Finite Group Databasebeta

← #31790308

Subgroups — #31790308

Order 17536.

490 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×#17984120C211yesmaximal, characteristic, complemented
2C2×SD16×D137C211yesmaximal, characteristic, complemented
3#22577971C211yesmaximal, characteristic
4#22577879C211yesmaximal, characteristic, complemented
5#22577628C211yesmaximal, characteristic, complemented
6#22577627C211yesmaximal, characteristic, complemented
7#22577622C211yesmaximal, characteristic, complemented
8#22577604C211yesmaximal, characteristic, complemented
9#22577603C211yesmaximal, characteristic, complemented
10#22577602C211yesmaximal, characteristic, complemented
11#22577601C211yesmaximal, characteristic, complemented
12#22577428C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#22577398C211yesmaximal, characteristic, complemented
15#464×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984120—24no—
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555499C2211yescharacteristic
22C2×#14555499C2211yescharacteristic, complemented
23C2×#14555498C2211yescharacteristic
24C2×#14555497C2211yescharacteristic
25C2×#14555497C2211yescharacteristic, complemented
26SD16×D137—24no—
27C2×#14555493C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic
29#17984085C2211yescharacteristic
30#17984084C2211yescharacteristic
31#17984077C2211yescharacteristic
32#17984061—21no—
33#17984061—21no—
34#17984058—21no—
35#17984058—21no—
36Q8×Dic137C2211yescharacteristic, complemented
37C4×Dic274—21no—
38C4×Dic274C2211yescharacteristic, complemented
39#17984053—21no—
40#17984053C2211yescharacteristic, complemented
41#17984046—21no—
42#17984046C2211yescharacteristic
43#17984045—21no—
44#17984045C2211yescharacteristic
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984017C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984012C2211yescharacteristic, complemented
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic, complemented
51#17984009C2211yescharacteristic
52#17984008C2211yescharacteristic
53#17984008C2211yescharacteristic, complemented
54C2×Q16×C137C2211yescharacteristic, complemented
55C2×SD16×C137C2211yescharacteristic
56#124×C137C2211yescharacteristic
57C4×Q8×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic
61C23×D137—21no—
62#14555511—22no—
63#14555511—22no—
64C2×Dic274—21no—
65C2×Dic274C2311yescharacteristic
66Q8×D137—22no—
67Q8×D137D414yescomplemented
68C2×D548—21no—
69C2×D548—21no—
70C2×D548—21no—
71C2×D548C2311yescharacteristic
72C2×D548C2311yescharacteristic
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76D4×D137—22no—
77D4×D137—22no—
78C2×C4×D137—21no—
79C2×C4×D137—21no—
80C2×C4×D137C2311yescharacteristic
81#14555500—22no—
82#14555499—22no—
83#14555499—22no—
84#14555498—22no—
85#14555497—22no—
86#14555497—22no—
87#14555493—22no—
88C2×#11764040C2311yescharacteristic
89C2×#11764040C2311yescharacteristic
90C8×D137—22no—
91C4⋊Dic137—21no—
92C4⋊Dic137C2311yescharacteristic
93#14555489—21no—
94#14555489—21no—
95#14555489D411yescharacteristic
96#14555489D411yescharacteristic
97#14555487—21no—
98#14555487—21no—
99#14555487—21no—
100#14555487D411yescharacteristic
101#14555487D411yescharacteristic
102C4×Dic137—21no—
103C4×Dic137—21no—
104C4×Dic137—21no—
105C4×Dic137—21no—
106C4×Dic137C2311yescharacteristic
107C2×Q8×C137C2311yescharacteristic
108C2×Q8×C137C2311yescharacteristic
109C2×D4×C137C2311yescharacteristic
110Q16×C137—22no—
111SD16×C137—22no—
112C2×C8×C137C2311yescharacteristic
113C2×C8×C137C2311yescharacteristic
114C4⋊C4×C137—21no—
115C4⋊C4×C137C2311yescharacteristic
116C22⋊C4×C137—21no—
117C22⋊C4×C137—21no—
118C42×C137—21no—
119C42×C137C2311yescharacteristic
120C22×D137—21no—
121C22×D137—21no—
122C22×D137—42no—
123C22×D137—81no—
124C22×D137—81no—
125C22×D137C2×D411yescharacteristic
126Dic274—21no—
127Dic274—24no—
128Dic274C2×D412yes—
129D548—21no—
130D548—21no—
131D548—21no—
132D548—21no—
133D548—42no—
134D548—81no—
135D548—81no—
136C22⋊D137—42no—
137C22⋊D137—81no—
138C22⋊D137—81no—
139C2×Dic137—21no—
140C2×Dic137—21no—
141C2×Dic137C2×D411yescharacteristic
142C2×Dic137C2×D411yescharacteristic
143C2×Dic137C2×D411yescharacteristic
144C4×D137—24no—
145C4×D137—42no—
146C4×D137C2×D412yes—
147C4×D137#4112yes—
148#11764040—21no—
149#11764040—21no—
150C23×C137—21no—
151Q8×C137—21no—
152Q8×C137—21no—
153Q8×C137—21no—
154Q8×C137C2×D412yes—
155D4×C137—21no—
156D4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137C2×D411yescharacteristic
161C2×C4×C137C2×D411yescharacteristic
162C2×C4×C137C2411yescharacteristic
163C8×C137—21no—
164C8×C137—21no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137C22×D411yescharacteristic
169C2×D137—21no—
170C2×D137—21no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—41no—
176C2×D137—41no—
177C2×D137—84no—
178C2×D137#12112yes—
179C4×C137—22no—
180C4×C137—21no—
181C4×C137—21no—
182C4×C137—41no—
183C4×C137—41no—
184C4×C137C22×D411yescharacteristic
185C4×C137C2×#4111yescharacteristic
186Dic137—22no—
187Dic137—21no—
188Dic137—21no—
189Dic137—41no—
190Dic137#12112yes—
191C2×C137—81no—
192C2×C137C2×#12111yescharacteristic
193C2×C137#57711yescharacteristic
194C2×C137#57711yescharacteristic
195D137—22no—
196D137—81no—
197D137—81no—
198C137#304311yescharacteristic, complemented, Sylow
199#3043—1371nomaximal, Sylow
200C2×#137—1371no—
201C22×SD16—1371no—
202#535—1371no—
203#519—1371no—
204#464—1371no—
205#464—1371no—
206#464—1371no—
207#464D13711yescharacteristic, complemented
208#460—1371no—
209#460—1371no—
210#460—1371no—
211#460—1371no—
212#424—1371no—
213#418—1371no—
214#416—1371no—
215C2×#41—1371no—
216C22×Q8—1371no—
217C22×D4—1371no—
218#137—2744no—
219C2×Q16—1371no—
220C2×Q16C2×D13711yescharacteristic, complemented
221C2×SD16—2744no—
222C2×SD16—1371no—
223C2×SD16—1371no—
224C2×SD16—1371no—
225C2×SD16—1371no—
226C2×SD16—1371no—
227C2×SD16C2×D13711yescharacteristic
228C2×M16—1371no—
229C22×C8—1371no—
230#127—1371no—
231#127—1371no—
232#124—1371no—
233#124C2×D13711yescharacteristic
234#121—2741no—
235#121—2741no—
236#120—2741no—
237#120—2741no—
238C4×Q8—2741no—
239C4×Q8—2741no—
240C4×Q8—1371no—
241C4×Q8—1371no—
242C4×Q8—1371no—
243C4×Q8C2×D13711yescharacteristic, complemented
244#117—2741no—
245#117—2741no—
246#117—1371no—
247#117—1371no—
248C4⋊C8—1371no—
249C4⋊C8—1371no—
250C4⋊C8—1371no—
251C4⋊C8C2×D13711yescharacteristic, complemented
252#103—1371no—
253#103C2×D13711yescharacteristic, complemented
254#102—1371no—
255#102—1371no—
256#102—1371no—
257#102—1371no—
258#102—1371no—
259#102C2×D13711yescharacteristic
260C24—2741no—
261#41—2742no—
262#41—2742no—
263C2×Q8—2741no—
264C2×Q8—2742no—
265C2×Q8—1374no—
266C2×Q8—1371no—
267C2×Q8C22×D13711yescharacteristic
268C2×Q8C22×D13711yescharacteristic
269C2×D4—2741no—
270C2×D4—2742no—
271C2×D4—2742no—
272C2×D4—2741no—
273C2×D4—2741no—
274C2×D4—2741no—
275C2×D4—2741no—
276C2×D4—2741no—
277C2×D4—1371no—
278C2×D4—1371no—
279C2×D4C22×D13711yescharacteristic
280C22×C4—2741no—
281C22×C4—2741no—
282C22×C4—1371no—
283Q16—22no—
284Q16—2742no—
285SD16—22no—
286SD16—2742no—
287SD16—2742no—
288SD16—2742no—
289SD16—2742no—
290SD16—2742no—
291M16—2742no—
292C2×C8—2742no—
293C2×C8—1371no—
294C2×C8—1371no—
295C2×C8C22×D13711yescharacteristic
296C2×C8C22×D13711yescharacteristic
297C4⋊C4—21no—
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—2741no—
308C22⋊C4—2741no—
309C22⋊C4—2741no—
310C22⋊C4—1371no—
311C22⋊C4—1371no—
312C42—21no—
313C42—2741no—
314C42—2741no—
315C42—2741no—
316C42—2741no—
317C42—1371no—
318C42C22×D13711yescharacteristic
319C23—21no—
320C23—2741no—
321C23—2741no—
322C23—5482no—
323C23—10961no—
324C23—10961no—
325C23—1371no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—2744no—
330Q8—2741no—
331Q8—1372no—
332Q8D4×D13712yes—
333D4—21no—
334D4—21no—
335D4—2741no—
336D4—2741no—
337D4—2741no—
338D4—2741no—
339D4—5482no—
340D4—5482no—
341D4—10961no—
342D4—10961no—
343D4—10961no—
344D4—10961no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—2744no—
349C2×C4—2741no—
350C2×C4—2741no—
351C2×C4—5482no—
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—41no—
365C22—41no—
366C22—41no—
367C22—2741no—
368C22—2741no—
369C22—5481no—
370C22—5481no—
371C22—5481no—
372C22—5481no—
373C22—5481no—
374C22—5481no—
375C22—10964no—
376C22—1372no—
377C22C2×D4×D13711yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—2741no—
384C4—2741no—
385C4—2742no—
386C4—5481no—
387C4—1372no—
388C4C2×D4×D13711yescharacteristic
389C4#41×D13711yescharacteristic
390C2—81no—
391C2—2742no—
392C2—10961no—
393C2—10961no—
394C2#121×D13711yescharacteristic
395C2#2257821011yescharacteristic
396C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#464×C137C2
socleC22×C137C22×D4
prime radical#22577971C2
hypercenter#464D137
hypocenterC137#3043
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#17984120C2semidirect
2. C2×SD16×D137C2semidirect
4. #22577879C2semidirect
5. #22577628C2semidirect
6. #22577627C2semidirect
7. #22577622C2semidirect
8. #22577604C2semidirect
9. #22577603C2semidirect
10. #22577602C2semidirect
11. #22577601C2semidirect
12. #22577428C2semidirect
13. #22577406C2semidirect
14. #22577398C2semidirect
15. #464×C137C2semidirect
20. C2×#14555500C22semidirect
22. C2×#14555499C22semidirect
25. C2×#14555497C22semidirect
36. Q8×Dic137C22semidirect
38. C4×Dic274C22semidirect
40. #17984053C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984017C22semidirect
47. #17984017C22semidirect
48. #17984012C22semidirect
50. #17984010C22semidirect
53. #17984008C22semidirect
54. C2×Q16×C137C22semidirect
57. C4×Q8×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #103×C137C22semidirect
67. Q8×D137D4semidirect
198. C137#3043semidirect
207. #464D137semidirect
220. C2×Q16C2×D137semidirect
243. C4×Q8C2×D137semidirect
251. C4⋊C8C2×D137semidirect
253. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841202
C2×SD16×D1372
#225779712
#225778792
#225776282
#225776272
#225776222
#225776042
#225776032
#225776022
#225776012
#225774282
#225774062
#225773982
#464×C1372
#3043137