Finite Group Databasebeta

← #31791034

Subgroups — #31791034

Order 17536.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578009C211yesmaximal, characteristic, complemented
2#22577875C211yesmaximal, characteristic, complemented
3#22577730C211yesmaximal, characteristic, complemented
4#22577729C211yesmaximal, characteristic, complemented
5#22577700C211yesmaximal, characteristic, complemented
6#22577599C211yesmaximal, characteristic, complemented
7#22577598C211yesmaximal, characteristic, complemented
8#22577538C211yesmaximal, characteristic, complemented
9#22577538C211yesmaximal, characteristic, complemented
10#22577514C211yesmaximal, characteristic, complemented
11#22577512C211yesmaximal, characteristic, complemented
12#22577485C211yesmaximal, characteristic, complemented
13#22577483C211yesmaximal, characteristic, complemented
14#22577463C211yesmaximal, characteristic, complemented
15#478×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C2×D1096C2211yescharacteristic
18C2×D1096C2211yescharacteristic
19C2×#14555495C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21#17984085C2211yescharacteristic, complemented
22#17984073—21no—
23#17984073—21no—
24#17984063—21no—
25#17984063—21no—
26#17984063C2211yescharacteristic
27#17984063C2211yescharacteristic
28C4×D548C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30#17984050C2211yescharacteristic
31#17984050C2211yescharacteristic, complemented
32#17984050C2211yescharacteristic, complemented
33C4×C22⋊D137—21no—
34#17984048—21no—
35#17984046C2211yescharacteristic
36#17984045C2211yescharacteristic
37C2×#14555487—21no—
38C22⋊C4×D137—21no—
39#17984021C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41C4⋊#11764040C2211yescharacteristic, complemented
42#17984010C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic, complemented
44#17984008C2211yescharacteristic
45#17984008C2211yescharacteristic
46#17984008C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic, complemented
48#17984000C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52#125×C137C2211yescharacteristic, complemented
53C4×D4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56C4⋊C8×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C23×D137—21no—
60C2×D548—21no—
61C2×D548—21no—
62C2×D548—21no—
63C2×D548—21no—
64C2×D548C2311yescharacteristic
65C2×D548C2311yescharacteristic
66C2×D548C2311yescharacteristic
67C2×C22⋊D137—21no—
68D4×D137—22no—
69D4×D137—22no—
70C22×Dic137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76D1096—22no—
77D1096—22no—
78#14555495—22no—
79#14555495—22no—
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489D411yescharacteristic
84#14555489D411yescharacteristic
85C22⋊Dic137—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C2×D4×C137C2311yescharacteristic
98C22×C4×C137—21no—
99C2×C8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C4⋊C4×C137—21no—
102C4⋊C4×C137—21no—
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—24no—
110C22×D137—21no—
111C22×D137—21no—
112C22×D137—81no—
113C22×D137—81no—
114C22×D137C2×D411yescharacteristic
115D548—21no—
116D548—21no—
117D548—21no—
118D548—21no—
119D548—21no—
120D548—81no—
121D548—81no—
122D548—81no—
123D548—81no—
124D548#4112yes—
125C22⋊D137—24no—
126C2×Dic137—21no—
127C2×Dic137—81no—
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C4×D137—21no—
132C4×D137—21no—
133C4×D137—81no—
134C4×D137—81no—
135C4×D137—81no—
136C4×D137—81no—
137#11764040—21no—
138#11764040—21no—
139C23×C137—21no—
140D4×C137—21no—
141D4×C137#4112yes—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—81no—
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2411yescharacteristic
149C8×C137—21no—
150C8×C137—21no—
151C22×C137—22no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137C22×D411yescharacteristic
155C2×D137—21no—
156C2×D137—22no—
157C2×D137—21no—
158C2×D137—21no—
159C2×D137—41no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—84no—
168C4×C137—21no—
169C4×C137—21no—
170C4×C137—41no—
171C4×C137—41no—
172C4×C137—41no—
173C4×C137C2×#4111yescharacteristic
174C4×C137C2×#4111yescharacteristic
175Dic137—21no—
176Dic137—21no—
177Dic137—21no—
178Dic137—41no—
179C2×C137—42no—
180C2×C137#52511yescharacteristic
181C2×C137#57611yescharacteristic
182C2×C137#57611yescharacteristic
183D137—41no—
184D137—42no—
185D137—81no—
186D137—81no—
187C137#312211yescharacteristic, complemented, Sylow
188#3122—1371nomaximal, Sylow
189#540—1371no—
190#518—1371no—
191#482—1371no—
192#482—1371no—
193#478—1371no—
194#478D13711yescharacteristic, complemented
195#459—1371no—
196#459—1371no—
197#447—1371no—
198#447—1371no—
199#442—1371no—
200#442—1371no—
201C4×D8—1371no—
202C4×D8—1371no—
203#433—1371no—
204C22×D4—1371no—
205C2×D8—1371no—
206C2×D8—1371no—
207C2×D8—1371no—
208C2×D8—1371no—
209#127—1371no—
210#125C2×D13711yescharacteristic, complemented
211#123—2741no—
212#123—2741no—
213#121—2741no—
214#121—2741no—
215#121—1371no—
216#121—1371no—
217C4×D4—2741no—
218C4×D4—2741no—
219C4×D4—1371no—
220C4×D4—1371no—
221C4×D4—1371no—
222C4×D4—1371no—
223C4×D4—1371no—
224C4×D4C2×D13711yescharacteristic
225#117—1371no—
226#117—1371no—
227C2×C22⋊C4—2741no—
228C2×C22⋊C4—2741no—
229#107—1371no—
230#107C2×D13711yescharacteristic, complemented
231#106—1371no—
232#106C2×D13711yescharacteristic, complemented
233C4⋊C8—1371no—
234C4⋊C8C2×D13711yescharacteristic, complemented
235#102—1371no—
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102C2×D13711yescharacteristic
242#102C2×D13711yescharacteristic
243C22⋊C8—1371no—
244C22⋊C8—1371no—
245#97—1371no—
246C4×C8—1371no—
247C24—2741no—
248C2×D4—2741no—
249C2×D4—2741no—
250C2×D4—2742no—
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4—1371no—
258C2×D4C22×D13711yescharacteristic
259C22×C4—21no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—2741no—
263C22×C4—2741no—
264C22×C4—2741no—
265C22×C4—1371no—
266D8—2742no—
267D8—2742no—
268D8—2742no—
269D8—2742no—
270C2×C8—1371no—
271C2×C8—1371no—
272C2×C8C22×D13711yescharacteristic
273C2×C8C22×D13711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1371no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4C22×D13711yescharacteristic
280C4⋊C4C22×D13711yescharacteristic
281C4⋊C4C22×D13711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—2741no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—1371no—
292C22⋊C4—1371no—
293C42—2741no—
294C42—1371no—
295C42C22×D13711yescharacteristic
296C23—21no—
297C23—2744no—
298C23—2741no—
299C23—2741no—
300C23—2741no—
301C23—10961no—
302C23—10961no—
303C23—1371no—
304D4—21no—
305D4—2744no—
306D4—2741no—
307D4—2741no—
308D4—2741no—
309D4—2741no—
310D4—2741no—
311D4—10961no—
312D4—10961no—
313D4—10961no—
314D4—10961no—
315D4—1372no—
316D4#1455551112yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—10961no—
325C2×C4—10961no—
326C2×C4—10961no—
327C2×C4—10961no—
328C2×C4—10961no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4D4×D13711yescharacteristic
334C2×C4C23×D13711yescharacteristic
335C8—21no—
336C8—21no—
337C8—2741no—
338C8—2741no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—2741no—
343C22—2741no—
344C22—2741no—
345C22—2742no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—5481no—
350C22—5481no—
351C22—5481no—
352C22—5481no—
353C22—5481no—
354C22—10964no—
355C22C2×D4×D13711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—2741no—
362C4—2741no—
363C4—2741no—
364C4—5481no—
365C4#41×D13711yescharacteristic
366C4C2×#1455551111yescharacteristic
367C2—42no—
368C2—5481no—
369C2—5482no—
370C2—10961no—
371C2—10961no—
372C2#2257791611yescharacteristic
373C2#2257820511yescharacteristic
374C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#478×C137C2
socleC22×C137C22×D4
hypercenter#478D137
hypocenterC137#3122
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22578009C2semidirect
2. #22577875C2semidirect
3. #22577730C2semidirect
4. #22577729C2semidirect
5. #22577700C2semidirect
6. #22577599C2semidirect
7. #22577598C2semidirect
8. #22577538C2semidirect
9. #22577538C2semidirect
10. #22577514C2semidirect
11. #22577512C2semidirect
12. #22577485C2semidirect
13. #22577483C2semidirect
14. #22577463C2semidirect
15. #478×C137C2semidirect
21. #17984085C22semidirect
31. #17984050C22semidirect
32. #17984050C22semidirect
39. #17984021C22semidirect
40. #17984019C22semidirect
41. C4⋊#11764040C22semidirect
42. #17984010C22semidirect
43. #17984010C22semidirect
46. #17984008C22semidirect
47. #17984008C22semidirect
50. #17983998C22semidirect
51. C8×Dic137C22semidirect
52. #125×C137C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. C4⋊C8×C137C22semidirect
187. C137#3122semidirect
194. #478D137semidirect
210. #125C2×D137semidirect
230. #107C2×D137semidirect
232. #106C2×D137semidirect
234. C4⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780092
#225778752
#225777302
#225777292
#225777002
#225775992
#225775982
#225775382
#225775142
#225775122
#225774852
#225774832
#225774632
#478×C1372
#3122137