Finite Group Databasebeta

← #27205592

Subgroups — #27205592

Order 13184.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15693049C211yesmaximal, characteristic, complemented
2C2×#15693047C211yesmaximal, characteristic, complemented
3C2×#15693046C211yesmaximal, characteristic, complemented
4#20268337C211yesmaximal, characteristic, complemented
5C2×#15693017C211yesmaximal, characteristic, complemented
6#20268179C212yesmaximal, complemented
7#20268177C212yesmaximal, complemented
8#20268166C212yesmaximal, complemented
9#20268162C212yesmaximal, complemented
10M16×Dic103C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C103C211yesmaximal, characteristic, complemented
12C2×#13110105C2211yescharacteristic
13C22×D412C2211yescharacteristic
14C22×C22⋊D103—22no—
15#15693049—24no—
16#15693047—24no—
17#15693046—24no—
18C2×#13110094C2212yescomplemented
19C2×#13110093C2212yescomplemented
20C2×#13110092C2212yescomplemented
21C2×D824C2212yescomplemented
22C2×#13110090C2212yescomplemented
23C2×#13110090C2212yescomplemented
24C2×#13110089C2211yescharacteristic
25#15693019C2211yescharacteristic, complemented
26#15693017—22no—
27#15693017C2212yes—
28#15693017C2212yes—
29#15693017C2211yescharacteristic, complemented
30#15693010C2211yescharacteristic, complemented
31#15693008C2211yescharacteristic, complemented
32#15692993—22no—
33C4×C22⋊D103—22no—
34C2×C4×Dic103C2211yescharacteristic
35#15692931C2212yescomplemented
36C8×Dic103C2212yescomplemented
37C2×#41×C103C2211yescharacteristic
38C22×D4×C103C2211yescharacteristic
39AGL(1,ℤ/8)×C103—24no—
40C2×SD16×C103C2212yescomplemented
41C2×D8×C103C2212yescomplemented
42C2×M16×C103C2211yescharacteristic
43C23×D103—21no—
44#13110105—22no—
45#13110105—22no—
46C2×Dic206C2311yescharacteristic
47C2×D412—22no—
48C2×D412—22no—
49C2×D412—21no—
50C2×D412C2312yes—
51C2×D412C2311yescharacteristic
52C2×C22⋊D103—21no—
53C2×C22⋊D103—24no—
54C2×C22⋊D103—24no—
55C2×C22⋊D103—22no—
56C2×C22⋊D103—48no—
57C22×Dic103D412yes—
58C2×C4×D103—21no—
59#13110094—24no—
60#13110093—24no—
61#13110092—24no—
62D824—24no—
63#13110090—24no—
64#13110090—24no—
65#13110089D414yescomplemented
66C2×C103⋊C8C2312yes—
67Dic103⋊C4—22no—
68C22⋊Dic103—22no—
69(C2×D103)⋊C4—22no—
70C4×Dic103D412yes—
71C4×Dic103C2311yescharacteristic
72C4×Dic103C2311yescharacteristic
73C24×C103—21no—
74#41×C103—22no—
75#41×C103—22no—
76C2×Q8×C103C2311yescharacteristic
77C2×D4×C103—21no—
78C2×D4×C103—22no—
79C2×D4×C103—22no—
80C2×D4×C103—21no—
81C2×D4×C103C2311yescharacteristic
82C2×D4×C103C2312yes—
83C22×C4×C103—21no—
84C22×C4×C103C2311yescharacteristic
85SD16×C103—24no—
86D8×C103—24no—
87M16×C103D414yescomplemented
88C2×C8×C103C2312yes—
89C23×C103—21no—
90C23×C103—22no—
91C23×C103—44no—
92C23×C103—42no—
93C23×C103C2×D411yescharacteristic
94C22×D103—21no—
95C22×D103—22no—
96C22×D103—44no—
97C22×D103—42no—
98Q8×C103—21no—
99Q8×C103—21no—
100D4×C103—21no—
101D4×C103—22no—
102D4×C103—22no—
103D4×C103—21no—
104D4×C103—42no—
105D4×C103—42no—
106C2×C4×C103—21no—
107C2×C4×C103—42no—
108C2×C4×C103C2×D411yescharacteristic
109C2×C4×C103C2×D412yes—
110C2×C4×C103C2×D412yes—
111C2×C4×C103C2411yescharacteristic
112C22⋊D103—48no—
113C22⋊D103—42no—
114C22⋊D103—44no—
115C22⋊D103—48no—
116C2×Dic103—24no—
117C2×Dic103—21no—
118C2×Dic103C2×D412yes—
119C2×Dic103C2×D412yes—
120D412—21no—
121D412—22no—
122D412—21no—
123D412—22no—
124D412—42no—
125C4×D103—42no—
126Dic206—21no—
127Dic206—21no—
128C8×C103C2×D414yescomplemented
129C103⋊C8C2×D414yescomplemented
130C22×C103—22no—
131C22×C103—41no—
132C22×C103—42no—
133C22×C103—44no—
134C22×C103—42no—
135C22×C103—84no—
136C22×C103#12712yes—
137C22×C103C22×D411yescharacteristic
138C2×D103—41no—
139C2×D103—42no—
140C2×D103—44no—
141C2×D103—42no—
142C2×D103—84no—
143C4×C103—41no—
144C4×C103#12712yes—
145C4×C103C22×D411yescharacteristic
146C4×C103C22×D411yescharacteristic
147Dic103—22no—
148Dic103—41no—
149Dic103AGL(1,ℤ/8)14yescomplemented
150C2×C103—22no—
151C2×C103—81no—
152C2×C103—82no—
153C2×C103C2×#12711yescharacteristic
154C2×C103C2×AGL(1,ℤ/8)12yes—
155D103—81no—
156D103—82no—
157#3151—1031nomaximal, Sylow
158C103#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—1031no—
160C2×AGL(1,ℤ/8)—1031no—
161C2×AGL(1,ℤ/8)—1031no—
162C2×AGL(1,ℤ/8)D10311yescharacteristic, complemented
163#532—1031no—
164C2×#127—1031no—
165#496—1032no—
166#496—1032no—
167#493—1032no—
168#492—1032no—
169C4×M16—1031no—
170C2×#41—1031no—
171C2×#41C2×D10311yescharacteristic
172C22×D4—1031no—
173C22×D4—2062no—
174C22×D4C2×D10311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2064no—
177AGL(1,ℤ/8)—2064no—
178AGL(1,ℤ/8)—2064no—
179C2×SD16—1032no—
180C2×SD16—1032no—
181C2×SD16—1032no—
182C2×SD16C2×D10312yescomplemented
183C2×D8—1032no—
184C2×D8—1032no—
185C2×D8—1032no—
186C2×D8C2×D10312yescomplemented
187C2×M16—1031no—
188C2×M16C2×D10311yescharacteristic
189C4⋊Q8—1031no—
190#127—1032no—
191#127—1032no—
192#127—1031no—
193#127—2062no—
194#124—1031no—
195#124—1031no—
196#121—2062no—
197C4×D4—2062no—
198C2×C42—1031no—
199#97—1032no—
200C4×C8—1032no—
201C24—21no—
202C24—2061no—
203#41—22no—
204#41—22no—
205#41—2062no—
206#41—2062no—
207C2×Q8—1031no—
208C2×Q8C22×D10311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—1031no—
214C2×D4—1032no—
215C2×D4—4128no—
216C2×D4—2062no—
217C2×D4—2064no—
218C2×D4—2064no—
219C2×D4—2061no—
220C2×D4—2061no—
221C2×D4—2062no—
222C2×D4—2062no—
223C2×D4C22×D10311yescharacteristic
224C2×D4C22×D10312yes—
225C22×C4—21no—
226C22×C4—1032no—
227C22×C4—2061no—
228C22×C4C22×D10311yescharacteristic
229SD16—24no—
230SD16—2064no—
231SD16—2064no—
232SD16—2064no—
233D8—24no—
234D8—2064no—
235D8—2064no—
236D8—2064no—
237M16—1034no—
238M16C22⋊D10314yescomplemented
239C2×C8—1032no—
240C2×C8C22×D10312yes—
241C4⋊C4—2062no—
242C22⋊C4—2062no—
243C22⋊C4—2062no—
244C42—1032no—
245C42—1031no—
246C42—1031no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—4124no—
252C23—4122no—
253C23—2061no—
254C23—2062no—
255C23D4×D10311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2061no—
259Q8—2061no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—4128no—
267D4—4124no—
268D4—4122no—
269D4—4128no—
270D4—4122no—
271D4—2061no—
272D4—2062no—
273D4—2061no—
274D4—2062no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1032no—
278C2×C4—1032no—
279C2×C4—4122no—
280C2×C4—2064no—
281C2×C4—2061no—
282C2×C4D4×D10312yes—
283C2×C4D4×D10311yescharacteristic
284C2×C4C2×C22⋊D10312yes—
285C2×C4C23×D10311yescharacteristic
286C8—1034no—
287C8C2×C22⋊D10314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—8244no—
295C22—4124no—
296C22—4122no—
297C22—4121no—
298C22—4122no—
299C22#1569301712yes—
300C22C2×D4×D10311yescharacteristic
301C4—41no—
302C4—1034no—
303C4—4121no—
304C4—2062no—
305C4#1569301712yes—
306C4C2×D4×D10311yescharacteristic
307C4C22×C22⋊D10311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—8242no—
312C2—8241no—
313C2C2×#1569301711yescharacteristic
314C2AGL(1,ℤ/8)×D10312yes—
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#3151
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×#15693049C2semidirect
2. C2×#15693047C2semidirect
3. C2×#15693046C2semidirect
4. #20268337C2semidirect
5. C2×#15693017C2semidirect
6. #20268179C2semidirect
7. #20268177C2semidirect
8. #20268166C2semidirect
9. #20268162C2semidirect
10. M16×Dic103C2semidirect
11. C2×AGL(1,ℤ/8)×C103C2semidirect
18. C2×#13110094C22semidirect
19. C2×#13110093C22semidirect
20. C2×#13110092C22semidirect
21. C2×D824C22semidirect
22. C2×#13110090C22semidirect
23. C2×#13110090C22semidirect
25. #15693019C22semidirect
29. #15693017C22semidirect
30. #15693010C22semidirect
31. #15693008C22semidirect
35. #15692931C22semidirect
36. C8×Dic103C22semidirect
40. C2×SD16×C103C22semidirect
41. C2×D8×C103C22semidirect
65. #13110089D4semidirect
87. M16×C103D4semidirect
128. C8×C103C2×D4semidirect
129. C103⋊C8C2×D4semidirect
149. Dic103AGL(1,ℤ/8)semidirect
158. C103#3151semidirect
162. C2×AGL(1,ℤ/8)D103semidirect
182. C2×SD16C2×D103semidirect
186. C2×D8C2×D103semidirect
238. M16C22⋊D103semidirect
287. C8C2×C22⋊D103semidirect

Maximal subgroups

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

subgroupindex
C2×#156930492
C2×#156930472
C2×#156930462
#202683372
C2×#156930172
#202681792
#202681772
#202681662
#202681622
M16×Dic1032
C2×AGL(1,ℤ/8)×C1032
#3151103