Finite Group Databasebeta

← #31792797

Subgroups — #31792797

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#22578074C211yesmaximal, characteristic, complemented
3#22577768C211yesmaximal, characteristic, complemented
4#22577728C211yesmaximal, characteristic, complemented
5#22577717C211yesmaximal, characteristic, complemented
6#22577711C211yesmaximal, characteristic, complemented
7#22577656C211yesmaximal, characteristic, complemented
8#22577639C211yesmaximal, characteristic, complemented
9#22577599C211yesmaximal, characteristic, complemented
10#22577550C211yesmaximal, characteristic, complemented
11#22577550C211yesmaximal, characteristic, complemented
12#22577505C211yesmaximal, characteristic, complemented
13#22577405C211yesmaximal, characteristic, complemented
14#22577400C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555510C2211yescharacteristic
18C2×#14555499C2211yescharacteristic
19C2×D1096C2211yescharacteristic
20C2×D1096C2211yescharacteristic
21#17984085C2211yescharacteristic
22#17984084—21no—
23#17984083—21no—
24#17984080C2211yescharacteristic
25#17984070—21no—
26#17984062C2211yescharacteristic
27#17984062C2211yescharacteristic
28#17984062C2211yescharacteristic
29#17984062C2211yescharacteristic
30#17984061—21no—
31#17984061—21no—
32#17984060—21no—
33C4×Dic274C2211yescharacteristic
34C4×D548C2211yescharacteristic
35#17984050—21no—
36#17984048—21no—
37#17984047C2211yescharacteristic
38C2×C4⋊Dic137C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984022C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic
52C2×M16×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic
55C4⋊C8×C137C2211yescharacteristic
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic
58#97×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—22no—
61#14555510—41no—
62#14555510—41no—
63C2×Dic274C2311yescharacteristic
64C2×D548—21no—
65C2×D548—21no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C22×Dic137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137—21no—
79#14555499—22no—
80D1096—41no—
81D1096—41no—
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90C22⋊Dic137—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C2×D4×C137D411yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101M16×C137—41no—
102C2×C8×C137—41no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137D411yescharacteristic
108C22⋊C4×C137D411yescharacteristic
109C22⋊C4×C137D411yescharacteristic
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—21no—
114Dic274—21no—
115Dic274#4112yes—
116D548—21no—
117D548—21no—
118D548—21no—
119D548—21no—
120D548—21no—
121D548—21no—
122D548—42no—
123D548—81no—
124C22⋊D137—42no—
125C22⋊D137—42no—
126C22⋊D137—42no—
127C22⋊D137—81no—
128C22⋊D137—81no—
129C22⋊D137—81no—
130C2×Dic137—21no—
131C2×Dic137—21no—
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—81no—
136C4×D137—42no—
137C4×D137—42no—
138C4×D137—81no—
139C23×C137C2×D411yescharacteristic
140C23×C137C2×D411yescharacteristic
141D4×C137—41no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2×D411yescharacteristic
150C2×C4×C137C2411yescharacteristic
151C8×C137—21no—
152C8×C137—21no—
153C8×C137—21no—
154C8×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137C22×D411yescharacteristic
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C4×C137—21no—
172C4×C137—21no—
173C4×C137—21no—
174C4×C137—21no—
175C4×C137C2×#4111yescharacteristic
176C4×C137H(5,𝔽2)11yescharacteristic
177Dic137—22no—
178Dic137—41no—
179Dic137—41no—
180Dic137—41no—
181Dic137—41no—
182C2×C137—41no—
183C2×C137—41no—
184C2×C137#54611yescharacteristic
185C2×C137#57511yescharacteristic
186C2×C137#57611yescharacteristic
187D137—81no—
188D137—81no—
189D137—81no—
190C137#330911yescharacteristic, complemented, Sylow
191#3309—1371nomaximal, Sylow
192#553—1371no—
193#550—1371no—
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#435D13711yescharacteristic, complemented
205#417—1371no—
206#416—1371no—
207C2×#41—1371no—
208C2×#41—1371no—
209C2×SD16—1371no—
210C2×D8—1371no—
211C2×D8—1371no—
212C2×M16C2×D13711yescharacteristic
213C22×C8C2×D13711yescharacteristic
214#127—2741no—
215#127—1371no—
216#126—2741no—
217#125—1371no—
218#123—2741no—
219#121—2741no—
220#121—2741no—
221#121—2741no—
222#121—1371no—
223#121—1371no—
224#121—1371no—
225#121—1371no—
226C4×Q8—1371no—
227C4×D4—2741no—
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⋊C8C2×D13711yescharacteristic
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⋊C8C2×D13711yescharacteristic
248C22⋊C8C2×D13711yescharacteristic
249#97C2×D13711yescharacteristic
250#41—2742no—
251#41—2742no—
252#41—5481no—
253#41—5481no—
254C2×Q8—1371no—
255C2×D4—2741no—
256C2×D4—2741no—
257C2×D4—2741no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—2741no—
262C2×D4—2741no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4—1371no—
266C2×D4D54811yescharacteristic
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4—2741no—
271C22×C4C22×D13711yescharacteristic
272C22×C4C22×D13711yescharacteristic
273SD16—2742no—
274D8—5481no—
275D8—5481no—
276M16—41no—
277C2×C8—41no—
278C2×C8C22×D13711yescharacteristic
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C2×C8C22×D13711yescharacteristic
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—2741no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4—1371no—
289C4⋊C4—1371no—
290C4⋊C4D54811yescharacteristic
291C22⋊C4—2741no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4D54811yescharacteristic
298C22⋊C4D54811yescharacteristic
299C42—2741no—
300C42—2741no—
301C42C22×D13711yescharacteristic
302C23—2741no—
303C23—2741no—
304C23—2741no—
305C23C2×D54811yescharacteristic
306C23C2×D54811yescharacteristic
307Q8—2741no—
308Q8—1372no—
309D4—41no—
310D4—2741no—
311D4—2741no—
312D4—2741no—
313D4—2741no—
314D4—2741no—
315D4—2741no—
316D4—5482no—
317D4—5482no—
318D4—5482no—
319D4—5482no—
320D4—10961no—
321D4—10961no—
322D4—10961no—
323D4—10961no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—2741no—
329C2×C4—2741no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—5482no—
334C2×C4—5482no—
335C2×C4—10961no—
336C2×C4—10961no—
337C2×C4C2×D54811yescharacteristic
338C2×C4C2×D54811yescharacteristic
339C2×C4C2×D54811yescharacteristic
340C2×C4C2×D54811yescharacteristic
341C2×C4C23×D13711yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—5481no—
353C22—5481no—
354C22—5481no—
355C22—5481no—
356C22—5481no—
357C22—5481no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22C22×D54811yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—2742no—
367C4—5481no—
368C4—5481no—
369C4—5481no—
370C4—5481no—
371C4#41×D13711yescharacteristic
372C4#1798413311yescharacteristic
373C2—41no—
374C2—41no—
375C2—10961no—
376C2—10961no—
377C2—10961no—
378C2#2257804811yescharacteristic
379C2#2257820311yescharacteristic
380C2#2257820811yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#435×C137C2
socleC22×C137C22×D4
hypercenter#435D137
hypocenterC137#3309
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #22578099C2semidirect
2. #22578074C2semidirect
3. #22577768C2semidirect
4. #22577728C2semidirect
5. #22577717C2semidirect
6. #22577711C2semidirect
7. #22577656C2semidirect
8. #22577639C2semidirect
9. #22577599C2semidirect
10. #22577550C2semidirect
11. #22577550C2semidirect
12. #22577505C2semidirect
13. #22577405C2semidirect
14. #22577400C2semidirect
15. #435×C137C2semidirect
190. C137#3309semidirect
204. #435D137semidirect

Maximal subgroups

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

subgroupindex
#225780992
#225780742
#225777682
#225777282
#225777172
#225777112
#225776562
#225776392
#225775992
#225775502
#225775052
#225774052
#225774002
#435×C1372
#3309137