Finite Group Databasebeta

← #31790299

Subgroups — #31790299

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic
2C2×SD16×D137C211yesmaximal, characteristic, complemented
3#22577973C211yesmaximal, characteristic, complemented
4#22577875C211yesmaximal, characteristic, complemented
5#22577617C211yesmaximal, characteristic, complemented
6#22577616C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577609C211yesmaximal, characteristic, complemented
9#22577608C211yesmaximal, characteristic, complemented
10#22577607C211yesmaximal, characteristic, complemented
11#22577606C211yesmaximal, characteristic, complemented
12#22577428C211yesmaximal, characteristic, complemented
13#22577410C211yesmaximal, characteristic, complemented
14#22577396C211yesmaximal, characteristic, complemented
15#463×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984112—24no—
20C2×#14555499C2211yescharacteristic
21C2×#14555499C2211yescharacteristic
22C2×#14555498C2211yescharacteristic
23C2×#14555498C2211yescharacteristic
24C2×#14555497C2211yescharacteristic
25SD16×D137—24no—
26C2×#14555495C2211yescharacteristic
27C2×#14555493C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic
29#17984088C2211yescharacteristic, complemented
30#17984086C2211yescharacteristic, complemented
31#17984075C2211yescharacteristic, complemented
32#17984070—21no—
33#17984070—21no—
34#17984065—21no—
35#17984065—21no—
36C4×D548C2211yescharacteristic
37D4×Dic137C2211yescharacteristic
38#17984050C2211yescharacteristic
39C4×C22⋊D137—21no—
40#17984048—21no—
41#17984046C2211yescharacteristic
42#17984045C2211yescharacteristic
43C2×#14555487—21no—
44C22⋊C4×D137—21no—
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984017C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984012C2211yescharacteristic, complemented
51#17984011C2211yescharacteristic, complemented
52#17984011C2211yescharacteristic, complemented
53#17984009C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic
55C2×D8×C137C2211yescharacteristic
56#124×C137C2211yescharacteristic, complemented
57C4×D4×C137C2211yescharacteristic
58C4⋊C8×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555509—22no—
63#14555509—22no—
64C2×Dic274—21no—
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67Q8×D137—22no—
68Q8×D137—22no—
69C2×D548C2311yescharacteristic
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72D4×D137—22no—
73D4×D137D414yes—
74C22×Dic137—21no—
75C22×Dic137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137—21no—
79C2×C4×D137C2311yescharacteristic
80#14555499—22no—
81#14555499—22no—
82#14555498—22no—
83#14555498—22no—
84#14555497—22no—
85#14555495—22no—
86#14555493—22no—
87C2×#11764040C2311yescharacteristic
88C2×#11764040C2311yescharacteristic
89C8×D137—22no—
90C4⋊Dic137—21no—
91C4⋊Dic137—21no—
92C4⋊Dic137C2311yescharacteristic
93#14555489—21no—
94#14555489—21no—
95#14555489D411yescharacteristic
96#14555489D411yescharacteristic
97C22⋊Dic137—21no—
98C22⋊Dic137—21no—
99C22⋊Dic137—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487D411yescharacteristic
104#14555487D411yescharacteristic
105C4×Dic137—21no—
106C4×Dic137C2311yescharacteristic
107C2×Q8×C137C2311yescharacteristic
108C2×D4×C137C2311yescharacteristic
109C2×D4×C137C2311yescharacteristic
110C22×C4×C137—21no—
111SD16×C137—22no—
112D8×C137—22no—
113C2×C8×C137C2311yescharacteristic
114C2×C8×C137C2311yescharacteristic
115C4⋊C4×C137C2311yescharacteristic
116C22⋊C4×C137—21no—
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C42×C137C2311yescharacteristic
120C22×D137—21no—
121C22×D137—24no—
122C22×D137—81no—
123C22×D137—81no—
124C22×D137C2×D411yescharacteristic
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128Dic274—21no—
129Dic274—42no—
130D548—21no—
131D548C2×D412yes—
132C22⋊D137—24no—
133C22⋊D137—42no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—42no—
138C2×Dic137—81no—
139C2×Dic137C2×D411yescharacteristic
140C2×Dic137C2×D411yescharacteristic
141C2×Dic137C2×D411yescharacteristic
142C4×D137—42no—
143C4×D137—81no—
144C4×D137—81no—
145C4×D137C2×D412yes—
146C4×D137#4112yes—
147#11764040—21no—
148#11764040—21no—
149C23×C137—21no—
150C23×C137—21no—
151Q8×C137—21no—
152Q8×C137—21no—
153D4×C137—21no—
154D4×C137—21no—
155D4×C137—21no—
156D4×C137C2×D412yes—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—81no—
160C2×C4×C137C2×D411yescharacteristic
161C2×C4×C137C2×D411yescharacteristic
162C2×C4×C137C2411yescharacteristic
163C8×C137—21no—
164C8×C137—21no—
165C22×C137—22no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137C22×D411yescharacteristic
172C2×D137—21no—
173C2×D137—21no—
174C2×D137—22no—
175C2×D137—41no—
176C2×D137—41no—
177C2×D137—84no—
178C2×D137#12112yes—
179C4×C137—21no—
180C4×C137—21no—
181C4×C137—41no—
182C4×C137—41no—
183C4×C137C22×D411yescharacteristic
184C4×C137C2×#4111yescharacteristic
185Dic137—21no—
186Dic137—21no—
187Dic137—41no—
188Dic137—41no—
189Dic137—41no—
190Dic137#12112yes—
191C2×C137—42no—
192C2×C137—81no—
193C2×C137C2×#12111yescharacteristic
194C2×C137#57711yescharacteristic
195C2×C137#57711yescharacteristic
196D137—22no—
197D137—42no—
198C137#304211yescharacteristic, complemented, Sylow
199#3042—1371nomaximal, Sylow
200C2×AGL(1,ℤ/8)—1371no—
201C22×SD16—1371no—
202#536—1371no—
203#518—1371no—
204#463—1371no—
205#463—1371no—
206#463—1371no—
207#463D13711yescharacteristic, complemented
208#461—1371no—
209#461—1371no—
210#461—1371no—
211#461—1371no—
212#424—1371no—
213#419—1371no—
214#416—1371no—
215C2×#41—1371no—
216C22×Q8—1371no—
217C22×D4—1371no—
218AGL(1,ℤ/8)—2744no—
219C2×SD16—2744no—
220C2×SD16—1371no—
221C2×SD16—1371no—
222C2×SD16—1371no—
223C2×SD16—1371no—
224C2×SD16—1371no—
225C2×SD16C2×D13711yescharacteristic
226C2×D8—1371no—
227C2×D8C2×D13711yescharacteristic
228C2×M16—1371no—
229C22×C8—1371no—
230C4⋊Q8—1371no—
231C4⋊Q8—1371no—
232#124—1371no—
233#124C2×D13711yescharacteristic, complemented
234#123—2741no—
235#123—2741no—
236C22⋊Q8—2741no—
237C22⋊Q8—2741no—
238C4×D4—2741no—
239C4×D4—2741no—
240C4×D4—1371no—
241C4×D4—1371no—
242C4×D4—1371no—
243C4×D4C2×D13711yescharacteristic
244#117—1371no—
245#117—1371no—
246C2×C22⋊C4—2741no—
247C2×C22⋊C4—2741no—
248C4⋊C8—1371no—
249C4⋊C8—1371no—
250C4⋊C8—1371no—
251C4⋊C8C2×D13711yescharacteristic, complemented
252#103—1371no—
253#103—1371no—
254#103—1371no—
255#103—1371no—
256#103—1371no—
257#103C2×D13711yescharacteristic, complemented
258#102—1371no—
259#102C2×D13711yescharacteristic, complemented
260C24—2741no—
261#41—2742no—
262#41—2742no—
263C2×Q8—2741no—
264C2×Q8—2742no—
265C2×Q8—2742no—
266C2×Q8—1371no—
267C2×Q8—1371no—
268C2×Q8C22×D13711yescharacteristic
269C2×D4—2741no—
270C2×D4—2742no—
271C2×D4—2741no—
272C2×D4—1374no—
273C2×D4—1371no—
274C2×D4C22×D13711yescharacteristic
275C2×D4C22×D13711yescharacteristic
276C22×C4—21no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—2741no—
280C22×C4—2741no—
281C22×C4—2741no—
282C22×C4—1371no—
283SD16—22no—
284SD16—2742no—
285SD16—2742no—
286SD16—2742no—
287SD16—2742no—
288SD16—2742no—
289D8—22no—
290D8—2742no—
291M16—2742no—
292C2×C8—2742no—
293C2×C8—1371no—
294C2×C8—1371no—
295C2×C8C22×D13711yescharacteristic
296C2×C8C22×D13711yescharacteristic
297C4⋊C4—2741no—
298C4⋊C4—2741no—
299C4⋊C4—2741no—
300C4⋊C4—2741no—
301C4⋊C4—1371no—
302C4⋊C4—1371no—
303C4⋊C4—1371no—
304C4⋊C4C22×D13711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—2741no—
309C22⋊C4—2741no—
310C22⋊C4—2741no—
311C22⋊C4—2741no—
312C22⋊C4—2741no—
313C22⋊C4—2741no—
314C22⋊C4—1371no—
315C22⋊C4—1371no—
316C42—2741no—
317C42—1371no—
318C42C22×D13711yescharacteristic
319C23—21no—
320C23—21no—
321C23—2744no—
322C23—2741no—
323C23—10961no—
324C23—10961no—
325C23—1371no—
326Q8—21no—
327Q8—21no—
328Q8—2741no—
329Q8—2741no—
330Q8—2741no—
331Q8—2741no—
332Q8—5482no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—2744no—
337D4—2741no—
338D4—5482no—
339D4—1372no—
340D4D4×D13712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—81no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—2741no—
347C2×C4—5482no—
348C2×C4—5482no—
349C2×C4—10961no—
350C2×C4—10961no—
351C2×C4—10961no—
352C2×C4—1371no—
353C2×C4—1371no—
354C2×C4—1371no—
355C2×C4—1372no—
356C2×C4—1372no—
357C2×C4D4×D13711yescharacteristic
358C2×C4D4×D13711yescharacteristic
359C2×C4C23×D13711yescharacteristic
360C8—21no—
361C8—21no—
362C8—2741no—
363C8—2741no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—2741no—
371C22—2741no—
372C22—2742no—
373C22—5481no—
374C22—5481no—
375C22—10964no—
376C22—1372no—
377C22C2×D4×D13711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—2741no—
383C4—2741no—
384C4—5481no—
385C4—5481no—
386C4—5481no—
387C4—1372no—
388C4C2×D4×D13711yescharacteristic
389C4#41×D13711yescharacteristic
390C2—42no—
391C2—81no—
392C2—2742no—
393C2—5482no—
394C2#121×D13711yescharacteristic
395C2#2257821011yescharacteristic
396C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#463×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984112C2
hypercenter#463D137
hypocenterC137#3042
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D137C2semidirect
3. #22577973C2semidirect
4. #22577875C2semidirect
5. #22577617C2semidirect
6. #22577616C2semidirect
7. #22577615C2semidirect
8. #22577609C2semidirect
9. #22577608C2semidirect
10. #22577607C2semidirect
11. #22577606C2semidirect
12. #22577428C2semidirect
13. #22577410C2semidirect
14. #22577396C2semidirect
15. #463×C137C2semidirect
29. #17984088C22semidirect
30. #17984086C22semidirect
31. #17984075C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984017C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
50. #17984012C22semidirect
51. #17984011C22semidirect
52. #17984011C22semidirect
53. #17984009C22semidirect
56. #124×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #103×C137C22semidirect
60. #102×C137C22semidirect
198. C137#3042semidirect
207. #463D137semidirect
233. #124C2×D137semidirect
251. C4⋊C8C2×D137semidirect
257. #103C2×D137semidirect
259. #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×SD16×D1372
#225779732
#225778752
#225776172
#225776162
#225776152
#225776092
#225776082
#225776072
#225776062
#225774282
#225774102
#225773962
#463×C1372
#3042137