Finite Group Databasebeta

← #31792635

Subgroups — #31792635

Order 17536.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578101C211yesmaximal, characteristic, complemented
2#22578047C211yesmaximal, characteristic, complemented
3#22577756C211yesmaximal, characteristic, complemented
4#22577738C211yesmaximal, characteristic, complemented
5#22577726C211yesmaximal, characteristic, complemented
6#22577705C211yesmaximal, characteristic, complemented
7#22577638C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577617C211yesmaximal, characteristic, complemented
10#22577549C211yesmaximal, characteristic, complemented
11#22577537C211yesmaximal, characteristic, complemented
12#22577483C211yesmaximal, characteristic, complemented
13C8×C22⋊D137C211yesmaximal, characteristic, complemented
14#22577396C211yesmaximal, characteristic, complemented
15C2×#102×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×#14555499C2211yescharacteristic, complemented
18C2×#14555495C2211yescharacteristic
19C2×#14555495C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic, complemented
21#17984083—21no—
22#17984079C2211yescharacteristic, complemented
23#17984075C2211yescharacteristic
24#17984073—21no—
25#17984069—21no—
26#17984067C2211yescharacteristic
27#17984063C2211yescharacteristic, complemented
28#17984062—21no—
29#17984062C2211yescharacteristic
30#17984060—21no—
31#17984059C2211yescharacteristic
32#17984057—21no—
33D4×Dic137C2211yescharacteristic
34#17984050C2211yescharacteristic, complemented
35C4×C22⋊D137—21no—
36C4×C22⋊D137C2211yescharacteristic
37#17984046C2211yescharacteristic, complemented
38C2×C22⋊Dic137—21no—
39#17984021C2211yescharacteristic, complemented
40#17984021C2211yescharacteristic, complemented
41#17984020C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C2×C4⋊C4×C137C2211yescharacteristic
55#102×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61C2×Dic274C2311yescharacteristic
62C2×D548—21no—
63C2×D548C2311yescharacteristic
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137D411yescharacteristic
68C22×Dic137—21no—
69C22×Dic137—21no—
70C2×C4×D137—21no—
71C2×C4×D137C2311yescharacteristic
72#14555499—41no—
73#14555495—22no—
74#14555495—41no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C8×D137—41no—
78C4⋊Dic137—21no—
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C24×C137—21no—
98C2×D4×C137—22no—
99C2×D4×C137—21no—
100C2×D4×C137—41no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105C2×C8×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115D548—21no—
116D548—21no—
117D548—81no—
118C22⋊D137—22no—
119C22⋊D137—42no—
120C22⋊D137—42no—
121C22⋊D137—81no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—42no—
126C2×Dic137—81no—
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—21no—
131C4×D137—21no—
132C4×D137—81no—
133#11764040—21no—
134#11764040—21no—
135C23×C137—21no—
136C23×C137—21no—
137C23×C137—42no—
138C23×C137—42no—
139C23×C137—81no—
140C23×C137C2×D411yescharacteristic
141D4×C137—21no—
142D4×C137—21no—
143D4×C137—21no—
144D4×C137—42no—
145D4×C137#4112yes—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—81no—
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2411yescharacteristic
153C8×C137—21no—
154C8×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—22no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—84no—
164C22×C137#13512yes—
165C22×C137C22×D411yescharacteristic
166C2×D137—21no—
167C2×D137—21no—
168C2×D137—21no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C4×C137—21no—
173C4×C137—41no—
174C4×C137—41no—
175C4×C137C2×#4111yescharacteristic
176C4×C137H(5,𝔽2)11yescharacteristic
177Dic137—21no—
178Dic137—21no—
179Dic137—21no—
180Dic137—41no—
181Dic137—41no—
182Dic137—41no—
183C2×C137—22no—
184C2×C137—42no—
185C2×C137—81no—
186C2×C137#54611yescharacteristic
187C2×C137C2×#13511yescharacteristic
188C2×C137#57611yescharacteristic
189D137—41no—
190D137—81no—
191C137#329711yescharacteristic, complemented, Sylow
192#3297—1371nomaximal, Sylow
193#553—1371no—
194#546—1371no—
195#488—1371no—
196#484—1371no—
197#482—1371no—
198#478—1371no—
199#466—1371no—
200#465—1371no—
201#463—1371no—
202#449—1371no—
203#447—1371no—
204C4×D8—1371no—
205C8×D4—1371no—
206#416—1371no—
207C2×#102D13711yescharacteristic, complemented
208C2×#41—1371no—
209C22×D4C2×D13711yescharacteristic
210C2×SD16—1371no—
211C2×D8—1371no—
212C2×D8—1371no—
213C22×C8—1371no—
214C22×C8C2×D13711yescharacteristic
215#126—2741no—
216#125—1371no—
217#124—1371no—
218#123—2741no—
219#123—2741no—
220C22⋊Q8—1371no—
221#121—2741no—
222#121—2741no—
223#121—1371no—
224#121—1371no—
225#121—1371no—
226#120—2741no—
227C4×D4—2741no—
228C4×D4—1371no—
229C4×D4—1371no—
230C4×D4—1371no—
231#117—1371no—
232C2×C4⋊C4C2×D13711yescharacteristic
233C2×C22⋊C4—2741no—
234#107—1371no—
235#107—1371no—
236#106—1371no—
237C4⋊C8—1371no—
238#103—1371no—
239#103—1371no—
240#103—1371no—
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102C2×D13711yescharacteristic
245#102C2×D13711yescharacteristic
246#102C2×D13711yescharacteristic
247#102C2×D13711yescharacteristic
248C22⋊C8—1371no—
249C22⋊C8—1371no—
250C4×C8—1371no—
251C24—21no—
252#41—2742no—
253#41—5481no—
254C2×Q8—1371no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—2741no—
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4C22×D13711yescharacteristic
265C2×D4C22×D13711yescharacteristic
266C22×C4—21no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4—1371no—
271C22×C4C22×D13711yescharacteristic
272SD16—5481no—
273D8—2742no—
274D8—5481no—
275C2×C8—41no—
276C2×C8—5481no—
277C2×C8—1371no—
278C2×C8—1371no—
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4C22×D13711yescharacteristic
289C4⋊C4C22×D13711yescharacteristic
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4—1371no—
300C22⋊C4—1371no—
301C42—2741no—
302C42—1371no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—2741no—
309C23—1371no—
310C23D4×D13711yescharacteristic
311Q8—2741no—
312Q8—2741no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—2742no—
318D4—2741no—
319D4—2741no—
320D4—5482no—
321D4—5482no—
322D4—10961no—
323D4—10961no—
324D4#1455551012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—81no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—5482no—
336C2×C4—10961no—
337C2×C4—10961no—
338C2×C4—1371no—
339C2×C4—1371no—
340C2×C4—1371no—
341C2×C4D4×D13711yescharacteristic
342C2×C4C23×D13711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2741no—
346C8—2741no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—84no—
356C22—2741no—
357C22—2741no—
358C22—2741no—
359C22—5481no—
360C22—5481no—
361C22—5481no—
362C22#1798410912yes—
363C22C2×D4×D13711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—2741no—
368C4—2741no—
369C4—2741no—
370C4—5481no—
371C4—5481no—
372C4—5481no—
373C4C2×#1455551011yescharacteristic
374C4#1798413311yescharacteristic
375C2—22no—
376C2—42no—
377C2—81no—
378C2—5481no—
379C2—10961no—
380C2#2257804311yescharacteristic
381C2C2×#1798410911yescharacteristic
382C2#2257820411yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
hypercenterC2×#102D137
hypocenterC137#3297
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22578101C2semidirect
2. #22578047C2semidirect
3. #22577756C2semidirect
4. #22577738C2semidirect
5. #22577726C2semidirect
6. #22577705C2semidirect
7. #22577638C2semidirect
8. #22577632C2semidirect
9. #22577617C2semidirect
10. #22577549C2semidirect
11. #22577537C2semidirect
12. #22577483C2semidirect
13. C8×C22⋊D137C2semidirect
14. #22577396C2semidirect
15. C2×#102×C137C2semidirect
17. C2×#14555499C22semidirect
20. C2×C8×D137C22semidirect
22. #17984079C22semidirect
27. #17984063C22semidirect
34. #17984050C22semidirect
37. #17984046C22semidirect
39. #17984021C22semidirect
40. #17984021C22semidirect
41. #17984020C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984011C22semidirect
45. #17984011C22semidirect
46. #17984010C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
191. C137#3297semidirect
207. C2×#102D137semidirect

Maximal subgroups

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

subgroupindex
#225781012
#225780472
#225777562
#225777382
#225777262
#225777052
#225776382
#225776322
#225776172
#225775492
#225775372
#225774832
C8×C22⋊D1372
#225773962
C2×#102×C1372
#3297137