Finite Group Databasebeta

← #31792575

Subgroups — #31792575

Order 17536.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#22578211C212yesmaximal, complemented
2#22578205C211yesmaximal, characteristic, complemented
3#22577788C211yesmaximal, characteristic, complemented
4#22577675C211yesmaximal, characteristic, complemented
5#22577673C211yesmaximal, characteristic, complemented
6#22577668C212yesmaximal, complemented
7#22577579C212yesmaximal, complemented
8#22577576C212yesmaximal, complemented
9#22577520C211yesmaximal, characteristic, complemented
10#22577518C211yesmaximal, characteristic, complemented
11#578×C137C211yesmaximal, characteristic, complemented
12#17984134C2212yes—
13#17984119—22no—
14#17984119C2212yes—
15#17984115—22no—
16#17984113—22no—
17#17984113C2212yes—
18#17984113C2212yescomplemented
19#17984111—21no—
20#17984108—22no—
21#17984108C2211yescharacteristic
22C2×#14555497—22no—
23C2×#14555497C2212yescomplemented
24C2×D1096C2211yescharacteristic
25D8×D137—21no—
26#17984095C2212yescomplemented
27#17984094C2211yescharacteristic
28#17984077C2212yescomplemented
29#17984024C2211yescharacteristic, complemented
30#17984023C2211yescharacteristic, complemented
31#17984015C2211yescharacteristic, complemented
32#17984015C2212yescomplemented
33#17984014C2211yescharacteristic
34#17984007C2212yescomplemented
35#17984004C2212yescomplemented
36C8×Dic137C2211yescharacteristic, complemented
372−1+4×C137C2212yescomplemented
38#137×C137—22no—
39#137×C137C2212yescomplemented
40#135×C137—21no—
41#135×C137C2211yescharacteristic
42C2×Q16×C137—21no—
43C2×Q16×C137C2211yescharacteristic, complemented
44#131×C137C2211yescharacteristic, complemented
45#14555511—22no—
46#14555510—22no—
47#14555510C2311yescharacteristic
48C2×D548—22no—
49C2×D548C2312yes—
50D4×D137—22no—
51D4×D137—42no—
52#14555500—22no—
53#14555500D411yescharacteristic
54#14555498D411yescharacteristic
55#14555497—22no—
56#14555497—22no—
57#14555497—22no—
58#14555497—22no—
59#14555497—22no—
60#14555497D411yescharacteristic
61D1096—21no—
62D1096—21no—
63#14555495—22no—
64#14555495—22no—
65#14555495D411yescharacteristic
66#14555494—22no—
67#14555494C2312yes—
68#14555494C2311yescharacteristic
69#14555493—21no—
70C2×#11764040—22no—
71C2×#11764040C2311yescharacteristic
72C8×D137—21no—
73#14555487—42no—
74C4×Dic137C2311yescharacteristic
75#41×C137—22no—
76#41×C137—22no—
77#41×C137—42no—
78#41×C137C2312yes—
79#41×C137C2311yescharacteristic
80C2×Q8×C137—22no—
81C2×Q8×C137—22no—
82C2×Q8×C137C2312yes—
83Q16×C137—22no—
84Q16×C137—21no—
85Q16×C137—21no—
86Q16×C137D411yescharacteristic
87SD16×C137—22no—
88SD16×C137D412yes—
89D8×C137D411yescharacteristic
90M16×C137—21no—
91M16×C137C2311yescharacteristic
92C2×C8×C137—21no—
93C2×C8×C137C2311yescharacteristic
94C22×D137—22no—
95C22×D137—42no—
96Dic274C2×D411yescharacteristic
97D548—22no—
98D548—22no—
99D548—42no—
100D548C2×D411yescharacteristic
101C22⋊D137—21no—
102C22⋊D137—42no—
103C2×Dic137—21no—
104C4×D137—21no—
105C4×D137—42no—
106#11764040—21no—
107#11764040—22no—
108#11764040C2×D411yescharacteristic
109#11764040C2×D411yescharacteristic
110Q8×C137—22no—
111Q8×C137—22no—
112Q8×C137—42no—
113Q8×C137C2×D411yescharacteristic
114Q8×C137C2×D412yes—
115D4×C137—21no—
116D4×C137—22no—
117D4×C137—42no—
118D4×C137C2×D411yescharacteristic
119D4×C137C2×D412yes—
120C2×C4×C137—21no—
121C2×C4×C137—22no—
122C2×C4×C137—22no—
123C2×C4×C137—42no—
124C2×C4×C137—42no—
125C2×C4×C137C2411yescharacteristic
126C8×C137—21no—
127C8×C137C2×D411yescharacteristic
128C8×C137C2×D411yescharacteristic
129C22×C137—21no—
130C22×C137—22no—
131C22×C137H(5,𝔽2)11yescharacteristic
132C2×D137—21no—
133C2×D137—42no—
134C2×D137—82no—
135C2×D137—162no—
136C4×C137—21no—
137C4×C137—22no—
138C4×C137—42no—
139C4×C137C22×D411yescharacteristic
140C4×C137C22×D411yescharacteristic
141Dic137—21no—
142Dic137—41no—
143C2×C137—21no—
144C2×C137—42no—
145C2×C137—41no—
146C2×C137D4211yescharacteristic
147D137—41no—
148D137—82no—
149C137#329011yescharacteristic, complemented, Sylow
150#3290—1371nomaximal, Sylow
151#578D13711yescharacteristic, complemented
152#577—1372no—
153#576—1371no—
154#495—1371no—
155#473—1371no—
156#472—1371no—
157#471—1372no—
158#455—1372no—
159#454—1372no—
160#443—1371no—
161#443—1371no—
1622−1+4C2×D13712yescomplemented
163H(5,𝔽2)—1372no—
164#137—22no—
165#137—2742no—
166#137—1372no—
167#137C2×D13712yescomplemented
168AGL(1,ℤ/8)—2742no—
169AGL(1,ℤ/8)—2742no—
170AGL(1,ℤ/8)—1372no—
171AGL(1,ℤ/8)—1372no—
172#135—21no—
173#135—2741no—
174#135—2742no—
175#135—1371no—
176#135C2×D13711yescharacteristic
177C2×Q16—21no—
178C2×Q16C2×D13711yescharacteristic, complemented
179C2×SD16—2742no—
180C2×SD16—1372no—
181C2×D8—2741no—
182C2×D8—1371no—
183#131—1372no—
184#131—1371no—
185#131C2×D13711yescharacteristic, complemented
186#124—1372no—
187#108—1371no—
188#108—1371no—
189#104—1371no—
190#104—1372no—
191#104—1371no—
192#101—1372no—
193#100—1372no—
194C4×C8—1371no—
195#41—22no—
196#41—22no—
197#41—42no—
198#41—2742no—
199#41—2742no—
200#41—1371no—
201#41C22×D13711yescharacteristic
202#41C22×D13712yes—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C22×D13712yes—
206C2×D4—2742no—
207C2×D4—2742no—
208C2×D4—5482no—
209C2×D4—1372no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—2742no—
214Q16—1371no—
215Q16C22⋊D13711yescharacteristic
216SD16—22no—
217SD16—2742no—
218SD16—2742no—
219SD16—2742no—
220SD16—2742no—
221SD16—2742no—
222SD16—1371no—
223SD16—1371no—
224SD16C22⋊D13712yes—
225D8—2742no—
226D8—2742no—
227D8—2741no—
228D8—2741no—
229D8—1371no—
230D8C22⋊D13711yescharacteristic
231M16—21no—
232M16—2742no—
233M16—2741no—
234M16—1372no—
235M16—1371no—
236M16C22×D13711yescharacteristic
237C2×C8—21no—
238C2×C8—2742no—
239C2×C8—2741no—
240C2×C8—1371no—
241C2×C8C22×D13711yescharacteristic
242C22⋊C4—5482no—
243C42—1371no—
244C23—2742no—
245C23—5482no—
246Q8—22no—
247Q8—22no—
248Q8—42no—
249Q8—1371no—
250Q8D4×D13711yescharacteristic
251Q8C2×C22⋊D13712yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—2741no—
256D4—2742no—
257D4—2742no—
258D4—5482no—
259D4—5482no—
260D4—1371no—
261D4D4×D13711yescharacteristic
262D4C2×C22⋊D13712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—2741no—
269C2×C4—2741no—
270C2×C4—5482no—
271C2×C4C23×D13711yescharacteristic
272C8—21no—
273C8—2742no—
274C8—2741no—
275C8—1371no—
276C8—1371no—
277C8C2×C22⋊D13711yescharacteristic
278C8C2×C22⋊D13711yescharacteristic
279C22—21no—
280C22—22no—
281C22—2741no—
282C22—5482no—
283C22—10962no—
284C22—21922no—
285C22#1798413311yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—2741no—
290C4—5481no—
291C4C2×D4×D13711yescharacteristic
292C4C22×C22⋊D13711yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—5481no—
297C2—10962no—
298C2D4×C22⋊D13711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#578×C137C2
socleC2×C137D42
hypercenter#578D137
hypocenterC137#3290
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #22578211C2semidirect
2. #22578205C2semidirect
3. #22577788C2semidirect
4. #22577675C2semidirect
5. #22577673C2semidirect
6. #22577668C2semidirect
7. #22577579C2semidirect
8. #22577576C2semidirect
9. #22577520C2semidirect
10. #22577518C2semidirect
11. #578×C137C2semidirect
18. #17984113C22semidirect
23. C2×#14555497C22semidirect
26. #17984095C22semidirect
28. #17984077C22semidirect
29. #17984024C22semidirect
30. #17984023C22semidirect
31. #17984015C22semidirect
32. #17984015C22semidirect
34. #17984007C22semidirect
35. #17984004C22semidirect
36. C8×Dic137C22semidirect
37. 2−1+4×C137C22semidirect
39. #137×C137C22semidirect
43. C2×Q16×C137C22semidirect
44. #131×C137C22semidirect
149. C137#3290semidirect
151. #578D137semidirect
162. 2−1+4C2×D137semidirect
167. #137C2×D137semidirect
178. C2×Q16C2×D137semidirect
185. #131C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782112
#225782052
#225777882
#225776752
#225776732
#225776682
#225775792
#225775762
#225775202
#225775182
#578×C1372
#3290137