Subgroups — #25251759

Order 11008.

580 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143448	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136783	C₂	1	2	yes	maximal, complemented
3	#19136749	C₂	1	2	yes	maximal, complemented
4	#19136627	C₂	1	2	yes	maximal, complemented
5	#19136506	C₂	1	2	yes	maximal, complemented
6	#19136469	C₂	1	2	yes	maximal, complemented
7	C₄×#13063266	C₂	1	2	yes	maximal, complemented
8	#19136402	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3176	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
11	#14583373	—	2	1	no	—
12	C₂×#13063266	C₂²	1	2	yes	—
13	C₂×#13063266	C₂²	1	2	yes	—
14	#14583012	—	2	1	no	—
15	#14583012	—	2	1	no	—
16	C₄×D₃₄₄	C₂²	1	2	yes	—
17	C₄×D₃₄₄	C₂²	1	2	yes	—
18	#14582937	C₂²	1	1	yes	characteristic
19	#14582905	—	2	1	no	—
20	#14582726	C₂²	1	2	yes	complemented
21	#14582724	C₂²	1	2	yes	complemented
22	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	complemented
23	#14582700	C₂²	1	2	yes	complemented
24	#14582700	C₂²	1	2	yes	complemented
25	#14582699	C₂²	1	2	yes	—
26	#14582699	C₂²	1	2	yes	—
27	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
28	#14582675	C₂²	1	1	yes	characteristic, complemented
29	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₄₃×#540	—	2	1	no	—
31	C₄₃×#499	C₂²	1	1	yes	characteristic, complemented
32	C₄₃×#493	C₂²	1	1	yes	characteristic, complemented
33	C₄₃×#490	—	2	1	no	—
34	C₄₃×#469	—	2	1	no	—
35	C₄₃×#466	C₂²	1	2	yes	—
36	C₄₃×#442	—	2	1	no	—
37	C₄×D₈×C₄₃	C₂²	1	2	yes	complemented
38	C₄₃×#405	C₂²	1	1	yes	characteristic
39	C₂²×D₁₇₂	—	2	1	no	—
40	C₂×D₃₄₄	—	2	1	no	—
41	C₂×D₃₄₄	—	2	4	no	—
42	C₂×D₃₄₄	C₂³	1	2	yes	—
43	C₄×D₁₇₂	—	2	2	no	—
44	C₄×D₁₇₂	—	2	2	no	—
45	#13063287	—	4	2	no	—
46	#13063284	—	2	1	no	—
47	C₂²⋊C₄×D₄₃	—	4	2	no	—
48	#13063266	—	2	4	no	—
49	#13063266	—	2	4	no	—
50	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	#13063259	C₂³	1	2	yes	—
53	#13063257	—	2	1	no	—
54	#13063247	—	2	2	no	—
55	#13063247	—	2	2	no	—
56	C₂×D₈×C₄₃	—	2	1	no	—
57	C₂×D₈×C₄₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	#127×C₄₃	—	2	1	no	—
61	#125×C₄₃	—	2	1	no	—
62	#123×C₄₃	—	4	2	no	—
63	#121×C₄₃	—	2	2	no	—
64	#121×C₄₃	—	4	2	no	—
65	C₄×D₄×C₄₃	—	2	2	no	—
66	#117×C₄₃	D₄	1	1	yes	characteristic
67	#107×C₄₃	C₂³	1	2	yes	—
68	#106×C₄₃	—	2	1	no	—
69	#102×C₄₃	—	2	2	no	—
70	#102×C₄₃	—	2	2	no	—
71	#97×C₄₃	D₄	1	1	yes	characteristic
72	#97×C₄₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂³×D₄₃	—	4	1	no	—
76	C₂×D₄×C₄₃	—	2	2	no	—
77	C₂×D₄×C₄₃	—	4	2	no	—
78	C₂×D₄×C₄₃	—	4	2	no	—
79	C₂²×C₄×C₄₃	—	4	2	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×D₁₇₂	—	2	4	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	4	2	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×C₄×D₄₃	—	4	2	no	—
86	C₂×C₄×D₄₃	—	4	2	no	—
87	D₈×C₄₃	—	2	2	no	—
88	D₈×C₄₃	—	2	2	no	—
89	D₈×C₄₃	—	4	2	no	—
90	M₁₆×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	4	2	no	—
98	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
99	C₂²⋊C₄×C₄₃	—	4	2	no	—
100	C₂²⋊C₄×C₄₃	—	4	2	no	—
101	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₄₃	—	4	1	no	—
105	(C₂×D₄₃)⋊C₄	—	4	2	no	—
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄×Dic₄₃	—	4	1	no	—
109	D₃₄₄	—	2	4	no	—
110	D₃₄₄	—	2	2	no	—
111	D₃₄₄	#41	1	4	yes	—
112	C₄₃⋊C₁₆	—	2	2	no	—
113	C₄₃⋊C₁₆	—	2	2	no	—
114	C₂³×C₄₃	—	4	2	no	—
115	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	4	4	no	—
118	C₂²×D₄₃	—	16	2	no	—
119	D₄×C₄₃	—	4	2	no	—
120	D₄×C₄₃	—	4	2	no	—
121	D₄×C₄₃	—	16	2	no	—
122	D₄×C₄₃	—	16	2	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	16	2	no	—
127	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	—	4	2	no	—
132	D₁₇₂	—	2	4	no	—
133	D₁₇₂	—	4	4	no	—
134	D₁₇₂	—	4	2	no	—
135	C₄×D₄₃	—	16	2	no	—
136	C₄×D₄₃	—	16	2	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	2	1	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	C₂×#41	1	2	yes	—
141	#33682	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	—	8	2	no	—
145	C₂²×C₄₃	—	8	2	no	—
146	C₂²×C₄₃	—	8	2	no	—
147	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₄₃	—	4	4	no	—
149	C₂×D₄₃	—	8	2	no	—
150	C₂×D₄₃	—	8	2	no	—
151	C₂×D₄₃	—	16	4	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	#525	1	1	yes	characteristic
157	C₄×C₄₃	#575	1	1	yes	characteristic
158	Dic₄₃	—	8	2	no	—
159	#3176	D₄₃	1	1	yes	characteristic, complemented
160	#2948	—	43	1	no	—
161	#2234	—	43	2	no	—
162	#2228	—	43	2	no	—
163	#2206	—	43	2	no	—
164	#2184	—	43	2	no	—
165	#2177	—	43	2	no	—
166	C₄×D₁₆	—	43	2	no	—
167	#2164	—	43	1	no	—
168	C₂×C₄₃	—	4	1	no	—
169	C₂×C₄₃	—	16	2	no	—
170	C₂×C₄₃	#3100	1	1	yes	characteristic
171	C₂×C₄₃	#3415	1	2	yes	—
172	D₄₃	—	8	4	no	—
173	C₂²×D₈	—	43	1	no	—
174	#540	—	2	1	no	—
175	#518	—	86	1	no	—
176	C₂×D₁₆	—	43	2	no	—
177	C₂×D₁₆	—	43	2	no	—
178	#499	C₂×D₄₃	1	1	yes	characteristic, complemented
179	#493	C₂×D₄₃	1	1	yes	characteristic, complemented
180	#490	—	2	1	no	—
181	#469	—	2	1	no	—
182	#466	C₂×D₄₃	1	2	yes	—
183	#442	—	2	1	no	—
184	#442	—	86	1	no	—
185	#442	—	86	1	no	—
186	C₄×D₈	—	43	2	no	—
187	C₄×D₈	—	43	2	no	—
188	C₄×D₈	C₂×D₄₃	1	2	yes	complemented
189	#427	—	43	1	no	—
190	#418	—	86	1	no	—
191	#405	C₂×D₄₃	1	1	yes	characteristic
192	#367	—	43	2	no	—
193	#366	—	43	2	no	—
194	C₄⋊C₁₆	—	43	2	no	—
195	#357	—	43	2	no	—
196	#357	—	43	2	no	—
197	#357	—	43	2	no	—
198	#357	—	43	2	no	—
199	C₂²⋊C₁₆	—	43	2	no	—
200	#346	—	43	1	no	—
201	C₄×C₁₆	—	43	1	no	—
202	C₄₃	#33682	1	1	yes	characteristic, complemented, Sylow
203	C₂²×D₄	—	86	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	43	2	no	—
206	C₂×D₈	—	86	4	no	—
207	C₂×D₈	—	86	1	no	—
208	C₂×D₈	C₂²×D₄₃	1	2	yes	—
209	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
211	#127	—	2	1	no	—
212	#125	—	2	1	no	—
213	#123	—	4	2	no	—
214	#121	—	2	2	no	—
215	#121	—	4	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	86	2	no	—
218	C₄×D₄	—	86	2	no	—
219	C₄×D₄	—	172	2	no	—
220	#117	—	86	1	no	—
221	#117	C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	172	2	no	—
223	D₁₆	—	86	4	no	—
224	D₁₆	—	86	4	no	—
225	C₂×C₁₆	—	43	2	no	—
226	C₂×C₁₆	—	43	2	no	—
227	#107	—	43	2	no	—
228	#107	C₂²×D₄₃	1	2	yes	—
229	#106	—	2	1	no	—
230	#106	—	86	1	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	86	2	no	—
234	#102	—	86	2	no	—
235	#97	C₂²⋊D₄₃	1	1	yes	characteristic
236	#97	C₂²⋊D₄₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₂⁴	—	172	1	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	86	4	no	—
244	C₂×D₄	—	86	2	no	—
245	C₂×D₄	—	172	2	no	—
246	C₂×D₄	—	172	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	172	2	no	—
249	C₂²×C₄	—	172	2	no	—
250	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	D₈	—	2	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	4	2	no	—
254	D₈	—	43	4	no	—
255	D₈	—	86	4	no	—
256	D₈	—	86	2	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	86	2	no	—
266	C₄⋊C₄	#10621882	1	2	yes	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	172	1	no	—
270	C₂²⋊C₄	—	172	2	no	—
271	C₂²⋊C₄	—	172	2	no	—
272	C₂²⋊C₄	#10621882	1	2	yes	—
273	C₄²	—	172	1	no	—
274	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₁₆	—	86	2	no	—
277	C₁₆	—	86	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	172	4	no	—
280	C₂³	—	172	2	no	—
281	C₂³	—	688	2	no	—
282	C₂³	C₂×#10621882	1	1	yes	characteristic
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	16	2	no	—
286	D₄	—	16	2	no	—
287	D₄	—	86	4	no	—
288	D₄	—	172	4	no	—
289	D₄	—	172	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	16	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	688	2	no	—
296	C₂×C₄	—	688	2	no	—
297	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
298	C₂×C₄	C₂×#10621882	1	2	yes	—
299	C₂×C₄	C₂×#10621882	1	2	yes	—
300	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	#41×D₄₃	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	344	2	no	—
311	C₂²	—	344	2	no	—
312	C₂²	—	172	4	no	—
313	C₂²	—	688	4	no	—
314	C₂²	C₂²×#10621882	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	344	2	no	—
320	C₄	#14583406	1	1	yes	characteristic
321	C₄	#14583697	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	16	2	no	—
324	C₂	—	344	4	no	—
325	C₂	#19144591	1	1	yes	characteristic
326	C₂	#19147734	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3176	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3176	D₄₃
hypocenter	C₄₃	#33682
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #19143448	C₂	semidirect
2. #19136783	C₂	semidirect
3. #19136749	C₂	semidirect
4. #19136627	C₂	semidirect
5. #19136506	C₂	semidirect
6. #19136469	C₂	semidirect
7. C₄×#13063266	C₂	semidirect
8. #19136402	C₂	semidirect
9. C₄₃×#3176	C₂	semidirect
20. #14582726	C₂²	semidirect
21. #14582724	C₂²	semidirect
22. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
23. #14582700	C₂²	semidirect
24. #14582700	C₂²	semidirect
28. #14582675	C₂²	semidirect
29. C₄×C₄₃⋊C₁₆	C₂²	semidirect
31. C₄₃×#499	C₂²	semidirect
32. C₄₃×#493	C₂²	semidirect
37. C₄×D₈×C₄₃	C₂²	semidirect
159. #3176	D₄₃	semidirect
178. #499	C₂×D₄₃	semidirect
179. #493	C₂×D₄₃	semidirect
188. C₄×D₈	C₂×D₄₃	semidirect
202. C₄₃	#33682	semidirect

Maximal subgroups

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

subgroup	index
#19143448	2
#19136783	2
#19136749	2
#19136627	2
#19136506	2
#19136469	2
C₄×#13063266	2
#19136402	2
C₄₃×#3176	2
#33682	43