Finite Group Databasebeta

← #27205593

Subgroups — #27205593

Order 13184.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15693048C212yesmaximal, complemented
2C2×#15693045C212yesmaximal, complemented
3#20268340C211yesmaximal, characteristic, complemented
4#127×D103C211yesmaximal, characteristic, complemented
5#20268181C211yesmaximal, characteristic, complemented
6#20268180C211yesmaximal, characteristic, complemented
7#20268178C211yesmaximal, characteristic, complemented
8#20268166C212yesmaximal, complemented
9#20268162C212yesmaximal, complemented
10#20267743C211yesmaximal, characteristic, complemented
11#496×C103C211yesmaximal, characteristic, complemented
12C2×#13110106C2211yescharacteristic
13C2×#13110104C2211yescharacteristic
14C2×D4×D103—22no—
15C2×D4×D103C2212yes—
16#15693048—28no—
17#15693045—28no—
18C2×#13110094C2212yescomplemented
19C2×#13110093C2212yescomplemented
20C2×#13110092C2212yescomplemented
21C2×D824C2212yescomplemented
22C2×#13110090C2212yescomplemented
23C2×#13110090C2212yescomplemented
24C2×#13110088C2212yes—
25#15693019C2211yescharacteristic, complemented
26#15693018C2211yescharacteristic
27#15693017—22no—
28#15693017C2212yes—
29#15693017C2211yescharacteristic, complemented
30#15693009C2211yescharacteristic, complemented
31#15692994—22no—
32(C22⋊D103)⋊C4—22no—
33C42×D103C2211yescharacteristic
34#15692932C2211yescharacteristic, complemented
35C8×Dic103C2212yescomplemented
36C2×SD16×C103C2212yescomplemented
37C2×D8×C103C2212yescomplemented
38#127×C103C2211yescharacteristic
39#124×C103C2211yescharacteristic, complemented
40#97×C103C2211yescharacteristic, complemented
41C23×D103—22no—
42#13110106—24no—
43#13110104—24no—
44C2×Dic206C2311yescharacteristic
45C2×D412—21no—
46C2×D412—22no—
47C2×D412C2312yes—
48C2×D412C2311yescharacteristic
49C2×C22⋊D103—21no—
50C2×C22⋊D103—24no—
51C2×C22⋊D103—22no—
52C2×C22⋊D103—22no—
53D4×D103—24no—
54D4×D103—24no—
55D4×D103—48no—
56C22×Dic103—21no—
57C2×C4×D103—21no—
58C2×C4×D103D412yes—
59C2×C4×D103C2311yescharacteristic
60#13110094—24no—
61#13110093—24no—
62#13110092—24no—
63D824—24no—
64#13110090—24no—
65#13110090—24no—
66#13110088D418yescomplemented
67C2×C103⋊C8C2312yes—
68Dic103⋊C4—22no—
69(C2×D103)⋊C4—22no—
70C4×Dic103D412yes—
71C4×Dic103C2311yescharacteristic
72C2×Q8×C103C2311yescharacteristic
73C2×D4×C103—22no—
74C2×D4×C103C2312yes—
75C2×D4×C103C2311yescharacteristic
76SD16×C103—24no—
77D8×C103—24no—
78C2×C8×C103C2312yes—
79C22⋊C4×C103—22no—
80C42×C103C2311yescharacteristic
81C23×C103—21no—
82C23×C103—22no—
83C22×D103—21no—
84C22×D103—22no—
85C22×D103—48no—
86C22×D103—44no—
87C22×D103C2×D411yescharacteristic
88Q8×C103—22no—
89D4×C103—22no—
90D4×C103—22no—
91D4×C103—22no—
92D4×C103—44no—
93C2×C4×C103—21no—
94C2×C4×C103C2×D412yes—
95C2×C4×C103C2411yescharacteristic
96C22⋊D103—42no—
97C22⋊D103—44no—
98C22⋊D103—48no—
99C22⋊D103—44no—
100C2×Dic103—21no—
101C2×Dic103—42no—
102C2×Dic103C2×D411yescharacteristic
103C2×Dic103C2×D412yes—
104D412—22no—
105D412—22no—
106D412—22no—
107D412—44no—
108D412—42no—
109C4×D103—24no—
110C4×D103—42no—
111C4×D103C2×D414yes—
112Dic206—22no—
113C8×C103C2×D414yescomplemented
114C103⋊C8C2×D414yescomplemented
115C22×C103—41no—
116C22×C103—42no—
117C22×C103—44no—
118C22×C103—42no—
119C22×C103C22×D411yescharacteristic
120C2×D103—22no—
121C2×D103—41no—
122C2×D103—42no—
123C2×D103—44no—
124C2×D103—42no—
125C2×D103—88no—
126C2×D103#12712yes—
127C4×C103—22no—
128C4×C103—41no—
129C4×C103C22×D412yes—
130Dic103—41no—
131Dic103#12712yes—
132Dic103AGL(1,ℤ/8)14yescomplemented
133C2×C103—82no—
134C2×C103—81no—
135C2×C103C2×#12711yescharacteristic
136C2×C103C2×AGL(1,ℤ/8)12yes—
137D103—22no—
138D103—81no—
139D103—82no—
140#3151—1031nomaximal, Sylow
141C103#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—1032no—
143C2×AGL(1,ℤ/8)—1032no—
144#532—1031no—
145C2×#127—1031no—
146#496—1031no—
147#496—1031no—
148#496—1031no—
149#496D10311yescharacteristic, complemented
150#493—1032no—
151#492—1032no—
152C4×M16—1031no—
153C2×#41—1031no—
154C2×#41—1031no—
155C22×D4—1032no—
156C22×D4—2062no—
157AGL(1,ℤ/8)—2068no—
158AGL(1,ℤ/8)—2068no—
159C2×SD16—1032no—
160C2×SD16—1032no—
161C2×SD16—1032no—
162C2×SD16C2×D10312yescomplemented
163C2×D8—1032no—
164C2×D8—1032no—
165C2×D8—1032no—
166C2×D8C2×D10312yescomplemented
167C2×M16—1032no—
168C4⋊Q8—1031no—
169#127—1032no—
170#127—1031no—
171#127—1031no—
172#127—2062no—
173#127C2×D10311yescharacteristic
174#124—1031no—
175#124C2×D10311yescharacteristic, complemented
176#121—2062no—
177C4×D4—2062no—
178C2×C42—1031no—
179#97—1031no—
180#97C2×D10311yescharacteristic, complemented
181C4×C8—1032no—
182C24—2062no—
183#41—2064no—
184#41—2064no—
185C2×Q8—1031no—
186C2×Q8C22×D10311yescharacteristic
187C2×D4—22no—
188C2×D4—1031no—
189C2×D4—1032no—
190C2×D4—4128no—
191C2×D4—2061no—
192C2×D4—2064no—
193C2×D4—2064no—
194C2×D4—2062no—
195C2×D4—2064no—
196C2×D4—2062no—
197C2×D4—2061no—
198C2×D4—2062no—
199C2×D4C22×D10311yescharacteristic
200C2×D4C22×D10312yes—
201C22×C4—1031no—
202C22×C4—1032no—
203C22×C4—2061no—
204C22×C4—2061no—
205SD16—24no—
206SD16—2064no—
207SD16—2064no—
208SD16—2064no—
209D8—24no—
210D8—2064no—
211D8—2064no—
212D8—2064no—
213M16—1038no—
214C2×C8—1032no—
215C2×C8C22×D10312yes—
216C4⋊C4—2062no—
217C22⋊C4—22no—
218C22⋊C4—2062no—
219C42—1032no—
220C42—1031no—
221C42C22×D10311yescharacteristic
222C23—21no—
223C23—22no—
224C23—1031no—
225C23—4124no—
226C23—4128no—
227C23—2061no—
228C23—2062no—
229Q8—22no—
230Q8—2062no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—4124no—
236D4—4122no—
237D4—4128no—
238D4—4124no—
239D4—4124no—
240D4—4122no—
241D4—2062no—
242D4—2062no—
243D4—2062no—
244C2×C4—21no—
245C2×C4—1032no—
246C2×C4—1031no—
247C2×C4—1034no—
248C2×C4—4122no—
249C2×C4—4122no—
250C2×C4—2064no—
251C2×C4—2061no—
252C2×C4D4×D10312yes—
253C2×C4C23×D10311yescharacteristic
254C8—1034no—
255C8D4×D10314yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—8248no—
261C22—1032no—
262C22—4124no—
263C22—4122no—
264C22—4121no—
265C22—4122no—
266C22—2062no—
267C22C2×D4×D10311yescharacteristic
268C4—22no—
269C4—41no—
270C4—1034no—
271C4—1032no—
272C4—4121no—
273C4C2×D4×D10312yes—
274C2—82no—
275C2—81no—
276C2—8242no—
277C2—8241no—
278C2—2062no—
279C2#127×D10311yescharacteristic
280C2AGL(1,ℤ/8)×D10312yes—
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#496×C103C2
socleC22×C103C22×D4
hypercenter#496D103
hypocenterC103#3151
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15693048C2semidirect
2. C2×#15693045C2semidirect
3. #20268340C2semidirect
4. #127×D103C2semidirect
5. #20268181C2semidirect
6. #20268180C2semidirect
7. #20268178C2semidirect
8. #20268166C2semidirect
9. #20268162C2semidirect
10. #20267743C2semidirect
11. #496×C103C2semidirect
18. C2×#13110094C22semidirect
19. C2×#13110093C22semidirect
20. C2×#13110092C22semidirect
21. C2×D824C22semidirect
22. C2×#13110090C22semidirect
23. C2×#13110090C22semidirect
25. #15693019C22semidirect
29. #15693017C22semidirect
30. #15693009C22semidirect
34. #15692932C22semidirect
35. C8×Dic103C22semidirect
36. C2×SD16×C103C22semidirect
37. C2×D8×C103C22semidirect
39. #124×C103C22semidirect
40. #97×C103C22semidirect
66. #13110088D4semidirect
113. C8×C103C2×D4semidirect
114. C103⋊C8C2×D4semidirect
132. Dic103AGL(1,ℤ/8)semidirect
141. C103#3151semidirect
149. #496D103semidirect
162. C2×SD16C2×D103semidirect
166. C2×D8C2×D103semidirect
175. #124C2×D103semidirect
180. #97C2×D103semidirect
255. C8D4×D103semidirect

Maximal subgroups

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

subgroupindex
C2×#156930482
C2×#156930452
#202683402
#127×D1032
#202681812
#202681802
#202681782
#202681662
#202681622
#202677432
#496×C1032
#3151103