Subgroups — #25249956

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#19143442	C₂	1	2	yes	maximal, complemented
3	#19136472	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136471	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136470	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13063266	C₂	1	2	yes	maximal, complemented
7	#19136439	C₂	1	2	yes	maximal, complemented
8	#19136378	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136309	C₂	1	2	yes	maximal, complemented
10	#19136193	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₃	C₂²	1	2	yes	—
13	#14583371	—	2	2	no	—
14	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
15	C₂×#13063266	C₂²	1	2	yes	—
16	D₁₆×D₄₃	C₄	1	8	yes	complemented
17	C₂×#13063264	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
19	#14583009	—	2	2	no	—
20	#14583008	—	2	2	no	—
21	C₄×D₃₄₄	C₂²	1	2	yes	complemented
22	D₈×Dic₄₃	C₂²	1	2	yes	complemented
23	#14582981	C₂²	1	2	yes	complemented
24	#14582936	C₂²	1	2	yes	—
25	#14582902	—	2	2	no	—
26	#14582853	C₂²	1	1	yes	characteristic
27	#14582727	C₂²	1	1	yes	characteristic, complemented
28	#14582724	C₂²	1	2	yes	complemented
29	#14582701	C₂²	1	2	yes	complemented
30	#14582700	C₂²	1	2	yes	complemented
31	#14582699	C₂²	1	2	yes	complemented
32	#14582674	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₃	C₂²	1	2	yes	complemented
37	C₄₃×#367	C₂²	1	1	yes	characteristic, complemented
38	C₄₃×#357	C₂²	1	2	yes	complemented
39	C₄₃×#346	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₄₃	—	2	2	no	—
41	C₂×D₃₄₄	C₂³	1	2	yes	—
42	C₂×#10621882	—	2	2	no	—
43	D₈×D₄₃	—	2	4	no	—
44	D₈×D₄₃	C₂×C₄	1	8	yes	complemented
45	C₂×#10621872	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
47	C₄×D₁₇₂	—	2	2	no	—
48	D₄×Dic₄₃	—	2	2	no	—
49	#13063287	—	2	2	no	—
50	C₄×C₂²⋊D₄₃	—	4	2	no	—
51	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
52	#13063283	—	2	2	no	—
53	#13063282	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
55	C₂²⋊C₄×D₄₃	—	4	2	no	—
56	D₆₈₈	C₂×C₄	1	4	yes	complemented
57	#13063266	C₂×C₄	1	8	yes	complemented
58	#13063264	—	2	2	no	—
59	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₄₃	C₂×C₄	1	4	yes	complemented
62	#13063259	C₂³	1	2	yes	—
63	#13063256	—	2	2	no	—
64	#13063247	—	2	2	no	—
65	#13063246	—	2	2	no	—
66	#13063245	—	2	2	no	—
67	#13063236	D₄	1	1	yes	characteristic
68	#13063235	D₄	1	1	yes	characteristic
69	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₄₃	C₂³	1	2	yes	—
71	C₄×D₄×C₄₃	—	2	2	no	—
72	D₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
75	#107×C₄₃	C₂³	1	2	yes	—
76	#102×C₄₃	—	2	2	no	—
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂³×D₄₃	—	4	2	no	—
79	C₂×D₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	4	2	no	—
81	C₂×C₂²⋊D₄₃	—	4	2	no	—
82	C₂²×Dic₄₃	—	4	2	no	—
83	D₄×D₄₃	—	2	8	no	—
84	D₄×D₄₃	—	4	4	no	—
85	C₂×D₁₇₂	—	2	2	no	—
86	C₂×C₄×D₄₃	—	4	2	no	—
87	C₂×C₄×D₄₃	—	4	2	no	—
88	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
89	D₈×C₄₃	—	2	2	no	—
90	D₈×C₄₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₃	—	2	2	no	—
94	C₂²⋊C₄×C₄₃	—	4	2	no	—
95	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₄₃	—	4	2	no	—
97	#10621882	—	2	8	no	—
98	(C₂×D₄₃)⋊C₄	—	4	2	no	—
99	(C₂×D₄₃)⋊C₄	—	4	2	no	—
100	(C₂×D₄₃)⋊C₄	—	4	2	no	—
101	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
102	C₄⋊Dic₄₃	—	2	2	no	—
103	Dic₄₃⋊C₄	D₈	1	2	yes	—
104	C₄×Dic₄₃	—	4	2	no	—
105	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
108	D₃₄₄	—	2	2	no	—
109	D₃₄₄	C₂²×C₄	1	4	yes	—
110	#10621872	—	2	2	no	—
111	C₈×D₄₃	C₂²×C₄	1	2	yes	—
112	C₈×D₄₃	#41	1	2	yes	—
113	C₁₆×C₄₃	—	2	1	no	—
114	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
115	C₄₃⋊C₁₆	—	2	1	no	—
116	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
117	C₂³×C₄₃	—	4	2	no	—
118	C₂²×D₄₃	—	4	2	no	—
119	C₂²×D₄₃	—	4	8	no	—
120	C₂²×D₄₃	—	16	2	no	—
121	C₂²×D₄₃	—	16	2	no	—
122	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
123	D₄×C₄₃	—	2	4	no	—
124	D₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	16	2	no	—
127	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₃	—	4	8	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	16	2	no	—
132	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
133	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
134	D₁₇₂	—	2	4	no	—
135	D₁₇₂	—	4	2	no	—
136	C₄×D₄₃	—	16	2	no	—
137	C₄×D₄₃	—	16	2	no	—
138	C₄×D₄₃	C₄×D₄	1	2	yes	—
139	C₄×D₄₃	#135	1	2	yes	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₄₃⋊C₈	—	2	1	no	—
144	C₄₃⋊C₈	C₄×D₄	1	2	yes	—
145	#33456	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	4	4	no	—
147	C₂²×C₄₃	—	8	2	no	—
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₄₃	—	2	2	no	—
151	C₂×D₄₃	—	4	4	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₂×D₄₃	—	8	2	no	—
154	C₂×D₄₃	—	16	8	no	—
155	C₂×D₄₃	C₄×D₈	1	2	yes	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	8	2	no	—
158	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
160	Dic₄₃	—	2	2	no	—
161	Dic₄₃	—	8	2	no	—
162	Dic₄₃	C₄×D₈	1	2	yes	—
163	C₂²×D₁₆	—	43	1	no	—
164	#2948	—	43	2	no	—
165	#2177	—	43	1	no	—
166	#2177	—	43	1	no	—
167	#2177	—	43	1	no	—
168	#2177	D₄₃	1	1	yes	characteristic, complemented
169	C₄×D₁₆	—	43	2	no	—
170	C₄×D₁₆	—	43	2	no	—
171	#2158	—	43	1	no	—
172	#2140	—	43	2	no	—
173	#2108	—	43	1	no	—
174	C₂×C₄₃	—	8	4	no	—
175	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
176	C₂×C₄₃	#3415	1	2	yes	—
177	D₄₃	—	2	2	no	—
178	D₄₃	—	8	4	no	—
179	C₂²×D₈	—	43	2	no	—
180	#518	—	86	2	no	—
181	C₂×D₁₆	—	43	2	no	—
182	C₂×D₁₆	—	43	8	no	—
183	C₂×D₁₆	—	43	1	no	—
184	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
185	C₂×M₃₂	—	43	1	no	—
186	C₂²×C₁₆	—	43	1	no	—
187	#442	—	86	2	no	—
188	#442	—	86	2	no	—
189	C₄×D₈	—	43	2	no	—
190	C₄×D₈	—	43	2	no	—
191	C₄×D₈	—	43	2	no	—
192	C₄×D₈	C₂×D₄₃	1	2	yes	complemented
193	#427	—	43	2	no	—
194	#418	—	86	2	no	—
195	#405	—	43	1	no	—
196	#367	—	43	1	no	—
197	#367	C₂×D₄₃	1	1	yes	characteristic, complemented
198	#366	—	43	2	no	—
199	#357	—	43	2	no	—
200	#357	—	43	2	no	—
201	#357	—	43	2	no	—
202	#357	C₂×D₄₃	1	2	yes	complemented
203	#346	—	43	1	no	—
204	#346	C₂×D₄₃	1	1	yes	characteristic, complemented
205	C₄×C₁₆	—	43	1	no	—
206	C₄×C₁₆	—	43	1	no	—
207	C₄₃	#33456	1	1	yes	characteristic, complemented, Sylow
208	C₂²×D₄	—	86	2	no	—
209	C₂×D₈	—	43	8	no	—
210	C₂×D₈	—	43	2	no	—
211	C₂×D₈	—	86	4	no	—
212	C₂×D₈	—	86	2	no	—
213	C₂×D₈	C₂²×D₄₃	1	2	yes	—
214	C₂×M₁₆	—	43	1	no	—
215	C₂²×C₈	—	43	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	86	2	no	—
218	C₄×D₄	—	86	2	no	—
219	C₄×D₄	—	86	2	no	—
220	C₄×D₄	—	172	2	no	—
221	C₄×D₄	—	172	2	no	—
222	#117	—	43	1	no	—
223	#117	—	86	2	no	—
224	C₂×C₂²⋊C₄	—	172	2	no	—
225	C₂×C₂²⋊C₄	—	172	2	no	—
226	D₁₆	—	43	8	no	—
227	D₁₆	—	43	4	no	—
228	D₁₆	C₄×D₄₃	1	4	yes	complemented
229	M₃₂	—	86	2	no	—
230	C₂×C₁₆	—	43	1	no	—
231	C₂×C₁₆	—	43	4	no	—
232	C₂×C₁₆	—	43	1	no	—
233	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
235	#107	—	43	2	no	—
236	#107	C₂²×D₄₃	1	2	yes	—
237	#106	—	86	2	no	—
238	#102	—	2	2	no	—
239	#102	—	86	2	no	—
240	#102	—	86	2	no	—
241	#102	—	86	2	no	—
242	#97	—	43	1	no	—
243	#97	—	43	1	no	—
244	C₄×C₈	—	43	1	no	—
245	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂⁴	—	172	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	86	8	no	—
249	C₂×D₄	—	86	2	no	—
250	C₂×D₄	—	172	4	no	—
251	C₂×D₄	—	172	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	172	2	no	—
255	C₂²×C₄	—	172	2	no	—
256	C₂²×C₄	—	172	2	no	—
257	D₈	—	2	2	no	—
258	D₈	—	43	4	no	—
259	D₈	—	86	8	no	—
260	D₈	—	86	2	no	—
261	D₈	C₂×C₄×D₄₃	1	4	yes	—
262	M₁₆	—	86	2	no	—
263	C₂×C₈	—	43	1	no	—
264	C₂×C₈	—	43	2	no	—
265	C₂×C₈	—	43	2	no	—
266	C₂×C₈	—	43	1	no	—
267	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	43	2	no	—
271	C₄⋊C₄	—	86	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	43	2	no	—
274	C₂²⋊C₄	—	172	2	no	—
275	C₂²⋊C₄	—	172	2	no	—
276	C₂²⋊C₄	—	172	2	no	—
277	C₂²⋊C₄	—	172	2	no	—
278	C₄²	—	43	1	no	—
279	C₄²	—	172	2	no	—
280	C₄²	D₄×D₄₃	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	43	2	no	—
283	C₁₆	—	86	1	no	—
284	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
285	C₂³	—	4	2	no	—
286	C₂³	—	43	1	no	—
287	C₂³	—	172	8	no	—
288	C₂³	—	172	2	no	—
289	C₂³	—	688	2	no	—
290	C₂³	—	688	2	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	86	4	no	—
294	D₄	—	172	8	no	—
295	D₄	—	172	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	43	2	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	43	2	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	—	688	2	no	—
304	C₂×C₄	—	688	2	no	—
305	C₂×C₄	—	688	2	no	—
306	C₂×C₄	D₈×D₄₃	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	43	2	no	—
310	C₈	—	86	1	no	—
311	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
312	C₈	#41×D₄₃	1	1	yes	characteristic
313	C₂²	—	4	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	43	2	no	—
317	C₂²	—	86	2	no	—
318	C₂²	—	344	2	no	—
319	C₂²	—	344	2	no	—
320	C₂²	—	172	4	no	—
321	C₂²	—	688	8	no	—
322	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	43	2	no	—
326	C₄	—	86	2	no	—
327	C₄	—	344	2	no	—
328	C₄	C₄×D₄×D₄₃	1	1	yes	characteristic
329	C₄	#135×D₄₃	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	—	86	2	no	—
332	C₂	—	344	4	no	—
333	C₂	C₄×D₈×D₄₃	1	1	yes	characteristic
334	C₂	#19147735	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2177	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×D₁₆×D₄₃	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2177	D₄₃
hypocenter	C₄₃	#33456
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #19143442	C₂	semidirect
3. #19136472	C₂	semidirect
4. #19136471	C₂	semidirect
5. #19136470	C₂	semidirect
6. C₄×#13063266	C₂	semidirect
7. #19136439	C₂	semidirect
8. #19136378	C₂	semidirect
9. #19136309	C₂	semidirect
10. #19136193	C₂	semidirect
11. C₄₃×#2177	C₂	semidirect
16. D₁₆×D₄₃	C₄	semidirect
21. C₄×D₃₄₄	C₂²	semidirect
22. D₈×Dic₄₃	C₂²	semidirect
23. #14582981	C₂²	semidirect
27. #14582727	C₂²	semidirect
28. #14582724	C₂²	semidirect
29. #14582701	C₂²	semidirect
30. #14582700	C₂²	semidirect
31. #14582699	C₂²	semidirect
32. #14582674	C₂²	semidirect
33. C₄×C₄₃⋊C₁₆	C₂²	semidirect
34. C₁₆×Dic₄₃	C₂²	semidirect
36. C₄×D₈×C₄₃	C₂²	semidirect
37. C₄₃×#367	C₂²	semidirect
38. C₄₃×#357	C₂²	semidirect
39. C₄₃×#346	C₂²	semidirect
44. D₈×D₄₃	C₂×C₄	semidirect
56. D₆₈₈	C₂×C₄	semidirect
57. #13063266	C₂×C₄	semidirect
61. C₁₆×D₄₃	C₂×C₄	semidirect
72. D₁₆×C₄₃	C₂×C₄	semidirect
168. #2177	D₄₃	semidirect
192. C₄×D₈	C₂×D₄₃	semidirect
197. #367	C₂×D₄₃	semidirect
202. #357	C₂×D₄₃	semidirect
204. #346	C₂×D₄₃	semidirect
207. C₄₃	#33456	semidirect
228. D₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₄₃	2
#19143442	2
#19136472	2
#19136471	2
#19136470	2
C₄×#13063266	2
#19136439	2
#19136378	2
#19136309	2
#19136193	2
C₄₃×#2177	2
#33456	43