Finite Group Databasebeta

← #31790966

Subgroups — #31790966

Order 17536.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578020C211yesmaximal, characteristic, complemented
2#22577878C211yesmaximal, characteristic, complemented
3#22577745C211yesmaximal, characteristic, complemented
4#22577724C211yesmaximal, characteristic, complemented
5#22577691C211yesmaximal, characteristic, complemented
6#22577681C211yesmaximal, characteristic, complemented
7#22577616C211yesmaximal, characteristic, complemented
8#22577616C211yesmaximal, characteristic, complemented
9#22577554C211yesmaximal, characteristic, complemented
10#22577515C211yesmaximal, characteristic, complemented
11#22577500C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13D8×Dic137C211yesmaximal, characteristic, complemented
14#22577461C211yesmaximal, characteristic, complemented
15#447×C137C211yesmaximal, characteristic
16C2×#14555498C2211yescharacteristic
17C2×#14555498C2211yescharacteristic
18#17984087C2211yescharacteristic, complemented
19#17984081—21no—
20#17984075—21no—
21#17984075C2211yescharacteristic, complemented
22#17984069—21no—
23#17984067C2211yescharacteristic
24#17984065—21no—
25#17984053C2211yescharacteristic, complemented
26D4×Dic137—21no—
27D4×Dic137—21no—
28D4×Dic137C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30D4×Dic137C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32D4×Dic137C2211yescharacteristic, complemented
33#17984047C2211yescharacteristic
34#17984047C2211yescharacteristic
35C2×C22⋊Dic137—21no—
36C2×C22⋊Dic137—21no—
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×D8×C137C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274C2311yescharacteristic
61C22×Dic137—21no—
62C22×Dic137—21no—
63C22×Dic137—21no—
64C22×Dic137—21no—
65C22×Dic137—21no—
66#14555498—22no—
67#14555498—22no—
68C2×#11764040C2311yescharacteristic
69C2×#11764040C2311yescharacteristic
70C4⋊Dic137D411yescharacteristic
71C4⋊Dic137D411yescharacteristic
72C4⋊Dic137C2311yescharacteristic
73C4⋊Dic137C2311yescharacteristic
74C4⋊Dic137C2311yescharacteristic
75#14555489—21no—
76#14555489—21no—
77#14555489—21no—
78C22⋊Dic137—21no—
79C22⋊Dic137—21no—
80C22⋊Dic137—21no—
81C22⋊Dic137—21no—
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137D411yescharacteristic
88C22⋊Dic137D411yescharacteristic
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137C2311yescharacteristic
92C4×Dic137C2311yescharacteristic
93C24×C137—21no—
94C2×D4×C137—22no—
95C2×D4×C137—22no—
96C2×D4×C137—21no—
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C2×D4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102D8×C137—22no—
103D8×C137—22no—
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108Dic274—21no—
109Dic274—21no—
110Dic274—81no—
111C2×Dic137—21no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—81no—
116C2×Dic137—81no—
117C2×Dic137—81no—
118C2×Dic137—81no—
119C2×Dic137—81no—
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124#11764040—21no—
125#11764040—21no—
126C23×C137—24no—
127C23×C137—21no—
128C23×C137—21no—
129C23×C137—21no—
130C23×C137—81no—
131C23×C137—81no—
132C23×C137C2×D411yescharacteristic
133D4×C137—24no—
134D4×C137—21no—
135D4×C137—21no—
136D4×C137—21no—
137D4×C137—21no—
138D4×C137—81no—
139D4×C137#4112yes—
140D4×C137#4112yes—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2411yescharacteristic
146C8×C137—21no—
147C8×C137—21no—
148C22×C137—21no—
149C22×C137—22no—
150C22×C137—22no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—84no—
161C22×C137C22×D411yescharacteristic
162C4×C137—21no—
163C4×C137—41no—
164C4×C137C2×#4111yescharacteristic
165C4×C137C2×#4111yescharacteristic
166Dic137—21no—
167Dic137—21no—
168Dic137—21no—
169Dic137—21no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173Dic137—41no—
174C2×C137—41no—
175C2×C137—42no—
176C2×C137—42no—
177C2×C137—81no—
178C2×C137#52511yescharacteristic
179C2×C137#57611yescharacteristic
180C2×C137#57711yescharacteristic
181C137#311811yescharacteristic, complemented, Sylow
182#3118—1371nomaximal, Sylow
183#542—1371no—
184#518—1371no—
185#485—1371no—
186#482—1371no—
187#476—1371no—
188#474—1371no—
189#463—1371no—
190#463—1371no—
191#450—1371no—
192#447D13711yescharacteristic
193#442—1371no—
194#440—1371no—
195C4×SD16—1371no—
196C4×D8—1371no—
197#433—1371no—
198C22×D4C2×D13711yescharacteristic
199C2×SD16—1371no—
200C2×SD16—1371no—
201C2×D8C2×D13711yescharacteristic
202C2×D8C2×D13711yescharacteristic
203C4⋊Q8—1371no—
204#126—2741no—
205#124—2741no—
206#124—1371no—
207#123—2741no—
208C22⋊Q8—2741no—
209C22⋊Q8—1371no—
210#121C2×D13711yescharacteristic
211C4×Q8—1371no—
212C4×D4—2741no—
213C4×D4—2741no—
214C4×D4—1371no—
215C4×D4—1371no—
216C4×D4—1371no—
217C4×D4—1371no—
218C4×D4—1371no—
219#117—1371no—
220#117—1371no—
221C2×C22⋊C4—2741no—
222C2×C22⋊C4—2741no—
223#107—1371no—
224#107—1371no—
225#106—1371no—
226#106—1371no—
227C4⋊C8—1371no—
228C4⋊C8—1371no—
229#103—1371no—
230#103—1371no—
231#102—1371no—
232#102—1371no—
233#102—1371no—
234#102—1371no—
235#102C2×D13711yescharacteristic
236#102C2×D13711yescharacteristic
237C22⋊C8—1371no—
238C22⋊C8C2×D13711yescharacteristic
239#97—1371no—
240C4×C8—1371no—
241C24—21no—
242C2×Q8—2741no—
243C2×Q8—1371no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22×D13711yescharacteristic
249C2×D4C22×D13711yescharacteristic
250C2×D4C22×D13711yescharacteristic
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—2741no—
254C22×C4—2741no—
255C22×C4—2741no—
256C22×C4C22×D13711yescharacteristic
257SD16—2742no—
258SD16—2742no—
259D8—22no—
260D8—22no—
261C2×C8—1371no—
262C2×C8—1371no—
263C2×C8C22×D13711yescharacteristic
264C2×C8C22×D13711yescharacteristic
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—1371no—
269C4⋊C4—1371no—
270C4⋊C4—1371no—
271C4⋊C4—1371no—
272C4⋊C4—1371no—
273C4⋊C4C22×D13711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—1371no—
285C22⋊C4—1371no—
286C42—2741no—
287C42—2741no—
288C42—1371no—
289C42—1371no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—81no—
295C23—81no—
296C23D4×D13711yescharacteristic
297Q8—2741no—
298Q8—2741no—
299Q8—10961no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—81no—
306D4#1455550912yes—
307D4#1455550912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—10961no—
316C2×C4—10961no—
317C2×C4—10961no—
318C2×C4—10961no—
319C2×C4—10961no—
320C2×C4—1371no—
321C2×C4—1371no—
322C2×C4—1371no—
323C2×C4—1371no—
324C2×C4D4×D13711yescharacteristic
325C2×C4C23×D13711yescharacteristic
326C8—21no—
327C8—21no—
328C8—2741no—
329C8—2741no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—84no—
343C22C2×D4×D13711yescharacteristic
344C4—21no—
345C4—41no—
346C4—2741no—
347C4—2741no—
348C4—2741no—
349C4—2741no—
350C4—5481no—
351C4—5481no—
352C4—5481no—
353C4—5481no—
354C4C2×#1455550911yescharacteristic
355C4C2×#1455550911yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—81no—
360C2#2257790911yescharacteristic
361C2#2257820411yescharacteristic
362C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radical#447×C137C2
socleC22×C137C22×D4
hypercenter#447D137
hypocenterC137#3118
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22578020C2semidirect
2. #22577878C2semidirect
3. #22577745C2semidirect
4. #22577724C2semidirect
5. #22577691C2semidirect
6. #22577681C2semidirect
7. #22577616C2semidirect
8. #22577616C2semidirect
9. #22577554C2semidirect
10. #22577515C2semidirect
11. #22577500C2semidirect
12. #22577494C2semidirect
13. D8×Dic137C2semidirect
14. #22577461C2semidirect
18. #17984087C22semidirect
21. #17984075C22semidirect
25. #17984053C22semidirect
32. D4×Dic137C22semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
47. #17984009C22semidirect
48. #17984009C22semidirect
50. #17983998C22semidirect
51. C8×Dic137C22semidirect
181. C137#3118semidirect

Maximal subgroups

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

subgroupindex
#225780202
#225778782
#225777452
#225777242
#225776912
#225776812
#225776162
#225775542
#225775152
#225775002
#225774942
D8×Dic1372
#225774612
#447×C1372
#3118137