Finite Group Databasebeta

← #31792334

Subgroups — #31792334

Order 17536.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984116C211yesmaximal, characteristic, complemented
2C2×#17984110C211yesmaximal, characteristic, complemented
3#22578074C211yesmaximal, characteristic, complemented
4#22578049C211yesmaximal, characteristic, complemented
5#22577792C211yesmaximal, characteristic, complemented
6#22577649C211yesmaximal, characteristic, complemented
7#22577639C211yesmaximal, characteristic, complemented
8#22577626C211yesmaximal, characteristic, complemented
9#22577599C211yesmaximal, characteristic, complemented
10#22577567C211yesmaximal, characteristic, complemented
11#22577550C211yesmaximal, characteristic, complemented
12#22577550C211yesmaximal, characteristic, complemented
13#22577539C211yesmaximal, characteristic, complemented
14#22577505C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555510C2211yescharacteristic
18C2×#14555510C2211yescharacteristic, complemented
19C22×D548C2211yescharacteristic
20#17984116—24no—
21#17984110—24no—
22C2×Dic548C2211yescharacteristic, complemented
23C2×#14555499C2211yescharacteristic
24C2×#14555499C2211yescharacteristic
25C2×#14555499C2211yescharacteristic, complemented
26C2×#14555499C2211yescharacteristic, complemented
27C2×D1096C2211yescharacteristic
28C2×D1096C2211yescharacteristic
29C2×D1096C2211yescharacteristic
30C2×D1096C2211yescharacteristic
31#17984085C2211yescharacteristic, complemented
32#17984084—21no—
33#17984076C2211yescharacteristic
34#17984073—21no—
35#17984067C2211yescharacteristic
36#17984062C2211yescharacteristic
37#17984062C2211yescharacteristic
38#17984062C2211yescharacteristic, complemented
39#17984061—21no—
40#17984061—21no—
41#17984061—21no—
42#17984056—21no—
43C4×Dic274C2211yescharacteristic, complemented
44C4×D548C2211yescharacteristic
45#17984050—21no—
46C22⋊C4×D137—21no—
47#17984022C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic
52#17984010C2211yescharacteristic, complemented
53#17984010C2211yescharacteristic, complemented
54C2×M16×C137C2211yescharacteristic
55C22×C8×C137C2211yescharacteristic, complemented
56C4×D4×C137C2211yescharacteristic
57C4⋊C8×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C22⋊C8×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555510—22no—
63#14555510—22no—
64#14555510—22no—
65#14555510—41no—
66#14555510—41no—
67#14555510—41no—
68C2×Dic274C2311yescharacteristic
69C2×Dic274C2311yescharacteristic
70C2×D548—22no—
71C2×D548—21no—
72C2×D548—21no—
73C2×D548—21no—
74C2×D548—41no—
75C2×D548C2311yescharacteristic
76C2×D548C2311yescharacteristic
77C2×D548C2311yescharacteristic
78C2×D548C2311yescharacteristic
79C2×C22⋊D137—21no—
80C2×C22⋊D137—21no—
81C2×C22⋊D137—21no—
82C2×C22⋊D137—21no—
83C2×C22⋊D137—21no—
84C2×C22⋊D137—21no—
85C2×C4×D137—21no—
86C2×C4×D137—21no—
87C2×C4×D137—21no—
88C2×C4×D137—21no—
89Dic548—22no—
90#14555499—22no—
91#14555499—22no—
92#14555499—41no—
93#14555499D414yescomplemented
94D1096—22no—
95D1096—22no—
96D1096—22no—
97D1096—41no—
98C4⋊Dic137C2311yescharacteristic
99C4⋊Dic137C2311yescharacteristic
100#14555489—21no—
101#14555489—21no—
102C22⋊Dic137—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109C4×Dic137—21no—
110C2×D4×C137D411yescharacteristic
111C22×C4×C137C2311yescharacteristic
112C22×C4×C137C2311yescharacteristic
113M16×C137—22no—
114C2×C8×C137—22no—
115C2×C8×C137C2311yescharacteristic
116C2×C8×C137C2311yescharacteristic
117C2×C8×C137C2311yescharacteristic
118C2×C8×C137C2311yescharacteristic
119C4⋊C4×C137D411yescharacteristic
120C22⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C42×C137C2311yescharacteristic
123C22×D137—21no—
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—81no—
130Dic274—21no—
131Dic274—21no—
132Dic274—21no—
133Dic274C2×D412yes—
134D548—21no—
135D548—21no—
136D548—21no—
137D548—21no—
138D548—21no—
139D548—21no—
140D548—21no—
141D548—42no—
142D548—42no—
143D548—81no—
144D548C2×D412yes—
145C22⋊D137—42no—
146C22⋊D137—42no—
147C22⋊D137—42no—
148C22⋊D137—81no—
149C22⋊D137—81no—
150C22⋊D137—81no—
151C2×Dic137—21no—
152C2×Dic137—21no—
153C2×Dic137—21no—
154C2×Dic137—21no—
155C4×D137—42no—
156C4×D137—42no—
157C4×D137—42no—
158C4×D137—81no—
159C23×C137C2×D411yescharacteristic
160C23×C137C2×D411yescharacteristic
161D4×C137—41no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2×D411yescharacteristic
169C2×C4×C137C2×D411yescharacteristic
170C2×C4×C137C2411yescharacteristic
171C8×C137—21no—
172C8×C137—21no—
173C8×C137—21no—
174C8×C137C2×D412yes—
175C22×C137—21no—
176C22×C137—21no—
177C22×C137—21no—
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—21no—
181C22×C137C22×D411yescharacteristic
182C2×D137—22no—
183C2×D137—41no—
184C2×D137—41no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—41no—
189C2×D137—41no—
190C2×D137—41no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—84no—
195C4×C137—21no—
196C4×C137—21no—
197C4×C137—21no—
198C4×C137—21no—
199C4×C137C22×D411yescharacteristic
200C4×C137H(5,𝔽2)11yescharacteristic
201Dic137—22no—
202Dic137—41no—
203Dic137—41no—
204Dic137—41no—
205C2×C137—41no—
206C2×C137—41no—
207C2×C137D4211yescharacteristic
208C2×C137#57511yescharacteristic
209C2×C137#57611yescharacteristic
210D137—42no—
211D137—81no—
212D137—81no—
213D137—81no—
214C137#326811yescharacteristic, complemented, Sylow
215#3268—1371nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1371no—
217C2×#135—1371no—
218#550—1371no—
219#546—1371no—
220#496—1371no—
221#468—1371no—
222#466—1371no—
223#464—1371no—
224#459—1371no—
225#452—1371no—
226#449—1371no—
227#449—1371no—
228#447—1371no—
229#440—1371no—
230#435D13711yescharacteristic, complemented
231C2×#41—1371no—
232C2×#41—1371no—
233C2×#41—1371no—
234C22×D4—1371no—
235AGL(1,ℤ/8)—2744no—
236#135—2744no—
237C2×Q16—1371no—
238C2×SD16—1371no—
239C2×SD16—1371no—
240C2×SD16—1371no—
241C2×SD16—1371no—
242C2×D8—1371no—
243C2×D8—1371no—
244C2×D8—1371no—
245C2×D8—1371no—
246C2×M16C2×D13711yescharacteristic
247C22×C8C2×D13711yescharacteristic, complemented
248#127—2741no—
249#127—1371no—
250#124—1371no—
251#123—2741no—
252C22⋊Q8—1371no—
253#121—2741no—
254#121—2741no—
255#121—2741no—
256#121—1371no—
257#121—1371no—
258#121—1371no—
259#120—2741no—
260C4×Q8—1371no—
261C4×D4—2741no—
262C4×D4—1371no—
263C4×D4C2×D13711yescharacteristic
264C2×C22⋊C4—2741no—
265#107—1371no—
266C4⋊C8C2×D13711yescharacteristic, complemented
267#103—1371no—
268#103—1371no—
269#102—1371no—
270#102—1371no—
271#102—1371no—
272#102—1371no—
273C22⋊C8C2×D13711yescharacteristic
274C22⋊C8C2×D13711yescharacteristic, complemented
275#97C2×D13711yescharacteristic, complemented
276C24—2741no—
277#41—2742no—
278#41—2742no—
279#41—2742no—
280#41—5481no—
281#41—5481no—
282#41—5481no—
283C2×Q8—1371no—
284C2×Q8—1371no—
285C2×D4—2742no—
286C2×D4—2741no—
287C2×D4—2741no—
288C2×D4—2741no—
289C2×D4—2741no—
290C2×D4—2741no—
291C2×D4—2741no—
292C2×D4—2741no—
293C2×D4—2741no—
294C2×D4—2741no—
295C2×D4—5481no—
296C2×D4—1371no—
297C2×D4—1371no—
298C2×D4—1371no—
299C2×D4—1371no—
300C2×D4D54811yescharacteristic
301C22×C4—2741no—
302C22×C4—2741no—
303C22×C4—2741no—
304C22×C4—2741no—
305C22×C4C22×D13711yescharacteristic
306C22×C4C22×D13711yescharacteristic
307Q16—2742no—
308SD16—2742no—
309SD16—2742no—
310SD16—5481no—
311SD16—1374no—
312D8—2742no—
313D8—2742no—
314D8—2742no—
315D8—5481no—
316M16—22no—
317C2×C8—22no—
318C2×C8C22×D13711yescharacteristic
319C2×C8C22×D13711yescharacteristic
320C2×C8C22×D13711yescharacteristic
321C2×C8C22×D13711yescharacteristic
322C4⋊C4—2741no—
323C4⋊C4—2741no—
324C4⋊C4—1371no—
325C4⋊C4—1371no—
326C4⋊C4D54811yescharacteristic
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4—2741no—
330C22⋊C4—2741no—
331C22⋊C4—2741no—
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C22⋊C4D54811yescharacteristic
335C22⋊C4D54811yescharacteristic
336C42—2741no—
337C42C22×D13711yescharacteristic
338C23—2741no—
339C23—2741no—
340C23—2741no—
341C23—2741no—
342C23—5482no—
343C23—5482no—
344C23—10961no—
345C23C2×D54811yescharacteristic
346C23C2×D54811yescharacteristic
347Q8—2741no—
348Q8—2741no—
349Q8—2741no—
350Q8—1372no—
351D4—41no—
352D4—2741no—
353D4—2741no—
354D4—2741no—
355D4—2741no—
356D4—2741no—
357D4—2741no—
358D4—2741no—
359D4—5482no—
360D4—5482no—
361D4—5482no—
362D4—5482no—
363D4—5482no—
364D4—10961no—
365D4—10961no—
366D4—10961no—
367D4—10961no—
368D4—1372no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—2741no—
374C2×C4—2741no—
375C2×C4—2741no—
376C2×C4—2741no—
377C2×C4—5482no—
378C2×C4—5482no—
379C2×C4—5482no—
380C2×C4—10961no—
381C2×C4C2×D54811yescharacteristic
382C2×C4C2×D54811yescharacteristic
383C2×C4C2×D54811yescharacteristic
384C2×C4C2×D54811yescharacteristic
385C2×C4C23×D13711yescharacteristic
386C8—21no—
387C8—21no—
388C8—21no—
389C8D4×D13712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—2742no—
397C22—5481no—
398C22—5481no—
399C22—5481no—
400C22—5481no—
401C22—5481no—
402C22—5481no—
403C22—5481no—
404C22—5481no—
405C22—5481no—
406C22—5481no—
407C22—5481no—
408C22—10964no—
409C22C22×D54811yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—2742no—
415C4—5481no—
416C4—5481no—
417C4—5481no—
418C4C2×D4×D13711yescharacteristic
419C4#1798413411yescharacteristic
420C2—41no—
421C2—41no—
422C2—5482no—
423C2—10961no—
424C2—10961no—
425C2—10961no—
426C2D4×D54811yescharacteristic
427C2#2257820311yescharacteristic
428C2#2257820811yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#435×C137C2
socleC22×C137C22×D4
hypercenter#435D137
hypocenterC137#3268
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. C2×#17984116C2semidirect
2. C2×#17984110C2semidirect
3. #22578074C2semidirect
4. #22578049C2semidirect
5. #22577792C2semidirect
6. #22577649C2semidirect
7. #22577639C2semidirect
8. #22577626C2semidirect
9. #22577599C2semidirect
10. #22577567C2semidirect
11. #22577550C2semidirect
12. #22577550C2semidirect
13. #22577539C2semidirect
14. #22577505C2semidirect
15. #435×C137C2semidirect
18. C2×#14555510C22semidirect
22. C2×Dic548C22semidirect
25. C2×#14555499C22semidirect
26. C2×#14555499C22semidirect
31. #17984085C22semidirect
38. #17984062C22semidirect
43. C4×Dic274C22semidirect
47. #17984022C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
52. #17984010C22semidirect
53. #17984010C22semidirect
55. C22×C8×C137C22semidirect
57. C4⋊C8×C137C22semidirect
59. C22⋊C8×C137C22semidirect
60. #97×C137C22semidirect
93. #14555499D4semidirect
214. C137#3268semidirect
230. #435D137semidirect
247. C22×C8C2×D137semidirect
266. C4⋊C8C2×D137semidirect
274. C22⋊C8C2×D137semidirect
275. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×#179841102
#225780742
#225780492
#225777922
#225776492
#225776392
#225776262
#225775992
#225775672
#225775502
#225775392
#225775052
#435×C1372
#3268137