Finite Group Databasebeta

← #447×D197 #38751885

Subgroups — #447×D197 #38751885

Order 25216.

1048 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1048 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×D197C211yesmaximal, characteristic, complemented
2C2×D8×D197C212yesmaximal, complemented
3#121×D197C211yesmaximal, characteristic, complemented
4#27161215C211yesmaximal, characteristic, complemented
5#27161214C212yesmaximal, complemented
6#27161213C211yesmaximal, characteristic, complemented
7#27161212C211yesmaximal, characteristic, complemented
8#27161211C212yesmaximal, complemented
9#102×D197C212yesmaximal, complemented
10C22⋊C8×D197C211yesmaximal, characteristic, complemented
11#447×C197C211yesmaximal, characteristic, complemented
12C24×D197—21no—
13C22×D788C2211yescharacteristic, complemented
14C22×C22⋊D197—21no—
15C2×D4×D197—28no—
16C2×D4×D197—21no—
17C2×D4×D197—21no—
18C2×D4×D197—21no—
19C2×D4×D197C2212yes—
20C2×D4×D197C2212yescomplemented
21C2×D4×D197C2211yescharacteristic, complemented
22C22×C4×D197C2211yescharacteristic
23C2×D1576C2212yescomplemented
24C2×#15691343C2212yescomplemented
25C2×#15691343C2212yescomplemented
26D8×D197—28no—
27C2×C8×D197C2212yescomplemented
28#20263301C2211yescharacteristic, complemented
29#20263300C2211yescharacteristic, complemented
30#20263299—21no—
31#20263298—21no—
32#20263297C2211yescharacteristic, complemented
33C4⋊C4×D197C2211yescharacteristic, complemented
34C22⋊C4×D197—21no—
35#20263248C2212yescomplemented
36#20263247C2212yescomplemented
37#20263246C2212yescomplemented
38C22⋊#13096802C2211yescharacteristic, complemented
39#20263238C2212yescomplemented
40C22×D4×C197C2211yescharacteristic, complemented
41C2×D8×C197C2212yescomplemented
42#121×C197C2211yescharacteristic, complemented
43#102×C197C2212yescomplemented
44C22⋊C8×C197C2211yescharacteristic, complemented
45C23×D197—28no—
46C23×D197—22no—
47C23×D197—22no—
48C23×D197—21no—
49C23×D197—21no—
50C23×D197—44no—
51C23×D197—44no—
52C23×D197D411yescharacteristic
53C2×D788—24no—
54C2×D788—21no—
55C2×D788—21no—
56C2×D788C2312yes—
57C2×D788C2311yescharacteristic, complemented
58C2×C22⋊D197—28no—
59C2×C22⋊D197—21no—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—22no—
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64D4×D197—28no—
65D4×D197—28no—
66D4×D197—24no—
67D4×D197—24no—
68D4×D197—24no—
69D4×D197—24no—
70D4×D197—44no—
71D4×D197D418yescomplemented
72C22×Dic197D411yescharacteristic
73C2×C4×D197—24no—
74C2×C4×D197—21no—
75C2×C4×D197D411yescharacteristic
76C2×C4×D197D411yescharacteristic
77C2×C4×D197C2311yescharacteristic
78C2×C4×D197C2311yescharacteristic
79D1576—24no—
80#15691343—24no—
81#15691343—24no—
82C2×#13096802C2312yescomplemented
83C8×D197—24no—
84C4⋊Dic197C2311yescharacteristic, complemented
85#15691337—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89C24×C197—21no—
90C2×D4×C197—24no—
91C2×D4×C197—21no—
92C2×D4×C197—21no—
93C2×D4×C197C2312yes—
94C2×D4×C197C2311yescharacteristic, complemented
95C22×C4×C197C2311yescharacteristic
96D8×C197—24no—
97C2×C8×C197C2312yescomplemented
98C4⋊C4×C197C2311yescharacteristic, complemented
99C22⋊C4×C197—21no—
100C22×D197—22no—
101C22×D197—22no—
102C22×D197—28no—
103C22×D197—24no—
104C22×D197—22no—
105C22×D197—28no—
106C22×D197—21no—
107C22×D197—44no—
108C22×D197—432no—
109C22×D197—42no—
110C22×D197—42no—
111C22×D197—44no—
112C22×D197—44no—
113C22×D197—44no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197D814yescomplemented
118C22×D197C2×D411yescharacteristic
119C22×D197C2×D411yescharacteristic
120D788—24no—
121D788—22no—
122D788—22no—
123D788—42no—
124D788C2×D414yescomplemented
125C22⋊D197—28no—
126C22⋊D197—28no—
127C22⋊D197—42no—
128C22⋊D197—42no—
129C22⋊D197—42no—
130C2×Dic197—22no—
131C2×Dic197—21no—
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C4×D197—22no—
135C4×D197—22no—
136C4×D197—22no—
137C4×D197—42no—
138C4×D197C2×D414yes—
139#13096802—22no—
140C23×C197—24no—
141C23×C197—22no—
142C23×C197—21no—
143C23×C197—42no—
144C23×C197—42no—
145C23×C197C2×D411yescharacteristic
146D4×C197—24no—
147D4×C197—22no—
148D4×C197—22no—
149D4×C197—42no—
150D4×C197C2×D414yescomplemented
151C2×C4×C197—22no—
152C2×C4×C197—21no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2411yescharacteristic
155C8×C197—22no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—24no—
159C22×C197—48no—
160C22×C197—42no—
161C22×C197—42no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197C2×D812yescomplemented
166C22×C197C22×D411yescharacteristic
167C2×D197—21no—
168C2×D197—21no—
169C2×D197—28no—
170C2×D197—24no—
171C2×D197—416no—
172C2×D197—48no—
173C2×D197—42no—
174C2×D197—42no—
175C2×D197—416no—
176C2×D197—41no—
177C2×D197—41no—
178C2×D197—41no—
179C2×D197—84no—
180C2×D197#12012yes—
181C2×D197C2×D812yes—
182C2×D197C2×D812yescomplemented
183C2×D197AGL(1,ℤ/8)12yes—
184C4×C197—21no—
185C4×C197—41no—
186C4×C197C22×D412yes—
187Dic197—21no—
188Dic197—41no—
189Dic197#12012yes—
190C2×C197—22no—
191C2×C197—44no—
192C2×C197—81no—
193C2×C197C2×#12011yescharacteristic
194C2×C197C22×D811yescharacteristic
195C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
196D197—22no—
197D197—44no—
198D197—81no—
199D197#44714yesdirect product factor
200C197C2×#44711yescharacteristic, complemented, Sylow
201C2×#447—1971nomaximal, Sylow
202C23×D4—1971no—
203C22×D8—1972no—
204C2×#121—1971no—
205#447—1971no—
206#447—1972no—
207#447—1971no—
208#447—1972no—
209#447—1971no—
210#447D19711yescharacteristic, direct product factor
211C2×#102—1972no—
212C2×C22⋊C8—1971no—
213C25—3941no—
214C22×D4—3941no—
215C22×D4—3948no—
216C22×D4—3941no—
217C22×D4—3941no—
218C22×D4—3941no—
219C22×D4—1972no—
220C22×D4—1971no—
221C22×D4—1972no—
222C22×D4—1971no—
223C22×D4C2×D19711yescharacteristic, complemented
224C23×C4—1971no—
225C2×D8—3948no—
226C2×D8—1972no—
227C2×D8—1972no—
228C2×D8—1972no—
229C2×D8C2×D19712yescomplemented
230C22×C8—1972no—
231#121—3941no—
232#121—3941no—
233#121—1971no—
234#121—1971no—
235#121—1971no—
236#121C2×D19711yescharacteristic, complemented
237C2×C4⋊C4—1971no—
238C2×C22⋊C4—3941no—
239#102—1972no—
240#102—1972no—
241#102—1972no—
242#102C2×D19712yescomplemented
243C22⋊C8—1971no—
244C22⋊C8—1972no—
245C22⋊C8C2×D19711yescharacteristic, complemented
246C24—21no—
247C24—7884no—
248C24—7884no—
249C24—3941no—
250C24—3948no—
251C24—3942no—
252C24—3942no—
253C24—3941no—
254C24—1971no—
255C2×D4—24no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—7884no—
259C2×D4—3941no—
260C2×D4—3944no—
261C2×D4—3948no—
262C2×D4—3948no—
263C2×D4—3944no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3948no—
267C2×D4—3944no—
268C2×D4—3944no—
269C2×D4—3944no—
270C2×D4—3941no—
271C2×D4—3942no—
272C2×D4—3942no—
273C2×D4—3941no—
274C2×D4—1978no—
275C2×D4—1971no—
276C2×D4—1972no—
277C2×D4C22×D19712yes—
278C2×D4C22×D19711yescharacteristic, complemented
279C22×C4—3941no—
280C22×C4—3944no—
281C22×C4—1971no—
282C22×C4—1971no—
283C22×C4—1971no—
284C22×C4—1971no—
285C22×C4—1971no—
286C22×C4C22×D19711yescharacteristic
287D8—24no—
288D8—3944no—
289D8—3944no—
290D8—3944no—
291C2×C8—3944no—
292C2×C8—1972no—
293C2×C8C22×D19712yescomplemented
294C4⋊C4—3941no—
295C4⋊C4—1971no—
296C4⋊C4C22×D19711yescharacteristic, complemented
297C22⋊C4—21no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C23—24no—
302C23—22no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—7882no—
307C23—78832no—
308C23—7884no—
309C23—7884no—
310C23—7884no—
311C23—7884no—
312C23—7882no—
313C23—7882no—
314C23—7882no—
315C23—7882no—
316C23—3944no—
317C23—3948no—
318C23—3948no—
319C23—3942no—
320C23—3942no—
321C23—3941no—
322C23—3942no—
323C23—1974no—
324C23—1971no—
325C23—1971no—
326C23D4×D19711yescharacteristic
327D4—24no—
328D4—22no—
329D4—22no—
330D4—42no—
331D4—7882no—
332D4—7882no—
333D4—7882no—
334D4—7882no—
335D4—3944no—
336D4—3948no—
337D4—3948no—
338D4—3942no—
339D4—3942no—
340D4—1974no—
341D4D4×D19714yescomplemented
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—7882no—
345C2×C4—3942no—
346C2×C4—3942no—
347C2×C4—3941no—
348C2×C4—3942no—
349C2×C4—3942no—
350C2×C4—1971no—
351C2×C4—1971no—
352C2×C4—1974no—
353C2×C4D4×D19711yescharacteristic
354C2×C4C23×D19711yescharacteristic
355C8—22no—
356C8—3942no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—48no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—7888no—
367C22—78816no—
368C22—78816no—
369C22—7882no—
370C22—7882no—
371C22—7881no—
372C22—7881no—
373C22—7881no—
374C22—15764no—
375C22—3948no—
376C22—3944no—
377C22—3941no—
378C22—3941no—
379C22—1972no—
380C22—1972no—
381C22—1972no—
382C22—1972no—
383C22D8×D19712yescomplemented
384C22C2×D4×D19711yescharacteristic
385C4—21no—
386C4—41no—
387C4—7881no—
388C4—3941no—
389C4—1972no—
390C4C2×D4×D19712yes—
391C2—22no—
392C2—44no—
393C2—81no—
394C2—7884no—
395C2—15761no—
396C2—3942no—
397C2—1974no—
398C2#120×D19711yescharacteristic
399C2C2×D8×D19711yescharacteristic
400C2AGL(1,ℤ/8)×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#447×C197C2
socleC22×C197C22×D4
hypercenter#447D197
hypocenterC197C2×#447
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×D197C2semidirect
2. C2×D8×D197C2semidirect
3. #121×D197C2semidirect
4. #27161215C2semidirect
5. #27161214C2semidirect
6. #27161213C2semidirect
7. #27161212C2semidirect
8. #27161211C2semidirect
9. #102×D197C2semidirect
10. C22⋊C8×D197C2semidirect
11. #447×C197C2semidirect
13. C22×D788C22semidirect
20. C2×D4×D197C22semidirect
21. C2×D4×D197C22semidirect
23. C2×D1576C22semidirect
24. C2×#15691343C22semidirect
25. C2×#15691343C22semidirect
27. C2×C8×D197C22semidirect
28. #20263301C22semidirect
29. #20263300C22semidirect
32. #20263297C22semidirect
33. C4⋊C4×D197C22semidirect
35. #20263248C22semidirect
36. #20263247C22semidirect
37. #20263246C22semidirect
38. C22⋊#13096802C22semidirect
39. #20263238C22semidirect
40. C22×D4×C197C22semidirect
41. C2×D8×C197C22semidirect
42. #121×C197C22semidirect
43. #102×C197C22semidirect
44. C22⋊C8×C197C22semidirect
57. C2×D788C23semidirect
71. D4×D197D4semidirect
82. C2×#13096802C23semidirect
84. C4⋊Dic197C23semidirect
94. C2×D4×C197C23semidirect
97. C2×C8×C197C23semidirect
98. C4⋊C4×C197C23semidirect
117. C22×D197D8semidirect
124. D788C2×D4semidirect
150. D4×C197C2×D4semidirect
165. C22×C197C2×D8semidirect
182. C2×D197C2×D8semidirect
199. D197#447direct (N × H)
200. C197C2×#447semidirect
210. #447D197direct (N × H)
223. C22×D4C2×D197semidirect
229. C2×D8C2×D197semidirect
236. #121C2×D197semidirect
242. #102C2×D197semidirect
245. C22⋊C8C2×D197semidirect
278. C2×D4C22×D197semidirect
293. C2×C8C22×D197semidirect
296. C4⋊C4C22×D197semidirect
341. D4D4×D197semidirect
383. C22D8×D197semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×D8×D1972
#121×D1972
#271612152
#271612142
#271612132
#271612122
#271612112
#102×D1972
C22⋊C8×D1972
#447×C1972
C2×#447197