Finite Group Databasebeta

← #31797288

Subgroups — #31797288

Order 17536.

532 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578355C211yesmaximal, characteristic
2#22578354C211yesmaximal, characteristic, complemented
3#22578353C211yesmaximal, characteristic, complemented
4#22578353C211yesmaximal, characteristic, complemented
5#22578353C211yesmaximal, characteristic, complemented
6#22578353C211yesmaximal, characteristic, complemented
7#22578352C211yesmaximal, characteristic, complemented
8#22578352C211yesmaximal, characteristic, complemented
9#22578351C211yesmaximal, characteristic
10#22578350C211yesmaximal, characteristic, complemented
11#22578345C211yesmaximal, characteristic, complemented
12C4⋊C4×#10587739C211yesmaximal, characteristic, complemented
13C22⋊C4×#10587739C211yesmaximal, characteristic, complemented
14C4×C22⋊#10587739C211yesmaximal, characteristic, complemented
15#123×D137C211yesmaximal, characteristic
16C23×#10587739C2211yescharacteristic
17C2×C4⋊#10587739C2211yescharacteristic
18C2×C4⋊#10587739C2211yescharacteristic
19C2×C4⋊#10587739C2211yescharacteristic
20C2×C4⋊#10587739C2211yescharacteristic
21C2×C4⋊#10587739C2211yescharacteristic
22C2×C22⋊#10587739C2211yescharacteristic
23C2×C22⋊#10587739C2211yescharacteristic
24C2×C22⋊#10587739C2211yescharacteristic
25C2×C22⋊#10587739C2211yescharacteristic
26C2×C22⋊#10587739C2211yescharacteristic
27C2×C22⋊#10587739C2211yescharacteristic
28C2×C22⋊#10587739C2211yescharacteristic
29C2×C4×#10587739C2211yescharacteristic
30C2×C4×#10587739C2211yescharacteristic
31C2×C4×#10587739C2211yescharacteristic
32C2×C4×#10587739C2211yescharacteristic
33C2×C4×#10587739C2211yescharacteristic
34(C2×#10587739)⋊C4C2211yescharacteristic
35(C2×#10587739)⋊C4C2211yescharacteristic
36(C2×#10587739)⋊C4C2211yescharacteristic
37(C2×#10587739)⋊C4C2211yescharacteristic
38(C2×#10587739)⋊C4C2211yescharacteristic
39(C2×#10587739)⋊C4C2211yescharacteristic
40(C2×#10587739)⋊C4C2211yescharacteristic
41(C2×#10587739)⋊C4C2211yescharacteristic
42(C2×#10587739)⋊C4C2211yescharacteristic
43(C2×#10587739)⋊C4C2211yescharacteristic
44C2×D4×D137C2211yescharacteristic
45C22×C4×D137C2211yescharacteristic
46#17984073C411yescharacteristic, complemented
47#17984073C411yescharacteristic, complemented
48#17984072C411yescharacteristic, complemented
49#17984071C411yescharacteristic, complemented
50#17984070C411yescharacteristic, complemented
51#17984070C411yescharacteristic, complemented
52#17984069C411yescharacteristic, complemented
53C4⋊C4×D137C2211yescharacteristic
54C4⋊C4×D137C2211yescharacteristic
55C22⋊C4×D137C2211yescharacteristic
56C22⋊C4×D137C2211yescharacteristic
57C22⋊C4×D137C2211yescharacteristic
58#123×C137C411yescharacteristic, complemented
59C22×#10587739—22no—
60C22×#10587739—41no—
61C22×#10587739—41no—
62C22×#10587739C2311yescharacteristic
63C22×#10587739C2311yescharacteristic
64C22×#10587739C2311yescharacteristic
65C22×#10587739C2311yescharacteristic
66C22×#10587739C2311yescharacteristic
67C22×#10587739C2311yescharacteristic
68C22×#10587739C2311yescharacteristic
69C22×#10587739C2311yescharacteristic
70C4⋊#10587739—22no—
71C4⋊#10587739—41no—
72C4⋊#10587739—41no—
73C4⋊#10587739—41no—
74C4⋊#10587739—41no—
75C22⋊#10587739—22no—
76C22⋊#10587739—22no—
77C22⋊#10587739—22no—
78C22⋊#10587739—41no—
79C22⋊#10587739—41no—
80C22⋊#10587739—41no—
81C22⋊#10587739—41no—
82C4×#10587739—22no—
83C4×#10587739—41no—
84C4×#10587739—41no—
85C4×#10587739—41no—
86C4×#10587739—41no—
87C23×D137C2311yescharacteristic
88C23×D137C2311yescharacteristic
89C2×D548C2×C411yescharacteristic
90C2×C22⋊D137C2×C411yescharacteristic
91C2×C22⋊D137C2×C411yescharacteristic
92D4×D137—41no—
93D4×D137—41no—
94C22×Dic137C2×C411yescharacteristic, complemented
95C2×C4×D137—41no—
96C2×C4×D137—41no—
97C2×C4×D137C2×C411yescharacteristic, complemented
98C2×C4×D137C2×C411yescharacteristic, complemented
99C2×C4×D137C2311yescharacteristic
100C2×C4×D137C2311yescharacteristic
101C2×C4×D137C2311yescharacteristic
102C2×C4×D137C2311yescharacteristic
103C2×C4×D137C2311yescharacteristic
104C4⋊Dic137C2×C411yescharacteristic, complemented
105C4⋊Dic137C2×C411yescharacteristic, complemented
106#14555489C2×C411yescharacteristic, complemented
107#14555489C2×C411yescharacteristic, complemented
108#14555489C2×C411yescharacteristic, complemented
109#14555489C2×C411yescharacteristic, complemented
110C22⋊Dic137C2×C411yescharacteristic
111C22⋊Dic137C2×C411yescharacteristic
112C22⋊Dic137C2×C411yescharacteristic, complemented
113#14555487C2×C411yescharacteristic
114#14555487C2×C411yescharacteristic
115#14555487C2×C411yescharacteristic
116#14555487C2×C411yescharacteristic
117#14555487C2×C411yescharacteristic, complemented
118#14555487C2×C411yescharacteristic, complemented
119C2×D4×C137C2×C411yescharacteristic
120C22×C4×C137C2×C411yescharacteristic, complemented
121C4⋊C4×C137C2×C411yescharacteristic, complemented
122C4⋊C4×C137C2×C411yescharacteristic, complemented
123C22⋊C4×C137C2×C411yescharacteristic
124C22⋊C4×C137C2×C411yescharacteristic
125C22⋊C4×C137C2×C411yescharacteristic, complemented
126C2×#10587739—21no—
127C2×#10587739—21no—
128C2×#10587739—21no—
129C2×#10587739—21no—
130C2×#10587739—21no—
131C2×#10587739—21no—
132C2×#10587739—21no—
133C2×#10587739—21no—
134C2×#10587739—21no—
135C2×#10587739—21no—
136C2×#10587739—21no—
137C2×#10587739—21no—
138C2×#10587739—21no—
139C2×#10587739—21no—
140C2×#10587739—21no—
141C2×#10587739—21no—
142C2×#10587739—21no—
143C2×#10587739—21no—
144C2×#10587739—21no—
145C2×#10587739—21no—
146C2×#10587739—21no—
147C2×#10587739—21no—
148C2×#10587739—44no—
149C2×#10587739#4112yes—
150C2×#10587739#4112yes—
151C22×D137—21no—
152C22×D137—21no—
153C22×D137—21no—
154C22×D137—21no—
155C22×D137—21no—
156C22×D137—21no—
157C22×D137—21no—
158C22×D137—21no—
159C22×D137—21no—
160C22×D137—21no—
161C22×D137—21no—
162C22×D137—21no—
163C22×D137C22×C411yescharacteristic
164C22×D137C22×C411yescharacteristic
165C22×D137C2411yescharacteristic
166D548—41no—
167C22⋊D137—41no—
168C22⋊D137—41no—
169C2×Dic137—41no—
170C2×Dic137C22×C411yescharacteristic
171C2×Dic137C22×C411yescharacteristic
172C2×Dic137C22×C411yescharacteristic
173C2×Dic137C22×C411yescharacteristic
174C2×Dic137C22×C411yescharacteristic
175C4×D137—21no—
176C4×D137—21no—
177C4×D137—21no—
178C4×D137—21no—
179C4×D137—21no—
180C4×D137—21no—
181C4×D137—21no—
182C4×D137—21no—
183C4×D137—21no—
184C4×D137—21no—
185C4×D137—41no—
186C4×D137—41no—
187C23×C137C22×C411yescharacteristic
188C23×C137C22×C411yescharacteristic
189D4×C137—41no—
190C2×C4×C137—41no—
191C2×C4×C137C22×C411yescharacteristic
192C2×C4×C137C22×C411yescharacteristic
193C2×C4×C137C22×C411yescharacteristic
194C2×C4×C137C22×C411yescharacteristic
195C2×C4×C137C22×C411yescharacteristic
196C22×C137—21no—
197C22×C137—21no—
198C22×C137—21no—
199C22×C137—21no—
200C22×C137—21no—
201C22×C137—21no—
202C22×C137C23×C411yescharacteristic
203C2×D137—21no—
204C2×D137—21no—
205C2×D137—21no—
206C2×D137—21no—
207C2×D137—21no—
208C2×D137—21no—
209C2×D137—41no—
210C2×D137—41no—
211C2×D137—41no—
212C2×D137—41no—
213C2×D137—41no—
214C2×D137—41no—
215C2×D137—41no—
216C2×D137—41no—
217C2×D137C2×#4111yescharacteristic
218C2×D137C2×#4111yescharacteristic
219C2×D137H(5,𝔽2)11yescharacteristic
220C2×D137H(5,𝔽2)11yescharacteristic
221C2×D137H(5,𝔽2)11yescharacteristic
222C2×D1372−1+411yescharacteristic
223#10587739—22no—
224#10587739—22no—
225#10587739—41no—
226#10587739—41no—
227#10587739—41no—
228#10587739—41no—
229#10587739—41no—
230#10587739—41no—
231C4×C137—21no—
232C4×C137—21no—
233C4×C137—21no—
234C4×C137—21no—
235C4×C137—21no—
236Dic137—21no—
237Dic137—21no—
238Dic137—21no—
239Dic137—21no—
240Dic137—21no—
241C2×C137—41no—
242C2×C137—41no—
243C2×C137C4×#4111yescharacteristic
244C2×C137#51811yescharacteristic
245C2×C137#52011yescharacteristic
246D137—41no—
247D137—41no—
248D137#53811yescharacteristic
249D137#53911yescharacteristic
250D137#54011yescharacteristic
251D137#54211yescharacteristic
252C137#233511yescharacteristic, complemented, Sylow
253#2335—1371nomaximal, Sylow
254C2×#123—1371no—
255#390—1371no—
256#389—1371no—
257#388—1371no—
258#388—1371no—
259#388—1371no—
260#388—1371no—
261#387—1371no—
262#387—1371no—
263#386—1371no—
264#385—1371no—
265#380—1371no—
266C4×C4⋊C4—1371no—
267C4×C22⋊C4—1371no—
268C4×C22⋊C4—1371no—
269C22×D4—1371no—
270C23×C4—1371no—
271C23×C4—1371no—
272#123—1371no—
273#123—1371no—
274#123—1371no—
275#123—1371no—
276#123—1371no—
277#123—1371no—
278#123—1371no—
279#123#1058773911yescharacteristic, complemented
280C2×C4⋊C4—1371no—
281C2×C4⋊C4—1371no—
282C2×C4⋊C4—1371no—
283C2×C4⋊C4—1371no—
284C2×C4⋊C4—1371no—
285C2×C4⋊C4—1371no—
286C2×C4⋊C4—1371no—
287C2×C22⋊C4—1371no—
288C2×C22⋊C4—1371no—
289C2×C22⋊C4—1371no—
290C2×C22⋊C4—1371no—
291C2×C22⋊C4—1371no—
292C2×C22⋊C4—1371no—
293C2×C22⋊C4—1371no—
294C2×C22⋊C4—1371no—
295C2×C22⋊C4—1371no—
296C2×C22⋊C4—1371no—
297C2×C42—1371no—
298C2×C42—1371no—
299C2×C42—1371no—
300C2×C42—1371no—
301C2×C42—1371no—
302#95—1371no—
303#95—1371no—
304#95—1371no—
305#95—1371no—
306#95—1371no—
307#95—1371no—
308#95—1371no—
309#95—1371no—
310#95—1371no—
311#95—1371no—
312C24—1371no—
313C24—1371no—
314C2×D4—5481no—
315C2×D4—5481no—
316C2×D4—1371no—
317C2×D4—1371no—
318C2×D4—1371no—
319C2×D4C2×#1058773911yescharacteristic
320C22×C4—2742no—
321C22×C4—5481no—
322C22×C4—5481no—
323C22×C4—5481no—
324C22×C4—5481no—
325C22×C4—1371no—
326C22×C4—1371no—
327C22×C4—1371no—
328C22×C4—1371no—
329C22×C4—1371no—
330C22×C4—1371no—
331C22×C4—1371no—
332C22×C4—1371no—
333C22×C4—1371no—
334C22×C4—1371no—
335C22×C4—1371no—
336C22×C4—1371no—
337C22×C4—1371no—
338C22×C4—1371no—
339C22×C4—1371no—
340C22×C4—1371no—
341C22×C4C2×#1058773911yescharacteristic, complemented
342C4⋊C4—2742no—
343C4⋊C4—5481no—
344C4⋊C4—5481no—
345C4⋊C4—5481no—
346C4⋊C4—5481no—
347C4⋊C4—1371no—
348C4⋊C4—1371no—
349C4⋊C4—1371no—
350C4⋊C4—1371no—
351C4⋊C4—1371no—
352C4⋊C4—1371no—
353C4⋊C4C2×#1058773911yescharacteristic, complemented
354C4⋊C4C2×#1058773911yescharacteristic, complemented
355C22⋊C4—2742no—
356C22⋊C4—2742no—
357C22⋊C4—2742no—
358C22⋊C4—5481no—
359C22⋊C4—5481no—
360C22⋊C4—5481no—
361C22⋊C4—5481no—
362C22⋊C4—1371no—
363C22⋊C4—1371no—
364C22⋊C4—1371no—
365C22⋊C4—1371no—
366C22⋊C4—1371no—
367C22⋊C4—1371no—
368C22⋊C4—1371no—
369C22⋊C4—1371no—
370C22⋊C4—1371no—
371C22⋊C4C2×#1058773911yescharacteristic
372C22⋊C4C2×#1058773911yescharacteristic
373C22⋊C4C2×#1058773911yescharacteristic, complemented
374C42—2742no—
375C42—5481no—
376C42—5481no—
377C42—5481no—
378C42—5481no—
379C23—2741no—
380C23—2741no—
381C23—2741no—
382C23—2741no—
383C23—2741no—
384C23—2741no—
385C23—2741no—
386C23—2741no—
387C23—2741no—
388C23—2741no—
389C23—2741no—
390C23—2741no—
391C23—1371no—
392C23—1371no—
393C23—1371no—
394C23C22×#1058773911yescharacteristic
395C23C22×#1058773911yescharacteristic
396D4—41no—
397D4—5481no—
398D4—5481no—
399D4—5481no—
400C2×C4—41no—
401C2×C4—2741no—
402C2×C4—2741no—
403C2×C4—2741no—
404C2×C4—2741no—
405C2×C4—2741no—
406C2×C4—2741no—
407C2×C4—2741no—
408C2×C4—2741no—
409C2×C4—2741no—
410C2×C4—2741no—
411C2×C4—2741no—
412C2×C4—2741no—
413C2×C4—2741no—
414C2×C4—2741no—
415C2×C4—2741no—
416C2×C4—2741no—
417C2×C4—2741no—
418C2×C4—2741no—
419C2×C4—2741no—
420C2×C4—2741no—
421C2×C4—2741no—
422C2×C4—2741no—
423C2×C4—2741no—
424C2×C4—2741no—
425C2×C4—2741no—
426C2×C4—2741no—
427C2×C4—2741no—
428C2×C4—2741no—
429C2×C4—2741no—
430C2×C4—2741no—
431C2×C4—2741no—
432C2×C4—2741no—
433C2×C4—5481no—
434C2×C4—5481no—
435C2×C4—5481no—
436C2×C4—5484no—
437C2×C4—1372no—
438C2×C4—1372no—
439C2×C4—1371no—
440C2×C4—1371no—
441C2×C4—1371no—
442C2×C4—1371no—
443C2×C4—1371no—
444C2×C4C22×#1058773911yescharacteristic
445C2×C4C22×#1058773911yescharacteristic
446C2×C4C22×#1058773911yescharacteristic
447C2×C4C22×#1058773911yescharacteristic
448C2×C4C22×#1058773911yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—2741no—
456C22—2741no—
457C22—2741no—
458C22—2741no—
459C22—2741no—
460C22—2741no—
461C22—5481no—
462C22—5481no—
463C22—5481no—
464C22—5481no—
465C22—5481no—
466C22—5481no—
467C22—5481no—
468C22—5481no—
469C22—1371no—
470C22—1371no—
471C22—1371no—
472C22—1371no—
473C22—1371no—
474C22—1371no—
475C22C23×#1058773911yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—2742no—
482C4—2742no—
483C4—2741no—
484C4—2741no—
485C4—2741no—
486C4—2741no—
487C4—2741no—
488C4—5481no—
489C4—5481no—
490C4—5481no—
491C4—5481no—
492C4—5481no—
493C4—5481no—
494C2—41no—
495C2—41no—
496C2—5481no—
497C2—5481no—
498C2—1371no—
499C2—1371no—
500C2—1371no—
501C2—1371no—
502C2#41×#1058773911yescharacteristic
503C2#2257847011yescharacteristic
504C2#2257847211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
Fitting#123×C137C4
prime radicalC2×D4×D137C22
cosocleC22×D137C24
hypercenter#123#10587739
hypocenterC137#2335
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #22578354C2semidirect
3. #22578353C2semidirect
4. #22578353C2semidirect
5. #22578353C2semidirect
6. #22578353C2semidirect
7. #22578352C2semidirect
8. #22578352C2semidirect
10. #22578350C2semidirect
11. #22578345C2semidirect
12. C4⋊C4×#10587739C2semidirect
13. C22⋊C4×#10587739C2semidirect
14. C4×C22⋊#10587739C2semidirect
46. #17984073C4semidirect
47. #17984073C4semidirect
48. #17984072C4semidirect
49. #17984071C4semidirect
50. #17984070C4semidirect
51. #17984070C4semidirect
52. #17984069C4semidirect
58. #123×C137C4semidirect
94. C22×Dic137C2×C4semidirect
97. C2×C4×D137C2×C4semidirect
98. C2×C4×D137C2×C4semidirect
104. C4⋊Dic137C2×C4semidirect
105. C4⋊Dic137C2×C4semidirect
106. #14555489C2×C4semidirect
107. #14555489C2×C4semidirect
108. #14555489C2×C4semidirect
109. #14555489C2×C4semidirect
112. C22⋊Dic137C2×C4semidirect
117. #14555487C2×C4semidirect
118. #14555487C2×C4semidirect
120. C22×C4×C137C2×C4semidirect
121. C4⋊C4×C137C2×C4semidirect
122. C4⋊C4×C137C2×C4semidirect
125. C22⋊C4×C137C2×C4semidirect
252. C137#2335semidirect
279. #123#10587739semidirect
341. C22×C4C2×#10587739semidirect
353. C4⋊C4C2×#10587739semidirect
354. C4⋊C4C2×#10587739semidirect
373. C22⋊C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783552
#225783542
#225783532
#225783522
#225783512
#225783502
#225783452
C4⋊C4×#105877392
C22⋊C4×#105877392
C4×C22⋊#105877392
#123×D1372
#2335137