Finite Group Databasebeta

← #31789667

Subgroups — #31789667

Order 17536.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984109C211yesmaximal, characteristic
2#22577885C211yesmaximal, characteristic, complemented
3#22577882C211yesmaximal, characteristic, complemented
4#22577513C211yesmaximal, characteristic, complemented
5#22577512C211yesmaximal, characteristic, complemented
6#22577508C211yesmaximal, characteristic, complemented
7#22577507C211yesmaximal, characteristic, complemented
8C4×#14555499C211yesmaximal, characteristic, complemented
9SD16×Dic137C211yesmaximal, characteristic, complemented
10#22577490C211yesmaximal, characteristic, complemented
11#106×D137C211yesmaximal, characteristic, complemented
12#22577403C211yesmaximal, characteristic, complemented
13#22577401C211yesmaximal, characteristic, complemented
14#22577357C211yesmaximal, characteristic, complemented
15C4×SD16×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18#17984109C418yescomplemented
19C2×#14555500C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21C2×#14555495C2211yescharacteristic
22C2×#14555493C2211yescharacteristic
23C2×C8×D137C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25C4×Dic274C2211yescharacteristic
26#17984053C2211yescharacteristic
27C4×D548—21no—
28C4×D548C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30#17984050—21no—
31#17984050C2211yescharacteristic
32C4×C22⋊D137—21no—
33#17984048—21no—
34#17984045—21no—
35#17984045C2211yescharacteristic
36#17984044—21no—
37C2×#14555489—21no—
38C4⋊C4×D137—21no—
39C4⋊C4×D137C2211yescharacteristic
40C4⋊C4×D137C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17983999C2211yescharacteristic
51#17983998C2211yescharacteristic
52C8×Dic137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C4×C8×C137C2211yescharacteristic
60#14555511—22no—
61#14555511C2×C414yescomplemented
62#14555509—22no—
63#14555509C2×C414yescomplemented
64C2×Dic274C2311yescharacteristic
65C2×D548—21no—
66C2×D548C2311yescharacteristic
67C2×C22⋊D137—21no—
68C22×Dic137—21no—
69C22×Dic137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76#14555500C2×C414yescomplemented
77#14555499C2×C414yescomplemented
78#14555495C2×C414yescomplemented
79#14555493—22no—
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C8×D137C2×C414yescomplemented
83C4⋊Dic137—21no—
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489D411yescharacteristic
91#14555489D411yescharacteristic
92C22⋊Dic137—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104SD16×C137C2×C414yescomplemented
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C42×C137—21no—
112C42×C137C2311yescharacteristic
113C22×D137—21no—
114C22×D137C2×D411yescharacteristic
115Dic274—21no—
116Dic274C22×C412yes—
117D548—24no—
118D548—21no—
119D548C22×C412yes—
120C22⋊D137—24no—
121C2×Dic137—21no—
122C2×Dic137—24no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—81no—
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C4×D137—24no—
130C4×D137—81no—
131C4×D137—81no—
132C4×D137—81no—
133C4×D137—81no—
134C4×D137C22×C412yes—
135C4×D137#4112yes—
136#11764040—21no—
137#11764040C22×C412yes—
138C23×C137—21no—
139Q8×C137—21no—
140Q8×C137C22×C412yes—
141D4×C137—21no—
142D4×C137C22×C412yes—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—81no—
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137C22×C412yes—
152C22×C137—22no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—22no—
159C2×D137—41no—
160C2×D137—41no—
161C2×D137C4×D412yes—
162C4×C137—22no—
163C4×C137—21no—
164C4×C137—21no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137C23×C411yescharacteristic
168C4×C137C2×#4111yescharacteristic
169Dic137—22no—
170Dic137—21no—
171Dic137—21no—
172Dic137—41no—
173Dic137—41no—
174Dic137C4×D412yes—
175C2×C137—42no—
176C2×C137C2×C4×D411yescharacteristic
177C2×C137#57711yescharacteristic
178C2×C137#57711yescharacteristic
179D137—22no—
180D137—42no—
181C137#295211yescharacteristic, complemented, Sylow
182#2952—1371nomaximal, Sylow
183C2×#135—1371no—
184#520—1371no—
185#520—1371no—
186#442—1371no—
187#442—1371no—
188#441—1371no—
189#441—1371no—
190C4×SD16—1371no—
191C4×SD16—1371no—
192C4×SD16—1371no—
193C4×SD16D13711yescharacteristic, complemented
194C2×#106—1371no—
195#417—1371no—
196#417—1371no—
197#405—1371no—
198C2×#41—1371no—
199C2×#41—1371no—
200#135—1378no—
201C2×Q16—1371no—
202C2×SD16—1371no—
203C2×SD16C2×D13711yescharacteristic
204C2×D8—1371no—
205C2×M16—1371no—
206C22×C8—1371no—
207C4×Q8—1371no—
208C4×Q8—1371no—
209C4×Q8—1371no—
210C4×Q8C2×D13711yescharacteristic
211C4×D4—2741no—
212C4×D4—2741no—
213C4×D4—2741no—
214C4×D4—2741no—
215C4×D4—1371no—
216C4×D4—1371no—
217C4×D4—1371no—
218C4×D4C2×D13711yescharacteristic
219#117—2741no—
220#117—2741no—
221#117—1371no—
222C2×C4⋊C4—2741no—
223C2×C4⋊C4—2741no—
224C2×C4⋊C4—1371no—
225C2×C4⋊C4—1371no—
226#106—1371no—
227#106—1371no—
228#106—1371no—
229#106C2×D13711yescharacteristic
230#103—1371no—
231#103—1371no—
232#103—1371no—
233#103C2×D13711yescharacteristic
234#102—1371no—
235#102—1371no—
236#102—1371no—
237#102C2×D13711yescharacteristic
238#97—1371no—
239#97—1371no—
240C4×C8—1371no—
241C4×C8C2×D13711yescharacteristic
242#41—2742no—
243#41—2742no—
244#41—1374no—
245#41—1374no—
246C2×Q8—1371no—
247C2×Q8C22×D13711yescharacteristic
248C2×D4—2741no—
249C2×D4—2741no—
250C2×D4—1371no—
251C2×D4C22×D13711yescharacteristic
252C22×C4—21no—
253C22×C4—2741no—
254C22×C4—2741no—
255C22×C4—2741no—
256C22×C4—2741no—
257C22×C4—2741no—
258C22×C4—2741no—
259C22×C4—2741no—
260C22×C4—1371no—
261Q16—1374no—
262SD16—1374no—
263SD16C4×D13714yescomplemented
264D8—1374no—
265M16—2742no—
266C2×C8—1371no—
267C2×C8—1371no—
268C2×C8—1374no—
269C2×C8C22×D13711yescharacteristic
270C2×C8C22×D13711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4C22×D13711yescharacteristic
282C4⋊C4C22×D13711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—1371no—
288C22⋊C4—1371no—
289C42—21no—
290C42—2741no—
291C42—2741no—
292C42—2741no—
293C42—1371no—
294C42C22×D13711yescharacteristic
295C23—21no—
296C23—2741no—
297C23—1371no—
298Q8—21no—
299Q8—2741no—
300Q8—1372no—
301Q8C2×C4×D13712yes—
302D4—21no—
303D4—2744no—
304D4—2744no—
305D4—2741no—
306D4—1372no—
307D4C2×C4×D13712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—2744no—
313C2×C4—2741no—
314C2×C4—2744no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—10961no—
318C2×C4—10961no—
319C2×C4—10961no—
320C2×C4—10961no—
321C2×C4—10961no—
322C2×C4—1371no—
323C2×C4—1371no—
324C2×C4—1371no—
325C2×C4—1372no—
326C2×C4—1372no—
327C2×C4D4×D13711yescharacteristic
328C2×C4D4×D13711yescharacteristic
329C2×C4C23×D13711yescharacteristic
330C8—21no—
331C8—2741no—
332C8—1372no—
333C8C2×C4×D13712yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—2741no—
338C22—2741no—
339C22—2742no—
340C22—5481no—
341C22—5481no—
342C22—1372no—
343C22C2×D4×D13711yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—2741no—
350C4—2741no—
351C4—2742no—
352C4—5481no—
353C4—5481no—
354C4—1372no—
355C4C22×C4×D13711yescharacteristic
356C4#41×D13711yescharacteristic
357C2—42no—
358C2—2742no—
359C2—5482no—
360C2C4×D4×D13711yescharacteristic
361C2#2257820911yescharacteristic
362C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
FittingC4×SD16×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984109C2
cosocleC2×C4×C137C24
hypercenterC4×SD16D137
hypocenterC137#2952
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #22577885C2semidirect
3. #22577882C2semidirect
4. #22577513C2semidirect
5. #22577512C2semidirect
6. #22577508C2semidirect
7. #22577507C2semidirect
8. C4×#14555499C2semidirect
9. SD16×Dic137C2semidirect
10. #22577490C2semidirect
11. #106×D137C2semidirect
12. #22577403C2semidirect
13. #22577401C2semidirect
14. #22577357C2semidirect
15. C4×SD16×C137C2semidirect
18. #17984109C4semidirect
61. #14555511C2×C4semidirect
63. #14555509C2×C4semidirect
76. #14555500C2×C4semidirect
77. #14555499C2×C4semidirect
78. #14555495C2×C4semidirect
82. C8×D137C2×C4semidirect
104. SD16×C137C2×C4semidirect
181. C137#2952semidirect
193. C4×SD16D137semidirect
263. SD16C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841092
#225778852
#225778822
#225775132
#225775122
#225775082
#225775072
C4×#145554992
SD16×Dic1372
#225774902
#106×D1372
#225774032
#225774012
#225773572
C4×SD16×C1372
#2952137