Finite Group Databasebeta

← #31792626

Subgroups — #31792626

Order 17536.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578093C211yesmaximal, characteristic, complemented
2#22578049C211yesmaximal, characteristic
3#22577758C211yesmaximal, characteristic, complemented
4#22577734C211yesmaximal, characteristic, complemented
5#22577728C211yesmaximal, characteristic, complemented
6#22577704C211yesmaximal, characteristic, complemented
7#22577644C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577626C211yesmaximal, characteristic, complemented
10#22577567C211yesmaximal, characteristic, complemented
11#22577556C211yesmaximal, characteristic, complemented
12C4×#14555499C211yesmaximal, characteristic, complemented
13#22577400C211yesmaximal, characteristic, complemented
14C2×#17984010C211yesmaximal, characteristic, complemented
15C8×D4×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×Dic548C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21#17984081—21no—
22#17984080C2211yescharacteristic, complemented
23#17984078—21no—
24#17984076C2211yescharacteristic
25#17984073—21no—
26#17984069—21no—
27#17984067C2211yescharacteristic
28#17984067C2211yescharacteristic
29#17984067C2211yescharacteristic, complemented
30#17984062C2211yescharacteristic
31#17984061—21no—
32#17984056—21no—
33C4×Dic274C2211yescharacteristic, complemented
34C4×D548C2211yescharacteristic
35#17984047C2211yescharacteristic, complemented
36#17984044—21no—
37C2×C4⋊Dic137C2211yescharacteristic
38C22⋊C4×D137—21no—
39#17984022C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984020C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984013C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic
52C22×C8×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic, complemented
54C4×D4×C137C2211yescharacteristic
55C4⋊C8×C137C2211yescharacteristic, complemented
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic, complemented
58C4×C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62C2×Dic274C2311yescharacteristic
63C2×Dic274C2311yescharacteristic
64C2×D548—22no—
65C2×D548—21no—
66C2×D548—41no—
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74Dic548—41no—
75#14555499—22no—
76#14555499—41no—
77C4⋊Dic137—21no—
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C2×D4×C137D411yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C2×C8×C137—41no—
102C2×C8×C137—41no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137D411yescharacteristic
108C22⋊C4×C137D411yescharacteristic
109C22⋊C4×C137D411yescharacteristic
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116Dic274—21no—
117Dic274—21no—
118Dic274—21no—
119Dic274—21no—
120D548—21no—
121D548—21no—
122D548—21no—
123D548—42no—
124D548#4112yes—
125C22⋊D137—42no—
126C22⋊D137—81no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
131C2×Dic137—21no—
132C2×Dic137—21no—
133C2×Dic137—42no—
134C4×D137—42no—
135C4×D137—81no—
136C23×C137C2×D411yescharacteristic
137C23×C137C2×D411yescharacteristic
138D4×C137—22no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137C2×D411yescharacteristic
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C8×C137—21no—
149C8×C137—21no—
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137#13512yes—
158C22×C137C22×D411yescharacteristic
159C2×D137—22no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—84no—
166C4×C137—21no—
167C4×C137—21no—
168C4×C137—21no—
169C4×C137—21no—
170C4×C137C2×#4111yescharacteristic
171C4×C137H(5,𝔽2)11yescharacteristic
172Dic137—41no—
173Dic137—41no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—22no—
179C2×C137—41no—
180C2×C137#54611yescharacteristic
181C2×C137C2×#13511yescharacteristic
182C2×C137#57711yescharacteristic
183D137—42no—
184D137—81no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483—1371no—
191#482—1371no—
192#478—1371no—
193#467—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198C4×SD16—1371no—
199C8×D4D13711yescharacteristic, complemented
200#416—1371no—
201C2×#102—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C22×C8C2×D13711yescharacteristic
208C22×C8C2×D13711yescharacteristic, complemented
209#126—2741no—
210#125—2741no—
211#125—1371no—
212#124—1371no—
213#123—2741no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8—1371no—
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—1371no—
222C4×D4—1371no—
223C4×D4C2×D13711yescharacteristic
224#117—2741no—
225#117—1371no—
226C2×C4⋊C4—1371no—
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106—1371no—
230#106—1371no—
231C4⋊C8C2×D13711yescharacteristic, complemented
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102—1371no—
242C22⋊C8C2×D13711yescharacteristic
243C22⋊C8C2×D13711yescharacteristic, complemented
244C4×C8C2×D13711yescharacteristic, complemented
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8—1371no—
250C2×D4—2742no—
251C2×D4—2741no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—5481no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4D54811yescharacteristic
258C22×C4—2741no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4C22×D13711yescharacteristic
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D13711yescharacteristic
269C2×C8C22×D13711yescharacteristic
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4—1371no—
283C4⋊C4D54811yescharacteristic
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4D54811yescharacteristic
293C22⋊C4D54811yescharacteristic
294C42—2741no—
295C42—2741no—
296C42C22×D13711yescharacteristic
297C23—2741no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23C2×D54811yescharacteristic
303C23C2×D54811yescharacteristic
304Q8—2741no—
305Q8—2741no—
306Q8—2741no—
307Q8—2741no—
308D4—22no—
309D4—2741no—
310D4—2741no—
311D4—2741no—
312D4—5482no—
313D4—5482no—
314D4—10961no—
315D4—1372no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—5482no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4C2×D54811yescharacteristic
330C2×C4C2×D54811yescharacteristic
331C2×C4C2×D54811yescharacteristic
332C2×C4C2×D54811yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—2742no—
344C22—5481no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—10964no—
350C22#1798411012yes—
351C22C22×D54811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—5481no—
357C4—5481no—
358C4—5481no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4#41×D13711yescharacteristic
363C4#1798413311yescharacteristic
364C2—22no—
365C2—41no—
366C2—5482no—
367C2—10961no—
368C2#2257804811yescharacteristic
369C2C2×#1798411011yescharacteristic
370C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC8×D4×C137C2
socleC22×C137C22×D4
prime radical#22578049C2
hypercenterC8×D4D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22578093C2semidirect
3. #22577758C2semidirect
4. #22577734C2semidirect
5. #22577728C2semidirect
6. #22577704C2semidirect
7. #22577644C2semidirect
8. #22577632C2semidirect
9. #22577626C2semidirect
10. #22577567C2semidirect
11. #22577556C2semidirect
12. C4×#14555499C2semidirect
13. #22577400C2semidirect
14. C2×#17984010C2semidirect
15. C8×D4×C137C2semidirect
18. C2×Dic548C22semidirect
22. #17984080C22semidirect
29. #17984067C22semidirect
33. C4×Dic274C22semidirect
35. #17984047C22semidirect
39. #17984022C22semidirect
40. #17984020C22semidirect
41. #17984020C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
45. #17984013C22semidirect
53. C22×C8×C137C22semidirect
55. C4⋊C8×C137C22semidirect
57. C22⋊C8×C137C22semidirect
58. C4×C8×C137C22semidirect
185. C137#3296semidirect
199. C8×D4D137semidirect
208. C22×C8C2×D137semidirect
231. C4⋊C8C2×D137semidirect
243. C22⋊C8C2×D137semidirect
244. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780932
#225780492
#225777582
#225777342
#225777282
#225777042
#225776442
#225776322
#225776262
#225775672
#225775562
C4×#145554992
#225774002
C2×#179840102
C8×D4×C1372
#3296137