Finite Group Databasebeta

← #17997252

Subgroups — #17997252

Order 4736.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924504C211yesmaximal, characteristic
2C2×C4×C22⋊D37C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D37C211yesmaximal, characteristic, complemented
4#14557436C211yesmaximal, characteristic, complemented
5#14557433C211yesmaximal, characteristic, complemented
6#14557425C211yesmaximal, characteristic, complemented
7#14557422C211yesmaximal, characteristic, complemented
8#14557421C211yesmaximal, characteristic, complemented
9#14557421C211yesmaximal, characteristic, complemented
10#14557412C211yesmaximal, characteristic, complemented
11#14557409C211yesmaximal, characteristic, complemented
12#14557407C211yesmaximal, characteristic, complemented
13#14557388C211yesmaximal, characteristic, complemented
14#14557380C211yesmaximal, characteristic, complemented
15C37×#386C211yesmaximal, characteristic, complemented
16C24×D37C2211yescharacteristic
17C22×C22⋊D37C2211yescharacteristic
18C22×C22⋊D37C2211yescharacteristic
19C22×C4×D37C2211yescharacteristic
20C22×C4×D37C2211yescharacteristic
21#11924504C418yescomplemented
22C4×C22⋊D37—24no—
23C2×C4⋊Dic37C2211yescharacteristic, complemented
24C2×#10596936C2211yescharacteristic, complemented
25C2×#10596936C2211yescharacteristic, complemented
26C2×C22⋊Dic37C2211yescharacteristic
27C2×C22⋊Dic37C2211yescharacteristic
28C2×C22⋊Dic37C2211yescharacteristic
29C2×C22⋊Dic37C2211yescharacteristic, complemented
30C2×C22⋊Dic37C2211yescharacteristic, complemented
31C2×#10596938C2211yescharacteristic
32C2×#10596938C2211yescharacteristic
33C2×#10596938C2211yescharacteristic
34C2×#10596938C2211yescharacteristic
35C2×#10596938C2211yescharacteristic
36C2×#10596938C2211yescharacteristic
37C2×#10596938C2211yescharacteristic
38C2×#10596938C2211yescharacteristic
39C22⋊C4×D37—24no—
40C2×C4×Dic37C2211yescharacteristic, complemented
41C2×C4×Dic37C2211yescharacteristic, complemented
42C2×C4×Dic37C2211yescharacteristic, complemented
43#11924443C2211yescharacteristic, complemented
44#11924443C2211yescharacteristic, complemented
45#11924443C2211yescharacteristic, complemented
46#11924443C2211yescharacteristic, complemented
47C22×D4×C37C2211yescharacteristic
48C23×C4×C37C2211yescharacteristic, complemented
49C2×C22⋊C4×C37C2211yescharacteristic
50C2×C22⋊C4×C37C2211yescharacteristic
51C2×C22⋊C4×C37C2211yescharacteristic, complemented
52#95×C37C2211yescharacteristic, complemented
53#95×C37C2211yescharacteristic, complemented
54C24×C37C2311yescharacteristic
55C24×C37C2311yescharacteristic
56C23×D37—22no—
57C23×D37—22no—
58C23×D37—41no—
59C23×D37—41no—
60C23×D37—41no—
61C23×D37—41no—
62C23×D37C2×C414yescomplemented
63C23×D37C2311yescharacteristic
64C23×D37C2311yescharacteristic
65C2×D4×C37—41no—
66C2×D4×C37—41no—
67C2×D4×C37C2×C414yescomplemented
68C22×C4×C37—22no—
69C22×C4×C37—41no—
70C22×C4×C37—41no—
71C22×C4×C37C2311yescharacteristic
72C22×C4×C37C2311yescharacteristic
73C22×C4×C37C2311yescharacteristic
74C22×C4×C37C2311yescharacteristic
75C22×C4×C37C2311yescharacteristic
76C2×C22⋊D37—22no—
77C2×C22⋊D37—22no—
78C2×C22⋊D37—41no—
79C2×C22⋊D37—41no—
80C2×C22⋊D37C2×C414yes—
81C2×C22⋊D37C2×C414yescomplemented
82C22×Dic37C2311yescharacteristic
83C22×Dic37C2311yescharacteristic
84C22×Dic37C2311yescharacteristic
85C22×Dic37C2311yescharacteristic
86C22×Dic37C2311yescharacteristic
87C22×Dic37C2311yescharacteristic
88C2×C4×D37—22no—
89C2×C4×D37—41no—
90C2×C4×D37—41no—
91C2×C4×D37—41no—
92C2×C4×D37—41no—
93C2×C4×D37D414yescomplemented
94C22⋊C4×C37—22no—
95C22⋊C4×C37—41no—
96C22⋊C4×C37—41no—
97C22⋊Dic37—22no—
98C22⋊Dic37—22no—
99C22⋊Dic37—41no—
100C22⋊Dic37—41no—
101C22⋊Dic37C2×C414yescomplemented
102#10596938—22no—
103#10596938—22no—
104#10596938—22no—
105#10596938—22no—
106#10596938—41no—
107#10596938—41no—
108#10596938C2×C414yescomplemented
109#10596938C2×C414yescomplemented
110C4⋊Dic37—41no—
111#10596936—22no—
112#10596936—41no—
113C4×Dic37—22no—
114C4×Dic37—41no—
115C4×Dic37—41no—
116C23×C37—21no—
117C23×C37—21no—
118C23×C37—21no—
119C23×C37—21no—
120C23×C37—21no—
121C23×C37—21no—
122C23×C37—21no—
123C23×C37—21no—
124C23×C37—21no—
125C23×C37—21no—
126C23×C37—21no—
127C23×C37—21no—
128C23×C37C22×C412yes—
129C23×C37C22×C412yes—
130C23×C37C2411yescharacteristic
131C22×D37—21no—
132C22×D37—21no—
133C22×D37—21no—
134C22×D37—21no—
135C22×D37—21no—
136C22×D37—21no—
137C22×D37—21no—
138C22×D37—21no—
139C22×D37—21no—
140C22×D37—21no—
141C22×D37—28no—
142C22×D37—44no—
143C22×D37—44no—
144C22×D37—44no—
145C22×D37—44no—
146C22×D37—44no—
147C22×D37—44no—
148C22×D37C22×C412yes—
149C22×D37C22×C412yes—
150C22×D37C2×D412yes—
151C22×D37#4112yes—
152D4×C37—44no—
153C2×C4×C37—21no—
154C2×C4×C37—21no—
155C2×C4×C37—21no—
156C2×C4×C37—21no—
157C2×C4×C37—21no—
158C2×C4×C37—21no—
159C2×C4×C37—21no—
160C2×C4×C37—21no—
161C2×C4×C37—21no—
162C2×C4×C37—21no—
163C2×C4×C37—21no—
164C2×C4×C37—21no—
165C2×C4×C37—21no—
166C2×C4×C37—44no—
167C2×C4×C37C22×C412yes—
168C2×C4×C37C2×D412yes—
169C22⋊D37—28no—
170C22⋊D37—44no—
171C2×Dic37—21no—
172C2×Dic37—21no—
173C2×Dic37—21no—
174C2×Dic37—21no—
175C2×Dic37—21no—
176C2×Dic37—21no—
177C2×Dic37—21no—
178C2×Dic37—21no—
179C2×Dic37—21no—
180C2×Dic37—21no—
181C2×Dic37—21no—
182C2×Dic37—21no—
183C2×Dic37—21no—
184C2×Dic37—21no—
185C2×Dic37—21no—
186C2×Dic37C22×C412yes—
187C2×Dic37C22×C412yes—
188C2×Dic37C2×D412yes—
189C4×D37—44no—
190C4×D37—44no—
191C22×C37—22no—
192C22×C37—22no—
193C22×C37—22no—
194C22×C37—22no—
195C22×C37—22no—
196C22×C37—22no—
197C22×C37—41no—
198C22×C37—41no—
199C22×C37—41no—
200C22×C37—41no—
201C22×C37—41no—
202C22×C37—41no—
203C22×C37—41no—
204C22×C37—41no—
205C22×C37C23×C411yescharacteristic
206C22×C37C22×D411yescharacteristic
207C22×C37C2×#4111yescharacteristic
208C22×C37H(5,𝔽2)11yescharacteristic
209C22×C37H(5,𝔽2)11yescharacteristic
210C22×C37H(5,𝔽2)11yescharacteristic
211C22×C37H(5,𝔽2)11yescharacteristic
212C2×D37—22no—
213C2×D37—22no—
214C2×D37—22no—
215C2×D37—22no—
216C2×D37—22no—
217C2×D37—22no—
218C2×D37—22no—
219C2×D37—22no—
220C2×D37—22no—
221C2×D37—416no—
222C2×D37—41no—
223C2×D37—41no—
224C2×D37—41no—
225C2×D37—41no—
226C2×D37C4×D414yes—
227C4×C37—22no—
228C4×C37—22no—
229C4×C37—41no—
230C4×C37—41no—
231C4×C37—41no—
232Dic37—22no—
233Dic37—22no—
234Dic37—22no—
235Dic37—41no—
236Dic37—41no—
237Dic37—41no—
238#2321—371nomaximal, Sylow
239C2×C37—42no—
240C2×C37—42no—
241C2×C37C2×C4×D411yescharacteristic
242C2×C37#51811yescharacteristic
243C2×C37#51811yescharacteristic
244C2×C37#53411yescharacteristic
245C2×C37#53511yescharacteristic
246C2×C37#53811yescharacteristic
247C2×C37#53811yescharacteristic
248D37—24no—
249D37—42no—
250C2×#120—371no—
251C2×C4×D4—371no—
252C22×C22⋊C4—371no—
253#390—371no—
254#390—371no—
255#388—371no—
256#388—371no—
257#388—371no—
258#388—371no—
259#386—371no—
260#386—371no—
261#386D3711yescharacteristic, complemented
262#385—371no—
263#379—371no—
264C4×C22⋊C4—371no—
265C37#232111yescharacteristic, complemented, Sylow
266C25—371no—
267C22×D4—371no—
268C22×D4—371no—
269C22×D4C2×D3711yescharacteristic
270C23×C4—371no—
271C23×C4—371no—
272C23×C4C2×D3711yescharacteristic, complemented
273#120—378no—
274C4×D4—744no—
275C2×C4⋊C4—371no—
276C2×C4⋊C4—371no—
277C2×C4⋊C4—371no—
278C2×C22⋊C4—371no—
279C2×C22⋊C4—371no—
280C2×C22⋊C4—371no—
281C2×C22⋊C4—371no—
282C2×C22⋊C4—371no—
283C2×C22⋊C4—371no—
284C2×C22⋊C4—371no—
285C2×C22⋊C4—371no—
286C2×C22⋊C4—371no—
287C2×C22⋊C4—371no—
288C2×C22⋊C4—371no—
289C2×C22⋊C4—371no—
290C2×C22⋊C4—371no—
291C2×C22⋊C4—744no—
292C2×C22⋊C4C2×D3711yescharacteristic
293C2×C22⋊C4C2×D3711yescharacteristic
294C2×C22⋊C4C2×D3711yescharacteristic, complemented
295C2×C42—371no—
296C2×C42—371no—
297C2×C42—371no—
298#95—371no—
299#95—371no—
300#95—371no—
301#95—371no—
302#95C2×D3711yescharacteristic, complemented
303#95C2×D3711yescharacteristic, complemented
304C24—371no—
305C24—371no—
306C24—374no—
307C24—742no—
308C24—742no—
309C24—1481no—
310C24—1481no—
311C24—1481no—
312C24—1481no—
313C24C22×D3711yescharacteristic
314C24C22×D3711yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—374no—
318C2×D4—374no—
319C2×D4—742no—
320C2×D4—742no—
321C2×D4—1481no—
322C2×D4—1481no—
323C2×D4C4×D3714yescomplemented
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—371no—
328C22×C4—371no—
329C22×C4—371no—
330C22×C4—374no—
331C22×C4—371no—
332C22×C4—371no—
333C22×C4—371no—
334C22×C4—742no—
335C22×C4—1481no—
336C22×C4—1481no—
337C22×C4—1481no—
338C22×C4—1481no—
339C22×C4C22×D3711yescharacteristic
340C22×C4C22×D3711yescharacteristic
341C22×C4C22×D3711yescharacteristic
342C22×C4C22×D3711yescharacteristic
343C22×C4C22×D3711yescharacteristic
344C4⋊C4—742no—
345C4⋊C4—1481no—
346C4⋊C4—1481no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—374no—
351C22⋊C4—374no—
352C22⋊C4—374no—
353C22⋊C4—742no—
354C22⋊C4—742no—
355C22⋊C4—742no—
356C22⋊C4—742no—
357C22⋊C4—742no—
358C22⋊C4—742no—
359C22⋊C4—1481no—
360C22⋊C4—1481no—
361C22⋊C4—1481no—
362C22⋊C4—1481no—
363C42—742no—
364C42—1481no—
365C42—1481no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—372no—
379C23—372no—
380C23—372no—
381C23—372no—
382C23—748no—
383C23—741no—
384C23—741no—
385C23—741no—
386C23—741no—
387C23—741no—
388C23—741no—
389C23—741no—
390C23—741no—
391C23—741no—
392C23—741no—
393C23—1484no—
394C23—1484no—
395C23—1484no—
396C23—1484no—
397C23—1484no—
398C23—1484no—
399C23C2×C4×D3712yes—
400C23C2×C4×D3712yes—
401C23C23×D3711yescharacteristic
402D4—44no—
403D4—748no—
404D4—1484no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—44no—
419C2×C4—372no—
420C2×C4—372no—
421C2×C4—372no—
422C2×C4—741no—
423C2×C4—741no—
424C2×C4—741no—
425C2×C4—741no—
426C2×C4—741no—
427C2×C4—741no—
428C2×C4—741no—
429C2×C4—741no—
430C2×C4—741no—
431C2×C4—741no—
432C2×C4—741no—
433C2×C4—741no—
434C2×C4—741no—
435C2×C4—741no—
436C2×C4—741no—
437C2×C4—1484no—
438C2×C4—1484no—
439C2×C4C2×C4×D3712yes—
440C2×C4D4×D3712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—374no—
456C22—742no—
457C22—742no—
458C22—742no—
459C22—742no—
460C22—742no—
461C22—742no—
462C22—742no—
463C22—742no—
464C22—742no—
465C22—14816no—
466C22—1481no—
467C22—1481no—
468C22—1481no—
469C22—1481no—
470C22C22×C4×D3711yescharacteristic
471C22C2×D4×D3711yescharacteristic
472C22#41×D3711yescharacteristic
473C22#1192458111yescharacteristic
474C22#1192458111yescharacteristic
475C22#1192458111yescharacteristic
476C22#1192458211yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—742no—
483C4—742no—
484C4—742no—
485C4—1481no—
486C4—1481no—
487C4—1481no—
488C2—42no—
489C2—42no—
490C2—744no—
491C2—1482no—
492C2C4×D4×D3711yescharacteristic
493C2#1455800511yescharacteristic
494C2#1455800611yescharacteristic
495C2#1455809311yescharacteristic
496C2#1455809911yescharacteristic
497C2#1455811511yescharacteristic
498C2#1455811911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#386C2
socle, cosocleC23×C37C24
prime radicalC2×#11924504C2
hypercenter#386D37
hypocenterC37#2321
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D37C2semidirect
3. C2×C22⋊C4×D37C2semidirect
4. #14557436C2semidirect
5. #14557433C2semidirect
6. #14557425C2semidirect
7. #14557422C2semidirect
8. #14557421C2semidirect
9. #14557421C2semidirect
10. #14557412C2semidirect
11. #14557409C2semidirect
12. #14557407C2semidirect
13. #14557388C2semidirect
14. #14557380C2semidirect
15. C37×#386C2semidirect
21. #11924504C4semidirect
23. C2×C4⋊Dic37C22semidirect
24. C2×#10596936C22semidirect
25. C2×#10596936C22semidirect
29. C2×C22⋊Dic37C22semidirect
30. C2×C22⋊Dic37C22semidirect
40. C2×C4×Dic37C22semidirect
41. C2×C4×Dic37C22semidirect
42. C2×C4×Dic37C22semidirect
43. #11924443C22semidirect
44. #11924443C22semidirect
45. #11924443C22semidirect
46. #11924443C22semidirect
48. C23×C4×C37C22semidirect
51. C2×C22⋊C4×C37C22semidirect
52. #95×C37C22semidirect
53. #95×C37C22semidirect
62. C23×D37C2×C4semidirect
67. C2×D4×C37C2×C4semidirect
81. C2×C22⋊D37C2×C4semidirect
93. C2×C4×D37D4semidirect
101. C22⋊Dic37C2×C4semidirect
108. #10596938C2×C4semidirect
109. #10596938C2×C4semidirect
261. #386D37semidirect
265. C37#2321semidirect
272. C23×C4C2×D37semidirect
294. C2×C22⋊C4C2×D37semidirect
302. #95C2×D37semidirect
303. #95C2×D37semidirect
323. C2×D4C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245042
C2×C4×C22⋊D372
C2×C22⋊C4×D372
#145574362
#145574332
#145574252
#145574222
#145574212
#145574122
#145574092
#145574072
#145573882
#145573802
C37×#3862
#232137