Finite Group Databasebeta

← #31792609

Subgroups — #31792609

Order 17536.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578108C211yesmaximal, characteristic, complemented
2D4×C22⋊D137C211yesmaximal, characteristic
3#22577748C211yesmaximal, characteristic, complemented
4#22577721C211yesmaximal, characteristic, complemented
5#22577720C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577632C211yesmaximal, characteristic, complemented
8#22577631C211yesmaximal, characteristic, complemented
9#22577602C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577554C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13C8×C22⋊D137C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15C2×#102×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137—21no—
17C2×D4×D137C2211yescharacteristic
18C2×#14555499C2211yescharacteristic, complemented
19C2×#14555498C2211yescharacteristic
20C2×#14555498C2211yescharacteristic, complemented
21C2×C8×D137C2211yescharacteristic, complemented
22#17984087C2211yescharacteristic, complemented
23#17984084C2211yescharacteristic
24#17984078—21no—
25#17984068C2211yescharacteristic, complemented
26#17984067—21no—
27#17984067C2211yescharacteristic, complemented
28#17984066—21no—
29#17984062C2211yescharacteristic
30#17984060—21no—
31#17984059C2211yescharacteristic
32#17984057—21no—
33#17984056—21no—
34#17984053C2211yescharacteristic, complemented
35D4×Dic137C2211yescharacteristic
36C4×C22⋊D137—21no—
37C4×C22⋊D137C2211yescharacteristic
38C4⋊C4×D137C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic, complemented
54C2×C4⋊C4×C137C2211yescharacteristic, complemented
55#102×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C2×Dic274—21no—
61C2×Dic274C2311yescharacteristic
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—44no—
69C2×C22⋊D137D411yescharacteristic
70D4×D137—22no—
71D4×D137—41no—
72C22×Dic137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75#14555499—41no—
76#14555498—22no—
77#14555498—41no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C4⋊Dic137—21no—
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137D411yescharacteristic
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96C4×Dic137—21no—
97C4×Dic137C2311yescharacteristic
98C24×C137—21no—
99C2×D4×C137—22no—
100C2×D4×C137—21no—
101C2×D4×C137—41no—
102C2×D4×C137C2311yescharacteristic
103C2×D4×C137C2311yescharacteristic
104C22×C4×C137—21no—
105C22×C4×C137C2311yescharacteristic
106C2×C8×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137—21no—
110C4⋊C4×C137C2311yescharacteristic
111C4⋊C4×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116C22×D137C2×D411yescharacteristic
117Dic274—21no—
118Dic274—21no—
119Dic274—81no—
120D548—21no—
121D548—21no—
122C22⋊D137—42no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C22⋊D137—42no—
126C22⋊D137—81no—
127C22⋊D137SD1614yescomplemented
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
131C2×Dic137—42no—
132C2×Dic137C2×D411yescharacteristic
133C2×Dic137C2×D411yescharacteristic
134C2×Dic137C2×D411yescharacteristic
135C4×D137—21no—
136C4×D137—21no—
137C4×D137—81no—
138#11764040—21no—
139#11764040—21no—
140C23×C137—21no—
141C23×C137—21no—
142C23×C137—42no—
143C23×C137—42no—
144C23×C137—81no—
145C23×C137C2×D411yescharacteristic
146D4×C137—21no—
147D4×C137—21no—
148D4×C137—21no—
149D4×C137—42no—
150D4×C137#4112yes—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—21no—
155C2×C4×C137—81no—
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C8×C137—21no—
159C8×C137—21no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—22no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—84no—
169C22×C137C2×SD1612yes—
170C22×C137C22×D411yescharacteristic
171C2×D137—21no—
172C2×D137—21no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—41no—
176C2×D137—84no—
177C2×D137C2×SD1612yes—
178C4×C137—21no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137C2×#4111yescharacteristic
182C4×C137H(5,𝔽2)11yescharacteristic
183Dic137—21no—
184Dic137—21no—
185Dic137—41no—
186Dic137—41no—
187Dic137—41no—
188Dic137C2×SD1612yes—
189C2×C137—22no—
190C2×C137—42no—
191C2×C137—81no—
192C2×C137#54611yescharacteristic
193C2×C137C22×SD1611yescharacteristic
194C2×C137#57611yescharacteristic
195D137—22no—
196D137—81no—
197C137#329511yescharacteristic, complemented, Sylow
198#3295—1371nomaximal, Sylow
199#554—1371no—
200D42—1371no—
201#486—1371no—
202#481—1371no—
203#481—1371no—
204#476—1371no—
205#465—1371no—
206#465—1371no—
207#460—1371no—
208#450—1371no—
209#450—1371no—
210C4×SD16—1371no—
211C8×D4—1371no—
212C2×#102—1371no—
213C2×#102D13711yescharacteristic, complemented
214C22×D4—2741no—
215C22×D4—1371no—
216C22×D4C2×D13711yescharacteristic
217C2×SD16—1371no—
218C2×SD16—1371no—
219C2×SD16—1371no—
220C22×C8—1371no—
221C22×C8C2×D13711yescharacteristic, complemented
222C4⋊Q8—1371no—
223#127—1371no—
224#125—2741no—
225C22⋊Q8—2741no—
226C22⋊Q8—2741no—
227C22⋊Q8—1371no—
228C22⋊Q8—1371no—
229#121—2741no—
230#121—1371no—
231#121—1371no—
232#120—2741no—
233#120—2741no—
234C4×Q8—1371no—
235C4×D4—2741no—
236C4×D4—1371no—
237C4×D4—1371no—
238C2×C4⋊C4—1371no—
239C2×C4⋊C4C2×D13711yescharacteristic, complemented
240#106—1371no—
241#106—1371no—
242#106—1371no—
243C4⋊C8—1371no—
244#103—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102C2×D13711yescharacteristic
251#102C2×D13711yescharacteristic
252#102C2×D13711yescharacteristic, complemented
253#102C2×D13711yescharacteristic, complemented
254C22⋊C8—1371no—
255C22⋊C8—1371no—
256C4×C8—1371no—
257C24—21no—
258C24—2741no—
259C2×Q8—2741no—
260C2×Q8—1371no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—2742no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—2741no—
268C2×D4—2741no—
269C2×D4—2741no—
270C2×D4—5481no—
271C2×D4—5484no—
272C2×D4—1371no—
273C2×D4—1371no—
274C2×D4C22×D13711yescharacteristic
275C2×D4C22×D13711yescharacteristic
276C22×C4—21no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—1371no—
280C22×C4C22×D13711yescharacteristic
281SD16—2742no—
282SD16—5481no—
283SD16—5481no—
284C2×C8—41no—
285C2×C8—5481no—
286C2×C8—1371no—
287C2×C8—1371no—
288C2×C8C22×D13711yescharacteristic
289C2×C8C22×D13711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2741no—
292C4⋊C4—2741no—
293C4⋊C4—2741no—
294C4⋊C4—2741no—
295C4⋊C4—2741no—
296C4⋊C4—1371no—
297C4⋊C4—1371no—
298C4⋊C4—1371no—
299C4⋊C4C22×D13711yescharacteristic
300C4⋊C4C22×D13711yescharacteristic
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4—2741no—
305C22⋊C4—2741no—
306C22⋊C4—1371no—
307C22⋊C4—1371no—
308C42—2741no—
309C42—1371no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—2741no—
316C23—5482no—
317C23—5482no—
318C23—10961no—
319C23—1371no—
320C23D4×D13711yescharacteristic
321Q8—2741no—
322Q8—2741no—
323Q8—10961no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—2741no—
329D4—2741no—
330D4—5482no—
331D4—5482no—
332D4—5482no—
333D4—5482no—
334D4—10961no—
335D4—1374no—
336D4#1455551012yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—81no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—2741no—
347C2×C4—5482no—
348C2×C4—10961no—
349C2×C4—1371no—
350C2×C4—1371no—
351C2×C4—1371no—
352C2×C4D4×D13711yescharacteristic
353C2×C4C23×D13711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2741no—
357C8—2741no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—2741no—
368C22—2741no—
369C22—5481no—
370C22—5481no—
371C22—5481no—
372C22—10964no—
373C22—1372no—
374C22SD16×D13712yes—
375C22C2×D4×D13711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—2741no—
380C4—2741no—
381C4—5481no—
382C4—5481no—
383C4—5481no—
384C4—1372no—
385C4C2×#1455551011yescharacteristic
386C4#1798413311yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—2742no—
391C2—10961no—
392C2#2257804311yescharacteristic
393C2C2×SD16×D13711yescharacteristic
394C2#2257820411yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
prime radicalD4×C22⋊D137C2
hypercenterC2×#102D137
hypocenterC137#3295
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. #22578108C2semidirect
3. #22577748C2semidirect
4. #22577721C2semidirect
5. #22577720C2semidirect
6. #22577691C2semidirect
7. #22577632C2semidirect
8. #22577631C2semidirect
9. #22577602C2semidirect
10. #22577555C2semidirect
11. #22577554C2semidirect
12. #22577494C2semidirect
13. C8×C22⋊D137C2semidirect
14. #102×D137C2semidirect
15. C2×#102×C137C2semidirect
18. C2×#14555499C22semidirect
20. C2×#14555498C22semidirect
21. C2×C8×D137C22semidirect
22. #17984087C22semidirect
25. #17984068C22semidirect
27. #17984067C22semidirect
34. #17984053C22semidirect
38. C4⋊C4×D137C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984019C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984010C22semidirect
46. #17984009C22semidirect
47. #17984008C22semidirect
48. #17984008C22semidirect
49. C22⋊#11764040C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
53. C22×C8×C137C22semidirect
54. C2×C4⋊C4×C137C22semidirect
57. #102×C137C22semidirect
58. #102×C137C22semidirect
127. C22⋊D137SD16semidirect
197. C137#3295semidirect
213. C2×#102D137semidirect
221. C22×C8C2×D137semidirect
239. C2×C4⋊C4C2×D137semidirect
252. #102C2×D137semidirect
253. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781082
D4×C22⋊D1372
#225777482
#225777212
#225777202
#225776912
#225776322
#225776312
#225776022
#225775552
#225775542
#225774942
C8×C22⋊D1372
#102×D1372
C2×#102×C1372
#3295137