Finite Group Databasebeta

← #31791030

Subgroups — #31791030

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#22578007C211yesmaximal, characteristic, complemented
2#22577875C211yesmaximal, characteristic, complemented
3#22577726C211yesmaximal, characteristic, complemented
4#22577725C211yesmaximal, characteristic, complemented
5#22577704C211yesmaximal, characteristic, complemented
6#22577703C211yesmaximal, characteristic, complemented
7#22577595C211yesmaximal, characteristic, complemented
8#22577535C211yesmaximal, characteristic, complemented
9#22577535C211yesmaximal, characteristic, complemented
10#22577515C211yesmaximal, characteristic, complemented
11#22577512C211yesmaximal, characteristic, complemented
12D8×Dic137C211yesmaximal, characteristic, complemented
13#22577483C211yesmaximal, characteristic, complemented
14#22577463C211yesmaximal, characteristic, complemented
15#459×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C2×#14555495C2211yescharacteristic
18C2×#14555495C2211yescharacteristic
19#17984080C2211yescharacteristic, complemented
20#17984069—21no—
21#17984069—21no—
22#17984060—21no—
23#17984060—21no—
24#17984059C2211yescharacteristic
25#17984059C2211yescharacteristic
26C4×D548C2211yescharacteristic
27D4×Dic137C2211yescharacteristic
28D4×Dic137C2211yescharacteristic, complemented
29D4×Dic137C2211yescharacteristic, complemented
30#17984050C2211yescharacteristic
31C4×C22⋊D137—21no—
32#17984048—21no—
33#17984046C2211yescharacteristic
34#17984045C2211yescharacteristic
35C2×#14555487—21no—
36C22⋊C4×D137—21no—
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41C4⋊#11764040C2211yescharacteristic, complemented
42#17984010C2211yescharacteristic
43#17984010C2211yescharacteristic
44#17984009C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic, complemented
48#17984000C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C2×D8×C137C2211yescharacteristic
53C2×D8×C137C2211yescharacteristic
54#127×C137C2211yescharacteristic, complemented
55C4×D4×C137C2211yescharacteristic
56C4⋊C8×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C2×D548C2311yescharacteristic
61C2×C22⋊D137—21no—
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64D4×D137—22no—
65D4×D137—22no—
66C22×Dic137—21no—
67C22×Dic137—21no—
68C22×Dic137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137C2311yescharacteristic
72#14555495—22no—
73#14555495—22no—
74C2×#11764040C2311yescharacteristic
75C2×#11764040C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489D411yescharacteristic
82#14555489D411yescharacteristic
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487D411yescharacteristic
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137C2311yescharacteristic
95C2×D4×C137—21no—
96C2×D4×C137—21no—
97C2×D4×C137C2311yescharacteristic
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C22×C4×C137—21no—
101D8×C137—22no—
102D8×C137—22no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—24no—
110C22×D137—81no—
111C22×D137—81no—
112C22×D137C2×D411yescharacteristic
113D548—21no—
114D548#4112yes—
115C22⋊D137—24no—
116C22⋊D137—81no—
117C22⋊D137—81no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—81no—
122C2×Dic137—81no—
123C2×Dic137—81no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—21no—
128C4×D137—21no—
129C4×D137—81no—
130C4×D137—81no—
131#11764040—21no—
132#11764040—21no—
133C23×C137—21no—
134C23×C137—21no—
135C23×C137—21no—
136D4×C137—21no—
137D4×C137—21no—
138D4×C137—21no—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—81no—
142D4×C137—81no—
143D4×C137#4112yes—
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×C137—41no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137C22×D411yescharacteristic
161C2×D137—21no—
162C2×D137—22no—
163C2×D137—21no—
164C2×D137—21no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—84no—
168C4×C137—21no—
169C4×C137—21no—
170C4×C137—41no—
171C4×C137C2×#4111yescharacteristic
172C4×C137C2×#4111yescharacteristic
173Dic137—21no—
174Dic137—21no—
175Dic137—21no—
176Dic137—41no—
177Dic137—41no—
178Dic137—41no—
179C2×C137—42no—
180C2×C137—81no—
181C2×C137—81no—
182C2×C137#52511yescharacteristic
183C2×C137#57611yescharacteristic
184C2×C137#57611yescharacteristic
185D137—41no—
186D137—42no—
187C137#312211yescharacteristic, complemented, Sylow
188#3122—1371nomaximal, Sylow
189#540—1371no—
190#518—1371no—
191#482—1371no—
192#482—1371no—
193#478—1371no—
194#478—1371no—
195#459—1371no—
196#459D13711yescharacteristic, complemented
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×D8C2×D13711yescharacteristic
208C2×D8C2×D13711yescharacteristic
209#127C2×D13711yescharacteristic, complemented
210#125—1371no—
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#107—1371no—
231#106—1371no—
232#106—1371no—
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, complemented
242#102C2×D13711yescharacteristic, complemented
243C22⋊C8—1371no—
244C22⋊C8—1371no—
245#97—1371no—
246C4×C8—1371no—
247C24—2741no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—2742no—
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4C22×D13711yescharacteristic
257C2×D4C22×D13711yescharacteristic
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—22no—
267D8—22no—
268D8—2742no—
269D8—2742no—
270C2×C8—1371no—
271C2×C8—1371no—
272C2×C8C22×D13711yescharacteristic
273C2×C8C22×D13711yescharacteristic
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—1371no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
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—21no—
298C23—21no—
299C23—2744no—
300C23—2741no—
301C23—10961no—
302C23—10961no—
303C23—1371no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—2744no—
312D4—2741no—
313D4—10961no—
314D4—10961no—
315D4—1372no—
316D4#1455550912yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—2741no—
320C2×C4—2741no—
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—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—2741no—
349C22—2741no—
350C22—2741no—
351C22—2742no—
352C22—5481no—
353C22—5481no—
354C22—10964no—
355C22C2×D4×D13711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—2741no—
360C4—2741no—
361C4—2741no—
362C4—5481no—
363C4—5481no—
364C4—5481no—
365C4#41×D13711yescharacteristic
366C4C2×#1455550911yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—5481no—
371C2—5482no—
372C2#2257791311yescharacteristic
373C2#2257820411yescharacteristic
374C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#459×C137C2
socleC22×C137C22×D4
hypercenter#459D137
hypocenterC137#3122
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22578007C2semidirect
2. #22577875C2semidirect
3. #22577726C2semidirect
4. #22577725C2semidirect
5. #22577704C2semidirect
6. #22577703C2semidirect
7. #22577595C2semidirect
8. #22577535C2semidirect
9. #22577535C2semidirect
10. #22577515C2semidirect
11. #22577512C2semidirect
12. D8×Dic137C2semidirect
13. #22577483C2semidirect
14. #22577463C2semidirect
15. #459×C137C2semidirect
19. #17984080C22semidirect
28. D4×Dic137C22semidirect
29. D4×Dic137C22semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. C4⋊#11764040C22semidirect
46. #17984009C22semidirect
47. #17984009C22semidirect
50. #17983998C22semidirect
51. C8×Dic137C22semidirect
54. #127×C137C22semidirect
56. C4⋊C8×C137C22semidirect
57. #102×C137C22semidirect
58. #102×C137C22semidirect
187. C137#3122semidirect
196. #459D137semidirect
209. #127C2×D137semidirect
234. C4⋊C8C2×D137semidirect
241. #102C2×D137semidirect
242. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780072
#225778752
#225777262
#225777252
#225777042
#225777032
#225775952
#225775352
#225775152
#225775122
D8×Dic1372
#225774832
#225774632
#459×C1372
#3122137