Finite Group Databasebeta

← #31792795

Subgroups — #31792795

Order 17536.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578099C211yesmaximal, characteristic, complemented
2#22577767C211yesmaximal, characteristic, complemented
3#22577724C211yesmaximal, characteristic, complemented
4#22577714C211yesmaximal, characteristic, complemented
5#22577710C211yesmaximal, characteristic, complemented
6#22577654C211yesmaximal, characteristic, complemented
7#22577637C211yesmaximal, characteristic, complemented
8#22577595C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic, complemented
10#22577548C211yesmaximal, characteristic, complemented
11#22577501C211yesmaximal, characteristic, complemented
12#22577475C211yesmaximal, characteristic, complemented
13#22577404C211yesmaximal, characteristic, complemented
14#22577399C211yesmaximal, characteristic, complemented
15#550×C137C211yesmaximal, characteristic, complemented
16C2×#14555497C2211yescharacteristic
17C2×#14555495C2211yescharacteristic
18C2×#14555495C2211yescharacteristic
19C2×#14555494C2211yescharacteristic
20C22×#11764040C2211yescharacteristic
21#17984083—21no—
22#17984080C2211yescharacteristic
23#17984070—21no—
24#17984062C2211yescharacteristic
25#17984062C2211yescharacteristic
26#17984060—21no—
27C4×D548C2211yescharacteristic
28#17984048—21no—
29#17984047C2211yescharacteristic
30C2×C4⋊Dic137C2211yescharacteristic
31#17984021C2211yescharacteristic
32#17984021C2211yescharacteristic
33#17984019C2211yescharacteristic
34C4⋊#11764040C2211yescharacteristic
35#17984012C2211yescharacteristic
36#17984012C2211yescharacteristic
37#17984012C2211yescharacteristic
38#17984009C2211yescharacteristic
39#17984009C2211yescharacteristic
40#17984009C2211yescharacteristic
41#17984009C2211yescharacteristic
42#17984009C2211yescharacteristic
43#17984009C2211yescharacteristic
44#17984008C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic
47#17983999C2211yescharacteristic
48C2×#41×C137C2211yescharacteristic
49C2×#41×C137C2211yescharacteristic
50#127×C137—21no—
51#127×C137C2211yescharacteristic
52#121×C137—21no—
53#121×C137—21no—
54#121×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56C4×Q8×C137C2211yescharacteristic
57C4×D4×C137—21no—
58C4×D4×C137C2211yescharacteristic
59C2×D548C2311yescharacteristic
60C2×C22⋊D137—21no—
61C2×C22⋊D137—21no—
62C22×Dic137—21no—
63C2×C4×D137—21no—
64#14555497—22no—
65#14555495—41no—
66#14555495—41no—
67#14555494—41no—
68C2×#11764040—41no—
69C2×#11764040C2311yescharacteristic
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79C22⋊Dic137—21no—
80#14555487—21no—
81#14555487—21no—
82#14555487—21no—
83C4×Dic137—21no—
84#41×C137—22no—
85#41×C137—22no—
86#41×C137—41no—
87#41×C137—41no—
88C2×Q8×C137C2311yescharacteristic
89C2×D4×C137—21no—
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137—21no—
93C2×D4×C137—21no—
94C2×D4×C137—21no—
95C2×D4×C137D411yescharacteristic
96C2×D4×C137C2311yescharacteristic
97C2×D4×C137C2311yescharacteristic
98C22×C4×C137—21no—
99C22×C4×C137—21no—
100C22×C4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102C4⋊C4×C137—21no—
103C4⋊C4×C137D411yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C22⋊C4×C137D411yescharacteristic
108C22⋊C4×C137D411yescharacteristic
109C42×C137—21no—
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112D548—21no—
113D548—21no—
114C22⋊D137—42no—
115C22⋊D137—81no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—81no—
120C4×D137—81no—
121#11764040—21no—
122#11764040—21no—
123#11764040—21no—
124#11764040—21no—
125C23×C137—21no—
126C23×C137—21no—
127C23×C137C2×D411yescharacteristic
128C23×C137C2×D411yescharacteristic
129Q8×C137—21no—
130Q8×C137#4112yes—
131D4×C137—21no—
132D4×C137—21no—
133D4×C137—21no—
134D4×C137—21no—
135D4×C137—41no—
136D4×C137—42no—
137D4×C137—42no—
138D4×C137—42no—
139D4×C137—81no—
140D4×C137—81no—
141D4×C137—81no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—42no—
149C2×C4×C137—42no—
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2×D411yescharacteristic
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2411yescharacteristic
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137C22×D411yescharacteristic
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C4×C137—21no—
172C4×C137—21no—
173C4×C137—22no—
174C4×C137—21no—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137C2×#4111yescharacteristic
178C4×C137H(5,𝔽2)11yescharacteristic
179Dic137—41no—
180Dic137—41no—
181Dic137—41no—
182C2×C137—41no—
183C2×C137—41no—
184C2×C137—81no—
185C2×C137—81no—
186C2×C137#54611yescharacteristic
187C2×C137#57511yescharacteristic
188C2×C137#57611yescharacteristic
189D137—81no—
190C137#330911yescharacteristic, complemented, Sylow
191#3309—1371nomaximal, Sylow
192#553—1371no—
193#550D13711yescharacteristic, complemented
194#490—1371no—
195#482—1371no—
196#480—1371no—
197#479—1371no—
198#469—1371no—
199#466—1371no—
200#459—1371no—
201#449—1371no—
202#449—1371no—
203#440—1371no—
204#435—1371no—
205#417—1371no—
206#416—1371no—
207C2×#41C2×D13711yescharacteristic
208C2×#41C2×D13711yescharacteristic
209C2×SD16—1371no—
210C2×D8—1371no—
211C2×D8—1371no—
212C2×M16—1371no—
213C22×C8—1371no—
214#127—21no—
215#127C2×D13711yescharacteristic
216#126—2741no—
217#125—1371no—
218#123—2741no—
219#121—21no—
220#121—21no—
221#121—2741no—
222#121—1371no—
223#121—1371no—
224#121C2×D13711yescharacteristic
225#121C2×D13711yescharacteristic
226C4×Q8C2×D13711yescharacteristic
227C4×D4—21no—
228C4×D4—2741no—
229C4×D4—1371no—
230C4×D4C2×D13711yescharacteristic
231#117—1371no—
232C2×C4⋊C4—1371no—
233#107—1371no—
234#107—1371no—
235#106—1371no—
236C4⋊C8—1371no—
237#103—1371no—
238#103—1371no—
239#103—1371no—
240#102—1371no—
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247C22⋊C8—1371no—
248C22⋊C8—1371no—
249#97—1371no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D13711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—2741no—
262C2×D4—2741no—
263C2×D4—1371no—
264C2×D4C22⋊D13711yescharacteristic
265C2×D4C22×D13711yescharacteristic
266C2×D4C22×D13711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2741no—
270C22×C4—2741no—
271C22×C4C22×D13711yescharacteristic
272C22×C4C22×D13711yescharacteristic
273SD16—2742no—
274D8—5481no—
275D8—5481no—
276M16—5481no—
277C2×C8—5481no—
278C2×C8—1371no—
279C2×C8—1371no—
280C2×C8—1371no—
281C2×C8—1371no—
282C4⋊C4—21no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—2741no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4—1371no—
289C4⋊C4C22⋊D13711yescharacteristic
290C4⋊C4C22×D13711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4C22⋊D13711yescharacteristic
298C22⋊C4C22⋊D13711yescharacteristic
299C42—21no—
300C42—2741no—
301C42C22×D13711yescharacteristic
302C23—21no—
303C23—21no—
304C23—2741no—
305C23C2×C22⋊D13711yescharacteristic
306C23C2×C22⋊D13711yescharacteristic
307Q8—21no—
308Q8#1455550912yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—2741no—
321D4—2741no—
322D4—5482no—
323D4—10961no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—10961no—
336C2×C4—10961no—
337C2×C4C2×C22⋊D13711yescharacteristic
338C2×C4C2×C22⋊D13711yescharacteristic
339C2×C4C2×C22⋊D13711yescharacteristic
340C2×C4C2×C22⋊D13711yescharacteristic
341C2×C4C23×D13711yescharacteristic
342C8—2741no—
343C8—2741no—
344C8—2741no—
345C8—2741no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22C22×C22⋊D13711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—5481no—
369C4—5481no—
370C4—5481no—
371C4C2×#1455550911yescharacteristic
372C4#1798413311yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—10961no—
378C2#2257804711yescharacteristic
379C2#2257820111yescharacteristic
380C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#550×C137C2
socleC22×C137C22×D4
hypercenter#550D137
hypocenterC137#3309
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #22578099C2semidirect
2. #22577767C2semidirect
3. #22577724C2semidirect
4. #22577714C2semidirect
5. #22577710C2semidirect
6. #22577654C2semidirect
7. #22577637C2semidirect
8. #22577595C2semidirect
9. #22577548C2semidirect
10. #22577548C2semidirect
11. #22577501C2semidirect
12. #22577475C2semidirect
13. #22577404C2semidirect
14. #22577399C2semidirect
15. #550×C137C2semidirect
190. C137#3309semidirect
193. #550D137semidirect

Maximal subgroups

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

subgroupindex
#225780992
#225777672
#225777242
#225777142
#225777102
#225776542
#225776372
#225775952
#225775482
#225775012
#225774752
#225774042
#225773992
#550×C1372
#3309137