Finite Group Databasebeta

← #31997247

Subgroups — #31997247

Order 18176.

1774 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×#17985961C211yesmaximal, characteristic
3#22590722C211yesmaximal, characteristic
4#22588429C214yesmaximal, complemented
5#22588427C211yesmaximal, characteristic, complemented
6#22588426C211yesmaximal, characteristic, complemented
7#22588424C211yesmaximal, characteristic, complemented
8#22588058C214yesmaximal, complemented
9C71×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D71—21no—
11C23×C4×D71C2211yescharacteristic
12C22×#11765848C2212yes—
13C2×#14556086C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×Dic71⋊C4—21no—
16C2×C4⋊C4×D71—24no—
17C2×C4⋊C4×D71—21no—
18C2×C4⋊C4×D71—21no—
19C2×C4⋊C4×D71C418yescomplemented
20C2×C4⋊C4×D71C2212yes—
21C2×C4⋊C4×D71C2212yes—
22#17985961—24no—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×(C2×D71)⋊C8C2212yes—
25#17985861—21no—
26#17985859C2211yescharacteristic
27#17985856C2212yes—
28#17985854—21no—
29C22⋊C4×Dic71—21no—
30#17985846—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—44no—
46C22×C4×D71—44no—
47C22×C4×D71C2×C414yes—
48C22×C4×D71D411yescharacteristic
49C22×C4×D71D411yescharacteristic
50C22×C4×D71D414yes—
51C22×C4×D71C2311yescharacteristic
52C22×C4×D71C2311yescharacteristic
53C2×#11765848—24no—
54C2×#11765848—28no—
55C22×C71⋊C8C2312yes—
56#14556086—28no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71C2×C414yes—
60C2×C4⋊Dic71C2312yes—
61C2×C4⋊Dic71C2312yes—
62C2×Dic71⋊C4—28no—
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×D71C2×C418yes—
70C4⋊C4×D71C2×C418yes—
71C2×C22⋊Dic71—21no—
72C2×(C2×D71)⋊C4—22no—
73C2×(C2×D71)⋊C4—22no—
74C2×C4×Dic71—22no—
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×C414yes—
85C2×C4⋊C4×C71C2312yes—
86C2×C4⋊C4×C71C2312yes—
87C2×C22⋊C4×C71—21no—
88C22⋊C8×C71—22no—
89#95×C71—21no—
90C23×D71—22no—
91C23×D71—22no—
92C23×D71—24no—
93C23×D71—24no—
94C23×D71C22⋊C414yes—
95C23×D71C2×D411yescharacteristic
96C23×D71C2×D411yescharacteristic
97C22×Dic71—24no—
98C22×Dic71—24no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—42no—
102C22×Dic71—42no—
103C22×Dic71C22⋊C414yes—
104C22×Dic71C2×D411yescharacteristic
105C22×Dic71C2×D411yescharacteristic
106C2×C4×D71—24no—
107C2×C4×D71—24no—
108C2×C4×D71—24no—
109C2×C4×D71—216no—
110C2×C4×D71—216no—
111C2×C4×D71—28no—
112C2×C4×D71—28no—
113C2×C4×D71—44no—
114C2×C4×D71—44no—
115C2×C4×D71—44no—
116C2×C4×D71—44no—
117C2×C4×D71—432no—
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—
125#11765848—48no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—24no—
128C4⋊Dic71—24no—
129C4⋊Dic71—22no—
130C4⋊Dic71—24no—
131C4⋊Dic71C22×C414yes—
132Dic71⋊C4—28no—
133Dic71⋊C4—28no—
134C22⋊Dic71—42no—
135(C2×D71)⋊C4—44no—
136(C2×D71)⋊C4—44no—
137C4×Dic71—44no—
138C24×C71C2×D411yescharacteristic
139C22×C4×C71—24no—
140C22×C4×C71—24no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—42no—
144C22×C4×C71—42no—
145C22×C4×C71C22×C412yes—
146C22×C4×C71C2×D411yescharacteristic
147C22×C4×C71C2×D412yes—
148C22×C4×C71C2411yescharacteristic
149C2×C8×C71—22no—
150C2×C8×C71—24no—
151C4⋊C4×C71—24no—
152C4⋊C4×C71—22no—
153C4⋊C4×C71—24no—
154C4⋊C4×C71C22×C414yes—
155C22⋊C4×C71—42no—
156C23×C71—22no—
157C23×C71—22no—
158C23×C71C2×C22⋊C412yes—
159C23×C71AGL(1,ℤ/8)12yes—
160C23×C71AGL(1,ℤ/8)12yes—
161C23×C71C22×D411yescharacteristic
162C22×D71—216no—
163C22×D71—216no—
164C22×D71—22no—
165C22×D71—21no—
166C22×D71—21no—
167C22×D71—21no—
168C22×D71—21no—
169C22×D71—22no—
170C22×D71—28no—
171C22×D71—28no—
172C22×D71—28no—
173C22×D71C2×C22⋊C412yes—
174C22×D71C2×C22⋊C412yes—
175C22×D71#12314yes—
176C22×D71#13714yes—
177C2×C4×C71—22no—
178C2×C4×C71—24no—
179C2×C4×C71—24no—
180C2×C4×C71—24no—
181C2×C4×C71—42no—
182C2×C4×C71—42no—
183C2×C4×C71—48no—
184C2×C4×C71—42no—
185C2×C4×C71—42no—
186C2×C4×C71—42no—
187C2×C4×C71C2×C22⋊C412yes—
188C2×C4×C71C23×C411yescharacteristic
189C2×C4×C71C22×D411yescharacteristic
190C2×C4×C71C2×#4114yes—
191C2×Dic71—22no—
192C2×Dic71—28no—
193C2×Dic71—24no—
194C2×Dic71—42no—
195C2×Dic71—42no—
196C2×Dic71—48no—
197C2×Dic71—42no—
198C2×Dic71—42no—
199C2×Dic71—42no—
200C2×Dic71C2×C22⋊C412yes—
201C2×Dic71C2×C22⋊C412yes—
202C2×Dic71#12314yes—
203C4×D71—28no—
204C4×D71—24no—
205C4×D71—24no—
206C4×D71—28no—
207C4×D71—416no—
208C4×D71—416no—
209C4×D71#11718yes—
210C4×D71#12318yes—
211C8×C71—42no—
212C71⋊C8—42no—
213C22×C71—24no—
214C22×C71—24no—
215C22×C71—22no—
216C22×C71—22no—
217C22×C71#41814yes—
218C22×C71C22×C22⋊C411yescharacteristic
219C22×C71C2×#12312yes—
220C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
221C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C71C2×#13712yes—
223C2×D71—232no—
224C2×D71—24no—
225C2×D71—24no—
226C2×D71—24no—
227C2×D71—24no—
228C2×D71—22no—
229C2×D71—22no—
230C2×D71—22no—
231C2×D71—22no—
232C2×D71#38114yes—
233C2×D71#41914yes—
234C4×C71—22no—
235C4×C71—44no—
236C4×C71—82no—
237C4×C71C2×#11712yes—
238C4×C71C2×#12312yes—
239Dic71—22no—
240Dic71—44no—
241Dic71—82no—
242Dic71#38114yes—
243#20700—711nomaximal, Sylow
244C2×C71—24no—
245C2×C71C2×#38111yescharacteristic
246C2×C71C2×#41811yescharacteristic
247C2×C71C2×#41911yescharacteristic
248C2×C71#309212yes—
249C2×C71#309212yes—
250D71—24no—
251D71—24no—
252C23×C4⋊C4—711no—
253C2×#407—711no—
254#2280—711no—
255#1824—714no—
256#1823—711no—
257#1823—711no—
258#1823—711no—
259#1823D7111yescharacteristic, complemented
260#1736—714no—
261C71#2070011yescharacteristic, complemented, Sylow
262C24×C4—711no—
263C24×C4—1421no—
264C22×M16—712no—
265C2×#117—712no—
266C22×C4⋊C4—711no—
267C22×C4⋊C4—718no—
268C22×C4⋊C4—712no—
269C22×C4⋊C4—712no—
270C22×C4⋊C4—1421no—
271C22×C4⋊C4—1421no—
272C22×C4⋊C4—1424no—
273C22×C4⋊C4—1421no—
274C22×C4⋊C4C2×D7111yescharacteristic
275#407—1424no—
276C2×C22⋊C8—711no—
277C2×C22⋊C8—712no—
278C2×C22⋊C8C2×D7111yescharacteristic
279#381—1421no—
280#380—711no—
281#380—712no—
282#380C2×D7111yescharacteristic
283#379—1421no—
284C4×C22⋊C4—1421no—
285C4×C22⋊C4—1421no—
286#340—714no—
287#340—714no—
288#340—714no—
289#340C2×D7114yescomplemented
290C25—711no—
291C23×C4—21no—
292C23×C4—2844no—
293C23×C4—2844no—
294C23×C4—711no—
295C23×C4—711no—
296C23×C4—714no—
297C23×C4—711no—
298C23×C4—711no—
299C23×C4—714no—
300C23×C4—711no—
301C23×C4—1422no—
302C23×C4—1422no—
303C23×C4—1428no—
304C23×C4—1428no—
305C23×C4—1421no—
306C23×C4C22×D7111yescharacteristic
307C2×M16—1424no—
308C2×M16—1428no—
309C22×C8—712no—
310C22×C8C22×D7112yes—
311#117—1428no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—714no—
315C2×C4⋊C4—712no—
316C2×C4⋊C4—718no—
317C2×C4⋊C4—718no—
318C2×C4⋊C4—712no—
319C2×C4⋊C4—1424no—
320C2×C4⋊C4—1424no—
321C2×C4⋊C4—1428no—
322C2×C4⋊C4—1428no—
323C2×C4⋊C4—1422no—
324C2×C4⋊C4—1422no—
325C2×C4⋊C4—1428no—
326C2×C4⋊C4—1422no—
327C2×C4⋊C4—1421no—
328C2×C4⋊C4C4×D7114yes—
329C2×C4⋊C4C22×D7112yes—
330C2×C4⋊C4C22×D7112yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1422no—
333C2×C22⋊C4—1422no—
334C2×C22⋊C4—1421no—
335C2×C42—1422no—
336C22⋊C8—22no—
337C22⋊C8—1424no—
338C22⋊C8—1422no—
339#95—21no—
340#95—1422no—
341#95—1421no—
342C24—711no—
343C24—711no—
344C24—714no—
345C24—1424no—
346C24—1424no—
347C24—1422no—
348C24—1422no—
349C24D4×D7111yescharacteristic
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—28432no—
357C22×C4—2844no—
358C22×C4—2844no—
359C22×C4—2844no—
360C22×C4—2844no—
361C22×C4—2842no—
362C22×C4—2842no—
363C22×C4—711no—
364C22×C4—712no—
365C22×C4—712no—
366C22×C4—718no—
367C22×C4—711no—
368C22×C4—714no—
369C22×C4—712no—
370C22×C4—712no—
371C22×C4—714no—
372C22×C4—714no—
373C22×C4—1428no—
374C22×C4—1428no—
375C22×C4—1424no—
376C22×C4—14216no—
377C22×C4—14216no—
378C22×C4—1424no—
379C22×C4—1422no—
380C22×C4—1422no—
381C22×C4—1424no—
382C22×C4—1424no—
383C22×C4—1424no—
384C22×C4C2×C4×D7112yes—
385C22×C4D4×D7112yes—
386C22×C4D4×D7111yescharacteristic
387C22×C4C23×D7111yescharacteristic
388M16—2848no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—1422no—
392C2×C8—1424no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—714no—
397C4⋊C4—1428no—
398C4⋊C4—1428no—
399C4⋊C4—1424no—
400C4⋊C4—1422no—
401C4⋊C4—1424no—
402C4⋊C4C2×C4×D7114yes—
403C22⋊C4—42no—
404C22⋊C4—2844no—
405C22⋊C4—2844no—
406C22⋊C4—2842no—
407C42—2844no—
408C23—22no—
409C23—22no—
410C23—712no—
411C23—712no—
412C23—714no—
413C23—714no—
414C23—1428no—
415C23—1428no—
416C23—14216no—
417C23—14216no—
418C23—1422no—
419C23—1428no—
420C23—1421no—
421C23—1421no—
422C23—1421no—
423C23—1421no—
424C23—1422no—
425C23C22⋊C4×D7112yes—
426C23#1455615212yes—
427C23#1455615512yes—
428C23C2×D4×D7111yescharacteristic
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—28416no—
440C2×C4—28416no—
441C2×C4—2848no—
442C2×C4—2842no—
443C2×C4—2842no—
444C2×C4—2842no—
445C2×C4—2842no—
446C2×C4—2842no—
447C2×C4—712no—
448C2×C4—714no—
449C2×C4—718no—
450C2×C4—718no—
451C2×C4—712no—
452C2×C4—1428no—
453C2×C4—1428no—
454C2×C4—1424no—
455C2×C4—1428no—
456C2×C4—1422no—
457C2×C4—1424no—
458C2×C4—1424no—
459C2×C4C22⋊C4×D7112yes—
460C2×C4C22×C4×D7111yescharacteristic
461C2×C4C2×D4×D7111yescharacteristic
462C2×C4#41×D7114yes—
463C8—42no—
464C8—2842no—
465C22—24no—
466C22—24no—
467C22—22no—
468C22—22no—
469C22—714no—
470C22—714no—
471C22—14232no—
472C22—1424no—
473C22—1424no—
474C22—1424no—
475C22—1424no—
476C22—1422no—
477C22—1422no—
478C22—1422no—
479C22—1422no—
480C22#1798600214yes—
481C22C2×C22⋊C4×D7111yescharacteristic
482C22#123×D7112yes—
483C22C2×#1455615211yescharacteristic
484C22C2×#1455615511yescharacteristic
485C22#137×D7112yes—
486C4—22no—
487C4—44no—
488C4—82no—
489C4—2844no—
490C4—5682no—
491C4—714no—
492C4—1422no—
493C4#117×D7112yes—
494C4#123×D7112yes—
495C2—24no—
496C2—1424no—
497C2—1424no—
498C2#381×D7111yescharacteristic
499C2C2×#1798600211yescharacteristic
500C2#419×D7111yescharacteristic
501C2#2259825912yes—
502C2#2259826412yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1823C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1823D71
hypocenterC71#20700
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #22588429C2semidirect
5. #22588427C2semidirect
6. #22588426C2semidirect
7. #22588424C2semidirect
8. #22588058C2semidirect
9. C71×#1823C2semidirect
19. C2×C4⋊C4×D71C4semidirect
31. #17985762C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
37. #340×C71C22semidirect
259. #1823D71semidirect
261. C71#20700semidirect
289. #340C2×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×#179859612
#225907222
#225884292
#225884272
#225884262
#225884242
#225880582
C71×#18232
#2070071