Finite Group Databasebeta

← #18003868

Subgroups — #18003868

Order 4736.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924581C211yesmaximal, characteristic
2C2×#11924565C211yesmaximal, characteristic, complemented
3C2×SD16×D37C211yesmaximal, characteristic, complemented
4#14558106C211yesmaximal, characteristic, complemented
5#14557697C211yesmaximal, characteristic, complemented
6#14557696C211yesmaximal, characteristic, complemented
7#14557693C211yesmaximal, characteristic, complemented
8#14557692C211yesmaximal, characteristic, complemented
9#14557685C211yesmaximal, characteristic, complemented
10#14557684C211yesmaximal, characteristic, complemented
11#14557682C211yesmaximal, characteristic, complemented
12#14557536C211yesmaximal, characteristic, complemented
13#14557526C211yesmaximal, characteristic, complemented
14#14557498C211yesmaximal, characteristic, complemented
15C37×#450C211yesmaximal, characteristic, complemented
16#11924581—24no—
17#11924581—24no—
18C2×#10596962C2211yescharacteristic
19C2×#10596962C2211yescharacteristic
20C2×#10596964—21no—
21C2×#10596964C2211yescharacteristic
22C2×#10596964C2211yescharacteristic
23C2×Q8×D37C2211yescharacteristic, complemented
24C22×C22⋊D37—21no—
25C22×C22⋊D37—21no—
26C2×D4×D37—21no—
27C2×D4×D37C2211yescharacteristic
28C2×D4×D37C2211yescharacteristic
29#11924565—24no—
30C2×Dic148C2211yescharacteristic, complemented
31C2×#10596942C2211yescharacteristic, complemented
32C2×#10596930C2211yescharacteristic, complemented
33C2×#10596940C2211yescharacteristic, complemented
34C2×#10596940C2211yescharacteristic, complemented
35C2×#10596941C2211yescharacteristic, complemented
36SD16×D37—24no—
37C2×#10596929C2211yescharacteristic, complemented
38C2×C8×D37C2211yescharacteristic, complemented
39#11924535—21no—
40#11924522—21no—
41#11924521—21no—
42#11924518—21no—
43#11924516C2211yescharacteristic, complemented
44#11924515C2211yescharacteristic, complemented
45#11924512C2211yescharacteristic, complemented
46#11924494C2211yescharacteristic, complemented
47#11924461C2211yescharacteristic, complemented
48#11924459C2211yescharacteristic, complemented
49#11924458C2211yescharacteristic, complemented
50#11924457C2211yescharacteristic, complemented
51#11924457C2211yescharacteristic, complemented
52#11924456C2211yescharacteristic, complemented
53C22⋊#64436C2211yescharacteristic, complemented
54#11924448C2211yescharacteristic, complemented
55#11924448C2211yescharacteristic, complemented
56C22×D4×C37C2211yescharacteristic
57C2×SD16×C37C2211yescharacteristic, complemented
58C2×SD16×C37C2211yescharacteristic, complemented
59C22⋊Q8×C37C2211yescharacteristic, complemented
60#102×C37C2211yescharacteristic, complemented
61#102×C37C2211yescharacteristic, complemented
62C22⋊C8×C37C2211yescharacteristic, complemented
63C24×C37—21no—
64C23×D37—21no—
65C23×D37—21no—
66C2×Q8×C37C2311yescharacteristic
67C2×D4×C37—22no—
68C2×D4×C37—22no—
69C2×D4×C37—21no—
70C2×D4×C37C2311yescharacteristic
71C2×D4×C37C2311yescharacteristic
72C22×C4×C37C2311yescharacteristic
73C2×C22⋊D37—28no—
74C2×C22⋊D37—28no—
75C2×C22⋊D37—21no—
76C2×C22⋊D37—21no—
77C2×C22⋊D37—21no—
78C2×C22⋊D37—21no—
79C2×C22⋊D37—21no—
80C2×C22⋊D37D411yescharacteristic
81C2×C22⋊D37D411yescharacteristic
82C22×Dic37—21no—
83C22×Dic37—21no—
84Q8×D37—22no—
85Q8×D37—22no—
86#10596964—28no—
87#10596964—22no—
88#10596964—22no—
89#10596964—22no—
90#10596964D414yes—
91D4×D37—28no—
92D4×D37—22no—
93D4×D37—22no—
94D4×D37—22no—
95D4×D37D414yes—
96#10596962—22no—
97#10596962—22no—
98#10596962—22no—
99#10596962—22no—
100C2×D148D411yescharacteristic
101C2×D148C2311yescharacteristic
102C2×C4×D37—21no—
103C2×C4×D37—21no—
104C2×C4×D37C2311yescharacteristic
105C2×C4×D37C2311yescharacteristic
106C2×Dic74—21no—
107C2×Dic74D411yescharacteristic
108C2×Dic74C2311yescharacteristic
109C2×Dic74C2311yescharacteristic
110SD16×C37—22no—
111SD16×C37—22no—
112C2×C8×C37C2311yescharacteristic
113C2×C8×C37C2311yescharacteristic
114C4⋊C4×C37—21no—
115C4⋊C4×C37C2311yescharacteristic
116C22⋊C4×C37—21no—
117C22⋊Dic37—21no—
118#10596942—22no—
119#10596941—22no—
120#10596940—22no—
121#10596940—22no—
122#10596938—21no—
123#10596938—21no—
124#10596938—21no—
125C4⋊Dic37—21no—
126C4⋊Dic37C2311yescharacteristic
127#10596936—21no—
128#10596936—21no—
129C4×Dic37—21no—
130C2×#64436C2311yescharacteristic
131C2×#64436C2311yescharacteristic
132Dic148—22no—
133#10596930—22no—
134#10596929—22no—
135C8×D37—22no—
136C23×C37—24no—
137C23×C37—21no—
138C23×C37—21no—
139C23×C37—81no—
140C23×C37—81no—
141C23×C37C2×D411yescharacteristic
142C22×D37—21no—
143C22×D37—24no—
144C22×D37—24no—
145C22×D37—81no—
146C22×D37—81no—
147C22×D37—81no—
148C22×D37—81no—
149C22×D37C2×D411yescharacteristic
150C22×D37C2×D411yescharacteristic
151Q8×C37—21no—
152Q8×C37—21no—
153D4×C37—24no—
154D4×C37—21no—
155D4×C37—21no—
156D4×C37C2×D412yes—
157D4×C37C2×D412yes—
158C2×C4×C37—21no—
159C2×C4×C37—21no—
160C2×C4×C37—21no—
161C2×C4×C37—21no—
162C2×C4×C37C2×D411yescharacteristic
163C2×C4×C37C2411yescharacteristic
164C22⋊D37—24no—
165C22⋊D37—24no—
166C22⋊D37—24no—
167C22⋊D37—22no—
168C22⋊D37—24no—
169C22⋊D37—22no—
170C22⋊D37—24no—
171C2×Dic37—24no—
172C2×Dic37—21no—
173C2×Dic37—24no—
174C2×Dic37—21no—
175C2×Dic37—21no—
176C2×Dic37C2×D411yescharacteristic
177C2×Dic37C2×D411yescharacteristic
178D148—22no—
179D148—21no—
180D148C2×D412yes—
181C4×D37—24no—
182C4×D37—21no—
183C4×D37—21no—
184C4×D37—42no—
185C4×D37C2×D412yes—
186C4×D37C2×D412yes—
187Dic74—21no—
188Dic74—22no—
189Dic74—21no—
190Dic74—21no—
191Dic74—42no—
192Dic74C2×D412yes—
193C8×C37—21no—
194C8×C37—21no—
195#64436—21no—
196#64436—21no—
197C22×C37—22no—
198C22×C37—22no—
199C22×C37—21no—
200C22×C37—21no—
201C22×C37—21no—
202C22×C37—41no—
203C22×C37—41no—
204C22×C37—41no—
205C22×C37—41no—
206C22×C37—84no—
207C22×C37C22×D411yescharacteristic
208C2×D37—21no—
209C2×D37—21no—
210C2×D37—21no—
211C2×D37—22no—
212C2×D37—21no—
213C2×D37—21no—
214C2×D37—41no—
215C2×D37—41no—
216C2×D37—84no—
217C2×D37—84no—
218C2×D37#12012yes—
219C4×C37—21no—
220C4×C37—41no—
221C4×C37—41no—
222C4×C37C22×D411yescharacteristic
223C4×C37C22×D411yescharacteristic
224Dic37—21no—
225Dic37—22no—
226Dic37—41no—
227Dic37—41no—
228Dic37#12012yes—
229#3009—371nomaximal, Sylow
230C2×C37—42no—
231C2×C37—42no—
232C2×C37—41no—
233C2×C37C2×#12011yescharacteristic
234C2×C37#57711yescharacteristic
235C2×C37#57711yescharacteristic
236D37—22no—
237D37—41no—
238D37—42no—
239C2×H(5,𝔽2)—371no—
240C2×#137—371no—
241C22×SD16—371no—
242#536—371no—
243#452—371no—
244#452—371no—
245#452—371no—
246#452—371no—
247#450—371no—
248#450—371no—
249#450—371no—
250#450D3711yescharacteristic, complemented
251#418—371no—
252#416—371no—
253#408—371no—
254C37#300911yescharacteristic, complemented, Sylow
255H(5,𝔽2)—744no—
256H(5,𝔽2)—744no—
257C2×#41—371no—
258C2×#41—371no—
259C2×#41—371no—
260C2×#41—371no—
261C2×#41—741no—
262C22×Q8—371no—
263C22×D4—371no—
264C22×D4—371no—
265C22×D4—741no—
266C22×D4—741no—
267C22×D4—741no—
268C22×D4C2×D3711yescharacteristic
269#137—744no—
270C2×Q16—371no—
271C2×Q16—371no—
272C2×SD16—371no—
273C2×SD16—371no—
274C2×SD16—371no—
275C2×SD16—371no—
276C2×SD16—744no—
277C2×SD16C2×D3711yescharacteristic, complemented
278C2×SD16C2×D3711yescharacteristic, complemented
279C2×M16—371no—
280C22×C8—371no—
281C4⋊Q8—741no—
282#124—741no—
283#123—741no—
284#123—741no—
285C22⋊Q8—371no—
286C22⋊Q8—371no—
287C22⋊Q8—371no—
288C22⋊Q8C2×D3711yescharacteristic, complemented
289#117—371no—
290#103—371no—
291#103—371no—
292#102—371no—
293#102—371no—
294#102—371no—
295#102—371no—
296#102C2×D3711yescharacteristic, complemented
297#102C2×D3711yescharacteristic, complemented
298C22⋊C8—371no—
299C22⋊C8—371no—
300C22⋊C8—371no—
301C22⋊C8C2×D3711yescharacteristic, complemented
302C24—21no—
303C24—741no—
304C24—741no—
305#41—374no—
306#41—742no—
307#41—742no—
308#41—742no—
309#41—742no—
310#41—742no—
311#41—742no—
312#41—748no—
313#41—742no—
314C2×Q8—371no—
315C2×Q8—371no—
316C2×Q8—371no—
317C2×Q8—741no—
318C2×Q8—742no—
319C2×Q8—742no—
320C2×Q8C22×D3711yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—374no—
325C2×D4—371no—
326C2×D4—371no—
327C2×D4—371no—
328C2×D4—371no—
329C2×D4—742no—
330C2×D4—741no—
331C2×D4—741no—
332C2×D4—742no—
333C2×D4—748no—
334C2×D4—741no—
335C2×D4—741no—
336C2×D4—748no—
337C2×D4—748no—
338C2×D4—741no—
339C2×D4—742no—
340C2×D4C22×D3711yescharacteristic
341C2×D4C22×D3711yescharacteristic
342C22×C4—371no—
343C22×C4—371no—
344C22×C4—741no—
345C22×C4—741no—
346C22×C4—741no—
347C22×C4—741no—
348C22×C4C22×D3711yescharacteristic
349Q16—742no—
350Q16—742no—
351SD16—22no—
352SD16—22no—
353SD16—742no—
354SD16—742no—
355SD16—742no—
356SD16—742no—
357M16—742no—
358C2×C8—371no—
359C2×C8—371no—
360C2×C8—742no—
361C2×C8C22×D3711yescharacteristic
362C2×C8C22×D3711yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—371no—
365C4⋊C4—741no—
366C4⋊C4—741no—
367C4⋊C4—741no—
368C4⋊C4C22×D3711yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—741no—
371C22⋊C4—741no—
372C22⋊C4—741no—
373C22⋊C4—741no—
374C42—741no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—371no—
381C23—371no—
382C23—2961no—
383C23—2961no—
384C23—2961no—
385C23—2961no—
386C23—744no—
387C23—744no—
388C23—741no—
389C23D4×D3711yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—372no—
393Q8—742no—
394Q8—741no—
395Q8—741no—
396Q8—741no—
397Q8—1482no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—372no—
402D4—742no—
403D4—742no—
404D4—744no—
405D4—742no—
406D4—744no—
407D4—744no—
408D4—744no—
409D4—744no—
410D4—741no—
411D4D4×D3712yes—
412D4D4×D3712yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—371no—
418C2×C4—372no—
419C2×C4—372no—
420C2×C4—371no—
421C2×C4—744no—
422C2×C4—744no—
423C2×C4—744no—
424C2×C4—741no—
425C2×C4—741no—
426C2×C4—741no—
427C2×C4—741no—
428C2×C4—741no—
429C2×C4—1482no—
430C2×C4D4×D3711yescharacteristic
431C2×C4C23×D3711yescharacteristic
432C8—21no—
433C8—21no—
434C8—741no—
435C8—741no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—372no—
447C22—2964no—
448C22—2964no—
449C22—741no—
450C22—741no—
451C22—742no—
452C22—741no—
453C22—741no—
454C22—741no—
455C22—1481no—
456C22—1481no—
457C22C2×D4×D3711yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—372no—
462C4—741no—
463C4—742no—
464C4—1481no—
465C4—1481no—
466C4C2×D4×D3711yescharacteristic
467C4C2×D4×D3711yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—742no—
472C2—1481no—
473C2—1482no—
474C2#120×D3711yescharacteristic
475C2#1455833911yescharacteristic
476C2#1455834311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#450C2
socleC22×C37C22×D4
prime radicalC2×#11924581C2
hypercenter#450D37
hypocenterC37#3009
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#11924565C2semidirect
3. C2×SD16×D37C2semidirect
4. #14558106C2semidirect
5. #14557697C2semidirect
6. #14557696C2semidirect
7. #14557693C2semidirect
8. #14557692C2semidirect
9. #14557685C2semidirect
10. #14557684C2semidirect
11. #14557682C2semidirect
12. #14557536C2semidirect
13. #14557526C2semidirect
14. #14557498C2semidirect
15. C37×#450C2semidirect
23. C2×Q8×D37C22semidirect
30. C2×Dic148C22semidirect
31. C2×#10596942C22semidirect
32. C2×#10596930C22semidirect
33. C2×#10596940C22semidirect
34. C2×#10596940C22semidirect
35. C2×#10596941C22semidirect
37. C2×#10596929C22semidirect
38. C2×C8×D37C22semidirect
43. #11924516C22semidirect
44. #11924515C22semidirect
45. #11924512C22semidirect
46. #11924494C22semidirect
47. #11924461C22semidirect
48. #11924459C22semidirect
49. #11924458C22semidirect
50. #11924457C22semidirect
51. #11924457C22semidirect
52. #11924456C22semidirect
53. C22⋊#64436C22semidirect
54. #11924448C22semidirect
55. #11924448C22semidirect
57. C2×SD16×C37C22semidirect
58. C2×SD16×C37C22semidirect
59. C22⋊Q8×C37C22semidirect
60. #102×C37C22semidirect
61. #102×C37C22semidirect
62. C22⋊C8×C37C22semidirect
250. #450D37semidirect
254. C37#3009semidirect
277. C2×SD16C2×D37semidirect
278. C2×SD16C2×D37semidirect
288. C22⋊Q8C2×D37semidirect
296. #102C2×D37semidirect
297. #102C2×D37semidirect
301. C22⋊C8C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245812
C2×#119245652
C2×SD16×D372
#145581062
#145576972
#145576962
#145576932
#145576922
#145576852
#145576842
#145576822
#145575362
#145575262
#145574982
C37×#4502
#300937