Finite Group Databasebeta

← #31792641

Subgroups — #31792641

Order 17536.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578099C211yesmaximal, characteristic, complemented
2#22578049C211yesmaximal, characteristic, complemented
3#22577758C211yesmaximal, characteristic, complemented
4#22577739C211yesmaximal, characteristic, complemented
5#22577728C211yesmaximal, characteristic, complemented
6#22577704C211yesmaximal, characteristic, complemented
7#22577639C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577619C211yesmaximal, characteristic, complemented
10#22577550C211yesmaximal, characteristic, complemented
11#22577539C211yesmaximal, characteristic, complemented
12C4×D1096C211yesmaximal, characteristic, complemented
13#22577400C211yesmaximal, characteristic, complemented
14C2×#17984010C211yesmaximal, characteristic, complemented
15C8×D4×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555499C2211yescharacteristic, complemented
19C2×D1096C2211yescharacteristic
20C2×D1096C2211yescharacteristic
21#17984083—21no—
22#17984080C2211yescharacteristic, complemented
23#17984076C2211yescharacteristic
24#17984073—21no—
25#17984070—21no—
26#17984067C2211yescharacteristic
27#17984062C2211yescharacteristic
28#17984062C2211yescharacteristic
29#17984062C2211yescharacteristic, complemented
30#17984061—21no—
31#17984060—21no—
32#17984056—21no—
33C4×D548C2211yescharacteristic
34C4×D548C2211yescharacteristic, complemented
35#17984048—21no—
36#17984047C2211yescharacteristic, complemented
37C2×C4⋊Dic137C2211yescharacteristic
38C22⋊C4×D137—21no—
39#17984022C2211yescharacteristic, complemented
40#17984022C2211yescharacteristic, complemented
41#17984020C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic, complemented
52C22×C8×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic, complemented
54C4×D4×C137C2211yescharacteristic
55C4⋊C8×C137C2211yescharacteristic, complemented
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic, complemented
58C4×C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62C2×Dic274C2311yescharacteristic
63C2×D548—22no—
64C2×D548—21no—
65C2×D548—41no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C22×Dic137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77#14555499—41no—
78D1096—22no—
79D1096—41no—
80C4⋊Dic137—21no—
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97C4×Dic137—21no—
98C2×D4×C137D411yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C2×C8×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—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—81no—
117Dic274—21no—
118Dic274—21no—
119D548—21no—
120D548—21no—
121D548—21no—
122D548—21no—
123D548—21no—
124D548—42no—
125D548#4112yes—
126C22⋊D137—42no—
127C22⋊D137—42no—
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—81no—
138C4×D137—81no—
139C23×C137C2×D411yescharacteristic
140C23×C137C2×D411yescharacteristic
141D4×C137—22no—
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#13512yes—
161C22×C137C22×D411yescharacteristic
162C2×D137—22no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—84no—
172C4×C137—21no—
173C4×C137—21no—
174C4×C137—21no—
175C4×C137—21no—
176C4×C137C2×#4111yescharacteristic
177C4×C137H(5,𝔽2)11yescharacteristic
178Dic137—41no—
179Dic137—41no—
180Dic137—41no—
181Dic137—41no—
182Dic137—41no—
183C2×C137—22no—
184C2×C137—41no—
185C2×C137#54611yescharacteristic
186C2×C137C2×#13511yescharacteristic
187C2×C137#57611yescharacteristic
188D137—42no—
189D137—81no—
190D137—81no—
191C137#329711yescharacteristic, complemented, Sylow
192#3297—1371nomaximal, Sylow
193#553—1371no—
194#546—1371no—
195#488—1371no—
196#484—1371no—
197#482—1371no—
198#478—1371no—
199#466—1371no—
200#465—1371no—
201#463—1371no—
202#449—1371no—
203#447—1371no—
204C4×D8—1371no—
205C8×D4D13711yescharacteristic, complemented
206#416—1371no—
207C2×#102—1371no—
208C2×#41—1371no—
209C22×D4—1371no—
210C2×SD16—1371no—
211C2×D8—1371no—
212C2×D8—1371no—
213C22×C8C2×D13711yescharacteristic
214C22×C8C2×D13711yescharacteristic, complemented
215#126—2741no—
216#125—1371no—
217#124—1371no—
218#123—2741no—
219#123—2741no—
220C22⋊Q8—1371no—
221#121—2741no—
222#121—2741no—
223#121—1371no—
224#121—1371no—
225#121—1371no—
226#120—2741no—
227C4×D4—2741no—
228C4×D4—1371no—
229C4×D4—1371no—
230C4×D4C2×D13711yescharacteristic
231#117—1371no—
232C2×C4⋊C4—1371no—
233C2×C22⋊C4—2741no—
234#107—1371no—
235#107—1371no—
236#106—1371no—
237C4⋊C8C2×D13711yescharacteristic, complemented
238#103—1371no—
239#103—1371no—
240#103—1371no—
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248C22⋊C8C2×D13711yescharacteristic
249C22⋊C8C2×D13711yescharacteristic, complemented
250C4×C8C2×D13711yescharacteristic, complemented
251C24—2741no—
252#41—2742no—
253#41—5481no—
254C2×Q8—1371no—
255C2×D4—2742no—
256C2×D4—2741no—
257C2×D4—2741no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—5481no—
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4D54811yescharacteristic
266C22×C4—2741no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4C22×D13711yescharacteristic
271C22×C4C22×D13711yescharacteristic
272SD16—5481no—
273D8—2742no—
274D8—5481no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D13711yescharacteristic
278C2×C8C22×D13711yescharacteristic
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C4⋊C4—2741no—
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4—1371no—
289C4⋊C4D54811yescharacteristic
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4D54811yescharacteristic
300C22⋊C4D54811yescharacteristic
301C42—2741no—
302C42C22×D13711yescharacteristic
303C23—2741no—
304C23—2741no—
305C23—2741no—
306C23—5482no—
307C23—5482no—
308C23—10961no—
309C23C2×D54811yescharacteristic
310C23C2×D54811yescharacteristic
311Q8—2741no—
312Q8—2741no—
313D4—22no—
314D4—2741no—
315D4—2741no—
316D4—2741no—
317D4—2741no—
318D4—2741no—
319D4—5482no—
320D4—5482no—
321D4—5482no—
322D4—10961no—
323D4—10961no—
324D4—1372no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—2741no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—5482no—
335C2×C4—10961no—
336C2×C4—10961no—
337C2×C4—10961no—
338C2×C4C2×D54811yescharacteristic
339C2×C4C2×D54811yescharacteristic
340C2×C4C2×D54811yescharacteristic
341C2×C4C2×D54811yescharacteristic
342C2×C4C23×D13711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—2742no—
353C22—5481no—
354C22—5481no—
355C22—5481no—
356C22—5481no—
357C22—5481no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22—10964no—
362C22#1798411012yes—
363C22C22×D54811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—5481no—
369C4—5481no—
370C4—5481no—
371C4—5481no—
372C4—5481no—
373C4#41×D13711yescharacteristic
374C4#1798413311yescharacteristic
375C2—22no—
376C2—41no—
377C2—5482no—
378C2—10961no—
379C2—10961no—
380C2#2257804811yescharacteristic
381C2C2×#1798411011yescharacteristic
382C2#2257820811yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC8×D4×C137C2
socleC22×C137C22×D4
hypercenterC8×D4D137
hypocenterC137#3297
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22578099C2semidirect
2. #22578049C2semidirect
3. #22577758C2semidirect
4. #22577739C2semidirect
5. #22577728C2semidirect
6. #22577704C2semidirect
7. #22577639C2semidirect
8. #22577632C2semidirect
9. #22577619C2semidirect
10. #22577550C2semidirect
11. #22577539C2semidirect
12. C4×D1096C2semidirect
13. #22577400C2semidirect
14. C2×#17984010C2semidirect
15. C8×D4×C137C2semidirect
18. C2×#14555499C22semidirect
22. #17984080C22semidirect
29. #17984062C22semidirect
34. C4×D548C22semidirect
36. #17984047C22semidirect
39. #17984022C22semidirect
40. #17984022C22semidirect
41. #17984020C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
51. #17984010C22semidirect
53. C22×C8×C137C22semidirect
55. C4⋊C8×C137C22semidirect
57. C22⋊C8×C137C22semidirect
58. C4×C8×C137C22semidirect
191. C137#3297semidirect
205. C8×D4D137semidirect
214. C22×C8C2×D137semidirect
237. C4⋊C8C2×D137semidirect
249. C22⋊C8C2×D137semidirect
250. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780992
#225780492
#225777582
#225777392
#225777282
#225777042
#225776392
#225776322
#225776192
#225775502
#225775392
C4×D10962
#225774002
C2×#179840102
C8×D4×C1372
#3297137