Finite Group Databasebeta

← #18005372

Subgroups — #18005372

Order 4736.

420 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14558043C211yesmaximal, characteristic
2C22⋊Q8×D37C211yesmaximal, characteristic, complemented
3#14557877C211yesmaximal, characteristic, complemented
4#14557873C211yesmaximal, characteristic, complemented
5#14557865C211yesmaximal, characteristic, complemented
6#14557845C211yesmaximal, characteristic, complemented
7#14557774C211yesmaximal, characteristic, complemented
8#14557772C211yesmaximal, characteristic, complemented
9#14557760C211yesmaximal, characteristic, complemented
10#14557696C211yesmaximal, characteristic, complemented
11#14557692C211yesmaximal, characteristic, complemented
12#14557690C211yesmaximal, characteristic, complemented
13#14557670C211yesmaximal, characteristic, complemented
14C2×#11924448C211yesmaximal, characteristic, complemented
15C37×#465C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic
17C2×Q8×D37C2211yescharacteristic
18C22×C4×D37C2211yescharacteristic
19C2×Dic148C2211yescharacteristic, complemented
20C2×#10596942C2211yescharacteristic, complemented
21C2×#10596940C2211yescharacteristic, complemented
22C22×#64436C2211yescharacteristic
23#11924518—21no—
24#11924516C2211yescharacteristic
25#11924516C2211yescharacteristic
26#11924515C2211yescharacteristic
27#11924515C2211yescharacteristic
28#11924514—21no—
29#11924513—21no—
30#11924512C2211yescharacteristic
31#11924507C2211yescharacteristic
32#11924504—21no—
33D4×Dic37—21no—
34#11924496—21no—
35#11924494C2211yescharacteristic
36C4⋊C4×D37—21no—
37C4⋊C4×D37C2211yescharacteristic
38C22⋊C4×D37—21no—
39#11924470C2211yescharacteristic, complemented
40#11924469C2211yescharacteristic, complemented
41#11924467C2211yescharacteristic, complemented
42#11924461C2211yescharacteristic, complemented
43#11924461C2211yescharacteristic, complemented
44#11924460C2211yescharacteristic, complemented
45#11924459C2211yescharacteristic, complemented
46#11924459C2211yescharacteristic, complemented
47#11924457C2211yescharacteristic, complemented
48#11924448C2211yescharacteristic
49#11924448C2211yescharacteristic
50#11924448C2211yescharacteristic
51#11924448C2211yescharacteristic
52C2×SD16×C37C2211yescharacteristic, complemented
53C22×C8×C37C2211yescharacteristic
54C22⋊Q8×C37C2211yescharacteristic
55#121×C37C2211yescharacteristic
56#106×C37C2211yescharacteristic, complemented
57#103×C37C2211yescharacteristic, complemented
58#102×C37C2211yescharacteristic, complemented
59C23×D37D411yescharacteristic
60C2×Q8×C37C2311yescharacteristic
61C2×D4×C37—21no—
62C2×D4×C37C2311yescharacteristic
63C22×C4×C37C2311yescharacteristic
64C2×C22⋊D37—21no—
65C2×C22⋊D37—21no—
66C22×Dic37—21no—
67C22×Dic37D411yescharacteristic
68Q8×D37—41no—
69Q8×D37—41no—
70#10596964—41no—
71#10596964—41no—
72C2×C4×D37—21no—
73C2×C4×D37—21no—
74C2×C4×D37—41no—
75C2×C4×D37—41no—
76C2×C4×D37D411yescharacteristic
77C2×C4×D37D411yescharacteristic
78C2×C4×D37C2311yescharacteristic
79C2×C4×D37C2311yescharacteristic
80C2×Dic74—21no—
81C2×Dic74C2311yescharacteristic
82C2×Dic74C2311yescharacteristic
83SD16×C37—41no—
84C2×C8×C37—41no—
85C2×C8×C37C2311yescharacteristic
86C2×C8×C37C2311yescharacteristic
87C4⋊C4×C37—21no—
88C4⋊C4×C37C2311yescharacteristic
89C4⋊C4×C37C2311yescharacteristic
90C22⋊C4×C37—21no—
91C22⋊C4×C37—21no—
92C22⋊Dic37—21no—
93C22⋊Dic37—21no—
94C22⋊Dic37—21no—
95#10596942—41no—
96#10596940—41no—
97#10596938—21no—
98#10596938—21no—
99#10596938—21no—
100#10596938—21no—
101C4⋊Dic37—21no—
102C4⋊Dic37—21no—
103C4⋊Dic37C2311yescharacteristic
104C4⋊Dic37C2311yescharacteristic
105#10596936—21no—
106#10596936—21no—
107#10596936—21no—
108#10596936—21no—
109C4×Dic37—21no—
110C2×#64436—41no—
111C2×#64436C2311yescharacteristic
112C2×#64436C2311yescharacteristic
113Dic148—41no—
114C23×C37—21no—
115C23×C37C2×D411yescharacteristic
116C22×D37—22no—
117C22×D37—22no—
118C22×D37—41no—
119C22×D37C2×D411yescharacteristic
120C22×D37C2×D411yescharacteristic
121Q8×C37—21no—
122Q8×C37—21no—
123D4×C37—21no—
124D4×C37—21no—
125D4×C37—42no—
126C2×C4×C37—21no—
127C2×C4×C37—21no—
128C2×C4×C37—21no—
129C2×C4×C37—21no—
130C2×C4×C37—21no—
131C2×C4×C37C2×D411yescharacteristic
132C2×C4×C37C2411yescharacteristic
133C22⋊D37—42no—
134C22⋊D37—42no—
135C2×Dic37—21no—
136C2×Dic37—21no—
137C2×Dic37—21no—
138C2×Dic37—21no—
139C2×Dic37—41no—
140C2×Dic37—42no—
141C2×Dic37C2×D411yescharacteristic
142C2×Dic37C2×D411yescharacteristic
143C4×D37—21no—
144C4×D37—21no—
145C4×D37—21no—
146C4×D37—21no—
147C4×D37—41no—
148C4×D37—41no—
149C4×D37—42no—
150C4×D37—42no—
151Dic74—21no—
152Dic74—21no—
153Dic74—21no—
154Dic74—21no—
155Dic74—42no—
156C8×C37—21no—
157C8×C37—21no—
158#64436—21no—
159#64436—21no—
160C22×C37—21no—
161C22×C37—21no—
162C22×C37—41no—
163C22×C37—41no—
164C22×C37—41no—
165C22×C37#13712yes—
166C22×C37C22×D411yescharacteristic
167C2×D37—21no—
168C2×D37—21no—
169C2×D37—21no—
170C2×D37—21no—
171C2×D37—21no—
172C2×D37—44no—
173C2×D37#13512yes—
174C4×C37—21no—
175C4×C37—41no—
176C4×C37—41no—
177C4×C37—41no—
178C4×C37H(5,𝔽2)11yescharacteristic
179C4×C37H(5,𝔽2)11yescharacteristic
180Dic37—21no—
181Dic37—21no—
182Dic37—41no—
183Dic37—41no—
184Dic37—41no—
185Dic37—41no—
186#3190—371nomaximal, Sylow
187C2×C37—22no—
188C2×C37—81no—
189C2×C37#53411yescharacteristic
190C2×C37C2×#13511yescharacteristic
191C2×C37C2×#13711yescharacteristic
192D37—22no—
193D37—41no—
194#525—371no—
195C2×C22⋊Q8—371no—
196#485—371no—
197#485—371no—
198#483—371no—
199#480—371no—
200#467—371no—
201#467—371no—
202#465—371no—
203#465D3711yescharacteristic, complemented
204#452—371no—
205#452—371no—
206#451—371no—
207#448—371no—
208C2×C22⋊C8—371no—
209C37#319011yescharacteristic, complemented, Sylow
210C2×#41—371no—
211C22×Q8—371no—
212C23×C4—371no—
213C2×Q16—371no—
214C2×Q16—371no—
215C2×SD16—371no—
216C2×SD16C2×D3711yescharacteristic, complemented
217C22×C8—371no—
218C22×C8C2×D3711yescharacteristic
219#123—741no—
220C22⋊Q8—371no—
221C22⋊Q8—371no—
222C22⋊Q8—371no—
223C22⋊Q8—371no—
224C22⋊Q8—371no—
225C22⋊Q8—741no—
226C22⋊Q8—741no—
227C22⋊Q8C2×D3711yescharacteristic
228#121—371no—
229#121C2×D3711yescharacteristic
230#120—741no—
231C4×D4—741no—
232C4×D4—741no—
233#117—371no—
234C2×C4⋊C4—371no—
235C2×C4⋊C4—741no—
236C2×C22⋊C4—741no—
237#107—371no—
238#107—371no—
239#106—371no—
240#106C2×D3711yescharacteristic, complemented
241#103—371no—
242#103—371no—
243#103—371no—
244#103—371no—
245#103—371no—
246#103C2×D3711yescharacteristic, complemented
247#102—371no—
248#102C2×D3711yescharacteristic, complemented
249C22⋊C8—371no—
250C22⋊C8—371no—
251C22⋊C8—371no—
252C22⋊C8—371no—
253C24—371no—
254#41—1481no—
255#41—1481no—
256C2×Q8—371no—
257C2×Q8—371no—
258C2×Q8—741no—
259C2×Q8—1481no—
260C2×Q8—1481no—
261C2×Q8C22×D3711yescharacteristic
262C2×D4—21no—
263C2×D4—741no—
264C2×D4—741no—
265C2×D4C22×D3711yescharacteristic
266C22×C4—371no—
267C22×C4—371no—
268C22×C4—371no—
269C22×C4—371no—
270C22×C4—371no—
271C22×C4—741no—
272C22×C4—741no—
273C22×C4—741no—
274C22×C4—1481no—
275C22×C4—1481no—
276C22×C4C22×D3711yescharacteristic
277Q16—1481no—
278Q16—1481no—
279SD16—41no—
280SD16—1481no—
281C2×C8—41no—
282C2×C8—371no—
283C2×C8—371no—
284C2×C8—1481no—
285C2×C8C22×D3711yescharacteristic
286C2×C8C22×D3711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—371no—
289C4⋊C4—371no—
290C4⋊C4—741no—
291C4⋊C4—741no—
292C4⋊C4—741no—
293C4⋊C4—741no—
294C4⋊C4—741no—
295C4⋊C4—741no—
296C4⋊C4C22×D3711yescharacteristic
297C4⋊C4C22×D3711yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—741no—
301C22⋊C4—741no—
302C22⋊C4—741no—
303C22⋊C4—741no—
304C22⋊C4—741no—
305C22⋊C4—741no—
306C22⋊C4—741no—
307C42—741no—
308C23—21no—
309C23—371no—
310C23—371no—
311C23—742no—
312C23—742no—
313C23—1481no—
314C23D4×D3711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—741no—
318Q8—741no—
319Q8—741no—
320Q8—741no—
321Q8—1482no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—1482no—
326D4—1482no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—371no—
333C2×C4—371no—
334C2×C4—741no—
335C2×C4—741no—
336C2×C4—741no—
337C2×C4—741no—
338C2×C4—741no—
339C2×C4—741no—
340C2×C4—741no—
341C2×C4—741no—
342C2×C4—1481no—
343C2×C4—1481no—
344C2×C4—1482no—
345C2×C4—1482no—
346C2×C4—1481no—
347C2×C4—1482no—
348C2×C4D4×D3711yescharacteristic
349C2×C4C23×D3711yescharacteristic
350C8—21no—
351C8—21no—
352C8—741no—
353C8—741no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—372no—
360C22—741no—
361C22—741no—
362C22—741no—
363C22—741no—
364C22—741no—
365C22—1484no—
366C22#1192456512yes—
367C22C2×D4×D3711yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—741no—
373C4—741no—
374C4—1481no—
375C4—1481no—
376C4—1481no—
377C4—1481no—
378C4#1192458111yescharacteristic
379C4#1192458211yescharacteristic
380C2—22no—
381C2—81no—
382C2—742no—
383C2—1481no—
384C2#1455809111yescharacteristic
385C2#135×D3711yescharacteristic
386C2C2×#1192456511yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#465C2
socleC22×C37C22×D4
prime radical#14558043C2
hypercenter#465D37
hypocenterC37#3190
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D37C2semidirect
3. #14557877C2semidirect
4. #14557873C2semidirect
5. #14557865C2semidirect
6. #14557845C2semidirect
7. #14557774C2semidirect
8. #14557772C2semidirect
9. #14557760C2semidirect
10. #14557696C2semidirect
11. #14557692C2semidirect
12. #14557690C2semidirect
13. #14557670C2semidirect
14. C2×#11924448C2semidirect
15. C37×#465C2semidirect
19. C2×Dic148C22semidirect
20. C2×#10596942C22semidirect
21. C2×#10596940C22semidirect
39. #11924470C22semidirect
40. #11924469C22semidirect
41. #11924467C22semidirect
42. #11924461C22semidirect
43. #11924461C22semidirect
44. #11924460C22semidirect
45. #11924459C22semidirect
46. #11924459C22semidirect
47. #11924457C22semidirect
52. C2×SD16×C37C22semidirect
56. #106×C37C22semidirect
57. #103×C37C22semidirect
58. #102×C37C22semidirect
203. #465D37semidirect
209. C37#3190semidirect
216. C2×SD16C2×D37semidirect
240. #106C2×D37semidirect
246. #103C2×D37semidirect
248. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580432
C22⋊Q8×D372
#145578772
#145578732
#145578652
#145578452
#145577742
#145577722
#145577602
#145576962
#145576922
#145576902
#145576702
C2×#119244482
C37×#4652
#319037