Finite Group Databasebeta

← #17997736

Subgroups — #17997736

Order 4736.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924506C211yesmaximal, characteristic
2#14557435C211yesmaximal, characteristic, complemented
3#14557434C211yesmaximal, characteristic, complemented
4#14557433C211yesmaximal, characteristic, complemented
5#14557426C211yesmaximal, characteristic, complemented
6#14557425C211yesmaximal, characteristic, complemented
7#14557424C211yesmaximal, characteristic, complemented
8#14557423C211yesmaximal, characteristic, complemented
9#14557422C211yesmaximal, characteristic, complemented
10#14557411C211yesmaximal, characteristic, complemented
11#14557411C211yesmaximal, characteristic, complemented
12#14557409C211yesmaximal, characteristic, complemented
13#14557388C211yesmaximal, characteristic, complemented
14#14557388C211yesmaximal, characteristic, complemented
15C37×#388C211yesmaximal, characteristic, complemented
16C24×D37C2211yescharacteristic
17C22×D148C2211yescharacteristic
18C22×D148C2211yescharacteristic
19C22×C22⋊D37C2211yescharacteristic
20C22×C4×D37C2211yescharacteristic, complemented
21#11924506C418yescomplemented
22C2×C4⋊Dic37C2211yescharacteristic, complemented
23C2×#10596936C2211yescharacteristic, complemented
24C2×C22⋊Dic37C2211yescharacteristic
25C2×C22⋊Dic37C2211yescharacteristic
26C2×#10596938C2211yescharacteristic
27C2×#10596938C2211yescharacteristic
28C2×#10596938C2211yescharacteristic
29C2×#10596938C2211yescharacteristic
30C2×#10596938C2211yescharacteristic
31C2×#10596938C2211yescharacteristic
32C2×#10596938C2211yescharacteristic
33C2×#10596938C2211yescharacteristic
34C2×#10596938C2211yescharacteristic
35C2×#10596938C2211yescharacteristic
36C2×#10596938C2211yescharacteristic, complemented
37C2×#10596938C2211yescharacteristic, complemented
38C2×#10596938C2211yescharacteristic, complemented
39C2×C4×Dic37C2211yescharacteristic, complemented
40C2×C4×Dic37C2211yescharacteristic, complemented
41#11924443C2211yescharacteristic, complemented
42#11924443C2211yescharacteristic, complemented
43#11924443C2211yescharacteristic, complemented
44#11924443C2211yescharacteristic, complemented
45C2×C4⋊C4×C37C2211yescharacteristic, complemented
46C2×C22⋊C4×C37C2211yescharacteristic
47C2×C22⋊C4×C37C2211yescharacteristic
48C2×C22⋊C4×C37C2211yescharacteristic
49C2×C42×C37C2211yescharacteristic, complemented
50#95×C37C2211yescharacteristic, complemented
51#95×C37C2211yescharacteristic, complemented
52C24×C37C2311yescharacteristic
53C23×D37—41no—
54C23×D37—41no—
55C23×D37—41no—
56C23×D37—41no—
57C23×D37—41no—
58C23×D37—41no—
59C23×D37C2×C414yescomplemented
60C23×D37C2311yescharacteristic
61C23×D37C2311yescharacteristic
62C23×D37C2311yescharacteristic
63C22×C4×C37C2311yescharacteristic
64C22×C4×C37C2311yescharacteristic
65C22×C4×C37C2311yescharacteristic
66C22×C4×C37C2311yescharacteristic
67C22×C4×C37C2311yescharacteristic
68C22×C4×C37C2311yescharacteristic
69C2×C22⋊D37—41no—
70C2×C22⋊D37—41no—
71C2×C22⋊D37C2×C414yes—
72C22×Dic37C2311yescharacteristic
73C22×Dic37C2311yescharacteristic
74C22×Dic37C2311yescharacteristic
75C22×Dic37C2311yescharacteristic
76C22×Dic37C2311yescharacteristic
77C2×D148—41no—
78C2×D148—41no—
79C2×D148—41no—
80C2×D148—41no—
81C2×D148C2×C414yes—
82C2×D148C2×C414yes—
83C2×C4×D37—41no—
84C2×C4×D37—41no—
85C2×C4×D37—41no—
86C4⋊C4×C37—41no—
87C22⋊C4×C37—41no—
88C22⋊C4×C37—41no—
89C22⋊C4×C37C2×C414yescomplemented
90C42×C37—41no—
91C22⋊Dic37—41no—
92C22⋊Dic37—41no—
93#10596938—41no—
94#10596938—41no—
95#10596938—41no—
96#10596938—41no—
97#10596938—41no—
98#10596938—41no—
99#10596938—41no—
100#10596938—41no—
101#10596938—41no—
102#10596938—41no—
103#10596938—41no—
104#10596938C2×C414yescomplemented
105#10596938C2×C414yescomplemented
106C4⋊Dic37—41no—
107#10596936—41no—
108C4×Dic37—41no—
109C4×Dic37—41no—
110C23×C37—21no—
111C23×C37—21no—
112C23×C37—21no—
113C23×C37—21no—
114C23×C37—21no—
115C23×C37—21no—
116C23×C37C22×C412yes—
117C23×C37C2411yescharacteristic
118C22×D37—21no—
119C22×D37—21no—
120C22×D37—21no—
121C22×D37—21no—
122C22×D37—21no—
123C22×D37—21no—
124C22×D37—21no—
125C22×D37—21no—
126C22×D37—21no—
127C22×D37—21no—
128C22×D37—21no—
129C22×D37—21no—
130C22×D37—21no—
131C22×D37—21no—
132C22×D37—21no—
133C22×D37—21no—
134C22×D37—21no—
135C22×D37—21no—
136C22×D37—44no—
137C22×D37—44no—
138C22×D37—44no—
139C22×D37—44no—
140C22×D37—44no—
141C22×D37—44no—
142C22×D37—44no—
143C22×D37C22×C412yes—
144C22×D37C22×C412yes—
145C22×D37C22×C412yes—
146C2×C4×C37—21no—
147C2×C4×C37—21no—
148C2×C4×C37—21no—
149C2×C4×C37—21no—
150C2×C4×C37—21no—
151C2×C4×C37—21no—
152C2×C4×C37—21no—
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×C37C22×C412yes—
163C2×C4×C37C22×C412yes—
164C22⋊D37—44no—
165C2×Dic37—21no—
166C2×Dic37—21no—
167C2×Dic37—21no—
168C2×Dic37—21no—
169C2×Dic37—21no—
170C2×Dic37—21no—
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×Dic37C22×C412yes—
180D148—44no—
181D148—44no—
182C4×D37—44no—
183C22×C37—22no—
184C22×C37—22no—
185C22×C37—22no—
186C22×C37—41no—
187C22×C37—41no—
188C22×C37—41no—
189C22×C37—41no—
190C22×C37C23×C411yescharacteristic
191C22×C37H(5,𝔽2)11yescharacteristic
192C22×C37H(5,𝔽2)11yescharacteristic
193C22×C37H(5,𝔽2)11yescharacteristic
194C22×C37H(5,𝔽2)11yescharacteristic
195C22×C37H(5,𝔽2)11yescharacteristic
196C22×C37H(5,𝔽2)11yescharacteristic
197C2×D37—22no—
198C2×D37—22no—
199C2×D37—22no—
200C2×D37—22no—
201C2×D37—22no—
202C2×D37—22no—
203C2×D37—22no—
204C2×D37—22no—
205C2×D37—22no—
206C2×D37—416no—
207C2×D37—41no—
208C2×D37—41no—
209C2×D37—41no—
210C2×D37—41no—
211C2×D37—41no—
212C2×D37—41no—
213C2×D37—41no—
214C2×D37—41no—
215C2×D37—41no—
216C2×D37—41no—
217C2×D37—41no—
218C2×D37—41no—
219C4×C37—22no—
220C4×C37—22no—
221C4×C37—41no—
222C4×C37—41no—
223C4×C37—41no—
224C4×C37—41no—
225Dic37—22no—
226Dic37—41no—
227Dic37—41no—
228Dic37—41no—
229Dic37—41no—
230#2375—371nomaximal, Sylow
231C2×C37—42no—
232C2×C37#51811yescharacteristic
233C2×C37#51811yescharacteristic
234C2×C37#51811yescharacteristic
235C2×C37#56011yescharacteristic
236C2×C37#56011yescharacteristic
237C2×C37#56011yescharacteristic
238C2×C37H(3,𝔽4)11yescharacteristic
239D37—42no—
240D37—42no—
241D37—42no—
242C2×#120—371no—
243#390—371no—
244#390—371no—
245#390—371no—
246#388—371no—
247#388—371no—
248#388—371no—
249#388—371no—
250#388—371no—
251#388D3711yescharacteristic, complemented
252#386—371no—
253#386—371no—
254#386—371no—
255#379—371no—
256#379—371no—
257C37#237511yescharacteristic, complemented, Sylow
258C25—371no—
259C22×D4—371no—
260C22×D4—371no—
261C22×D4—371no—
262C23×C4—371no—
263#120—378no—
264C2×C4⋊C4—371no—
265C2×C4⋊C4—371no—
266C2×C4⋊C4C2×D3711yescharacteristic, complemented
267C2×C22⋊C4—371no—
268C2×C22⋊C4—371no—
269C2×C22⋊C4—371no—
270C2×C22⋊C4—371no—
271C2×C22⋊C4—371no—
272C2×C22⋊C4—371no—
273C2×C22⋊C4—371no—
274C2×C22⋊C4—371no—
275C2×C22⋊C4—371no—
276C2×C22⋊C4—371no—
277C2×C22⋊C4—371no—
278C2×C22⋊C4—371no—
279C2×C22⋊C4—371no—
280C2×C22⋊C4—371no—
281C2×C22⋊C4—371no—
282C2×C22⋊C4C2×D3711yescharacteristic
283C2×C22⋊C4C2×D3711yescharacteristic
284C2×C22⋊C4C2×D3711yescharacteristic
285C2×C42—371no—
286C2×C42—371no—
287C2×C42C2×D3711yescharacteristic, complemented
288#95—371no—
289#95—371no—
290#95—371no—
291#95—371no—
292#95C2×D3711yescharacteristic, complemented
293#95C2×D3711yescharacteristic, complemented
294C24—371no—
295C24—371no—
296C24—371no—
297C24—374no—
298C24—1481no—
299C24—1481no—
300C24—1481no—
301C24—1481no—
302C24—1481no—
303C24—1481no—
304C24C22×D3711yescharacteristic
305C2×D4—374no—
306C2×D4—374no—
307C2×D4—374no—
308C2×D4—1481no—
309C2×D4—1481no—
310C2×D4—1481no—
311C2×D4—1481no—
312C2×D4—1481no—
313C2×D4—1481no—
314C22×C4—371no—
315C22×C4—371no—
316C22×C4—371no—
317C22×C4—371no—
318C22×C4—371no—
319C22×C4—1481no—
320C22×C4—1481no—
321C22×C4—1481no—
322C22×C4C22×D3711yescharacteristic
323C22×C4C22×D3711yescharacteristic
324C22×C4C22×D3711yescharacteristic
325C22×C4C22×D3711yescharacteristic
326C22×C4C22×D3711yescharacteristic
327C22×C4C22×D3711yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—1481no—
330C4⋊C4—1481no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—374no—
334C22⋊C4—374no—
335C22⋊C4—1481no—
336C22⋊C4—1481no—
337C22⋊C4—1481no—
338C22⋊C4—1481no—
339C22⋊C4—1481no—
340C22⋊C4—1481no—
341C22⋊C4—1481no—
342C22⋊C4—1481no—
343C22⋊C4—1481no—
344C22⋊C4—1481no—
345C22⋊C4—1481no—
346C22⋊C4—1481no—
347C22⋊C4—1481no—
348C22⋊C4C4×D3714yescomplemented
349C42—41no—
350C42—1481no—
351C42—1481no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—372no—
359C23—372no—
360C23—372no—
361C23—741no—
362C23—741no—
363C23—741no—
364C23—741no—
365C23—741no—
366C23—741no—
367C23—741no—
368C23—741no—
369C23—741no—
370C23—741no—
371C23—741no—
372C23—741no—
373C23—741no—
374C23—741no—
375C23—741no—
376C23—741no—
377C23—741no—
378C23—741no—
379C23—1484no—
380C23—1484no—
381C23—1484no—
382C23—1484no—
383C23—1484no—
384C23—1484no—
385C23—1484no—
386C23C2×C4×D3712yes—
387C23C23×D3711yescharacteristic
388D4—1484no—
389D4—1484no—
390D4—1484no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—372no—
408C2×C4—741no—
409C2×C4—741no—
410C2×C4—741no—
411C2×C4—741no—
412C2×C4—741no—
413C2×C4—741no—
414C2×C4—741no—
415C2×C4—741no—
416C2×C4—741no—
417C2×C4—741no—
418C2×C4—741no—
419C2×C4—741no—
420C2×C4—741no—
421C2×C4—741no—
422C2×C4—1484no—
423C2×C4C2×C4×D3712yes—
424C2×C4C2×C4×D3712yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—742no—
433C22—742no—
434C22—742no—
435C22—742no—
436C22—742no—
437C22—742no—
438C22—742no—
439C22—742no—
440C22—742no—
441C22—14816no—
442C22—1481no—
443C22—1481no—
444C22—1481no—
445C22—1481no—
446C22—1481no—
447C22—1481no—
448C22—1481no—
449C22—1481no—
450C22—1481no—
451C22—1481no—
452C22—1481no—
453C22—1481no—
454C22C22×C4×D3711yescharacteristic
455C22#1192458111yescharacteristic
456C22#1192458111yescharacteristic
457C22#1192458211yescharacteristic
458C22#1192458211yescharacteristic
459C22#1192458211yescharacteristic
460C22#1192458211yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—742no—
468C4—1481no—
469C4—1481no—
470C4—1481no—
471C4—1481no—
472C2—42no—
473C2—1482no—
474C2—1482no—
475C2—1482no—
476C2#1455800511yescharacteristic
477C2#1455800511yescharacteristic
478C2#1455800711yescharacteristic
479C2#1455827111yescharacteristic
480C2#1455827311yescharacteristic
481C2#1455827411yescharacteristic
482C2#1455827511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#388C2
socle, cosocleC23×C37C24
prime radicalC2×#11924506C2
hypercenter#388D37
hypocenterC37#2375
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #14557435C2semidirect
3. #14557434C2semidirect
4. #14557433C2semidirect
5. #14557426C2semidirect
6. #14557425C2semidirect
7. #14557424C2semidirect
8. #14557423C2semidirect
9. #14557422C2semidirect
10. #14557411C2semidirect
11. #14557411C2semidirect
12. #14557409C2semidirect
13. #14557388C2semidirect
14. #14557388C2semidirect
15. C37×#388C2semidirect
20. C22×C4×D37C22semidirect
21. #11924506C4semidirect
22. C2×C4⋊Dic37C22semidirect
23. C2×#10596936C22semidirect
36. C2×#10596938C22semidirect
37. C2×#10596938C22semidirect
38. C2×#10596938C22semidirect
39. C2×C4×Dic37C22semidirect
40. C2×C4×Dic37C22semidirect
41. #11924443C22semidirect
42. #11924443C22semidirect
43. #11924443C22semidirect
44. #11924443C22semidirect
45. C2×C4⋊C4×C37C22semidirect
49. C2×C42×C37C22semidirect
50. #95×C37C22semidirect
51. #95×C37C22semidirect
59. C23×D37C2×C4semidirect
89. C22⋊C4×C37C2×C4semidirect
104. #10596938C2×C4semidirect
105. #10596938C2×C4semidirect
251. #388D37semidirect
257. C37#2375semidirect
266. C2×C4⋊C4C2×D37semidirect
287. C2×C42C2×D37semidirect
292. #95C2×D37semidirect
293. #95C2×D37semidirect
348. C22⋊C4C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245062
#145574352
#145574342
#145574332
#145574262
#145574252
#145574242
#145574232
#145574222
#145574112
#145574092
#145573882
C37×#3882
#237537