Finite Group Databasebeta

← #450×D103 #27204323

Subgroups — #450×D103 #27204323

Order 13184.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×D4×D103C211yesmaximal, characteristic
2C2×SD16×D103C212yesmaximal, complemented
3C22⋊Q8×D103C211yesmaximal, characteristic, complemented
4#20267944C211yesmaximal, characteristic, complemented
5#20267943C212yesmaximal, complemented
6#20267942C211yesmaximal, characteristic, complemented
7#20267941C211yesmaximal, characteristic, complemented
8#20267940C212yesmaximal, complemented
9#102×D103C212yesmaximal, complemented
10C22⋊C8×D103C211yesmaximal, characteristic, complemented
11#450×C103C211yesmaximal, characteristic, complemented
12C24×D103—21no—
13C2×Q8×D103C2211yescharacteristic, complemented
14C22×D412C2211yescharacteristic
15C22×C22⋊D103—21no—
16C2×D4×D103—28no—
17C2×D4×D103—21no—
18C2×D4×D103—21no—
19C2×D4×D103C2212yes—
20C2×D4×D103C2212yes—
21C22×C4×D103C2211yescharacteristic
22C2×#13110094C2212yescomplemented
23C2×#13110093C2212yescomplemented
24C2×#13110092C2212yescomplemented
25SD16×D103—28no—
26C2×C8×D103C2212yescomplemented
27#15693001C2211yescharacteristic, complemented
28#15693000C2211yescharacteristic, complemented
29#15692999—21no—
30#15692998—21no—
31#15692997C2211yescharacteristic, complemented
32C4⋊C4×D103—21no—
33C4⋊C4×D103C2211yescharacteristic, complemented
34C22⋊C4×D103—21no—
35#15692943C2212yescomplemented
36#15692942C2212yescomplemented
37#15692941C2212yescomplemented
38C22⋊(C103⋊C8)C2211yescharacteristic, complemented
39(C2×D103)⋊C8C2212yescomplemented
40C22×D4×C103C2211yescharacteristic
41C2×SD16×C103C2212yescomplemented
42C22⋊Q8×C103C2211yescharacteristic, complemented
43#102×C103C2212yescomplemented
44C22⋊C8×C103C2211yescharacteristic, complemented
45C23×D103—28no—
46C23×D103—21no—
47C23×D103—22no—
48C23×D103—22no—
49C23×D103—44no—
50C23×D103—44no—
51C23×D103D411yescharacteristic
52C2×Dic206—21no—
53C2×Dic206C2311yescharacteristic, complemented
54Q8×D103—24no—
55C2×D412—21no—
56C2×D412—24no—
57C2×D412C2312yes—
58C2×C22⋊D103—28no—
59C2×C22⋊D103—22no—
60C2×C22⋊D103—22no—
61D4×D103—28no—
62D4×D103—28no—
63D4×D103—24no—
64D4×D103—24no—
65D4×D103—24no—
66D4×D103D418yes—
67C22×Dic103D411yescharacteristic
68C2×C4×D103—24no—
69C2×C4×D103—21no—
70C2×C4×D103—21no—
71C2×C4×D103D411yescharacteristic
72C2×C4×D103D411yescharacteristic
73C2×C4×D103C2311yescharacteristic
74C2×C4×D103C2311yescharacteristic
75#13110094—24no—
76#13110093—24no—
77#13110092—24no—
78C2×C103⋊C8C2312yescomplemented
79C8×D103—24no—
80C4⋊Dic103—21no—
81C4⋊Dic103C2311yescharacteristic, complemented
82Dic103⋊C4—21no—
83Dic103⋊C4—21no—
84Dic103⋊C4—21no—
85C22⋊Dic103—21no—
86(C2×D103)⋊C4—21no—
87(C2×D103)⋊C4—21no—
88C24×C103—21no—
89C2×Q8×C103C2311yescharacteristic, complemented
90C2×D4×C103—24no—
91C2×D4×C103—21no—
92C2×D4×C103C2312yes—
93C22×C4×C103C2311yescharacteristic
94SD16×C103—24no—
95C2×C8×C103C2312yescomplemented
96C4⋊C4×C103—21no—
97C4⋊C4×C103C2311yescharacteristic, complemented
98C22⋊C4×C103—21no—
99C23×C103—24no—
100C23×C103—22no—
101C23×C103—42no—
102C23×C103—42no—
103C23×C103C2×D411yescharacteristic
104C22×D103—22no—
105C22×D103—22no—
106C22×D103—28no—
107C22×D103—24no—
108C22×D103—22no—
109C22×D103—28no—
110C22×D103—44no—
111C22×D103—432no—
112C22×D103—42no—
113C22×D103—42no—
114C22×D103—44no—
115C22×D103—44no—
116C22×D103—44no—
117C22×D103SD1614yescomplemented
118C22×D103C2×D411yescharacteristic
119C22×D103C2×D411yescharacteristic
120Q8×C103—22no—
121D4×C103—24no—
122D4×C103—22no—
123D4×C103C2×D414yes—
124C2×C4×C103—22no—
125C2×C4×C103—21no—
126C2×C4×C103—21no—
127C2×C4×C103C2×D411yescharacteristic
128C2×C4×C103C2411yescharacteristic
129C22⋊D103—28no—
130C22⋊D103—28no—
131C2×Dic103—22no—
132C2×Dic103—21no—
133C2×Dic103—21no—
134C2×Dic103C2×D411yescharacteristic
135C2×Dic103C2×D411yescharacteristic
136D412—24no—
137D412—22no—
138D412C2×D414yes—
139C4×D103—22no—
140C4×D103—22no—
141C4×D103—22no—
142C4×D103—42no—
143C4×D103—42no—
144C4×D103C2×D414yes—
145Dic206—22no—
146Dic206—42no—
147C8×C103—22no—
148C103⋊C8—22no—
149C22×C103—21no—
150C22×C103—21no—
151C22×C103—24no—
152C22×C103—48no—
153C22×C103—42no—
154C22×C103—42no—
155C22×C103C2×SD1612yescomplemented
156C22×C103C22×D411yescharacteristic
157C2×D103—21no—
158C2×D103—21no—
159C2×D103—28no—
160C2×D103—24no—
161C2×D103—416no—
162C2×D103—48no—
163C2×D103—42no—
164C2×D103—42no—
165C2×D103—416no—
166C2×D103#12012yes—
167C2×D103C2×SD1612yes—
168C2×D103C2×SD1612yescomplemented
169C2×D103AGL(1,ℤ/8)12yes—
170C4×C103—21no—
171C4×C103—41no—
172C4×C103—41no—
173C4×C103C22×D412yes—
174Dic103—21no—
175Dic103—41no—
176Dic103—41no—
177Dic103#12012yes—
178C2×C103—22no—
179C2×C103—44no—
180C2×C103C2×#12011yescharacteristic
181C2×C103C22×SD1611yescharacteristic
182C2×C103C2×AGL(1,ℤ/8)11yescharacteristic
183D103—22no—
184D103—44no—
185D103#45014yesdirect product factor
186C2×#450—1031nomaximal, Sylow
187C103C2×#45011yescharacteristic, complemented, Sylow
188C23×D4—1031no—
189C22×SD16—1032no—
190C2×C22⋊Q8—1031no—
191#450—1031no—
192#450—1032no—
193#450—1031no—
194#450—1032no—
195#450—1031no—
196#450D10311yescharacteristic, direct product factor
197C2×#102—1032no—
198C2×C22⋊C8—1031no—
199C25—2061no—
200C22×Q8—1031no—
201C22×D4—1032no—
202C22×D4—1032no—
203C22×D4—1031no—
204C22×D4—2061no—
205C22×D4—2061no—
206C22×D4—2061no—
207C22×D4—2068no—
208C22×D4C2×D10311yescharacteristic
209C23×C4—1031no—
210C2×SD16—1032no—
211C2×SD16—1032no—
212C2×SD16—1032no—
213C2×SD16—2068no—
214C2×SD16C2×D10312yescomplemented
215C22×C8—1032no—
216C22⋊Q8—1031no—
217C22⋊Q8—1031no—
218C22⋊Q8—1031no—
219C22⋊Q8—2061no—
220C22⋊Q8—2061no—
221C22⋊Q8C2×D10311yescharacteristic, complemented
222C2×C4⋊C4—1031no—
223C2×C4⋊C4—2061no—
224C2×C22⋊C4—2061no—
225#102—1032no—
226#102—1032no—
227#102—1032no—
228#102C2×D10312yescomplemented
229C22⋊C8—1031no—
230C22⋊C8—1032no—
231C22⋊C8C2×D10311yescharacteristic, complemented
232C24—21no—
233C24—1031no—
234C24—4124no—
235C24—4124no—
236C24—2062no—
237C24—2062no—
238C24—2068no—
239C24—2061no—
240C2×Q8—1031no—
241C2×Q8—2061no—
242C2×Q8—2064no—
243C2×Q8C22×D10311yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—1038no—
247C2×D4—1032no—
248C2×D4—2064no—
249C2×D4—2064no—
250C2×D4—2062no—
251C2×D4—2062no—
252C2×D4—2064no—
253C2×D4—2068no—
254C2×D4—2068no—
255C2×D4—2068no—
256C2×D4—2064no—
257C2×D4—2061no—
258C2×D4C22×D10312yes—
259C22×C4—1031no—
260C22×C4—1031no—
261C22×C4—1031no—
262C22×C4—1031no—
263C22×C4—1031no—
264C22×C4—2064no—
265C22×C4—2061no—
266C22×C4—2061no—
267C22×C4C22×D10311yescharacteristic
268SD16—24no—
269SD16—2064no—
270SD16—2064no—
271SD16—2064no—
272C2×C8—1032no—
273C2×C8—2064no—
274C2×C8C22×D10312yescomplemented
275C4⋊C4—21no—
276C4⋊C4—1031no—
277C4⋊C4—2061no—
278C4⋊C4—2061no—
279C4⋊C4—2061no—
280C4⋊C4—2061no—
281C4⋊C4C22×D10311yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—2061no—
284C22⋊C4—2061no—
285C22⋊C4—2061no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—1031no—
291C23—1031no—
292C23—1034no—
293C23—4122no—
294C23—4124no—
295C23—4124no—
296C23—4124no—
297C23—41232no—
298C23—4124no—
299C23—4122no—
300C23—2068no—
301C23—2062no—
302C23—2062no—
303C23—2064no—
304C23—2068no—
305C23—2062no—
306C23D4×D10311yescharacteristic
307Q8—22no—
308Q8—4122no—
309Q8—2062no—
310D4—24no—
311D4—22no—
312D4—1034no—
313D4—2064no—
314D4—2068no—
315D4—2068no—
316D4—2062no—
317D4D4×D10314yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—1031no—
322C2×C4—1034no—
323C2×C4—1031no—
324C2×C4—4122no—
325C2×C4—4122no—
326C2×C4—2062no—
327C2×C4—2062no—
328C2×C4—2062no—
329C2×C4—2061no—
330C2×C4—2061no—
331C2×C4—2062no—
332C2×C4D4×D10311yescharacteristic
333C2×C4C23×D10311yescharacteristic
334C8—22no—
335C8—2062no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—1032no—
343C22—1032no—
344C22—1032no—
345C22—1032no—
346C22—4128no—
347C22—41216no—
348C22—41216no—
349C22—4122no—
350C22—4122no—
351C22—2068no—
352C22—2064no—
353C22—2061no—
354C22—2061no—
355C22SD16×D10312yescomplemented
356C22C2×D4×D10311yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—1032no—
361C4—4121no—
362C4—4121no—
363C4—2061no—
364C4C2×D4×D10312yes—
365C2—22no—
366C2—44no—
367C2—1034no—
368C2—4124no—
369C2—2062no—
370C2#120×D10311yescharacteristic
371C2C2×SD16×D10311yescharacteristic
372C2AGL(1,ℤ/8)×D10311yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#450×C103C2
socleC22×C103C22×D4
prime radicalC22×D4×D103C2
hypercenter#450D103
hypocenterC103C2×#450
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D103C2semidirect
3. C22⋊Q8×D103C2semidirect
4. #20267944C2semidirect
5. #20267943C2semidirect
6. #20267942C2semidirect
7. #20267941C2semidirect
8. #20267940C2semidirect
9. #102×D103C2semidirect
10. C22⋊C8×D103C2semidirect
11. #450×C103C2semidirect
13. C2×Q8×D103C22semidirect
22. C2×#13110094C22semidirect
23. C2×#13110093C22semidirect
24. C2×#13110092C22semidirect
26. C2×C8×D103C22semidirect
27. #15693001C22semidirect
28. #15693000C22semidirect
31. #15692997C22semidirect
33. C4⋊C4×D103C22semidirect
35. #15692943C22semidirect
36. #15692942C22semidirect
37. #15692941C22semidirect
38. C22⋊(C103⋊C8)C22semidirect
39. (C2×D103)⋊C8C22semidirect
41. C2×SD16×C103C22semidirect
42. C22⋊Q8×C103C22semidirect
43. #102×C103C22semidirect
44. C22⋊C8×C103C22semidirect
53. C2×Dic206C23semidirect
78. C2×C103⋊C8C23semidirect
81. C4⋊Dic103C23semidirect
89. C2×Q8×C103C23semidirect
95. C2×C8×C103C23semidirect
97. C4⋊C4×C103C23semidirect
117. C22×D103SD16semidirect
155. C22×C103C2×SD16semidirect
168. C2×D103C2×SD16semidirect
185. D103#450direct (N × H)
187. C103C2×#450semidirect
196. #450D103direct (N × H)
214. C2×SD16C2×D103semidirect
221. C22⋊Q8C2×D103semidirect
228. #102C2×D103semidirect
231. C22⋊C8C2×D103semidirect
243. C2×Q8C22×D103semidirect
274. C2×C8C22×D103semidirect
281. C4⋊C4C22×D103semidirect
355. C22SD16×D103semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1032
C2×SD16×D1032
C22⋊Q8×D1032
#202679442
#202679432
#202679422
#202679412
#202679402
#102×D1032
C22⋊C8×D1032
#450×C1032
C2×#450103