Finite Group Databasebeta

← #31792587

Subgroups — #31792587

Order 17536.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22578208C211yesmaximal, characteristic, complemented
2#22578207C212yesmaximal, complemented
3#22578204C211yesmaximal, characteristic, complemented
4#22577778C211yesmaximal, characteristic, complemented
5#22577674C211yesmaximal, characteristic, complemented
6#22577672C212yesmaximal, complemented
7#22577573C212yesmaximal, complemented
8#22577569C212yesmaximal, complemented
9#22577519C211yesmaximal, characteristic, complemented
10#22577518C211yesmaximal, characteristic, complemented
11#472×C137C211yesmaximal, characteristic, complemented
12#17984134C2212yescomplemented
13#17984133C2212yescomplemented
14#17984116—22no—
15#17984116C2212yescomplemented
16#17984116C2212yescomplemented
17#17984115—22no—
18#17984114C2212yescomplemented
19#17984112—22no—
20#17984112—22no—
21#17984110—21no—
22#17984110C2211yescharacteristic
23#17984109—22no—
24#17984108C2211yescharacteristic
25#17984107—21no—
26C2×D1096—21no—
27C2×D1096C2211yescharacteristic, complemented
28C2×#14555495C2212yescomplemented
29D8×D137—21no—
30D8×D137—22no—
31#17984096C2212yescomplemented
32#17984094C2211yescharacteristic, complemented
33#17984084C2212yescomplemented
34#17984023C2211yescharacteristic, complemented
35#17984015C2211yescharacteristic, complemented
36#17984014C2211yescharacteristic, complemented
37#17984014C2212yescomplemented
38#17984004C2212yescomplemented
39C8×Dic137C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C137C2212yescomplemented
41C2×D8×C137C2211yescharacteristic, complemented
42#131×C137C2211yescharacteristic, complemented
43#108×C137C2211yescharacteristic, complemented
44#100×C137C2212yescomplemented
45#14555511—22no—
46#14555510—22no—
47#14555510C2311yescharacteristic
48#14555510C2312yes—
49#14555509—22no—
50#14555509—22no—
51C2×D548—22no—
52C2×D548C2312yes—
53C2×C22⋊D137—42no—
54C2×C22⋊D137—42no—
55D4×D137—22no—
56D4×D137—22no—
57D4×D137—22no—
58D4×D137—42no—
59Dic548D411yescharacteristic, complemented
60#14555500D411yescharacteristic, complemented
61#14555499—22no—
62#14555499—22no—
63#14555499D412yescomplemented
64#14555498—22no—
65#14555498D411yescharacteristic, complemented
66#14555497D411yescharacteristic, complemented
67D1096—22no—
68D1096—22no—
69D1096—21no—
70D1096—21no—
71D1096D411yescharacteristic, complemented
72#14555495—22no—
73#14555495—22no—
74#14555495—22no—
75#14555495D411yescharacteristic, complemented
76#14555494C2311yescharacteristic
77#14555493—22no—
78#14555493—21no—
79C2×#11764040C2311yescharacteristic
80C8×D137—22no—
81C8×D137—21no—
82C4×Dic137C2311yescharacteristic
83#41×C137C2311yescharacteristic
84C2×D4×C137C2312yes—
85SD16×C137—22no—
86D8×C137—22no—
87D8×C137—21no—
88D8×C137—21no—
89M16×C137—21no—
90M16×C137C2312yes—
91M16×C137C2311yescharacteristic
92C2×C8×C137—21no—
93C2×C8×C137C2311yescharacteristic
94C22×D137—22no—
95C22×D137—42no—
96C22×D137—42no—
97Dic274C2×D412yes—
98Dic274C2×D411yescharacteristic
99D548—22no—
100D548—22no—
101D548—42no—
102D548C2×D412yes—
103D548C2×D411yescharacteristic
104C22⋊D137—22no—
105C22⋊D137—21no—
106C22⋊D137—42no—
107C22⋊D137—42no—
108C22⋊D137—42no—
109C22⋊D137—84no—
110C2×Dic137—21no—
111C2×Dic137—42no—
112C4×D137—21no—
113C4×D137—22no—
114C4×D137—42no—
115#11764040—21no—
116#11764040C2×D411yescharacteristic
117#11764040C2×D411yescharacteristic
118C23×C137—22no—
119Q8×C137C2×D411yescharacteristic
120D4×C137—21no—
121D4×C137—22no—
122D4×C137—22no—
123D4×C137C2×D411yescharacteristic
124C2×C4×C137—21no—
125C2×C4×C137C2411yescharacteristic
126C8×C137—21no—
127C8×C137—22no—
128C8×C137C2×D411yescharacteristic
129C8×C137C2×D411yescharacteristic
130C22×C137—21no—
131C22×C137—42no—
132C22×C137—82no—
133C22×C137H(5,𝔽2)11yescharacteristic
134C2×D137—21no—
135C2×D137—22no—
136C2×D137—42no—
137C2×D137—82no—
138C2×D137—162no—
139C2×D137—162no—
140C4×C137—21no—
141C4×C137C22×D411yescharacteristic
142C4×C137C22×D411yescharacteristic
143Dic137—22no—
144Dic137—22no—
145Dic137—21no—
146C2×C137—21no—
147C2×C137—41no—
148C2×C137—82no—
149C2×C137D4211yescharacteristic
150D137—41no—
151D137—42no—
152D137—82no—
153C137#329211yescharacteristic, complemented, Sylow
154#3292—1371nomaximal, Sylow
155#576—1372no—
156#576—1371no—
157#576—1371no—
158#493—1371no—
159#472—1371no—
160#472—1372no—
161#472D13711yescharacteristic, complemented
162#453—1372no—
163#453—1372no—
164#443—1371no—
165#443—1371no—
166H(5,𝔽2)—1372no—
167H(5,𝔽2)—1372no—
168AGL(1,ℤ/8)—2742no—
169AGL(1,ℤ/8)—2742no—
170AGL(1,ℤ/8)—2742no—
171AGL(1,ℤ/8)—2742no—
172AGL(1,ℤ/8)—1372no—
173AGL(1,ℤ/8)—1372no—
174AGL(1,ℤ/8)—1372no—
175AGL(1,ℤ/8)C2×D13712yescomplemented
176#135—2741no—
177#135—2741no—
178#135—2742no—
179#135—1371no—
180#135—1371no—
181C2×D8—2741no—
182C2×D8—2742no—
183C2×D8—2741no—
184C2×D8—1372no—
185C2×D8—1371no—
186C2×D8C2×D13711yescharacteristic, complemented
187#131—1372no—
188#131—1371no—
189#131C2×D13711yescharacteristic, complemented
190#127—1372no—
191#108—1371no—
192#108C2×D13711yescharacteristic, complemented
193#104—1371no—
194#104—1372no—
195#104—1371no—
196#100—1372no—
197#100C2×D13712yescomplemented
198C4×C8—1371no—
199#41—2742no—
200#41—2742no—
201#41—2742no—
202#41—2742no—
203#41—1371no—
204#41—1372no—
205#41C22×D13711yescharacteristic
206C2×D4—2742no—
207C2×D4—2742no—
208C2×D4—2742no—
209C2×D4—2742no—
210C2×D4—5482no—
211C2×D4—5482no—
212C2×D4—5482no—
213C2×D4—1372no—
214C2×D4C22×D13712yes—
215Q16—1371no—
216Q16—1371no—
217SD16—22no—
218SD16—2742no—
219SD16—2742no—
220SD16—2742no—
221SD16—1371no—
222SD16—1371no—
223SD16—1372no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2742no—
228D8—2742no—
229D8—2742no—
230D8—2742no—
231D8—2742no—
232D8—2741no—
233D8—2741no—
234D8—1371no—
235D8—1371no—
236M16—21no—
237M16—2742no—
238M16—2741no—
239M16—1371no—
240M16C22×D13712yes—
241M16C22×D13711yescharacteristic
242C2×C8—21no—
243C2×C8—2742no—
244C2×C8—2741no—
245C2×C8—1371no—
246C2×C8C22×D13711yescharacteristic
247C42—1371no—
248C23—22no—
249C23—2742no—
250C23—5482no—
251C23—5482no—
252Q8—1371no—
253Q8—1372no—
254Q8D4×D13711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2742no—
259D4—2741no—
260D4—2742no—
261D4—2742no—
262D4—5482no—
263D4—5482no—
264D4—5482no—
265D4—5482no—
266D4—10964no—
267D4—1371no—
268D4—1372no—
269D4D4×D13711yescharacteristic
270C2×C4—21no—
271C2×C4—2741no—
272C2×C4—2742no—
273C2×C4—2741no—
274C2×C4—5482no—
275C2×C4—5482no—
276C2×C4C23×D13711yescharacteristic
277C8—21no—
278C8—22no—
279C8—2741no—
280C8—1371no—
281C8—1371no—
282C8D4×D13711yescharacteristic
283C8D4×D13711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—2742no—
288C22—2741no—
289C22—5482no—
290C22—10962no—
291C22—21922no—
292C22—21922no—
293C22#1798413411yescharacteristic
294C4—21no—
295C4—2742no—
296C4—2742no—
297C4—2741no—
298C4C2×D4×D13711yescharacteristic
299C4C2×D4×D13711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—5482no—
304C2—5481no—
305C2—10962no—
306C2#2257803711yescharacteristic
subgroupG/H
centerC2#22578037
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#472×C137C2
socleC2×C137D42
hypercenter#472D137
hypocenterC137#3292
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22578208C2semidirect
2. #22578207C2semidirect
3. #22578204C2semidirect
4. #22577778C2semidirect
5. #22577674C2semidirect
6. #22577672C2semidirect
7. #22577573C2semidirect
8. #22577569C2semidirect
9. #22577519C2semidirect
10. #22577518C2semidirect
11. #472×C137C2semidirect
12. #17984134C22semidirect
13. #17984133C22semidirect
15. #17984116C22semidirect
16. #17984116C22semidirect
18. #17984114C22semidirect
27. C2×D1096C22semidirect
28. C2×#14555495C22semidirect
31. #17984096C22semidirect
32. #17984094C22semidirect
33. #17984084C22semidirect
34. #17984023C22semidirect
35. #17984015C22semidirect
36. #17984014C22semidirect
37. #17984014C22semidirect
38. #17984004C22semidirect
39. C8×Dic137C22semidirect
40. AGL(1,ℤ/8)×C137C22semidirect
41. C2×D8×C137C22semidirect
42. #131×C137C22semidirect
43. #108×C137C22semidirect
44. #100×C137C22semidirect
59. Dic548D4semidirect
60. #14555500D4semidirect
63. #14555499D4semidirect
65. #14555498D4semidirect
66. #14555497D4semidirect
71. D1096D4semidirect
75. #14555495D4semidirect
153. C137#3292semidirect
161. #472D137semidirect
175. AGL(1,ℤ/8)C2×D137semidirect
186. C2×D8C2×D137semidirect
189. #131C2×D137semidirect
192. #108C2×D137semidirect
197. #100C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782082
#225782072
#225782042
#225777782
#225776742
#225776722
#225775732
#225775692
#225775192
#225775182
#472×C1372
#3292137