Finite Group Databasebeta

← #31792826

Subgroups — #31792826

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578121C211yesmaximal, characteristic, complemented
2#22578047C211yesmaximal, characteristic
3#22577761C211yesmaximal, characteristic, complemented
4#22577744C211yesmaximal, characteristic, complemented
5#22577726C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577661C211yesmaximal, characteristic, complemented
8#22577631C211yesmaximal, characteristic, complemented
9#22577624C211yesmaximal, characteristic, complemented
10#22577566C211yesmaximal, characteristic, complemented
11#22577554C211yesmaximal, characteristic, complemented
12#22577500C211yesmaximal, characteristic, complemented
13#22577472C211yesmaximal, characteristic, complemented
14#22577396C211yesmaximal, characteristic, complemented
15#418×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Dic548C2211yescharacteristic, complemented
18C2×#14555498C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic, complemented
21#17984087C2211yescharacteristic, complemented
22#17984082—21no—
23#17984081—21no—
24#17984075C2211yescharacteristic
25#17984074—21no—
26#17984069—21no—
27#17984068C2211yescharacteristic, complemented
28#17984067C2211yescharacteristic
29#17984067C2211yescharacteristic
30#17984060—21no—
31#17984059C2211yescharacteristic
32#17984057—21no—
33C4×Dic274—21no—
34#17984053C2211yescharacteristic, complemented
35D4×Dic137C2211yescharacteristic
36C4×C22⋊D137C2211yescharacteristic
37#17984046C2211yescharacteristic, complemented
38C2×C22⋊Dic137—21no—
39#17984021C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984011C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×M16×C137C2211yescharacteristic
54#117×C137C2211yescharacteristic
55#102×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61C2×Dic274—21no—
62C2×Dic274C2311yescharacteristic
63C2×Dic274C2311yescharacteristic
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137D411yescharacteristic
67C22×Dic137—21no—
68C22×Dic137—21no—
69C2×C4×D137C2311yescharacteristic
70Dic548—41no—
71#14555498—22no—
72#14555498—41no—
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C8×D137—41no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137D411yescharacteristic
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C24×C137—21no—
98C2×D4×C137—22no—
99C2×D4×C137—21no—
100C2×D4×C137—41no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104M16×C137—41no—
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C22⋊C4×C137—21no—
110C42×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117C22⋊D137—41no—
118C22⋊D137—42no—
119C22⋊D137—81no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—42no—
125C2×Dic137—81no—
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C4×D137—21no—
130C4×D137—21no—
131#11764040—21no—
132#11764040—21no—
133C23×C137—21no—
134C23×C137—21no—
135C23×C137—42no—
136C23×C137—42no—
137C23×C137—81no—
138C23×C137C2×D411yescharacteristic
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—21no—
142D4×C137—42no—
143D4×C137#4112yes—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—22no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—84no—
162C22×C137C22×D411yescharacteristic
163C2×D137—21no—
164C2×D137—21no—
165C2×D137—21no—
166C4×C137—21no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137H(5,𝔽2)11yescharacteristic
171Dic137—21no—
172Dic137—21no—
173Dic137—21no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—41no—
179C2×C137—42no—
180C2×C137—81no—
181C2×C137#54611yescharacteristic
182C2×C137#57511yescharacteristic
183C2×C137#57711yescharacteristic
184D137—41no—
185C137#331111yescharacteristic, complemented, Sylow
186#3311—1371nomaximal, Sylow
187#556—1371no—
188#546—1371no—
189#489—1371no—
190#485—1371no—
191#482—1371no—
192#476—1371no—
193#470—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198#440—1371no—
199#435—1371no—
200#418D13711yescharacteristic, complemented
201#416—1371no—
202C2×#41—1371no—
203C22×D4C2×D13711yescharacteristic
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C2×M16C2×D13711yescharacteristic
208C22×C8—1371no—
209C4⋊Q8—1371no—
210#126—2741no—
211#126—2741no—
212#124—2741no—
213#124—1371no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8—1371no—
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—2741no—
222C4×Q8—1371no—
223C4×D4—1371no—
224C4×D4—1371no—
225#117—1371no—
226#117C2×D13711yescharacteristic
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106—1371no—
230#106—1371no—
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#102—1371no—
237#102—1371no—
238#102C2×D13711yescharacteristic
239#102C2×D13711yescharacteristic
240#102C2×D13711yescharacteristic
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244#97—1371no—
245C24—21no—
246#41—2742no—
247#41—5481no—
248C2×Q8—2741no—
249C2×Q8—1371no—
250C2×Q8—1371no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—2741no—
255C2×D4—2741no—
256C2×D4—1371no—
257C2×D4C22×D13711yescharacteristic
258C2×D4C22×D13711yescharacteristic
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266M16—41no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
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—21no—
294C42—2741no—
295C42—2741no—
296C42—1371no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1371no—
303C23D4×D13711yescharacteristic
304Q8—2741no—
305Q8—2741no—
306Q8—2741no—
307Q8—2741no—
308Q8—5482no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—5481no—
314D4—5482no—
315D4—10961no—
316D4#1455551012yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—2741no—
349C22—2741no—
350C22—2741no—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—2741no—
356C4—2741no—
357C4—2741no—
358C4—5481no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4C2×#1455551011yescharacteristic
363C4#1798413311yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—5481no—
368C2#2257804311yescharacteristic
369C2#2257819911yescharacteristic
370C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#418×C137C2
socleC22×C137C22×D4
prime radical#22578047C2
hypercenter#418D137
hypocenterC137#3311
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #22578121C2semidirect
3. #22577761C2semidirect
4. #22577744C2semidirect
5. #22577726C2semidirect
6. #22577691C2semidirect
7. #22577661C2semidirect
8. #22577631C2semidirect
9. #22577624C2semidirect
10. #22577566C2semidirect
11. #22577554C2semidirect
12. #22577500C2semidirect
13. #22577472C2semidirect
14. #22577396C2semidirect
15. #418×C137C2semidirect
17. C2×Dic548C22semidirect
20. C2×C8×D137C22semidirect
21. #17984087C22semidirect
27. #17984068C22semidirect
34. #17984053C22semidirect
37. #17984046C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984019C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
45. #17984011C22semidirect
46. #17984011C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
185. C137#3311semidirect
200. #418D137semidirect

Maximal subgroups

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

subgroupindex
#225781212
#225780472
#225777612
#225777442
#225777262
#225776912
#225776612
#225776312
#225776242
#225775662
#225775542
#225775002
#225774722
#225773962
#418×C1372
#3311137