Finite Group Databasebeta

← #447×D71 #22597513

Subgroups — #447×D71 #22597513

Order 9088.

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×D71C211yesmaximal, characteristic, complemented
2C2×D8×D71C212yesmaximal, complemented
3#121×D71C211yesmaximal, characteristic, complemented
4#17986135C211yesmaximal, characteristic, complemented
5#17986134C212yesmaximal, complemented
6#17986133C211yesmaximal, characteristic, complemented
7#17986132C211yesmaximal, characteristic, complemented
8#17986131C212yesmaximal, complemented
9#102×D71C212yesmaximal, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#447×C71C211yesmaximal, characteristic, complemented
12C24×D71—21no—
13C22×D284C2211yescharacteristic, complemented
14C22×C22⋊D71—21no—
15C2×D4×D71—28no—
16C2×D4×D71—21no—
17C2×D4×D71—21no—
18C2×D4×D71—21no—
19C2×D4×D71C2212yes—
20C2×D4×D71C2212yescomplemented
21C2×D4×D71C2211yescharacteristic, complemented
22C22×C4×D71C2211yescharacteristic
23C2×D568C2212yescomplemented
24C2×#11765850C2212yescomplemented
25C2×#11765850C2212yescomplemented
26D8×D71—28no—
27C2×C8×D71C2212yescomplemented
28#14556103C2211yescharacteristic, complemented
29#14556102C2211yescharacteristic, complemented
30#14556101—21no—
31#14556100—21no—
32#14556099C2211yescharacteristic, complemented
33C4⋊C4×D71C2211yescharacteristic, complemented
34C22⋊C4×D71—21no—
35#14556050C2212yescomplemented
36#14556049C2212yescomplemented
37#14556048C2212yescomplemented
38C22⋊(C71⋊C8)C2211yescharacteristic, complemented
39(C2×D71)⋊C8C2212yescomplemented
40C22×D4×C71C2211yescharacteristic, complemented
41C2×D8×C71C2212yescomplemented
42#121×C71C2211yescharacteristic, complemented
43#102×C71C2212yescomplemented
44C22⋊C8×C71C2211yescharacteristic, complemented
45C23×D71—28no—
46C23×D71—21no—
47C23×D71—22no—
48C23×D71—22no—
49C23×D71—21no—
50C23×D71—44no—
51C23×D71—44no—
52C23×D71D411yescharacteristic
53C2×D284—21no—
54C2×D284—24no—
55C2×D284—21no—
56C2×D284C2312yes—
57C2×D284C2311yescharacteristic, complemented
58C2×C22⋊D71—28no—
59C2×C22⋊D71—21no—
60C2×C22⋊D71—21no—
61C2×C22⋊D71—22no—
62C2×C22⋊D71—22no—
63C2×C22⋊D71—21no—
64D4×D71—28no—
65D4×D71—28no—
66D4×D71—24no—
67D4×D71—24no—
68D4×D71—24no—
69D4×D71—24no—
70D4×D71—44no—
71D4×D71D418yescomplemented
72C22×Dic71D411yescharacteristic
73C2×C4×D71—24no—
74C2×C4×D71—21no—
75C2×C4×D71D411yescharacteristic
76C2×C4×D71D411yescharacteristic
77C2×C4×D71C2311yescharacteristic
78C2×C4×D71C2311yescharacteristic
79D568—24no—
80#11765850—24no—
81#11765850—24no—
82C2×C71⋊C8C2312yescomplemented
83C8×D71—24no—
84C4⋊Dic71C2311yescharacteristic, complemented
85Dic71⋊C4—21no—
86C22⋊Dic71—21no—
87(C2×D71)⋊C4—21no—
88(C2×D71)⋊C4—21no—
89C24×C71—21no—
90C2×D4×C71—24no—
91C2×D4×C71—21no—
92C2×D4×C71—21no—
93C2×D4×C71C2312yes—
94C2×D4×C71C2311yescharacteristic, complemented
95C22×C4×C71C2311yescharacteristic
96D8×C71—24no—
97C2×C8×C71C2312yescomplemented
98C4⋊C4×C71C2311yescharacteristic, complemented
99C22⋊C4×C71—21no—
100C23×C71—24no—
101C23×C71—22no—
102C23×C71—21no—
103C23×C71—42no—
104C23×C71—42no—
105C23×C71C2×D411yescharacteristic
106C22×D71—22no—
107C22×D71—22no—
108C22×D71—28no—
109C22×D71—24no—
110C22×D71—22no—
111C22×D71—28no—
112C22×D71—21no—
113C22×D71—44no—
114C22×D71—432no—
115C22×D71—42no—
116C22×D71—42no—
117C22×D71—44no—
118C22×D71—44no—
119C22×D71—44no—
120C22×D71—42no—
121C22×D71—42no—
122C22×D71—42no—
123C22×D71D814yescomplemented
124C22×D71C2×D411yescharacteristic
125C22×D71C2×D411yescharacteristic
126D4×C71—24no—
127D4×C71—22no—
128D4×C71—22no—
129D4×C71—42no—
130D4×C71C2×D414yescomplemented
131C2×C4×C71—22no—
132C2×C4×C71—21no—
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2411yescharacteristic
135C22⋊D71—28no—
136C22⋊D71—28no—
137C22⋊D71—42no—
138C22⋊D71—42no—
139C22⋊D71—42no—
140C2×Dic71—22no—
141C2×Dic71—21no—
142C2×Dic71C2×D411yescharacteristic
143C2×Dic71C2×D411yescharacteristic
144D284—24no—
145D284—22no—
146D284—22no—
147D284—42no—
148D284C2×D414yescomplemented
149C4×D71—22no—
150C4×D71—22no—
151C4×D71—22no—
152C4×D71—42no—
153C4×D71C2×D414yes—
154C8×C71—22no—
155C71⋊C8—22no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—24no—
159C22×C71—48no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71C2×D812yescomplemented
166C22×C71C22×D411yescharacteristic
167C2×D71—21no—
168C2×D71—21no—
169C2×D71—28no—
170C2×D71—24no—
171C2×D71—416no—
172C2×D71—48no—
173C2×D71—42no—
174C2×D71—42no—
175C2×D71—416no—
176C2×D71—41no—
177C2×D71—41no—
178C2×D71—41no—
179C2×D71—84no—
180C2×D71#12012yes—
181C2×D71C2×D812yes—
182C2×D71C2×D812yescomplemented
183C2×D71AGL(1,ℤ/8)12yes—
184C4×C71—21no—
185C4×C71—41no—
186C4×C71C22×D412yes—
187Dic71—21no—
188Dic71—41no—
189Dic71#12012yes—
190C2×C71—22no—
191C2×C71—44no—
192C2×C71—81no—
193C2×C71C2×#12011yescharacteristic
194C2×C71C22×D811yescharacteristic
195C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
196D71—22no—
197D71—44no—
198D71—81no—
199D71#44714yesdirect product factor
200C2×#447—711nomaximal, Sylow
201C71C2×#44711yescharacteristic, complemented, Sylow
202C23×D4—711no—
203C22×D8—712no—
204C2×#121—711no—
205#447—711no—
206#447—712no—
207#447—711no—
208#447—712no—
209#447—711no—
210#447D7111yescharacteristic, direct product factor
211C2×#102—712no—
212C2×C22⋊C8—711no—
213C25—1421no—
214C22×D4—712no—
215C22×D4—712no—
216C22×D4—711no—
217C22×D4—711no—
218C22×D4—1421no—
219C22×D4—1421no—
220C22×D4—1421no—
221C22×D4—1421no—
222C22×D4—1428no—
223C22×D4C2×D7111yescharacteristic, complemented
224C23×C4—711no—
225C2×D8—712no—
226C2×D8—712no—
227C2×D8—712no—
228C2×D8—1428no—
229C2×D8C2×D7112yescomplemented
230C22×C8—712no—
231#121—711no—
232#121—711no—
233#121—711no—
234#121—1421no—
235#121—1421no—
236#121C2×D7111yescharacteristic, complemented
237C2×C4⋊C4—711no—
238C2×C22⋊C4—1421no—
239#102—712no—
240#102—712no—
241#102—712no—
242#102C2×D7112yescomplemented
243C22⋊C8—711no—
244C22⋊C8—712no—
245C22⋊C8C2×D7111yescharacteristic, complemented
246C24—21no—
247C24—2844no—
248C24—2844no—
249C24—711no—
250C24—1421no—
251C24—1422no—
252C24—1422no—
253C24—1428no—
254C24—1421no—
255C2×D4—24no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2844no—
259C2×D4—718no—
260C2×D4—712no—
261C2×D4—711no—
262C2×D4—1421no—
263C2×D4—1424no—
264C2×D4—1424no—
265C2×D4—1424no—
266C2×D4—1421no—
267C2×D4—1422no—
268C2×D4—1422no—
269C2×D4—1424no—
270C2×D4—1428no—
271C2×D4—1428no—
272C2×D4—1421no—
273C2×D4—1421no—
274C2×D4—1428no—
275C2×D4—1424no—
276C2×D4—1421no—
277C2×D4C22×D7112yes—
278C2×D4C22×D7111yescharacteristic, complemented
279C22×C4—711no—
280C22×C4—711no—
281C22×C4—711no—
282C22×C4—711no—
283C22×C4—711no—
284C22×C4—1421no—
285C22×C4—1424no—
286C22×C4C22×D7111yescharacteristic
287D8—24no—
288D8—1424no—
289D8—1424no—
290D8—1424no—
291C2×C8—712no—
292C2×C8—1424no—
293C2×C8C22×D7112yescomplemented
294C4⋊C4—711no—
295C4⋊C4—1421no—
296C4⋊C4C22×D7111yescharacteristic, complemented
297C22⋊C4—21no—
298C22⋊C4—1421no—
299C22⋊C4—1421no—
300C22⋊C4—1421no—
301C23—24no—
302C23—22no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—2842no—
307C23—2844no—
308C23—2844no—
309C23—2844no—
310C23—28432no—
311C23—2844no—
312C23—2842no—
313C23—2842no—
314C23—2842no—
315C23—2842no—
316C23—711no—
317C23—711no—
318C23—714no—
319C23—1428no—
320C23—1424no—
321C23—1428no—
322C23—1422no—
323C23—1422no—
324C23—1421no—
325C23—1422no—
326C23D4×D7111yescharacteristic
327D4—24no—
328D4—22no—
329D4—22no—
330D4—42no—
331D4—2842no—
332D4—2842no—
333D4—2842no—
334D4—2842no—
335D4—714no—
336D4—1424no—
337D4—1428no—
338D4—1428no—
339D4—1422no—
340D4—1422no—
341D4D4×D7114yescomplemented
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—2842no—
345C2×C4—711no—
346C2×C4—714no—
347C2×C4—711no—
348C2×C4—1422no—
349C2×C4—1422no—
350C2×C4—1421no—
351C2×C4—1422no—
352C2×C4—1422no—
353C2×C4D4×D7111yescharacteristic
354C2×C4C23×D7111yescharacteristic
355C8—22no—
356C8—1422no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—48no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—2848no—
367C22—28416no—
368C22—28416no—
369C22—2842no—
370C22—2842no—
371C22—2841no—
372C22—2841no—
373C22—2841no—
374C22—5684no—
375C22—712no—
376C22—712no—
377C22—712no—
378C22—712no—
379C22—1428no—
380C22—1424no—
381C22—1421no—
382C22—1421no—
383C22D8×D7112yescomplemented
384C22C2×D4×D7111yescharacteristic
385C4—21no—
386C4—41no—
387C4—2841no—
388C4—712no—
389C4—1421no—
390C4C2×D4×D7112yes—
391C2—22no—
392C2—44no—
393C2—81no—
394C2—2844no—
395C2—5681no—
396C2—714no—
397C2—1422no—
398C2#120×D7111yescharacteristic
399C2C2×D8×D7111yescharacteristic
400C2AGL(1,ℤ/8)×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#447×C71C2
socleC22×C71C22×D4
hypercenter#447D71
hypocenterC71C2×#447
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×D71C2semidirect
2. C2×D8×D71C2semidirect
3. #121×D71C2semidirect
4. #17986135C2semidirect
5. #17986134C2semidirect
6. #17986133C2semidirect
7. #17986132C2semidirect
8. #17986131C2semidirect
9. #102×D71C2semidirect
10. C22⋊C8×D71C2semidirect
11. #447×C71C2semidirect
13. C22×D284C22semidirect
20. C2×D4×D71C22semidirect
21. C2×D4×D71C22semidirect
23. C2×D568C22semidirect
24. C2×#11765850C22semidirect
25. C2×#11765850C22semidirect
27. C2×C8×D71C22semidirect
28. #14556103C22semidirect
29. #14556102C22semidirect
32. #14556099C22semidirect
33. C4⋊C4×D71C22semidirect
35. #14556050C22semidirect
36. #14556049C22semidirect
37. #14556048C22semidirect
38. C22⋊(C71⋊C8)C22semidirect
39. (C2×D71)⋊C8C22semidirect
40. C22×D4×C71C22semidirect
41. C2×D8×C71C22semidirect
42. #121×C71C22semidirect
43. #102×C71C22semidirect
44. C22⋊C8×C71C22semidirect
57. C2×D284C23semidirect
71. D4×D71D4semidirect
82. C2×C71⋊C8C23semidirect
84. C4⋊Dic71C23semidirect
94. C2×D4×C71C23semidirect
97. C2×C8×C71C23semidirect
98. C4⋊C4×C71C23semidirect
123. C22×D71D8semidirect
130. D4×C71C2×D4semidirect
148. D284C2×D4semidirect
165. C22×C71C2×D8semidirect
182. C2×D71C2×D8semidirect
199. D71#447direct (N × H)
201. C71C2×#447semidirect
210. #447D71direct (N × H)
223. C22×D4C2×D71semidirect
229. C2×D8C2×D71semidirect
236. #121C2×D71semidirect
242. #102C2×D71semidirect
245. C22⋊C8C2×D71semidirect
278. C2×D4C22×D71semidirect
293. C2×C8C22×D71semidirect
296. C4⋊C4C22×D71semidirect
341. D4D4×D71semidirect
383. C22D8×D71semidirect