Subgroups — #24990381

Order 11008.

438 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134754	C₂	1	1	yes	maximal, characteristic
2	#19134752	C₂	1	1	yes	maximal, characteristic
3	#19134751	C₂	1	1	yes	maximal, characteristic
4	#19134751	C₂	1	1	yes	maximal, characteristic
5	#19134722	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582621	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1834	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063258	—	2	2	no	—
9	C₂×#13063257	—	2	2	no	—
10	C₂×#13063256	—	2	2	no	—
11	C₂×#13063256	—	2	2	no	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	#14582769	C₂²	1	1	yes	characteristic
15	#14582769	C₂²	1	1	yes	characteristic
16	#14582661	—	2	2	no	—
17	#14582621	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#384	C₂²	1	1	yes	characteristic
20	C₄₃×#384	C₂²	1	1	yes	characteristic
21	C₄₃×#340	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
23	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
24	C₂×C₄⋊Dic₄₃	—	2	1	no	—
25	C₂×C₄⋊Dic₄₃	—	2	1	no	—
26	C₂×C₄⋊Dic₄₃	—	2	1	no	—
27	C₂×C₄⋊Dic₄₃	—	2	1	no	—
28	C₂×C₄⋊Dic₄₃	—	2	1	no	—
29	C₂×C₄⋊Dic₄₃	—	2	1	no	—
30	#13063258	—	4	4	no	—
31	#13063257	—	4	4	no	—
32	#13063256	—	4	4	no	—
33	#13063256	—	4	4	no	—
34	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
37	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
38	C₂²×C₈×C₄₃	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₃	Q₈	1	2	yes	—
48	C₂²×C₄×C₄₃	—	4	1	no	—
49	C₂²×C₄×C₄₃	—	4	1	no	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	4	1	no	—
54	C₂²×Dic₄₃	—	4	1	no	—
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
58	C₄⋊C₄×C₄₃	—	4	2	no	—
59	C₄⋊C₄×C₄₃	—	4	2	no	—
60	C₄⋊C₄×C₄₃	—	4	2	no	—
61	C₄⋊C₄×C₄₃	—	4	2	no	—
62	C₄⋊C₄×C₄₃	—	4	2	no	—
63	C₄⋊C₄×C₄₃	—	4	2	no	—
64	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
68	C₄⋊Dic₄₃	—	4	2	no	—
69	C₄⋊Dic₄₃	—	4	2	no	—
70	C₄⋊Dic₄₃	—	4	2	no	—
71	C₄⋊Dic₄₃	—	4	2	no	—
72	C₄⋊Dic₄₃	—	4	2	no	—
73	C₄⋊Dic₄₃	—	4	2	no	—
74	C₂×C₄₃⋊C₈	—	2	2	no	—
75	C₂×C₄₃⋊C₈	—	2	2	no	—
76	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
77	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
78	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	—
79	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	—
80	C₂³×C₄₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₄₃	—	8	1	no	—
82	C₂×C₄×C₄₃	—	8	1	no	—
83	C₂×C₄×C₄₃	—	8	1	no	—
84	C₂×C₄×C₄₃	—	8	1	no	—
85	C₂×C₄×C₄₃	—	8	1	no	—
86	C₂×C₄×C₄₃	—	8	1	no	—
87	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₄₃	#137	1	2	yes	—
98	C₂×C₄×C₄₃	#137	1	2	yes	—
99	C₂×Dic₄₃	—	8	1	no	—
100	C₂×Dic₄₃	—	8	1	no	—
101	C₂×Dic₄₃	—	8	1	no	—
102	C₂×Dic₄₃	—	8	1	no	—
103	C₂×Dic₄₃	—	8	1	no	—
104	C₂×Dic₄₃	—	8	1	no	—
105	C₈×C₄₃	—	2	4	no	—
106	C₄₃⋊C₈	—	2	4	no	—
107	C₄₃⋊C₈	#106	1	8	yes	complemented
108	#8819	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	#384	1	1	yes	characteristic
110	C₂²×C₄₃	#492	1	1	yes	characteristic
111	C₂²×C₄₃	#492	1	1	yes	characteristic
112	C₂²×C₄₃	#496	1	1	yes	characteristic
113	C₂²×C₄₃	#496	1	1	yes	characteristic
114	C₂²×C₄₃	#497	1	1	yes	characteristic
115	C₂²×C₄₃	#497	1	1	yes	characteristic
116	C₄×C₄₃	—	16	1	no	—
117	C₄×C₄₃	—	16	1	no	—
118	C₄×C₄₃	C₂×#106	1	2	yes	—
119	C₄×C₄₃	#428	1	2	yes	—
120	C₄×C₄₃	#428	1	2	yes	—
121	C₄×C₄₃	#498	1	2	yes	—
122	C₄×C₄₃	#501	1	2	yes	—
123	C₄×C₄₃	#501	1	2	yes	—
124	Dic₄₃	—	16	1	no	—
125	Dic₄₃	—	16	1	no	—
126	#1845	—	43	1	no	—
127	#1844	—	43	1	no	—
128	#1844	—	43	1	no	—
129	#1844	—	43	1	no	—
130	#1834	—	43	1	no	—
131	#1834	D₄₃	1	1	yes	characteristic
132	C₂×#322	—	43	1	no	—
133	C₂×C₄₃	#1714	1	1	yes	characteristic
134	C₂×C₄₃	#1714	1	1	yes	characteristic
135	C₂×C₄₃	#1715	1	1	yes	characteristic
136	C₂×C₄₃	#1716	1	1	yes	characteristic
137	C₂×C₄₃	#1844	1	1	yes	characteristic
138	C₂×C₄₃	#1847	1	1	yes	characteristic
139	C₂×C₄₃	#1847	1	1	yes	characteristic
140	C₂×#107	—	86	2	no	—
141	C₂×#106	—	86	2	no	—
142	C₂×#106	—	86	2	no	—
143	C₂×#106	—	86	2	no	—
144	C₂×C₄×C₈	—	43	1	no	—
145	C₂×C₄×C₈	—	43	1	no	—
146	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
147	#384	—	43	1	no	—
148	#384	—	43	1	no	—
149	#384	C₂×D₄₃	1	1	yes	characteristic
150	#384	C₂×D₄₃	1	1	yes	characteristic
151	#340	—	2	2	no	—
152	#340	—	86	2	no	—
153	#322	—	43	4	no	—
154	C₄₃	#8819	1	1	yes	characteristic, complemented, Sylow
155	C₂²×C₈	—	43	2	no	—
156	C₂²×C₈	—	43	2	no	—
157	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	86	1	no	—
165	C₂×C₄⋊C₄	—	86	1	no	—
166	C₂×C₄⋊C₄	—	86	1	no	—
167	C₂×C₄⋊C₄	—	86	1	no	—
168	C₂×C₄⋊C₄	—	86	1	no	—
169	C₂×C₄⋊C₄	—	86	1	no	—
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	#107	—	172	4	no	—
172	#106	—	172	4	no	—
173	#106	—	172	4	no	—
174	#106	—	172	4	no	—
175	C₄×C₈	—	43	2	no	—
176	C₄×C₈	—	43	2	no	—
177	C₄×C₈	—	43	2	no	—
178	C₄×C₈	—	43	2	no	—
179	C₄×C₈	Dic₈₆	1	2	yes	—
180	C₄×C₈	C₄×D₄₃	1	2	yes	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	172	1	no	—
184	C₂²×C₄	—	172	1	no	—
185	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
186	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	43	4	no	—
191	C₂×C₈	—	43	4	no	—
192	C₂×C₈	—	43	4	no	—
193	C₂×C₈	—	43	4	no	—
194	C₂×C₈	—	86	2	no	—
195	C₂×C₈	—	86	2	no	—
196	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	172	2	no	—
204	C₄⋊C₄	—	172	2	no	—
205	C₄⋊C₄	—	172	2	no	—
206	C₄⋊C₄	—	172	2	no	—
207	C₄⋊C₄	—	172	2	no	—
208	C₄⋊C₄	—	172	2	no	—
209	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
210	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
211	C₄²	Q₈×D₄₃	1	1	yes	characteristic
212	C₄²	Q₈×D₄₃	1	1	yes	characteristic
213	C₂³	#13063321	1	1	yes	characteristic
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	344	1	no	—
221	C₂×C₄	—	344	1	no	—
222	C₂×C₄	—	344	1	no	—
223	C₂×C₄	—	344	1	no	—
224	C₂×C₄	—	344	1	no	—
225	C₂×C₄	—	344	1	no	—
226	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
228	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#13063323	1	1	yes	characteristic
230	C₂×C₄	#13063324	1	1	yes	characteristic
231	C₂×C₄	#13063324	1	1	yes	characteristic
232	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
233	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
234	C₂×C₄	#13063349	1	2	yes	—
235	C₂×C₄	#13063350	1	2	yes	—
236	C₂×C₄	#13063355	1	2	yes	—
237	C₂×C₄	#13063357	1	2	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	43	8	no	—
240	C₈	—	86	4	no	—
241	C₂²	#14582768	1	1	yes	characteristic
242	C₂²	#14583270	1	1	yes	characteristic
243	C₂²	#14583270	1	1	yes	characteristic
244	C₂²	#14583287	1	1	yes	characteristic
245	C₂²	#14583289	1	1	yes	characteristic
246	C₂²	#14583292	1	1	yes	characteristic
247	C₂²	#14583294	1	1	yes	characteristic
248	C₄	—	16	1	no	—
249	C₄	—	16	1	no	—
250	C₄	—	688	1	no	—
251	C₄	—	688	1	no	—
252	C₄	#106×D₄₃	1	2	yes	—
253	C₄	#14582940	1	2	yes	—
254	C₄	#14582942	1	2	yes	—
255	C₄	#14583296	1	2	yes	—
256	C₄	#14583307	1	2	yes	—
257	C₄	#14583309	1	2	yes	—
258	C₂	#19134216	1	1	yes	characteristic
259	C₂	#19134219	1	1	yes	characteristic
260	C₂	#19134225	1	1	yes	characteristic
261	C₂	#19134230	1	1	yes	characteristic
262	C₂	#19134750	1	1	yes	characteristic
263	C₂	#19134762	1	1	yes	characteristic
264	C₂	#19134764	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1834	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1834	D₄₃
hypocenter	C₄₃	#8819
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
17. #14582621	C₄	semidirect
107. C₄₃⋊C₈	#106	semidirect
154. C₄₃	#8819	semidirect

Maximal subgroups

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

subgroup	index
#19134754	2
#19134752	2
#19134751	2
#19134722	2
C₂×#14582621	2
C₄₃×#1834	2
#8819	43