Finite Group Databasebeta

← #1824×D71 #31997235

Subgroups — #1824×D71 #31997235

Order 18176.

1878 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×C22⋊C8×D71C211yesmaximal, characteristic
3#380×D71C211yesmaximal, characteristic
4#22588431C211yesmaximal, characteristic, complemented
5#22588430C211yesmaximal, characteristic, complemented
6#22588429C214yesmaximal, complemented
7#22588428C211yesmaximal, characteristic, complemented
8#340×D71C214yesmaximal, complemented
9C71×#1824C211yesmaximal, characteristic, complemented
10C23×C4×D71—21no—
11C23×C4×D71C2211yescharacteristic
12C22×C8×D71C2212yes—
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4—21no—
15C2×C4⋊C4×D71—24no—
16C2×C4⋊C4×D71—21no—
17C2×C4⋊C4×D71—21no—
18C2×C4⋊C4×D71C418yescomplemented
19C2×C4⋊C4×D71C2212yes—
20C2×C4⋊C4×D71C2212yes—
21C2×C4⋊C4×D71C2212yes—
22C2×C22⋊C4×D71—21no—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×(C2×D71)⋊C8C2212yes—
25C22⋊C8×D71—24no—
26#17985859C2211yescharacteristic
27#17985858—21no—
28#17985857—21no—
29#17985856C2212yes—
30#95×D71—21no—
31#17985762C2214yescomplemented
32#17985761C2214yescomplemented
33#17985761C2214yescomplemented
34C22×C4⋊C4×C71C2211yescharacteristic
35C2×C22⋊C8×C71C2211yescharacteristic
36#380×C71C2211yescharacteristic
37#340×C71C2214yescomplemented
38C24×D71D411yescharacteristic
39C23×Dic71—21no—
40C23×Dic71D411yescharacteristic
41C22×C4×D71—28no—
42C22×C4×D71—28no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—22no—
46C22×C4×D71—48no—
47C22×C4×D71C2×C414yes—
48C22×C4×D71D411yescharacteristic
49C22×C4×D71D411yescharacteristic
50C22×C4×D71D414yes—
51C22×C4×D71C2311yescharacteristic
52C22×C4×D71C2311yescharacteristic
53C22×C71⋊C8C2312yes—
54C2×C8×D71—24no—
55C2×C8×D71—28no—
56C2×C4⋊Dic71—21no—
57C2×C4⋊Dic71—22no—
58C2×C4⋊Dic71C2×C414yescomplemented
59C2×C4⋊Dic71C2312yes—
60C2×C4⋊Dic71C2312yes—
61C2×Dic71⋊C4—28no—
62C2×Dic71⋊C4—22no—
63C2×Dic71⋊C4—22no—
64C2×Dic71⋊C4—22no—
65C4⋊C4×D71—28no—
66C4⋊C4×D71—28no—
67C4⋊C4×D71—24no—
68C4⋊C4×D71—24no—
69C4⋊C4×D71—28no—
70C4⋊C4×D71C2×C418yes—
71C4⋊C4×D71C2×C418yescomplemented
72C2×C22⋊Dic71—21no—
73C2×(C2×D71)⋊C4—22no—
74C22⋊C4×D71—44no—
75C22⋊(C71⋊C8)—22no—
76(C2×D71)⋊C8—24no—
77C71⋊#95—22no—
78C71⋊#95—21no—
79C23×C4×C71—21no—
80C23×C4×C71C2311yescharacteristic
81C22×C8×C71C2312yes—
82C2×C4⋊C4×C71—21no—
83C2×C4⋊C4×C71—22no—
84C2×C4⋊C4×C71C2×C414yescomplemented
85C2×C4⋊C4×C71C2312yes—
86C2×C4⋊C4×C71C2312yes—
87C2×C22⋊C4×C71—21no—
88C22⋊C8×C71—22no—
89#95×C71—21no—
90C23×D71—24no—
91C23×D71—24no—
92C23×D71C22⋊C414yes—
93C23×D71SD1614yes—
94C23×D71Q1614yes—
95C23×D71C2×D411yescharacteristic
96C23×D71C2×D411yescharacteristic
97C22×Dic71—24no—
98C22×Dic71—24no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—44no—
102C22×Dic71C22⋊C414yes—
103C22×Dic71C2×D411yescharacteristic
104C22×Dic71C2×D411yescharacteristic
105C2×C4×D71—24no—
106C2×C4×D71—24no—
107C2×C4×D71—24no—
108C2×C4×D71—216no—
109C2×C4×D71—216no—
110C2×C4×D71—28no—
111C2×C4×D71—28no—
112C2×C4×D71—48no—
113C2×C4×D71—432no—
114C2×C4×D71—48no—
115C2×C4×D71—44no—
116C2×C4×D71—44no—
117C2×C4×D71—44no—
118C2×C4×D71C22⋊C414yes—
119C2×C4×D71C22⋊C414yes—
120C2×C4×D71C22×C412yes—
121C2×C4×D71C22×C412yes—
122C2×C4×D71C2×D412yes—
123C2×C4×D71C2×D412yes—
124C2×C4×D71#4118yes—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—24no—
127C8×D71—48no—
128C4⋊Dic71—24no—
129C4⋊Dic71—22no—
130C4⋊Dic71—24no—
131C4⋊Dic71C22×C414yes—
132Dic71⋊C4—28no—
133Dic71⋊C4—28no—
134Dic71⋊C4—44no—
135C22⋊Dic71—42no—
136(C2×D71)⋊C4—44no—
137C24×C71C2×D411yescharacteristic
138C22×C4×C71—24no—
139C22×C4×C71—24no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—44no—
143C22×C4×C71C22×C412yes—
144C22×C4×C71C2×D411yescharacteristic
145C22×C4×C71C2×D412yes—
146C22×C4×C71C2411yescharacteristic
147C2×C8×C71—22no—
148C2×C8×C71—24no—
149C4⋊C4×C71—24no—
150C4⋊C4×C71—22no—
151C4⋊C4×C71—24no—
152C4⋊C4×C71C22×C414yes—
153C22⋊C4×C71—42no—
154C23×C71—22no—
155C23×C71—22no—
156C23×C71C2×C22⋊C412yes—
157C23×C71C2×SD1612yes—
158C23×C71C2×Q1612yes—
159C23×C71C22×D411yescharacteristic
160C22×D71—22no—
161C22×D71—22no—
162C22×D71—28no—
163C22×D71—28no—
164C22×D71—216no—
165C22×D71—216no—
166C22×D71#103116yes—
167C22×D71C2×C22⋊C412yes—
168C22×D71C2×C22⋊C412yes—
169C22×D71#12314yes—
170C22×D71C2×SD1612yes—
171C22×D71C2×SD1612yes—
172C22×D71C2×Q1612yes—
173C22×D71C2×Q1612yes—
174C22×D71AGL(1,ℤ/8)14yes—
175C2×C4×C71—22no—
176C2×C4×C71—24no—
177C2×C4×C71—24no—
178C2×C4×C71—24no—
179C2×C4×C71—48no—
180C2×C4×C71—44no—
181C2×C4×C71—42no—
182C2×C4×C71—42no—
183C2×C4×C71—42no—
184C2×C4×C71C2×C22⋊C412yes—
185C2×C4×C71C23×C411yescharacteristic
186C2×C4×C71C22×D411yescharacteristic
187C2×C4×C71C2×#4114yes—
188C2×Dic71—22no—
189C2×Dic71—28no—
190C2×Dic71—24no—
191C2×Dic71—48no—
192C2×Dic71—44no—
193C2×Dic71—42no—
194C2×Dic71—42no—
195C2×Dic71—42no—
196C2×Dic71C2×C22⋊C412yes—
197C2×Dic71C2×C22⋊C412yes—
198C2×Dic71#12314yes—
199C4×D71—28no—
200C4×D71—24no—
201C4×D71—24no—
202C4×D71—28no—
203C4×D71—416no—
204C4×D71—416no—
205C4×D71—88no—
206C4×D71#11718yes—
207C4×D71#12318yes—
208C8×C71—42no—
209C71⋊C8—42no—
210C22×C71—24no—
211C22×C71—24no—
212C22×C71—22no—
213C22×C71—22no—
214C22×C71C2×#10314yes—
215C22×C71C22×C22⋊C411yescharacteristic
216C22×C71C2×#12312yes—
217C22×C71C22×SD1611yescharacteristic
218C22×C71C22×Q1611yescharacteristic
219C22×C71C2×AGL(1,ℤ/8)12yes—
220C2×D71—24no—
221C2×D71—24no—
222C2×D71—22no—
223C2×D71—22no—
224C2×D71—232no—
225C2×D71#38114yes—
226C2×D71C2×#10314yes—
227C2×D71C2×#10314yes—
228C2×D71#41814yes—
229C2×D71#48118yes—
230C2×D71#48418yes—
231C4×C71—22no—
232C4×C71—44no—
233C4×C71—82no—
234C4×C71C2×#11712yes—
235C4×C71C2×#12312yes—
236Dic71—22no—
237Dic71—44no—
238Dic71—82no—
239Dic71#38114yes—
240C2×#1824—711nomaximal, Sylow
241C2×C71—24no—
242C2×C71C2×#38111yescharacteristic
243C2×C71C22×#10311yescharacteristic
244C2×C71C2×#41811yescharacteristic
245C2×C71C2×#48112yes—
246C2×C71C2×#48412yes—
247D71—24no—
248D71#182418yesdirect product factor
249C23×C4⋊C4—711no—
250C22×C22⋊C8—711no—
251C2×#380—711no—
252#1824—711no—
253#1824—711no—
254#1824—714no—
255#1824—711no—
256#1824D7111yescharacteristic, direct product factor
257C2×#340—714no—
258C71C2×#182411yescharacteristic, complemented, Sylow
259C24×C4—711no—
260C24×C4—1421no—
261C23×C8—712no—
262C22×C4⋊C4—711no—
263C22×C4⋊C4—718no—
264C22×C4⋊C4—712no—
265C22×C4⋊C4—712no—
266C22×C4⋊C4—712no—
267C22×C4⋊C4—1421no—
268C22×C4⋊C4—1421no—
269C22×C4⋊C4—1424no—
270C22×C4⋊C4—1421no—
271C22×C4⋊C4C2×D7111yescharacteristic
272C22×C22⋊C4—1421no—
273C2×C22⋊C8—711no—
274C2×C22⋊C8—712no—
275C2×C22⋊C8—1424no—
276C2×C22⋊C8C2×D7111yescharacteristic
277#380—711no—
278#380—712no—
279#380—1421no—
280#380—1421no—
281#380C2×D7111yescharacteristic
282C2×#95—1421no—
283#340—714no—
284#340—714no—
285#340—714no—
286#340C2×D7114yescomplemented
287C25—711no—
288C23×C4—21no—
289C23×C4—2848no—
290C23×C4—711no—
291C23×C4—711no—
292C23×C4—714no—
293C23×C4—711no—
294C23×C4—711no—
295C23×C4—714no—
296C23×C4—711no—
297C23×C4—1422no—
298C23×C4—1422no—
299C23×C4—1422no—
300C23×C4—1428no—
301C23×C4—1428no—
302C23×C4—1421no—
303C23×C4C22×D7111yescharacteristic
304C22×C8—712no—
305C22×C8—1424no—
306C22×C8—1428no—
307C22×C8C22×D7112yes—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—714no—
311C2×C4⋊C4—712no—
312C2×C4⋊C4—718no—
313C2×C4⋊C4—718no—
314C2×C4⋊C4—712no—
315C2×C4⋊C4—1428no—
316C2×C4⋊C4—1424no—
317C2×C4⋊C4—1424no—
318C2×C4⋊C4—1428no—
319C2×C4⋊C4—1428no—
320C2×C4⋊C4—1422no—
321C2×C4⋊C4—1422no—
322C2×C4⋊C4—1422no—
323C2×C4⋊C4—1428no—
324C2×C4⋊C4—1422no—
325C2×C4⋊C4—1421no—
326C2×C4⋊C4C4×D7114yescomplemented
327C2×C4⋊C4C22×D7112yes—
328C2×C4⋊C4C22×D7112yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—2844no—
331C2×C22⋊C4—1422no—
332C2×C22⋊C4—1421no—
333C22⋊C8—22no—
334C22⋊C8—1424no—
335C22⋊C8—1422no—
336#95—21no—
337#95—1422no—
338#95—1421no—
339C24—711no—
340C24—711no—
341C24—714no—
342C24—714no—
343C24—714no—
344C24—1424no—
345C24—1424no—
346C24D4×D7111yescharacteristic
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—44no—
352C22×C4—2848no—
353C22×C4—28432no—
354C22×C4—2848no—
355C22×C4—2844no—
356C22×C4—2844no—
357C22×C4—2844no—
358C22×C4—2844no—
359C22×C4—711no—
360C22×C4—712no—
361C22×C4—712no—
362C22×C4—718no—
363C22×C4—711no—
364C22×C4—714no—
365C22×C4—712no—
366C22×C4—712no—
367C22×C4—714no—
368C22×C4—714no—
369C22×C4—1428no—
370C22×C4—1428no—
371C22×C4—1424no—
372C22×C4—14216no—
373C22×C4—14216no—
374C22×C4—1424no—
375C22×C4—1422no—
376C22×C4—1422no—
377C22×C4—1424no—
378C22×C4—1424no—
379C22×C4—1424no—
380C22×C4C2×C4×D7112yes—
381C22×C4D4×D7112yes—
382C22×C4D4×D7111yescharacteristic
383C22×C4C23×D7111yescharacteristic
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—2848no—
387C2×C8—1422no—
388C2×C8—1424no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—2844no—
393C4⋊C4—714no—
394C4⋊C4—1428no—
395C4⋊C4—1428no—
396C4⋊C4—1424no—
397C4⋊C4—1422no—
398C4⋊C4—1424no—
399C4⋊C4C2×C4×D7114yes—
400C22⋊C4—42no—
401C22⋊C4—2844no—
402C22⋊C4—2842no—
403C23—22no—
404C23—22no—
405C23—712no—
406C23—714no—
407C23—712no—
408C23—712no—
409C23—714no—
410C23—712no—
411C23—7116no—
412C23—712no—
413C23—712no—
414C23—1428no—
415C23—1428no—
416C23—14216no—
417C23—14216no—
418C23—1422no—
419C23—1422no—
420C23C22⋊C4×D7112yes—
421C23SD16×D7112yes—
422C23Q16×D7112yes—
423C23C2×D4×D7111yescharacteristic
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—28416no—
434C2×C4—28416no—
435C2×C4—2848no—
436C2×C4—2844no—
437C2×C4—2842no—
438C2×C4—2842no—
439C2×C4—2842no—
440C2×C4—5688no—
441C2×C4—712no—
442C2×C4—714no—
443C2×C4—718no—
444C2×C4—718no—
445C2×C4—712no—
446C2×C4—1428no—
447C2×C4—1428no—
448C2×C4—1424no—
449C2×C4—1428no—
450C2×C4—1422no—
451C2×C4—1424no—
452C2×C4—1424no—
453C2×C4C22⋊C4×D7112yes—
454C2×C4C22×C4×D7111yescharacteristic
455C2×C4C2×D4×D7111yescharacteristic
456C2×C4#41×D7114yes—
457C8—42no—
458C8—2842no—
459C22—24no—
460C22—24no—
461C22—22no—
462C22—22no—
463C22—718no—
464C22—718no—
465C22—714no—
466C22—714no—
467C22—714no—
468C22—714no—
469C22—14232no—
470C22—1424no—
471C22—1424no—
472C22—1422no—
473C22—1422no—
474C22#103×D7114yes—
475C22C2×C22⋊C4×D7111yescharacteristic
476C22#123×D7112yes—
477C22C2×SD16×D7111yescharacteristic
478C22C2×Q16×D7111yescharacteristic
479C22AGL(1,ℤ/8)×D7112yes—
480C4—22no—
481C4—44no—
482C4—82no—
483C4—2844no—
484C4—5682no—
485C4—714no—
486C4—1422no—
487C4#117×D7112yes—
488C4#123×D7112yes—
489C2—24no—
490C2—718no—
491C2—1424no—
492C2#381×D7111yescharacteristic
493C2C2×#103×D7111yescharacteristic
494C2#418×D7111yescharacteristic
495C2#481×D7112yes—
496C2#484×D7112yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1824C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1824D71
hypocenterC71C2×#1824
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #22588431C2semidirect
5. #22588430C2semidirect
6. #22588429C2semidirect
7. #22588428C2semidirect
8. #340×D71C2semidirect
9. C71×#1824C2semidirect
18. C2×C4⋊C4×D71C4semidirect
31. #17985762C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
37. #340×C71C22semidirect
58. C2×C4⋊Dic71C2×C4semidirect
71. C4⋊C4×D71C2×C4semidirect
84. C2×C4⋊C4×C71C2×C4semidirect
248. D71#1824direct (N × H)
256. #1824D71direct (N × H)
258. C71C2×#1824semidirect
286. #340C2×D71semidirect
326. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D712
C2×C22⋊C8×D712
#380×D712
#225884312
#225884302
#225884292
#225884282
#340×D712
C71×#18242
C2×#182471