	—	86	4	no	—
193	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
195	C₄⋊C₄	—	43	4	no	—
196	C₄⋊C₄	—	43	4	no	—
197	C₄⋊C₄	—	86	4	no	—
198	C₄⋊C₄	—	86	4	no	—
199	C₄⋊C₄	—	86	2	no	—
200	C₄⋊C₄	—	86	2	no	—
201	C₂²⋊C₄	—	86	4	no	—
202	C₄²	—	86	4	no	—
203	C₄²	—	86	4	no	—
204	C₄²	—	86	4	no	—
205	C₄²	—	172	2	no	—
206	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
207	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
208	C₄²	D₄×D₄₃	1	1	yes	characteristic
209	C₄²	D₄×D₄₃	1	1	yes	characteristic
210	C₂³	C₂×#10621902	1	1	yes	characteristic
211	Q₈	—	172	4	no	—
212	Q₈	—	172	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	86	4	no	—
216	C₂×C₄	—	86	4	no	—
217	C₂×C₄	—	86	4	no	—
218	C₂×C₄	—	172	2	no	—
219	C₂×C₄	—	172	2	no	—
220	C₂×C₄	#13063287	1	1	yes	characteristic
221	C₂×C₄	#13063287	1	1	yes	characteristic
222	C₂×C₄	#13063287	1	1	yes	characteristic
223	C₂×C₄	#13063287	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
226	C₂×C₄	#13063372	1	4	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	4	4	no	—
230	C₈	—	86	2	no	—
231	C₈	—	86	2	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×#13063287	1	1	yes	characteristic
235	C₂²	#14583512	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	172	4	no	—
239	C₄	—	172	4	no	—
240	C₄	—	172	4	no	—
241	C₄	#14583015	1	2	yes	—
242	C₄	#14583020	1	2	yes	—
243	C₄	#14583376	1	2	yes	—
244	C₄	#14583469	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	#19137312	1	1	yes	characteristic
247	C₂	C₂×#14583015	1	1	yes	characteristic
248	C₂	C₂×#14583020	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1837	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1837	D₄₃
hypocenter	C₄₃	#22256
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
8. #14583304	C₄	semidirect
9. #14583304	C₄	semidirect
10. #14583301	C₄	semidirect
11. #14583297	C₄	semidirect
150. C₄₃	#22256	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19143525	2
#19135571	2
#19135193	2
#19134340	2
C₄₃×#1837	2
#22256	43