Subgroups — #25252196

Order 11008.

532 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	#19145238	C₂	1	2	yes	maximal, complemented
2	#19136872	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136843	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136835	C₂	1	2	yes	maximal, complemented
5	#19136696	C₂	1	2	yes	maximal, complemented
6	#19136681	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136468	C₂	1	2	yes	maximal, complemented
8	D₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136441	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136191	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2215	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583502	—	2	2	no	—
13	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
14	C₂×#13063266	C₂²	1	2	yes	—
15	C₂×#13063265	C₂²	1	1	yes	characteristic
16	#14583300	C₂²	1	2	yes	—
17	#14583274	C₂²	1	2	yes	—
18	#14583262	—	2	2	no	—
19	#14583150	—	2	2	no	—
20	#14583135	C₂²	1	2	yes	—
21	#14583008	—	2	2	no	—
22	D₈×Dic₄₃	C₂²	1	2	yes	—
23	#14582981	C₂²	1	2	yes	—
24	#14582854	C₂²	1	1	yes	characteristic
25	#14582726	C₂²	1	2	yes	—
26	#14582725	C₂²	1	1	yes	characteristic
27	#14582701	C₂²	1	2	yes	—
28	#14582700	C₂²	1	2	yes	—
29	#14582699	C₂²	1	2	yes	—
30	#14582674	C₂²	1	1	yes	characteristic
31	#14582674	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
36	C₄₃×#466	C₂²	1	2	yes	—
37	C₄₃×#366	C₂²	1	1	yes	characteristic
38	C₄₃×#357	C₂²	1	2	yes	—
39	C₂×D₃₄₄	C₂³	1	2	yes	—
40	C₂×#10621882	—	2	2	no	—
41	C₂×#10621877	D₄	1	1	yes	characteristic
42	#13063321	—	2	2	no	—
43	#13063316	—	2	2	no	—
44	#13063309	—	4	2	no	—
45	#13063307	—	4	2	no	—
46	#13063299	—	2	2	no	—
47	#13063297	—	4	2	no	—
48	#13063296	—	4	2	no	—
49	D₄×Dic₄₃	—	2	2	no	—
50	#13063287	—	2	2	no	—
51	#13063284	D₄	1	1	yes	characteristic
52	D₆₈₈	—	2	2	no	—
53	#13063266	—	2	4	no	—
54	#13063265	—	2	2	no	—
55	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
57	#13063259	C₂³	1	2	yes	—
58	#13063256	—	2	2	no	—
59	#13063247	—	2	2	no	—
60	#13063246	—	2	2	no	—
61	#13063245	—	2	2	no	—
62	#13063235	D₄	1	1	yes	characteristic
63	#13063235	D₄	1	1	yes	characteristic
64	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
65	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₄₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	#121×C₄₃	—	2	2	no	—
69	D₁₆×C₄₃	—	2	2	no	—
70	C₂×C₁₆×C₄₃	—	2	2	no	—
71	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
73	#107×C₄₃	C₂³	1	2	yes	—
74	#102×C₄₃	—	2	2	no	—
75	C₂×D₄×C₄₃	—	2	2	no	—
76	C₂×D₄×C₄₃	—	4	2	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₃	—	4	2	no	—
79	C₂×C₂²⋊D₄₃	—	4	2	no	—
80	C₂×C₂²⋊D₄₃	—	4	2	no	—
81	C₂²×Dic₄₃	—	4	2	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×C₄×D₄₃	—	4	2	no	—
84	D₈×C₄₃	—	2	2	no	—
85	D₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	2	1	no	—
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₂²⋊C₄×C₄₃	—	4	2	no	—
92	C₂²⋊Dic₄₃	—	4	2	no	—
93	C₂²⋊Dic₄₃	D₈	1	1	yes	characteristic
94	C₂²⋊Dic₄₃	D₈	1	1	yes	characteristic
95	#10621882	—	4	4	no	—
96	(C₂×D₄₃)⋊C₄	—	4	2	no	—
97	(C₂×D₄₃)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	4	2	no	—
100	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
101	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
102	Dic₄₃⋊C₄	—	4	2	no	—
103	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
105	#10621877	—	2	2	no	—
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	—	2	2	no	—
110	C₁₆×C₄₃	—	2	1	no	—
111	C₁₆×C₄₃	#41	1	2	yes	—
112	C₄₃⋊C₁₆	—	2	1	no	—
113	C₄₃⋊C₁₆	#41	1	2	yes	—
114	C₂³×C₄₃	—	4	2	no	—
115	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₄₃	—	4	2	no	—
117	D₄×C₄₃	—	4	2	no	—
118	D₄×C₄₃	—	4	2	no	—
119	D₄×C₄₃	—	16	2	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	16	2	no	—
126	C₂²⋊D₄₃	—	16	2	no	—
127	C₂²⋊D₄₃	—	16	2	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	16	2	no	—
130	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
134	D₁₇₂	—	4	2	no	—
135	D₁₇₂	—	4	2	no	—
136	C₄×D₄₃	—	16	2	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₄₃⋊C₈	—	2	1	no	—
141	C₄₃⋊C₈	#135	1	2	yes	—
142	#33740	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	8	2	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	—	8	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	#532	1	1	yes	characteristic
156	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	2	1	no	—
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	8	2	no	—
162	#3176	—	43	2	no	—
163	#2253	—	43	1	no	—
164	#2246	—	43	1	no	—
165	#2244	—	43	2	no	—
166	#2218	—	43	2	no	—
167	#2215	—	43	1	no	—
168	#2215	D₄₃	1	1	yes	characteristic, complemented
169	#2177	—	43	2	no	—
170	C₄×D₁₆	—	43	1	no	—
171	C₄×D₁₆	—	43	1	no	—
172	#2108	—	43	1	no	—
173	C₂×C₄₃	—	4	1	no	—
174	C₂×C₄₃	—	16	2	no	—
175	C₂×C₄₃	#3169	1	1	yes	characteristic
176	C₂×C₄₃	#3415	1	1	yes	characteristic
177	C₂×C₄₃	#3415	1	1	yes	characteristic
178	D₄₃	—	16	2	no	—
179	#540	—	86	2	no	—
180	C₂×D₁₆	—	43	2	no	—
181	C₂×D₁₆	—	43	1	no	—
182	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
183	C₂×M₃₂	—	43	1	no	—
184	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
185	#499	—	43	2	no	—
186	#493	—	43	2	no	—
187	#490	—	86	2	no	—
188	#469	—	86	2	no	—
189	#466	—	43	2	no	—
190	#466	C₂×D₄₃	1	2	yes	—
191	#442	—	86	2	no	—
192	C₄×D₈	—	43	2	no	—
193	C₄×D₈	—	43	2	no	—
194	#405	—	43	1	no	—
195	#367	—	43	2	no	—
196	#366	—	43	1	no	—
197	#366	C₂×D₄₃	1	1	yes	characteristic
198	#357	—	43	2	no	—
199	#357	—	43	2	no	—
200	#357	—	43	2	no	—
201	#357	C₂×D₄₃	1	2	yes	—
202	#346	—	43	1	no	—
203	#346	—	43	1	no	—
204	C₄×C₁₆	—	43	1	no	—
205	C₄×C₁₆	—	43	1	no	—
206	C₄₃	#33740	1	1	yes	characteristic, complemented, Sylow
207	C₂×D₈	—	43	2	no	—
208	C₂×D₈	—	86	2	no	—
209	C₂×D₈	C₂²×D₄₃	1	2	yes	—
210	C₂×M₁₆	—	43	1	no	—
211	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
212	#127	—	86	2	no	—
213	#125	—	86	2	no	—
214	#123	—	172	2	no	—
215	#123	—	172	2	no	—
216	#121	—	2	2	no	—
217	#121	—	86	2	no	—
218	#121	—	172	2	no	—
219	#121	—	172	2	no	—
220	C₄×D₄	—	86	2	no	—
221	C₄×D₄	—	86	2	no	—
222	#117	—	43	1	no	—
223	D₁₆	—	2	2	no	—
224	D₁₆	—	86	4	no	—
225	D₁₆	—	86	2	no	—
226	M₃₂	—	86	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	C₂×C₁₆	—	43	1	no	—
229	C₂×C₁₆	—	43	1	no	—
230	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
232	#107	—	43	2	no	—
233	#107	C₂²×D₄₃	1	2	yes	—
234	#106	—	86	2	no	—
235	#102	—	2	2	no	—
236	#102	—	86	2	no	—
237	#102	—	86	2	no	—
238	#102	—	86	2	no	—
239	#97	—	43	1	no	—
240	#97	—	43	1	no	—
241	C₄×C₈	—	43	1	no	—
242	C₄×C₈	—	43	1	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	86	2	no	—
246	C₂×D₄	—	172	2	no	—
247	C₂×D₄	—	172	2	no	—
248	C₂×D₄	—	172	2	no	—
249	C₂²×C₄	—	172	2	no	—
250	C₂²×C₄	—	172	2	no	—
251	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	86	2	no	—
255	D₈	—	86	2	no	—
256	D₈	—	172	4	no	—
257	M₁₆	—	86	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	43	1	no	—
261	C₂×C₈	—	43	1	no	—
262	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	43	1	no	—
266	C₄⋊C₄	—	43	1	no	—
267	C₄⋊C₄	—	86	2	no	—
268	C₄⋊C₄	—	172	2	no	—
269	C₄⋊C₄	—	172	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	43	1	no	—
272	C₂²⋊C₄	—	43	1	no	—
273	C₂²⋊C₄	—	172	2	no	—
274	C₂²⋊C₄	—	172	2	no	—
275	C₂²⋊C₄	—	172	2	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	43	1	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	—	43	2	no	—
280	C₁₆	—	86	1	no	—
281	C₁₆	#10621900	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	—	172	2	no	—
284	C₂³	D₈×D₄₃	1	1	yes	characteristic
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	16	2	no	—
288	D₄	—	172	2	no	—
289	D₄	—	172	2	no	—
290	D₄	—	688	2	no	—
291	D₄	—	688	2	no	—
292	D₄	—	688	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	43	1	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	43	1	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	688	2	no	—
302	C₂×C₄	—	688	2	no	—
303	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	43	2	no	—
307	C₈	—	86	1	no	—
308	C₈	#41×D₄₃	1	1	yes	characteristic
309	C₈	C₂×#10621900	1	1	yes	characteristic
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	344	2	no	—
317	C₂²	—	344	2	no	—
318	C₂²	—	344	2	no	—
319	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	86	1	no	—
323	C₄	—	86	1	no	—
324	C₄	—	86	1	no	—
325	C₄	—	86	1	no	—
326	C₄	—	344	2	no	—
327	C₄	#14583444	1	1	yes	characteristic
328	C₄	#135×D₄₃	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	16	2	no	—
331	C₂	—	688	2	no	—
332	C₂	#19145169	1	1	yes	characteristic
333	C₂	#19147732	1	1	yes	characteristic
334	C₂	#19147733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2215	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#2215	D₄₃
hypocenter	C₄₃	#33740
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #19145238	C₂	semidirect
2. #19136872	C₂	semidirect
3. #19136843	C₂	semidirect
4. #19136835	C₂	semidirect
5. #19136696	C₂	semidirect
6. #19136681	C₂	semidirect
7. #19136468	C₂	semidirect
8. D₁₆×Dic₄₃	C₂	semidirect
9. #19136441	C₂	semidirect
10. #19136191	C₂	semidirect
11. C₄₃×#2215	C₂	semidirect
168. #2215	D₄₃	semidirect
206. C₄₃	#33740	semidirect

Maximal subgroups

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

subgroup	index
#19145238	2
#19136872	2
#19136843	2
#19136835	2
#19136696	2
#19136681	2
#19136468	2
D₁₆×Dic₄₃	2
#19136441	2
#19136191	2
C₄₃×#2215	2
#33740	43