Finite Group Databasebeta

← #31792303

Subgroups — #31792303

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984117C211yesmaximal, characteristic, complemented
2C22×#14555498C211yesmaximal, characteristic, complemented
3#22578066C211yesmaximal, characteristic, complemented
4#22578047C211yesmaximal, characteristic
5#22577794C211yesmaximal, characteristic, complemented
6#22577648C211yesmaximal, characteristic, complemented
7#22577642C211yesmaximal, characteristic, complemented
8#22577624C211yesmaximal, characteristic, complemented
9#22577606C211yesmaximal, characteristic, complemented
10#22577566C211yesmaximal, characteristic, complemented
11#22577561C211yesmaximal, characteristic, complemented
12#22577540C211yesmaximal, characteristic, complemented
13#22577500C211yesmaximal, characteristic, complemented
14#22577471C211yesmaximal, characteristic, complemented
15#450×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18C2×Q8×D137C2211yescharacteristic, complemented
19#17984117—24no—
20C2×Dic548C2211yescharacteristic, complemented
21C2×Dic548C2211yescharacteristic, complemented
22C2×#14555500C2211yescharacteristic, complemented
23C2×#14555498—24no—
24C2×#14555498C2211yescharacteristic
25C2×#14555498C2211yescharacteristic
26C2×#14555498C2211yescharacteristic
27C2×#14555498C2211yescharacteristic
28C2×#14555493C2211yescharacteristic, complemented
29C22×#11764040C2211yescharacteristic
30#17984087—21no—
31#17984086C2211yescharacteristic, complemented
32#17984075C2211yescharacteristic
33#17984069—21no—
34#17984068C2211yescharacteristic, complemented
35#17984067C2211yescharacteristic
36#17984066—21no—
37#17984065—21no—
38#17984060—21no—
39#17984059C2211yescharacteristic
40#17984057—21no—
41#17984053C2211yescharacteristic, complemented
42D4×Dic137C2211yescharacteristic
43C4×C22⋊D137C2211yescharacteristic
44#17984048—21no—
45C2×C22⋊Dic137—21no—
46#17984021C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984012C2211yescharacteristic, complemented
51#17984011C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983998C2211yescharacteristic, complemented
54C22×D4×C137C2211yescharacteristic
55C2×SD16×C137C2211yescharacteristic
56C2×SD16×C137C2211yescharacteristic
57C22⋊Q8×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#102×C137C2211yescharacteristic
60C22⋊C8×C137C2211yescharacteristic
61#14555509—22no—
62#14555509—41no—
63C2×Dic274—21no—
64C2×Dic274—22no—
65C2×Dic274—21no—
66C2×Dic274—41no—
67C2×Dic274C2311yescharacteristic
68C2×Dic274C2311yescharacteristic
69C2×Dic274C2311yescharacteristic
70Q8×D137—22no—
71Q8×D137—41no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137D411yescharacteristic
75C22×Dic137—21no—
76C22×Dic137—21no—
77C22×Dic137—21no—
78C2×C4×D137—21no—
79C2×C4×D137C2311yescharacteristic
80Dic548—22no—
81Dic548—41no—
82#14555500—22no—
83#14555498—22no—
84#14555498—22no—
85#14555498—22no—
86#14555498D414yescomplemented
87#14555493—22no—
88C2×#11764040—22no—
89C2×#11764040C2311yescharacteristic
90C2×#11764040C2311yescharacteristic
91C4⋊Dic137—21no—
92C4⋊Dic137C2311yescharacteristic
93#14555489—21no—
94#14555489—21no—
95#14555489—21no—
96#14555489—21no—
97#14555489D411yescharacteristic
98C22⋊Dic137—21no—
99C22⋊Dic137—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137—21no—
102C22⋊Dic137—21no—
103C22⋊Dic137D411yescharacteristic
104#14555487—21no—
105#14555487D411yescharacteristic
106C4×Dic137—21no—
107C4×Dic137C2311yescharacteristic
108C24×C137—21no—
109C2×Q8×C137C2311yescharacteristic
110C2×D4×C137—22no—
111C2×D4×C137—21no—
112C2×D4×C137—41no—
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C22×C4×C137C2311yescharacteristic
116SD16×C137—22no—
117SD16×C137—41no—
118C2×C8×C137C2311yescharacteristic
119C2×C8×C137C2311yescharacteristic
120C4⋊C4×C137—21no—
121C4⋊C4×C137C2311yescharacteristic
122C22⋊C4×C137—21no—
123C22×D137C2×D411yescharacteristic
124Dic274—21no—
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128Dic274—21no—
129Dic274—42no—
130Dic274—42no—
131Dic274C2×D412yes—
132C22⋊D137—22no—
133C22⋊D137—42no—
134C22⋊D137—81no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—21no—
138C2×Dic137—21no—
139C2×Dic137—42no—
140C2×Dic137—42no—
141C2×Dic137—81no—
142C2×Dic137C2×D411yescharacteristic
143C2×Dic137C2×D411yescharacteristic
144C2×Dic137C2×D411yescharacteristic
145C4×D137—21no—
146C4×D137—21no—
147C4×D137—42no—
148#11764040—21no—
149#11764040C2×D412yes—
150C23×C137—21no—
151C23×C137—21no—
152C23×C137—42no—
153C23×C137—42no—
154C23×C137—81no—
155C23×C137C2×D411yescharacteristic
156Q8×C137—21no—
157Q8×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—42no—
162D4×C137C2×D412yes—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137—21no—
171C22×C137—21no—
172C22×C137—21no—
173C22×C137—22no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—84no—
180C22×C137#13712yescomplemented
181C22×C137C22×D411yescharacteristic
182C2×D137—21no—
183C2×D137—21no—
184C2×D137—21no—
185C4×C137—21no—
186C4×C137—41no—
187C4×C137—41no—
188C4×C137C22×D411yescharacteristic
189C4×C137H(5,𝔽2)11yescharacteristic
190Dic137—21no—
191Dic137—22no—
192Dic137—21no—
193Dic137—21no—
194Dic137—41no—
195Dic137—41no—
196Dic137—41no—
197C2×C137—22no—
198C2×C137—42no—
199C2×C137—81no—
200C2×C137D4211yescharacteristic
201C2×C137C2×#13711yescharacteristic
202C2×C137#57711yescharacteristic
203D137—41no—
204C137#326611yescharacteristic, complemented, Sylow
205#3266—1371nomaximal, Sylow
206C2×#137—1371no—
207C22×SD16—1371no—
208D4×Q8—1371no—
209#546—1371no—
210#497—1371no—
211#468—1371no—
212#467—1371no—
213#464—1371no—
214#461—1371no—
215#452—1371no—
216#451—1371no—
217#450D13711yescharacteristic, complemented
218#448—1371no—
219#440—1371no—
220#435—1371no—
221C2×#41—1371no—
222C22×Q8—1371no—
223C22×Q8—1371no—
224C22×D4C2×D13711yescharacteristic
225#137—2744no—
226C2×Q16—1371no—
227C2×Q16—1371no—
228C2×Q16—1371no—
229C2×SD16—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16—1371no—
233C2×SD16—1371no—
234C2×SD16C2×D13711yescharacteristic
235C2×SD16C2×D13711yescharacteristic
236C2×M16—1371no—
237C22×C8—1371no—
238C4⋊Q8—2741no—
239C4⋊Q8—1371no—
240#124—1371no—
241#123—2741no—
242C22⋊Q8—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8—1371no—
246C22⋊Q8C2×D13711yescharacteristic
247#121—2741no—
248#121—1371no—
249#120—2741no—
250C4×Q8—1371no—
251C4×D4—2741no—
252C4×D4—1371no—
253C4×D4—1371no—
254C2×C22⋊C4—2741no—
255#107—1371no—
256C4⋊C8—1371no—
257#103—1371no—
258#103—1371no—
259#103—1371no—
260#103—1371no—
261#102C2×D13711yescharacteristic
262#102C2×D13711yescharacteristic
263C22⋊C8—1371no—
264C22⋊C8C2×D13711yescharacteristic
265#97—1371no—
266C24—21no—
267#41—2742no—
268#41—5481no—
269C2×Q8—2741no—
270C2×Q8—2742no—
271C2×Q8—2741no—
272C2×Q8—2742no—
273C2×Q8—5481no—
274C2×Q8—5481no—
275C2×Q8—1371no—
276C2×Q8—1371no—
277C2×Q8—1371no—
278C2×Q8C22×D13711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2741no—
283C2×D4—2741no—
284C2×D4—1371no—
285C2×D4C22×D13711yescharacteristic
286C2×D4C22×D13711yescharacteristic
287C22×C4—2741no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—1371no—
292C22×C4C22×D13711yescharacteristic
293Q16—2742no—
294Q16—2742no—
295Q16—5481no—
296SD16—22no—
297SD16—41no—
298SD16—2742no—
299SD16—2742no—
300SD16—2742no—
301SD16—1374no—
302M16—2742no—
303C2×C8—2742no—
304C2×C8—1371no—
305C2×C8—1371no—
306C2×C8C22×D13711yescharacteristic
307C2×C8C22×D13711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2741no—
310C4⋊C4—2741no—
311C4⋊C4—2741no—
312C4⋊C4—2741no—
313C4⋊C4—2741no—
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4C22×D13711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—1371no—
325C22⋊C4—1371no—
326C42—2741no—
327C42—1371no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1371no—
334C23D4×D13711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2741no—
338Q8—2741no—
339Q8—2741no—
340Q8—2741no—
341Q8—2741no—
342Q8—5482no—
343Q8—5482no—
344Q8—1372no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—2742no—
350D4—5482no—
351D4—10961no—
352D4D4×D13712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—2741no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—2741no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—5482no—
364C2×C4—5482no—
365C2×C4—5482no—
366C2×C4—10961no—
367C2×C4—1371no—
368C2×C4—1371no—
369C2×C4—1371no—
370C2×C4D4×D13711yescharacteristic
371C2×C4C23×D13711yescharacteristic
372C8—21no—
373C8—21no—
374C8—2741no—
375C8—1372no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2741no—
386C22—2741no—
387C22—2741no—
388C22#1798411712yescomplemented
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—2741no—
394C4—2741no—
395C4—2741no—
396C4—2742no—
397C4—5481no—
398C4—5481no—
399C4—5481no—
400C4C2×D4×D13711yescharacteristic
401C4#1798413311yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—5481no—
406C2#2257803511yescharacteristic
407C2C2×#1798411711yescharacteristic
408C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
prime radical#22578047C2
hypercenter#450D137
hypocenterC137#3266
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#17984117C2semidirect
2. C22×#14555498C2semidirect
3. #22578066C2semidirect
5. #22577794C2semidirect
6. #22577648C2semidirect
7. #22577642C2semidirect
8. #22577624C2semidirect
9. #22577606C2semidirect
10. #22577566C2semidirect
11. #22577561C2semidirect
12. #22577540C2semidirect
13. #22577500C2semidirect
14. #22577471C2semidirect
15. #450×C137C2semidirect
18. C2×Q8×D137C22semidirect
20. C2×Dic548C22semidirect
21. C2×Dic548C22semidirect
22. C2×#14555500C22semidirect
28. C2×#14555493C22semidirect
31. #17984086C22semidirect
34. #17984068C22semidirect
41. #17984053C22semidirect
46. #17984021C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
50. #17984012C22semidirect
51. #17984011C22semidirect
52. #17984000C22semidirect
53. #17983998C22semidirect
86. #14555498D4semidirect
180. C22×C137#137semidirect
204. C137#3266semidirect
217. #450D137semidirect
388. C22#17984117semidirect

Maximal subgroups

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

subgroupindex
C2×#179841172
C22×#145554982
#225780662
#225780472
#225777942
#225776482
#225776422
#225776242
#225776062
#225775662
#225775612
#225775402
#225775002
#225774712
#450×C1372
#3266137