Finite Group Databasebeta

← #31793924

Subgroups — #31793924

Order 17536.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984111C211yesmaximal, characteristic, complemented
2#135×D137C211yesmaximal, characteristic, complemented
3#22577845C212yesmaximal, complemented
4#22577844C211yesmaximal, characteristic, complemented
5#22577841C211yesmaximal, characteristic, complemented
6#22577840C212yesmaximal, complemented
7#22577838C211yesmaximal, characteristic, complemented
8#22577836C212yesmaximal, complemented
9#22577834C212yesmaximal, complemented
10M32×D137C211yesmaximal, characteristic, complemented
11#510×C137C211yesmaximal, characteristic, complemented
12C2×#14555511—21no—
13#41×D137—21no—
14#17984111—21no—
15#17984111—21no—
16#17984111C2212yes—
17#17984111C2211yescharacteristic, complemented
18#17984111C2212yescomplemented
19#17984110C2211yescharacteristic
20#17984109—21no—
21#17984108—21no—
22#17984107C2211yescharacteristic, complemented
23Q16×D137C2211yescharacteristic, complemented
24C2×#14555497—21no—
25SD16×D137—21no—
26C2×D1096C2211yescharacteristic, complemented
27D8×D137C2211yescharacteristic, complemented
28C2×C8×D137C2211yescharacteristic
29#17984034C2212yescomplemented
30#17984033C2212yescomplemented
31#17984031C2212yescomplemented
32#17984031C2212yescomplemented
33D2192C2212yescomplemented
34#17984029C2212yescomplemented
35#17984028C2211yescharacteristic, complemented
36#17984027C2212yescomplemented
37C16×D137C2212yescomplemented
38#135×C137C2211yescharacteristic
39C2×Q16×C137C2211yescharacteristic, complemented
40Q32×C137C2212yescomplemented
41SD32×C137C2212yescomplemented
42M32×C137C2211yescharacteristic, complemented
43#14555511—21no—
44#14555511—22no—
45#14555511—22no—
46#14555511—41no—
47#14555511—41no—
48#14555510—21no—
49#14555510—41no—
50#14555509—21no—
51#14555509—41no—
52Q8×D137—21no—
53C2×D548—21no—
54C2×D548—41no—
55D4×D137—21no—
56D4×D137—41no—
57C2×C4×D137—41no—
58C2×C4×D137—41no—
59C2×C4×D137D411yescharacteristic
60Dic548C2311yescharacteristic
61#14555500—21no—
62#14555499—21no—
63#14555498—21no—
64#14555497—21no—
65#14555497—22no—
66#14555497—22no—
67D1096—21no—
68D1096C2312yes—
69D1096C2311yescharacteristic
70#14555495—21no—
71C2×#11764040D411yescharacteristic
72C8×D137D411yescharacteristic
73C8×D137D411yescharacteristic
74C8×D137C2311yescharacteristic
75C8×D137C2311yescharacteristic
76#14555485C2312yes—
77#41×C137—21no—
78C2×Q8×C137—21no—
79Q16×C137—21no—
80Q16×C137C2311yescharacteristic
81Q16×C137C2312yes—
82SD16×C137—21no—
83D8×C137C2311yescharacteristic
84C2×C8×C137C2311yescharacteristic
85C16×C137C2312yes—
86C22×D137—41no—
87C22×D137—41no—
88C22×D137D811yescharacteristic
89Dic274—21no—
90Dic274—41no—
91D548—21no—
92D548—22no—
93D548—41no—
94D548—42no—
95D548—42no—
96D548—41no—
97C22⋊D137—41no—
98C22⋊D137—41no—
99C22⋊D137—41no—
100C2×Dic137—41no—
101C2×Dic137D811yescharacteristic
102C4×D137—41no—
103C4×D137—41no—
104C4×D137—41no—
105C4×D137—42no—
106C4×D137—42no—
107C4×D137D811yescharacteristic
108C4×D137D811yescharacteristic
109C4×D137C2×D411yescharacteristic
110C4×D137C2×D411yescharacteristic
111#11764040C2×D411yescharacteristic
112#11764040C2×D411yescharacteristic
113Q8×C137—21no—
114Q8×C137—22no—
115Q8×C137—41no—
116D4×C137—21no—
117D4×C137—41no—
118C2×C4×C137—41no—
119C2×C4×C137—41no—
120C2×C4×C137C2×D411yescharacteristic
121C8×C137C2×D411yescharacteristic
122C8×C137C2411yescharacteristic
123C22×C137—41no—
124C22×C137C2×D811yescharacteristic
125C2×D137—41no—
126C2×D137—41no—
127C2×D137—42no—
128C2×D137—82no—
129C2×D137—82no—
130C2×D137C2×D811yescharacteristic
131C2×D137C2×D811yescharacteristic
132C4×C137—41no—
133C4×C137—42no—
134C4×C137C2×D811yescharacteristic
135C4×C137C22×D411yescharacteristic
136Dic137—41no—
137Dic137C2×D811yescharacteristic
138Dic137C2×D811yescharacteristic
139C2×C137—21no—
140C2×C137—81no—
141C2×C137C22×D811yescharacteristic
142D137—21no—
143D137—21no—
144D137—81no—
145D137—82no—
146C137#341411yescharacteristic, complemented, Sylow
147#3414—1371nomaximal, Sylow
148C2×#135—1371no—
149C2×#135—1371no—
150#510—1371no—
151#510—1372no—
152#510D13711yescharacteristic, complemented
153#509—1371no—
154#509—1372no—
155#509—1371no—
156#508—1372no—
157#508—1372no—
158C2×M32—1371no—
159C2×#41—2741no—
160C2×#41—2741no—
161#135—2741no—
162#135—2741no—
163#135—2741no—
164#135—2741no—
165#135—1372no—
166#135—1371no—
167#135—1372no—
168#135—1371no—
169#135—1371no—
170#135C2×D13711yescharacteristic
171C2×Q16—1371no—
172C2×Q16C2×D13711yescharacteristic, complemented
173C2×SD16—2741no—
174C2×SD16—2741no—
175C2×D8—1371no—
176C2×D8—1371no—
177C22×C8—1371no—
178Q32—1372no—
179Q32C2×D13712yescomplemented
180SD32—1372no—
181SD32—1372no—
182SD32—1372no—
183SD32C2×D13712yescomplemented
184D16—1372no—
185D16—1372no—
186M32—1371no—
187M32—1372no—
188M32C2×D13711yescharacteristic, complemented
189C2×C16—1372no—
190#41—21no—
191#41—2741no—
192#41—2742no—
193#41—2741no—
194#41—2742no—
195#41—2741no—
196#41—5481no—
197#41—5481no—
198#41—5481no—
199#41—5481no—
200C2×Q8—21no—
201C2×Q8—2741no—
202C2×D4—2741no—
203C2×D4—2741no—
204C2×D4—5481no—
205C2×D4—5481no—
206C22×C4—5481no—
207C22×C4—5481no—
208C22×C4—1371no—
209Q16—21no—
210Q16—2741no—
211Q16—1371no—
212Q16C22×D13711yescharacteristic
213Q16C22×D13712yes—
214SD16—21no—
215SD16—2741no—
216SD16—2742no—
217SD16—2741no—
218SD16—2742no—
219SD16—2741no—
220D8—2741no—
221D8—2741no—
222D8—1372no—
223D8—1371no—
224D8C22×D13711yescharacteristic
225C2×C8—1371no—
226C2×C8—1371no—
227C2×C8—1371no—
228C2×C8—1371no—
229C2×C8—1371no—
230C2×C8C22×D13711yescharacteristic
231C16—1372no—
232C16C22×D13712yes—
233C23—5481no—
234C23—5481no—
235C23—1371no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—2741no—
240Q8—5481no—
241D4—21no—
242D4—41no—
243D4—2741no—
244D4—2742no—
245D4—5481no—
246D4—5482no—
247D4—5482no—
248D4—5481no—
249D4—5481no—
250D4—5481no—
251D4—5481no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—5481no—
255C2×C4—5481no—
256C2×C4—5482no—
257C2×C4—5482no—
258C2×C4—5481no—
259C2×C4—5481no—
260C2×C4—1371no—
261C2×C4—1371no—
262C2×C4—1371no—
263C2×C4—1371no—
264C2×C4—1371no—
265C2×C4D4×D13711yescharacteristic
266C8—1371no—
267C8—1371no—
268C8D4×D13711yescharacteristic
269C8C23×D13711yescharacteristic
270C22—41no—
271C22—5481no—
272C22—5482no—
273C22—5481no—
274C22—10962no—
275C22—10962no—
276C22—1371no—
277C22—1371no—
278C22D8×D13711yescharacteristic
279C4—41no—
280C4—42no—
281C4—5481no—
282C4—1371no—
283C4—1371no—
284C4D8×D13711yescharacteristic
285C4C2×D4×D13711yescharacteristic
286C2—21no—
287C2—81no—
288C2—2741no—
289C2—2741no—
290C2—10962no—
291C2—10961no—
292C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
Fitting#510×C137C2
socleC2×C137C22×D8
hypercenter#510D137
hypocenterC137#3414
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#17984111C2semidirect
2. #135×D137C2semidirect
3. #22577845C2semidirect
4. #22577844C2semidirect
5. #22577841C2semidirect
6. #22577840C2semidirect
7. #22577838C2semidirect
8. #22577836C2semidirect
9. #22577834C2semidirect
10. M32×D137C2semidirect
11. #510×C137C2semidirect
17. #17984111C22semidirect
18. #17984111C22semidirect
22. #17984107C22semidirect
23. Q16×D137C22semidirect
26. C2×D1096C22semidirect
27. D8×D137C22semidirect
29. #17984034C22semidirect
30. #17984033C22semidirect
31. #17984031C22semidirect
32. #17984031C22semidirect
33. D2192C22semidirect
34. #17984029C22semidirect
35. #17984028C22semidirect
36. #17984027C22semidirect
37. C16×D137C22semidirect
39. C2×Q16×C137C22semidirect
40. Q32×C137C22semidirect
41. SD32×C137C22semidirect
42. M32×C137C22semidirect
146. C137#3414semidirect
152. #510D137semidirect
172. C2×Q16C2×D137semidirect
179. Q32C2×D137semidirect
183. SD32C2×D137semidirect
188. M32C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841112
#135×D1372
#225778452
#225778442
#225778412
#225778402
#225778382
#225778362
#225778342
M32×D1372
#510×C1372
#3414137