Finite Group Databasebeta

← #31790795

Subgroups — #31790795

Order 17536.

448 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577957C211yesmaximal, characteristic, complemented
2C4×D4×D137C211yesmaximal, characteristic
3#22577735C211yesmaximal, characteristic, complemented
4#22577732C211yesmaximal, characteristic, complemented
5#22577692C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577607C211yesmaximal, characteristic, complemented
8#22577606C211yesmaximal, characteristic, complemented
9#22577555C211yesmaximal, characteristic
10#22577552C211yesmaximal, characteristic
11C4×#14555499C211yesmaximal, characteristic, complemented
12C4×#14555497C211yesmaximal, characteristic, complemented
13C4×#14555498C211yesmaximal, characteristic, complemented
14#22577454C211yesmaximal, characteristic, complemented
15C4×SD16×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C22×C4×D137—21no—
18C2×#14555499C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×#14555497C2211yescharacteristic
21#17984087C2211yescharacteristic, complemented
22#17984086C2211yescharacteristic, complemented
23#17984078—21no—
24#17984068C2211yescharacteristic
25#17984066—21no—
26#17984064C2211yescharacteristic
27C4×Dic274—21no—
28C4×Dic274C2211yescharacteristic, complemented
29C4×D548C2211yescharacteristic
30D4×Dic137C2211yescharacteristic
31#17984050C2211yescharacteristic
32C4×C22⋊D137—21no—
33#17984048—21no—
34#17984045—21no—
35C4⋊C4×D137C2211yescharacteristic
36C22⋊C4×D137—21no—
37C42×D137C2211yescharacteristic
38#17984020C2211yescharacteristic, complemented
39#17984019C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17984000C2211yescharacteristic
51C4×#11764040C2211yescharacteristic, complemented
52C2×SD16×C137C2211yescharacteristic
53C4×Q8×C137C2211yescharacteristic, complemented
54C4×D4×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58C4×C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C2×Dic274—21no—
61C2×Dic274C2311yescharacteristic
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64D4×D137—22no—
65D4×D137—22no—
66C22×Dic137—21no—
67C2×C4×D137—21no—
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—44no—
71C2×C4×D137D411yescharacteristic
72C2×C4×D137D411yescharacteristic
73C2×C4×D137C2311yescharacteristic
74#14555499—22no—
75#14555498—22no—
76#14555497—22no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C4⋊Dic137—21no—
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93C4×Dic137—21no—
94C4×Dic137D411yescharacteristic
95C4×Dic137D411yescharacteristic
96C4×Dic137C2311yescharacteristic
97C2×Q8×C137C2311yescharacteristic
98C2×D4×C137C2311yescharacteristic
99C22×C4×C137—21no—
100SD16×C137—22no—
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C4⋊C4×C137—21no—
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C42×C137—21no—
108C42×C137C2311yescharacteristic
109C22×D137—24no—
110C22×D137—21no—
111C22×D137—42no—
112C22×D137—42no—
113C22×D137C2×D411yescharacteristic
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117D548—21no—
118D548#4112yes—
119C22⋊D137—24no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—42no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—21no—
128C4×D137—21no—
129C4×D137—41no—
130C4×D137—42no—
131C4×D137—42no—
132C4×D137—42no—
133C4×D137SD1614yescomplemented
134#11764040—21no—
135#11764040—21no—
136C23×C137—21no—
137Q8×C137—21no—
138Q8×C137—21no—
139D4×C137—21no—
140D4×C137#4112yes—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—42no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C8×C137—21no—
149C8×C137—21no—
150C22×C137—22no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137C22×D411yescharacteristic
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—22no—
157C2×D137—48no—
158C2×D137—41no—
159C2×D137—41no—
160C2×D137C2×SD1612yes—
161C4×C137—21no—
162C4×C137—41no—
163C4×C137—41no—
164C4×C137—41no—
165C4×C137C2×SD1612yes—
166C4×C137C2×#4111yescharacteristic
167C4×C137C2×#4111yescharacteristic
168Dic137—21no—
169Dic137—21no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173Dic137C2×SD1612yes—
174C2×C137—42no—
175C2×C137#52511yescharacteristic
176C2×C137C22×SD1611yescharacteristic
177C2×C137#57511yescharacteristic
178D137—22no—
179D137—42no—
180C137#310111yescharacteristic, complemented, Sylow
181#3101—1371nomaximal, Sylow
182#533—1371no—
183C2×C4×D4—1371no—
184#483—1371no—
185#483—1371no—
186#476—1371no—
187#476—1371no—
188#461—1371no—
189#461—1371no—
190#450—1371no—
191#450—1371no—
192C4×SD16—1371no—
193C4×SD16—1371no—
194C4×SD16—1371no—
195C4×SD16D13711yescharacteristic, complemented
196#431—1371no—
197C22×D4—1371no—
198C23×C4—2741no—
199C2×SD16—1371no—
200C2×SD16—1371no—
201C2×SD16—1371no—
202C2×SD16C2×D13711yescharacteristic
203C4⋊Q8—1371no—
204C4⋊Q8—1371no—
205#125—2741no—
206C22⋊Q8—2741no—
207C22⋊Q8—1371no—
208C22⋊Q8—1371no—
209C4×Q8—2741no—
210C4×Q8—1371no—
211C4×Q8C2×D13711yescharacteristic, complemented
212C4×D4—2741no—
213C4×D4—2741no—
214C4×D4—1371no—
215C4×D4—1371no—
216C4×D4—1371no—
217C4×D4C2×D13711yescharacteristic
218#117—2741no—
219C2×C4⋊C4—1371no—
220C2×C22⋊C4—2741no—
221C2×C42—1371no—
222#106—1371no—
223#106—1371no—
224#106—1371no—
225#106C2×D13711yescharacteristic, complemented
226C4⋊C8—1371no—
227C4⋊C8—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103C2×D13711yescharacteristic, complemented
232#102—1371no—
233#102—1371no—
234#102—1371no—
235#102C2×D13711yescharacteristic
236C22⋊C8—1371no—
237C22⋊C8—1371no—
238C4×C8—1371no—
239C4×C8C2×D13711yescharacteristic, complemented
240C24—2741no—
241C2×Q8—2741no—
242C2×Q8—1371no—
243C2×Q8C22×D13711yescharacteristic
244C2×D4—2742no—
245C2×D4—2742no—
246C2×D4—2741no—
247C2×D4—1371no—
248C2×D4C22×D13711yescharacteristic
249C22×C4—21no—
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—2741no—
254C22×C4—5484no—
255C22×C4—1371no—
256C22×C4—1371no—
257C22×C4—1371no—
258SD16—22no—
259SD16—2742no—
260SD16—2742no—
261SD16—2742no—
262C2×C8—1371no—
263C2×C8—1371no—
264C2×C8C22×D13711yescharacteristic
265C2×C8C22×D13711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4C22×D13711yescharacteristic
277C4⋊C4C22×D13711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C42—21no—
285C42—2741no—
286C42—1371no—
287C42—1371no—
288C42—1371no—
289C42C22×D13711yescharacteristic
290C23—21no—
291C23—2744no—
292C23—2741no—
293C23—5482no—
294C23—5482no—
295C23—1371no—
296Q8—21no—
297Q8—21no—
298Q8—2741no—
299Q8—2741no—
300Q8—5482no—
301D4—21no—
302D4—2744no—
303D4—2741no—
304D4—1372no—
305D4#1455551012yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—5481no—
316C2×C4—5482no—
317C2×C4—5482no—
318C2×C4—5482no—
319C2×C4—5482no—
320C2×C4—1371no—
321C2×C4—1371no—
322C2×C4—1371no—
323C2×C4—1374no—
324C2×C4D4×D13711yescharacteristic
325C2×C4D4×D13711yescharacteristic
326C2×C4C23×D13711yescharacteristic
327C8—21no—
328C8—21no—
329C8—2741no—
330C8—2741no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—2741no—
335C22—2741no—
336C22—2742no—
337C22—5488no—
338C22—5481no—
339C22—5481no—
340C22—1372no—
341C22C2×D4×D13711yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2741no—
347C4—2741no—
348C4—5481no—
349C4—5481no—
350C4—5481no—
351C4—1372no—
352C4SD16×D13712yes—
353C4#41×D13711yescharacteristic
354C4C2×#1455551011yescharacteristic
355C2—42no—
356C2—2742no—
357C2—5482no—
358C2#2257791211yescharacteristic
359C2C2×SD16×D13711yescharacteristic
360C2#2257820011yescharacteristic
subgroupG/H
centerC2×C4D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×SD16×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenterC4×SD16D137
hypocenterC137#3101
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22577957C2semidirect
3. #22577735C2semidirect
4. #22577732C2semidirect
5. #22577692C2semidirect
6. #22577691C2semidirect
7. #22577607C2semidirect
8. #22577606C2semidirect
11. C4×#14555499C2semidirect
12. C4×#14555497C2semidirect
13. C4×#14555498C2semidirect
14. #22577454C2semidirect
15. C4×SD16×C137C2semidirect
21. #17984087C22semidirect
22. #17984086C22semidirect
28. C4×Dic274C22semidirect
38. #17984020C22semidirect
39. #17984019C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984012C22semidirect
45. #17984011C22semidirect
51. C4×#11764040C22semidirect
53. C4×Q8×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
58. C4×C8×C137C22semidirect
133. C4×D137SD16semidirect
180. C137#3101semidirect
195. C4×SD16D137semidirect
211. C4×Q8C2×D137semidirect
225. #106C2×D137semidirect
231. #103C2×D137semidirect
239. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779572
C4×D4×D1372
#225777352
#225777322
#225776922
#225776912
#225776072
#225776062
#225775552
#225775522
C4×#145554992
C4×#145554972
C4×#145554982
#225774542
C4×SD16×C1372
#3101137