Finite Group Databasebeta

← #31790466

Subgroups — #31790466

Order 17536.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic, complemented
2C2×#17984096C211yesmaximal, characteristic, complemented
3#22577982C211yesmaximal, characteristic, complemented
4#22577969C211yesmaximal, characteristic, complemented
5#22577655C211yesmaximal, characteristic, complemented
6#22577649C211yesmaximal, characteristic, complemented
7#22577648C211yesmaximal, characteristic, complemented
8#22577637C211yesmaximal, characteristic, complemented
9#22577636C211yesmaximal, characteristic, complemented
10#22577633C211yesmaximal, characteristic, complemented
11#22577630C211yesmaximal, characteristic, complemented
12#22577442C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#469×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984112—24no—
20C2×#14555499C2211yescharacteristic
21C2×#14555498C2211yescharacteristic
22C2×#14555495C2211yescharacteristic
23#17984096—24no—
24C2×#14555493C2211yescharacteristic
25C2×#14555493C2211yescharacteristic, complemented
26C22×#11764040C2211yescharacteristic, complemented
27C2×C8×D137C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic, complemented
29#17984084—21no—
30#17984074—21no—
31#17984068C2211yescharacteristic, complemented
32#17984067C2211yescharacteristic, complemented
33#17984065—21no—
34#17984063C2211yescharacteristic
35#17984062C2211yescharacteristic
36#17984061—21no—
37#17984060—21no—
38#17984059—21no—
39#17984059C2211yescharacteristic
40#17984059C2211yescharacteristic, complemented
41#17984058—21no—
42#17984056—21no—
43#17984046C2211yescharacteristic
44C4⋊C4×D137C2211yescharacteristic
45#17984022C2211yescharacteristic, complemented
46#17984021C2211yescharacteristic, complemented
47#17984019C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984011C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic
51#17984009C2211yescharacteristic
52#17984009C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55C2×M16×C137C2211yescharacteristic, complemented
56#121×C137C2211yescharacteristic
57#121×C137C2211yescharacteristic, complemented
58#106×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555510—22no—
63#14555510—22no—
64#14555509—22no—
65#14555509—22no—
66C2×Dic274D411yescharacteristic
67C2×Dic274C2311yescharacteristic
68C2×D548—21no—
69C2×D548D411yescharacteristic
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137D411yescharacteristic
77C2×C22⋊D137D411yescharacteristic
78D4×D137—22no—
79D4×D137—22no—
80C22×Dic137—21no—
81C2×C4×D137—21no—
82C2×C4×D137C2311yescharacteristic
83C2×C4×D137C2311yescharacteristic
84#14555499—22no—
85#14555498—22no—
86#14555495—22no—
87#14555493—22no—
88#14555493D414yescomplemented
89C2×#11764040—22no—
90C2×#11764040C2311yescharacteristic
91C2×#11764040C2311yescharacteristic
92C8×D137—22no—
93C8×D137—22no—
94C4⋊Dic137—21no—
95C4⋊Dic137C2311yescharacteristic
96C4⋊Dic137C2311yescharacteristic
97#14555489—21no—
98#14555489—21no—
99C22⋊Dic137—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106C4×Dic137—21no—
107C2×D4×C137—21no—
108C2×D4×C137—21no—
109C2×D4×C137C2311yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C22×C4×C137C2311yescharacteristic
112D8×C137—22no—
113M16×C137—22no—
114C2×C8×C137C2311yescharacteristic
115C2×C8×C137C2311yescharacteristic
116C4⋊C4×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22×D137—21no—
121C22×D137—42no—
122C22×D137—81no—
123C22×D137—81no—
124C22×D137C2×D411yescharacteristic
125C22×D137C2×D411yescharacteristic
126Dic274—22no—
127Dic274—21no—
128Dic274—21no—
129D548—22no—
130D548—21no—
131D548—21no—
132D548—81no—
133C22⋊D137—22no—
134C22⋊D137—22no—
135C22⋊D137—42no—
136C22⋊D137—42no—
137C22⋊D137—81no—
138C22⋊D137—81no—
139C22⋊D137—81no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—42no—
144C2×Dic137C2×D411yescharacteristic
145C2×Dic137C2×D411yescharacteristic
146C4×D137—21no—
147C4×D137—21no—
148C4×D137—81no—
149C4×D137C2×D412yes—
150C4×D137#4112yes—
151#11764040—21no—
152#11764040C2×D412yes—
153C23×C137—21no—
154C23×C137—21no—
155C23×C137C2×D411yescharacteristic
156D4×C137—21no—
157D4×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—81no—
161D4×C137—81no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2411yescharacteristic
168C8×C137—21no—
169C8×C137C2×D412yes—
170C22×C137—21no—
171C22×C137—21no—
172C22×C137—21no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137C22×D411yescharacteristic
180C2×D137—21no—
181C2×D137—21no—
182C2×D137—21no—
183C2×D137—21no—
184C2×D137—21no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—84no—
189C2×D137#12112yes—
190C4×C137—21no—
191C4×C137—41no—
192C4×C137—41no—
193C4×C137C22×D411yescharacteristic
194C4×C137C2×#4111yescharacteristic
195Dic137—21no—
196Dic137—41no—
197Dic137—41no—
198Dic137—41no—
199Dic137#12112yes—
200C2×C137—41no—
201C2×C137—81no—
202C2×C137—81no—
203C2×C137C2×#12111yescharacteristic
204C2×C137#57611yescharacteristic
205C2×C137#57711yescharacteristic
206D137—22no—
207D137—41no—
208D137—81no—
209C137#306211yescharacteristic, complemented, Sylow
210#3062—1371nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1371no—
212C2×#131—1371no—
213#537—1371no—
214#535—1371no—
215#469—1371no—
216#469D13711yescharacteristic, complemented
217#468—1371no—
218#468—1371no—
219#466—1371no—
220#466—1371no—
221#465—1371no—
222#465—1371no—
223#428—1371no—
224#416—1371no—
225C2×#102—1371no—
226C2×#41—1371no—
227C2×#41—1371no—
228C22×D4—1371no—
229AGL(1,ℤ/8)—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×D8—1371no—
233C2×D8C2×D13711yescharacteristic
234#131—2744no—
235C2×M16—1371no—
236C2×M16—1371no—
237C2×M16C2×D13711yescharacteristic, complemented
238C22×C8—1371no—
239C22×C8—1371no—
240C22×C8—1371no—
241#127—2741no—
242#124—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8—1371no—
246#121—2741no—
247#121—2741no—
248#121—2741no—
249#121—1371no—
250#121—1371no—
251#121—1371no—
252#121—1371no—
253#121C2×D13711yescharacteristic
254#121C2×D13711yescharacteristic, complemented
255#120—2741no—
256#120—2741no—
257#117—1371no—
258C2×C4⋊C4—1371no—
259#107—1371no—
260#107—1371no—
261#106—1371no—
262#106C2×D13711yescharacteristic, complemented
263#103—1371no—
264#103—1371no—
265#102—1371no—
266#102—1371no—
267#102—1371no—
268#102—1371no—
269#102C2×D13711yescharacteristic
270#102C2×D13711yescharacteristic, complemented
271C24—2741no—
272#41—2742no—
273#41—2742no—
274#41—2742no—
275#41—2742no—
276C2×Q8—1371no—
277C2×Q8—1371no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—2741no—
281C2×D4—2741no—
282C2×D4—2741no—
283C2×D4—2741no—
284C2×D4—2741no—
285C2×D4—2742no—
286C2×D4—2742no—
287C2×D4—2741no—
288C2×D4—1371no—
289C2×D4—1371no—
290C2×D4—1371no—
291C2×D4—1371no—
292C2×D4C22×D13711yescharacteristic
293C2×D4C22×D13711yescharacteristic
294C22×C4—2741no—
295C22×C4—2741no—
296C22×C4—1371no—
297C22×C4—1371no—
298C22×C4C22×D13711yescharacteristic
299SD16—2742no—
300SD16—2742no—
301D8—22no—
302D8—2742no—
303M16—22no—
304M16—2742no—
305M16—1374no—
306C2×C8—2742no—
307C2×C8—2742no—
308C2×C8—2742no—
309C2×C8—1371no—
310C2×C8—1371no—
311C2×C8C22×D13711yescharacteristic
312C2×C8C22×D13711yescharacteristic
313C4⋊C4—2741no—
314C4⋊C4—2741no—
315C4⋊C4—2741no—
316C4⋊C4—1371no—
317C4⋊C4—1371no—
318C4⋊C4C22×D13711yescharacteristic
319C4⋊C4C22×D13711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C22⋊C4—2741no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C42—2741no—
330C23—21no—
331C23—21no—
332C23—2741no—
333C23—5482no—
334C23—10961no—
335C23—10961no—
336C23—1371no—
337C23—1371no—
338C23D4×D13711yescharacteristic
339Q8—2742no—
340Q8—2741no—
341Q8—2741no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—2742no—
349D4—2742no—
350D4—2742no—
351D4—2741no—
352D4—2741no—
353D4—5482no—
354D4—5482no—
355D4—10961no—
356D4—10961no—
357D4—10961no—
358D4—10961no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—2741no—
364C2×C4—2741no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—5482no—
369C2×C4—10961no—
370C2×C4—1371no—
371C2×C4—1371no—
372C2×C4—1372no—
373C2×C4—1372no—
374C2×C4D4×D13711yescharacteristic
375C2×C4C23×D13711yescharacteristic
376C8—21no—
377C8—2741no—
378C8—1372no—
379C8D4×D13712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—2741no—
390C22—2741no—
391C22—2741no—
392C22—2741no—
393C22—2741no—
394C22—5481no—
395C22—5481no—
396C22—5481no—
397C22—10964no—
398C22—1372no—
399C22C2×D4×D13711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—2741no—
404C4—5481no—
405C4—5481no—
406C4—5481no—
407C4—1372no—
408C4C2×D4×D13711yescharacteristic
409C4#41×D13711yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—2742no—
414C2—5481no—
415C2—10961no—
416C2#121×D13711yescharacteristic
417C2#2257820711yescharacteristic
418C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3062
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×#17984096C2semidirect
3. #22577982C2semidirect
4. #22577969C2semidirect
5. #22577655C2semidirect
6. #22577649C2semidirect
7. #22577648C2semidirect
8. #22577637C2semidirect
9. #22577636C2semidirect
10. #22577633C2semidirect
11. #22577630C2semidirect
12. #22577442C2semidirect
13. #22577396C2semidirect
14. #102×D137C2semidirect
15. #469×C137C2semidirect
25. C2×#14555493C22semidirect
26. C22×#11764040C22semidirect
28. C2×C8×D137C22semidirect
31. #17984068C22semidirect
32. #17984067C22semidirect
40. #17984059C22semidirect
45. #17984022C22semidirect
46. #17984021C22semidirect
47. #17984019C22semidirect
48. #17984013C22semidirect
49. #17984011C22semidirect
52. #17984009C22semidirect
55. C2×M16×C137C22semidirect
57. #121×C137C22semidirect
58. #106×C137C22semidirect
60. #102×C137C22semidirect
88. #14555493D4semidirect
209. C137#3062semidirect
216. #469D137semidirect
237. C2×M16C2×D137semidirect
254. #121C2×D137semidirect
262. #106C2×D137semidirect
270. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×#179840962
#225779822
#225779692
#225776552
#225776492
#225776482
#225776372
#225776362
#225776332
#225776302
#225774422
#225773962
#102×D1372
#469×C1372
#3062137