Subgroups — #24990620

Order 11008.

420 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143157	C₂	1	1	yes	maximal, characteristic
2	#19134576	C₂	1	1	yes	maximal, characteristic
3	#19133365	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132907	C₂	1	1	yes	maximal, characteristic
5	#19132491	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₄₃	—	2	1	no	—
9	C₂×#13063282	C₂²	1	1	yes	characteristic
10	#14582952	C₂²	1	1	yes	characteristic
11	#14582924	—	2	2	no	—
12	#14582924	—	2	2	no	—
13	#14582923	C₂²	1	1	yes	characteristic
14	#14582922	—	2	1	no	—
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
17	#14582653	—	2	1	no	—
18	#14582649	—	2	2	no	—
19	#14582624	C₄	1	2	yes	—
20	#14582622	C₄	1	2	yes	—
21	C₄₃×#431	C₂²	1	1	yes	characteristic
22	C₄₃×#423	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#334	—	2	2	no	—
25	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
26	C₂×#10621877	—	2	1	no	—
27	C₂×#10621872	—	2	2	no	—
28	M₁₆×D₄₃	—	4	2	no	—
29	M₁₆×D₄₃	—	4	2	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	C₂×C₈×D₄₃	—	2	2	no	—
32	#13063282	—	2	1	no	—
33	#13063282	—	2	1	no	—
34	#13063282	—	2	1	no	—
35	#13063282	—	2	1	no	—
36	C₂×Dic₄₃⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₄₃⋊C₄	Q₈	1	1	yes	characteristic
38	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
41	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
46	Dic₄₃⋊C₈	—	2	4	no	—
47	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
48	(C₂×D₄₃)⋊C₈	—	4	2	no	—
49	C₄×C₄₃⋊C₈	—	2	1	no	—
50	C₂×M₁₆×C₄₃	—	2	1	no	—
51	C₂²×C₈×C₄₃	—	2	1	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	—	2	1	no	—
54	C₄⋊C₈×C₄₃	—	2	1	no	—
55	C₄⋊C₈×C₄₃	—	2	1	no	—
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
58	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₃	—	2	1	no	—
60	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
72	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
73	M₁₆×C₄₃	—	4	1	no	—
74	M₁₆×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₃	#41	1	1	yes	characteristic
83	C₄²×C₄₃	#41	1	1	yes	characteristic
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	Dic₄₃⋊C₄	—	2	1	no	—
89	Dic₄₃⋊C₄	—	2	1	no	—
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
92	C₄×Dic₄₃	—	2	1	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	#10621877	—	4	1	no	—
95	#10621877	—	4	1	no	—
96	C₂×C₄₃⋊C₈	—	2	2	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	4	1	no	—
99	C₂×C₄₃⋊C₈	—	4	1	no	—
100	C₂×C₄₃⋊C₈	—	4	1	no	—
101	#10621872	—	4	4	no	—
102	C₈×D₄₃	—	4	4	no	—
103	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₄₃	—	2	1	no	—
105	C₂²×D₄₃	—	2	1	no	—
106	C₂²×D₄₃	—	4	1	no	—
107	C₂²×D₄₃	—	4	1	no	—
108	C₂²×D₄₃	#106	1	2	yes	—
109	C₂×C₄×C₄₃	—	4	1	no	—
110	C₂×C₄×C₄₃	—	4	1	no	—
111	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	4	1	no	—
126	C₂×Dic₄₃	—	4	1	no	—
127	C₂×Dic₄₃	#106	1	2	yes	—
128	C₄×D₄₃	—	2	2	no	—
129	C₄×D₄₃	—	2	2	no	—
130	C₄×D₄₃	—	2	2	no	—
131	C₄×D₄₃	—	2	2	no	—
132	C₈×C₄₃	—	4	2	no	—
133	C₈×C₄₃	—	4	2	no	—
134	C₈×C₄₃	—	8	1	no	—
135	C₄₃⋊C₈	—	4	2	no	—
136	C₄₃⋊C₈	—	4	2	no	—
137	C₄₃⋊C₈	—	8	1	no	—
138	#8862	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	#385	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#106	1	1	yes	characteristic
143	C₂²×C₄₃	#428	1	1	yes	characteristic
144	C₂×D₄₃	—	2	2	no	—
145	C₂×D₄₃	—	4	1	no	—
146	C₂×D₄₃	—	4	1	no	—
147	C₂×D₄₃	—	4	2	no	—
148	C₂×D₄₃	—	8	1	no	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	#448	1	1	yes	characteristic
154	C₄×C₄₃	#450	1	1	yes	characteristic
155	C₄×C₄₃	#481	1	1	yes	characteristic
156	C₄×C₄₃	#483	1	1	yes	characteristic
157	Dic₄₃	—	2	2	no	—
158	Dic₄₃	—	4	1	no	—
159	Dic₄₃	—	4	1	no	—
160	#2905	—	43	1	no	—
161	#1798	—	43	1	no	—
162	#1570	—	43	1	no	—
163	#1570	D₄₃	1	1	yes	characteristic, complemented
164	#1458	—	43	1	no	—
165	#1301	—	43	1	no	—
166	#1301	—	43	1	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	#1853	1	1	yes	characteristic
170	C₂×C₄₃	#1858	1	1	yes	characteristic
171	C₂×C₄₃	#1868	1	1	yes	characteristic
172	D₄₃	—	4	2	no	—
173	C₂²×M₁₆	—	86	1	no	—
174	C₂×#117	—	43	1	no	—
175	#431	—	43	1	no	—
176	#431	C₂×D₄₃	1	1	yes	characteristic
177	#424	—	86	2	no	—
178	#424	—	86	2	no	—
179	#423	—	43	1	no	—
180	#423	—	86	1	no	—
181	#423	C₂×D₄₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	43	1	no	—
183	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	—	86	1	no	—
185	#336	—	86	1	no	—
186	#334	—	2	2	no	—
187	#334	—	86	2	no	—
188	#324	—	43	2	no	—
189	#323	—	43	2	no	—
190	C₄₃	#8862	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	43	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	86	2	no	—
194	C₂×M₁₆	—	86	1	no	—
195	C₂×M₁₆	—	172	2	no	—
196	C₂×M₁₆	—	172	2	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	86	2	no	—
199	C₂²×C₈	—	86	1	no	—
200	#117	—	86	1	no	—
201	#117	—	86	1	no	—
202	#117	—	86	1	no	—
203	#117	—	86	1	no	—
204	C₂×C₄⋊C₄	—	43	1	no	—
205	C₂×C₄⋊C₄	—	43	1	no	—
206	C₂×C₂²⋊C₄	—	43	1	no	—
207	C₂×C₂²⋊C₄	—	43	1	no	—
208	C₂×C₄²	—	43	1	no	—
209	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	43	2	no	—
215	C₄⋊C₈	—	86	4	no	—
216	C₄⋊C₈	—	86	1	no	—
217	C₄⋊C₈	—	86	1	no	—
218	C₄⋊C₈	—	86	1	no	—
219	C₄⋊C₈	—	86	1	no	—
220	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
221	C₂²⋊C₈	—	43	2	no	—
222	C₂²⋊C₈	—	172	2	no	—
223	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	86	1	no	—
226	C₂⁴	—	43	1	no	—
227	C₂²×C₄	—	43	1	no	—
228	C₂²×C₄	—	43	2	no	—
229	C₂²×C₄	—	43	2	no	—
230	C₂²×C₄	—	43	1	no	—
231	C₂²×C₄	—	43	1	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	—	86	1	no	—
234	C₂²×C₄	—	86	1	no	—
235	C₂²×C₄	—	86	1	no	—
236	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	172	4	no	—
242	M₁₆	—	172	1	no	—
243	M₁₆	—	172	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	86	2	no	—
250	C₂×C₈	—	86	2	no	—
251	C₂×C₈	—	172	4	no	—
252	C₂×C₈	—	172	1	no	—
253	C₂×C₈	—	172	1	no	—
254	C₂×C₈	—	172	1	no	—
255	C₄⋊C₄	—	86	1	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	1	no	—
259	C₂²⋊C₄	—	86	1	no	—
260	C₂²⋊C₄	—	86	1	no	—
261	C₂²⋊C₄	—	86	1	no	—
262	C₂²⋊C₄	—	86	1	no	—
263	C₄²	—	86	1	no	—
264	C₄²	—	86	1	no	—
265	C₄²	#10621898	1	1	yes	characteristic
266	C₄²	D₄×D₄₃	1	1	yes	characteristic
267	C₄²	#10621902	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
269	C₂³	—	43	2	no	—
270	C₂³	—	86	1	no	—
271	C₂³	—	86	1	no	—
272	C₂³	—	172	1	no	—
273	C₂³	—	172	1	no	—
274	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	43	2	no	—
278	C₂×C₄	—	86	1	no	—
279	C₂×C₄	—	86	1	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	86	2	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	86	2	no	—
284	C₂×C₄	—	86	1	no	—
285	C₂×C₄	—	86	1	no	—
286	C₂×C₄	—	172	1	no	—
287	C₂×C₄	—	172	1	no	—
288	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
289	C₂×C₄	#13063287	1	1	yes	characteristic
290	C₂×C₄	#13063293	1	1	yes	characteristic
291	C₂×C₄	#13063301	1	1	yes	characteristic
292	C₂×C₄	#13063303	1	1	yes	characteristic
293	C₂×C₄	#13063310	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
295	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
296	C₂×C₄	#13063352	1	1	yes	characteristic
297	C₂×C₄	#13063354	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	344	1	no	—
302	C₈	—	172	2	no	—
303	C₈	—	172	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	86	2	no	—
307	C₂²	—	344	1	no	—
308	C₂²	—	172	2	no	—
309	C₂²	—	172	1	no	—
310	C₂²	—	172	1	no	—
311	C₂²	#14582770	1	1	yes	characteristic
312	C₂²	#106×D₄₃	1	1	yes	characteristic
313	C₂²	#14582938	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	86	2	no	—
319	C₄	—	172	1	no	—
320	C₄	—	172	1	no	—
321	C₄	#14583036	1	1	yes	characteristic
322	C₄	#14583048	1	1	yes	characteristic
323	C₄	#14583219	1	1	yes	characteristic
324	C₄	#14583231	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	172	2	no	—
328	C₂	#19134786	1	1	yes	characteristic
329	C₂	#19134813	1	1	yes	characteristic
330	C₂	#19134879	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1570	C₂
socle	C₂²×C₄₃	#385
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1570	D₄₃
hypocenter	C₄₃	#8862
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

Complements

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

N	H ≅ G/N	type
3. #19133365	C₂	semidirect
5. #19132491	C₂	semidirect
6. #19132490	C₂	semidirect
7. C₄₃×#1570	C₂	semidirect
163. #1570	D₄₃	semidirect
190. C₄₃	#8862	semidirect

Maximal subgroups

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

subgroup	index
#19143157	2
#19134576	2
#19133365	2
#19132907	2
#19132491	2
#19132490	2
C₄₃×#1570	2
#8862	43