Finite Group Databasebeta

← #27205607

Subgroups — #27205607

Order 13184.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15693053C212yesmaximal, complemented
2C2×#15693048C212yesmaximal, complemented
3#20268342C211yesmaximal, characteristic, complemented
4#124×D103C211yesmaximal, characteristic, complemented
5#20268183C211yesmaximal, characteristic, complemented
6#20268181C211yesmaximal, characteristic, complemented
7#20268180C211yesmaximal, characteristic, complemented
8#20268176C212yesmaximal, complemented
9#20268172C212yesmaximal, complemented
10#20267743C211yesmaximal, characteristic, complemented
11#497×C103C211yesmaximal, characteristic, complemented
12C2×#13110106C2212yes—
13C2×Q8×D103C2211yescharacteristic
14C2×D4×D103C2211yescharacteristic
15#15693053—28no—
16#15693048—28no—
17C2×#13110095C2212yescomplemented
18C2×#13110094C2212yescomplemented
19C2×#13110092C2212yes—
20C2×#13110092C2212yescomplemented
21C2×D824C2212yes—
22C2×#13110090C2212yes—
23C2×#13110088C2212yes—
24#15693018C2211yescharacteristic, complemented
25#15693010C2211yescharacteristic
26#15693010C2212yescomplemented
27#15693009C2211yescharacteristic
28#15693008C2211yescharacteristic
29#15693007—22no—
30#15692996—22no—
31#15692983—22no—
32C22⋊C4×D103—22no—
33C42×D103C2211yescharacteristic
34#15692932C2211yescharacteristic, complemented
35C8×Dic103C2212yescomplemented
36C2×Q16×C103C2212yescomplemented
37C2×SD16×C103C2212yes—
38C4⋊Q8×C103C2211yescharacteristic, complemented
39#124×C103C2211yescharacteristic
40#97×C103C2211yescharacteristic, complemented
41C23×D103—21no—
42#13110106—24no—
43#13110106—24no—
44C2×Dic206—21no—
45C2×Dic206C2311yescharacteristic
46Q8×D103—24no—
47C2×D412—22no—
48C2×D412—22no—
49C2×D412C2311yescharacteristic
50C2×D412C2312yes—
51C2×C22⋊D103—21no—
52D4×D103—24no—
53C2×C4×D103—21no—
54C2×C4×D103—22no—
55C2×C4×D103D412yes—
56C2×C4×D103C2311yescharacteristic
57#13110095—24no—
58#13110094—24no—
59#13110092—24no—
60#13110092—24no—
61D824—24no—
62#13110090—24no—
63#13110088D418yescomplemented
64C2×C103⋊C8C2312yes—
65C22⋊Dic103—22no—
66(C2×D103)⋊C4—22no—
67(C2×D103)⋊C4—22no—
68(C2×D103)⋊C4—24no—
69C4×Dic103D412yes—
70C4×Dic103C2311yescharacteristic
71C2×Q8×C103C2311yescharacteristic
72C2×Q8×C103C2312yes—
73C2×D4×C103C2311yescharacteristic
74Q16×C103—24no—
75SD16×C103—24no—
76C2×C8×C103C2312yes—
77C4⋊C4×C103—22no—
78C22⋊C4×C103—22no—
79C42×C103C2311yescharacteristic
80C23×C103—21no—
81C22×D103—21no—
82C22×D103—22no—
83C22×D103—44no—
84C22×D103—42no—
85C22×D103C2×D411yescharacteristic
86Q8×C103—22no—
87Q8×C103—22no—
88Q8×C103—22no—
89D4×C103—22no—
90C2×C4×C103—21no—
91C2×C4×C103—22no—
92C2×C4×C103C2×D412yes—
93C2×C4×C103C2411yescharacteristic
94C22⋊D103—42no—
95C2×Dic103—21no—
96C2×Dic103C2×D412yes—
97C2×Dic103C2×D411yescharacteristic
98D412—22no—
99D412—22no—
100D412—22no—
101D412—44no—
102D412—44no—
103C4×D103—42no—
104C4×D103—42no—
105C4×D103—44no—
106C4×D103C2×D414yes—
107Dic206—22no—
108Dic206—42no—
109C8×C103C2×D414yes—
110C103⋊C8C2×D414yes—
111C22×C103—41no—
112C22×C103—42no—
113C22×C103C22×D411yescharacteristic
114C2×D103—22no—
115C2×D103—42no—
116C2×D103—41no—
117C2×D103—44no—
118C2×D103—42no—
119C2×D103—84no—
120C2×D103#12712yes—
121C4×C103—22no—
122C4×C103—41no—
123C4×C103—42no—
124C4×C103C22×D412yes—
125Dic103—22no—
126Dic103—41no—
127Dic103#12712yes—
128C2×C103—81no—
129C2×C103C2×#12711yescharacteristic
130C2×C103#57512yes—
131D103—22no—
132D103—82no—
133D103—81no—
134#3153—1031nomaximal, Sylow
135C103#315311yescharacteristic, complemented, Sylow
136C2×#137—1032no—
137C2×AGL(1,ℤ/8)—1032no—
138#532—1031no—
139C2×#124—1031no—
140#497—1031no—
141#497D10311yescharacteristic, complemented
142#496—1031no—
143#496—1031no—
144#495—1032no—
145#495—1032no—
146C4×M16—1031no—
147C2×#41—1032no—
148C22×Q8—1031no—
149C22×D4—1031no—
150#137—2068no—
151AGL(1,ℤ/8)—2068no—
152C2×Q16—1032no—
153C2×Q16C2×D10312yescomplemented
154C2×SD16—1032no—
155C2×SD16—1032no—
156C2×SD16—1032no—
157C2×SD16C2×D10312yes—
158C2×D8—1032no—
159C2×D8—1032no—
160C2×M16—1032no—
161C4⋊Q8C2×D10311yescharacteristic, complemented
162#127—1031no—
163#124—1031no—
164#124—1032no—
165#124—1031no—
166#124—1031no—
167#124—2062no—
168#124C2×D10311yescharacteristic
169#121—2062no—
170C4×D4—2062no—
171C2×C22⋊C4—2062no—
172C2×C42—1031no—
173#97—1031no—
174#97C2×D10311yescharacteristic, complemented
175C4×C8—1032no—
176C24—2061no—
177#41—2064no—
178#41—2064no—
179C2×Q8—1031no—
180C2×Q8—2061no—
181C2×Q8—2064no—
182C2×Q8C22×D10311yescharacteristic
183C2×Q8C22×D10312yes—
184C2×D4—1031no—
185C2×D4—1032no—
186C2×D4—2062no—
187C2×D4—2064no—
188C2×D4—2061no—
189C2×D4—2062no—
190C2×D4C22×D10311yescharacteristic
191C22×C4—1031no—
192C22×C4—1032no—
193C22×C4—2062no—
194C22×C4—2061no—
195Q16—24no—
196Q16—2064no—
197SD16—24no—
198SD16—2064no—
199SD16—2064no—
200SD16—2064no—
201D8—2064no—
202D8—2064no—
203M16—1038no—
204C2×C8—1032no—
205C2×C8C22×D10312yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—2062no—
209C22⋊C4—2064no—
210C22⋊C4—2062no—
211C22⋊C4—2062no—
212C42—1032no—
213C42—1031no—
214C42C22×D10311yescharacteristic
215C23—21no—
216C23—1031no—
217C23—4122no—
218C23—4124no—
219C23—2061no—
220C23—2062no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—4122no—
225Q8—2062no—
226D4—22no—
227D4—4124no—
228D4—4124no—
229D4—4122no—
230D4—2062no—
231D4—2062no—
232D4—2062no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—1031no—
236C2×C4—1034no—
237C2×C4—1032no—
238C2×C4—4122no—
239C2×C4—4124no—
240C2×C4—4122no—
241C2×C4—2061no—
242C2×C4D4×D10312yes—
243C2×C4C23×D10311yescharacteristic
244C8—1034no—
245C8D4×D10314yes—
246C22—41no—
247C22—42no—
248C22—8244no—
249C22—1032no—
250C22—4122no—
251C22—4121no—
252C22—4124no—
253C22—4122no—
254C22—2062no—
255C22C2×D4×D10311yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—1032no—
260C4—4121no—
261C4—2062no—
262C4C2×D4×D10312yes—
263C2—81no—
264C2—8241no—
265C2—8242no—
266C2—2062no—
267C2#127×D10311yescharacteristic
268C2#2026859012yes—
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#497×C103C2
socleC22×C103C22×D4
hypercenter#497D103
hypocenterC103#3153
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#15693053C2semidirect
2. C2×#15693048C2semidirect
3. #20268342C2semidirect
4. #124×D103C2semidirect
5. #20268183C2semidirect
6. #20268181C2semidirect
7. #20268180C2semidirect
8. #20268176C2semidirect
9. #20268172C2semidirect
10. #20267743C2semidirect
11. #497×C103C2semidirect
17. C2×#13110095C22semidirect
18. C2×#13110094C22semidirect
20. C2×#13110092C22semidirect
24. #15693018C22semidirect
26. #15693010C22semidirect
34. #15692932C22semidirect
35. C8×Dic103C22semidirect
36. C2×Q16×C103C22semidirect
38. C4⋊Q8×C103C22semidirect
40. #97×C103C22semidirect
63. #13110088D4semidirect
135. C103#3153semidirect
141. #497D103semidirect
153. C2×Q16C2×D103semidirect
161. C4⋊Q8C2×D103semidirect
174. #97C2×D103semidirect

Maximal subgroups

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

subgroupindex
C2×#156930532
C2×#156930482
#202683422
#124×D1032
#202681832
#202681812
#202681802
#202681762
#202681722
#202677432
#497×C1032
#3153103