Finite Group Databasebeta

← #27205606

Subgroups — #27205606

Order 13184.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15693054C211yesmaximal, characteristic, complemented
2C2×#15693051C211yesmaximal, characteristic, complemented
3C2×#15693047C211yesmaximal, characteristic, complemented
4#20268337C211yesmaximal, characteristic, complemented
5C2×#15693008C211yesmaximal, characteristic, complemented
6#20268182C212yesmaximal, complemented
7#20268179C212yesmaximal, complemented
8#20268174C212yesmaximal, complemented
9#20268172C212yesmaximal, complemented
10M16×Dic103C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C103C211yesmaximal, characteristic, complemented
12C2×#13110105C2211yescharacteristic
13C22×Dic206C2211yescharacteristic
14#15693054—24no—
15#15693051—24no—
16#15693047—24no—
17C2×Dic412C2212yescomplemented
18C2×#13110095C2212yescomplemented
19C2×#13110094C2212yescomplemented
20C2×#13110093C2212yes—
21C2×#13110093C2212yescomplemented
22C2×#13110090C2212yes—
23C2×#13110089C2211yescharacteristic
24#15693019C2211yescharacteristic, complemented
25#15693017C2211yescharacteristic, complemented
26#15693010C2211yescharacteristic, complemented
27#15693008—22no—
28#15693008C2212yes—
29#15693008C2212yes—
30#15693008C2211yescharacteristic, complemented
31#15692993—22no—
32C4×C22⋊D103—22no—
33C2×C22⋊Dic103—22no—
34C2×C4×Dic103C2211yescharacteristic
35#15692931C2212yescomplemented
36C8×Dic103C2212yescomplemented
37C2×#41×C103C2211yescharacteristic
38C22×D4×C103C2211yescharacteristic
39AGL(1,ℤ/8)×C103—24no—
40C2×SD16×C103C2212yes—
41C2×D8×C103C2212yes—
42C2×M16×C103C2211yescharacteristic
43#13110105—22no—
44#13110105—22no—
45C2×Dic206—22no—
46C2×Dic206—22no—
47C2×Dic206—21no—
48C2×Dic206C2312yes—
49C2×Dic206C2311yescharacteristic
50C2×D412C2311yescharacteristic
51C2×C22⋊D103—21no—
52C2×C22⋊D103—22no—
53C22×Dic103—21no—
54C22×Dic103D412yes—
55C2×C4×D103—21no—
56Dic412—24no—
57#13110095—24no—
58#13110094—24no—
59#13110093—24no—
60#13110093—24no—
61#13110090—24no—
62#13110089D414yescomplemented
63C2×C103⋊C8C2312yes—
64Dic103⋊C4—22no—
65C22⋊Dic103—24no—
66C22⋊Dic103—24no—
67C22⋊Dic103—22no—
68(C2×D103)⋊C4—22no—
69C4×Dic103D412yes—
70C4×Dic103C2311yescharacteristic
71C4×Dic103C2311yescharacteristic
72C24×C103—21no—
73#41×C103—22no—
74#41×C103—22no—
75C2×Q8×C103C2311yescharacteristic
76C2×D4×C103—21no—
77C2×D4×C103—22no—
78C2×D4×C103—22no—
79C2×D4×C103—21no—
80C2×D4×C103C2311yescharacteristic
81C2×D4×C103C2312yes—
82C22×C4×C103—21no—
83C22×C4×C103C2311yescharacteristic
84SD16×C103—24no—
85D8×C103—24no—
86M16×C103D414yescomplemented
87C2×C8×C103C2312yes—
88C23×C103—21no—
89C23×C103—22no—
90C23×C103—44no—
91C23×C103—42no—
92C23×C103C2×D411yescharacteristic
93C22×D103—21no—
94Q8×C103—21no—
95Q8×C103—21no—
96D4×C103—21no—
97D4×C103—22no—
98D4×C103—22no—
99D4×C103—21no—
100D4×C103—42no—
101D4×C103—42no—
102C2×C4×C103—21no—
103C2×C4×C103—42no—
104C2×C4×C103C2×D411yescharacteristic
105C2×C4×C103C2×D412yes—
106C2×C4×C103C2×D412yes—
107C2×C4×C103C2411yescharacteristic
108C22⋊D103—44no—
109C22⋊D103—42no—
110C2×Dic103—22no—
111C2×Dic103—21no—
112C2×Dic103—42no—
113C2×Dic103—42no—
114C2×Dic103C2×D412yes—
115C2×Dic103C2×D412yes—
116D412—21no—
117D412—21no—
118C4×D103—42no—
119Dic206—21no—
120Dic206—22no—
121Dic206—22no—
122Dic206—21no—
123Dic206—42no—
124C8×C103C2×D414yes—
125C103⋊C8C2×D414yes—
126C22×C103—22no—
127C22×C103—41no—
128C22×C103—42no—
129C22×C103—42no—
130C22×C103—44no—
131C22×C103—84no—
132C22×C103#12712yes—
133C22×C103C22×D411yescharacteristic
134C2×D103—41no—
135C2×D103—42no—
136C4×C103—41no—
137C4×C103#12712yes—
138C4×C103C22×D411yescharacteristic
139C4×C103C22×D411yescharacteristic
140Dic103—22no—
141Dic103—22no—
142Dic103—42no—
143Dic103—41no—
144C2×C103—22no—
145C2×C103—81no—
146C2×C103—82no—
147C2×C103C2×#12711yescharacteristic
148C2×C103#57512yes—
149D103—81no—
150#3153—1031nomaximal, Sylow
151C103#315311yescharacteristic, complemented, Sylow
152C2×#137—1031no—
153C2×#137—1031no—
154C2×AGL(1,ℤ/8)—1031no—
155C2×AGL(1,ℤ/8)D10311yescharacteristic, complemented
156#532—1031no—
157C2×#124—1031no—
158#497—1032no—
159#496—1032no—
160#495—1032no—
161#495—1032no—
162C4×M16—1031no—
163C2×#41—1031no—
164C2×#41C2×D10311yescharacteristic
165C22×Q8—1031no—
166C22×D4C2×D10311yescharacteristic
167#137—2064no—
168#137—2064no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2064no—
171C2×Q16—1032no—
172C2×Q16—1032no—
173C2×SD16—1032no—
174C2×SD16—1032no—
175C2×SD16—1032no—
176C2×SD16C2×D10312yes—
177C2×D8—1032no—
178C2×D8C2×D10312yes—
179C2×M16—1031no—
180C2×M16C2×D10311yescharacteristic
181C4⋊Q8—1031no—
182#127—1031no—
183#124—1031no—
184#124—1032no—
185#124—1032no—
186#124—1031no—
187#124—2062no—
188#121—2062no—
189C4×D4—2062no—
190C2×C22⋊C4—2062no—
191C2×C42—1031no—
192#97—1032no—
193C4×C8—1032no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2062no—
198#41—2062no—
199C2×Q8—1032no—
200C2×Q8—1031no—
201C2×Q8—2062no—
202C2×Q8—2062no—
203C2×Q8—2061no—
204C2×Q8C22×D10311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1031no—
210C2×D4—2062no—
211C2×D4—2061no—
212C2×D4C22×D10312yes—
213C2×D4C22×D10311yescharacteristic
214C22×C4—21no—
215C22×C4—1032no—
216C22×C4—2061no—
217C22×C4—2061no—
218C22×C4C22×D10311yescharacteristic
219Q16—2064no—
220Q16—2064no—
221SD16—24no—
222SD16—2064no—
223SD16—2064no—
224SD16—2064no—
225D8—24no—
226D8—2064no—
227M16—1034no—
228M16C22⋊D10314yescomplemented
229C2×C8—1032no—
230C2×C8C22×D10312yes—
231C4⋊C4—2062no—
232C22⋊C4—2062no—
233C22⋊C4—2064no—
234C22⋊C4—2064no—
235C22⋊C4—2062no—
236C42—1032no—
237C42—1031no—
238C42—1031no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2061no—
244C23D4×D10311yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—4122no—
248Q8—2061no—
249Q8—2062no—
250Q8—2062no—
251Q8—2061no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—4122no—
259D4—4124no—
260D4—2061no—
261D4—2061no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—1032no—
265C2×C4—1032no—
266C2×C4—4122no—
267C2×C4—4122no—
268C2×C4—4122no—
269C2×C4—2062no—
270C2×C4—2061no—
271C2×C4D4×D10312yes—
272C2×C4D4×D10311yescharacteristic
273C2×C4C2×C22⋊D10312yes—
274C2×C4C23×D10311yescharacteristic
275C8—1034no—
276C8C2×C22⋊D10314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—4121no—
284C22—4122no—
285C22#1569301712yes—
286C22C2×D4×D10311yescharacteristic
287C4—41no—
288C4—4121no—
289C4—4122no—
290C4—2062no—
291C4—2062no—
292C4#1569301712yes—
293C4C2×D4×D10311yescharacteristic
294C4C22×C22⋊D10311yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—8241no—
299C2C2×#1569301711yescharacteristic
300C2#2026858712yes—
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
FittingC2×AGL(1,ℤ/8)×C103C2
socleC22×C103C22×D4
hypercenterC2×AGL(1,ℤ/8)D103
hypocenterC103#3153
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15693054C2semidirect
2. C2×#15693051C2semidirect
3. C2×#15693047C2semidirect
4. #20268337C2semidirect
5. C2×#15693008C2semidirect
6. #20268182C2semidirect
7. #20268179C2semidirect
8. #20268174C2semidirect
9. #20268172C2semidirect
10. M16×Dic103C2semidirect
11. C2×AGL(1,ℤ/8)×C103C2semidirect
17. C2×Dic412C22semidirect
18. C2×#13110095C22semidirect
19. C2×#13110094C22semidirect
21. C2×#13110093C22semidirect
24. #15693019C22semidirect
25. #15693017C22semidirect
26. #15693010C22semidirect
30. #15693008C22semidirect
35. #15692931C22semidirect
36. C8×Dic103C22semidirect
62. #13110089D4semidirect
86. M16×C103D4semidirect
151. C103#3153semidirect
155. C2×AGL(1,ℤ/8)D103semidirect
228. M16C22⋊D103semidirect

Maximal subgroups

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

subgroupindex
C2×#156930542
C2×#156930512
C2×#156930472
#202683372
C2×#156930082
#202681822
#202681792
#202681742
#202681722
M16×Dic1032
C2×AGL(1,ℤ/8)×C1032
#3153103