Finite Group Databasebeta

← #32158557

Subgroups — #32158557

Order 18176.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986475C211yesmaximal, characteristic
2#22591301C213yesmaximal, complemented
3#22591293C213yesmaximal, complemented
4#22591258C211yesmaximal, characteristic, complemented
5Q8×Dic71⋊C4C211yesmaximal, characteristic, complemented
6#22591229C213yesmaximal
7#22591221C213yesmaximal
8Q8×(C2×D71)⋊C4C211yesmaximal, characteristic
9Q8×(C2×D71)⋊C4C211yesmaximal, characteristic
10#22590992C213yesmaximal, complemented
11#22590945C213yesmaximal
12#22590835C213yesmaximal
13#22590814C213yesmaximal, complemented
14C4×Q8×Dic71C211yesmaximal, characteristic, complemented
15C71×#2276C211yesmaximal, characteristic, complemented
16C22×Q8×D71C2211yescharacteristic
17#17986475C4116yescomplemented
18C2×Q8×Dic71C2211yescharacteristic
19C2×Q8×Dic71C2211yescharacteristic
20C2×Q8×Dic71C2211yescharacteristic
21C2×C4×Dic142C2213yes—
22C2×#14556093C2213yes—
23C2×#14556086C2213yes—
24C2×#14556085C2213yes—
25#17985905C2213yes—
26#17985905C2213yes—
27#17985904C2213yes—
28#17985904C2213yes—
29#17985897C2213yes—
30#17985897C2213yes—
31#17985895C2213yes—
32#17985895C2213yes—
33#17985890C2216yes—
34#17985888C2211yescharacteristic
35#17985888C2213yes—
36#17985888C2213yes—
37#17985888C2213yes—
38#17985883C2216yes—
39#17985881C2213yes—
40#17985881C2216yes—
41#17985881C2213yes—
42#17985881C2213yes—
43#17985881C2213yes—
44#17985868C2213yes—
45#17985866C2216yes—
46#17985864C2213yes—
47#17985863C2216yes—
48#17985856C2213yes—
49#17985856C2213yes—
50C4⋊C4×Dic71C2213yes—
51C4⋊C4×Dic71C2213yes—
52C4×Dic71⋊C4C2213yes—
53C4×Dic71⋊C4C2213yes—
54C4×Dic71⋊C4C2216yes—
55C4×C4⋊Dic71C2213yes—
56C4×(C2×D71)⋊C4C2213yes—
57C4×(C2×D71)⋊C4C2213yes—
58#17985845C2216yes—
59C42×Dic71C2213yes—
60C2×C4×Q8×C71C2211yescharacteristic
61C2×C4×Q8×C71C2211yescharacteristic
62C2×C4×Q8×C71C2211yescharacteristic
63C4×C4⋊C4×C71C2213yes—
64C4×C4⋊C4×C71C2213yes—
65C4×C4⋊C4×C71C2213yes—
66#376×C71C2213yes—
67C22×Dic142C2313yes—
68C2×Q8×D71—24no—
69C2×Q8×D71—24no—
70C2×Q8×D71C2×C418yes—
71C22×C4×D71C2313yes—
72Q8×Dic71—24no—
73Q8×Dic71—24no—
74Q8×Dic71C2×C418yes—
75C4×Dic142C2×C4124yes—
76#14556093C2×C4124yes—
77#14556086C2×C4124yes—
78#14556085C2×C4124yes—
79C2×C4⋊Dic71C2313yes—
80C2×C4⋊Dic71C2313yes—
81C2×C4⋊Dic71C2313yes—
82C2×Dic71⋊C4C2311yescharacteristic
83C2×Dic71⋊C4C2313yes—
84C2×Dic71⋊C4C2313yes—
85C2×Dic71⋊C4C2316yes—
86C2×Dic71⋊C4C2313yes—
87C2×Dic71⋊C4C2313yes—
88C2×Dic71⋊C4C2313yes—
89C2×(C2×D71)⋊C4C2311yescharacteristic
90C2×(C2×D71)⋊C4C2311yescharacteristic
91C2×(C2×D71)⋊C4C2313yes—
92C2×(C2×D71)⋊C4C2313yes—
93C2×(C2×D71)⋊C4C2311yescharacteristic
94C2×(C2×D71)⋊C4C2313yes—
95C2×C4×Dic71C2311yescharacteristic
96C2×C4×Dic71C2313yes—
97C2×C4×Dic71C2313yes—
98C2×C4×Dic71C2313yes—
99C2×C4×Dic71C2313yes—
100C2×C4×Dic71C2313yes—
101C2×C4×Dic71C2316yes—
102C2×C4×Dic71C2313yes—
103C71⋊#95C2313yes—
104C71⋊#95C2313yes—
105C71⋊#95C2316yes—
106C71⋊#95C2316yes—
107C71⋊#95C2313yes—
108C71⋊#95C2313yes—
109C71⋊#95C2313yes—
110C71⋊#95C2316yes—
111C71⋊#95C2311yescharacteristic
112C71⋊#95C2316yes—
113C71⋊#95C2313yes—
114C71⋊#95C2313yes—
115C22×Q8×C71C2311yescharacteristic
116C4×Q8×C71—24no—
117C4×Q8×C71—24no—
118C4×Q8×C71C2×C418yes—
119C2×C4⋊C4×C71C2313yes—
120C2×C4⋊C4×C71C2313yes—
121C2×C4⋊C4×C71C2313yes—
122C2×C42×C71C2313yes—
123C2×C42×C71C2313yes—
124C2×C42×C71C2316yes—
125C2×C42×C71C2313yes—
126#95×C71C2313yes—
127#95×C71C2313yes—
128#95×C71C2311yescharacteristic
129#95×C71C2313yes—
130C23×D71C2411yescharacteristic
131C2×Dic142—26no—
132C2×Dic142—26no—
133C2×Dic142C22×C4112yes—
134Q8×D71—232no—
135C22×Dic71C2413yes—
136C22×Dic71C2413yes—
137C22×Dic71C2413yes—
138C22×Dic71C2411yescharacteristic
139C22×Dic71C2413yes—
140C22×Dic71C2411yescharacteristic
141C22×Dic71C2411yescharacteristic
142C2×C4×D71—26no—
143C2×C4×D71—26no—
144C2×C4×D71C22×C4112yes—
145C4⋊Dic71—26no—
146C4⋊Dic71—26no—
147C4⋊Dic71C22×C4112yes—
148Dic71⋊C4—22no—
149Dic71⋊C4—26no—
150Dic71⋊C4—43no—
151Dic71⋊C4—43no—
152Dic71⋊C4—46no—
153Dic71⋊C4C22×C4112yes—
154Dic71⋊C4C22×C4112yes—
155(C2×D71)⋊C4—22no—
156(C2×D71)⋊C4—22no—
157(C2×D71)⋊C4—43no—
158(C2×D71)⋊C4—43no—
159(C2×D71)⋊C4C22×C414yes—
160(C2×D71)⋊C4C22×C4112yes—
161C4×Dic71—26no—
162C4×Dic71—26no—
163C4×Dic71—22no—
164C4×Dic71—26no—
165C4×Dic71—43no—
166C4×Dic71—43no—
167C4×Dic71C22×C4112yes—
168C4×Dic71C22×C4124yes—
169C2×Q8×C71C22×C414yes—
170C2×Q8×C71#4114yes—
171C2×Q8×C71#4114yes—
172C22×C4×C71C2411yescharacteristic
173C22×C4×C71C2411yescharacteristic
174C22×C4×C71C2413yes—
175C22×C4×C71C2413yes—
176C22×C4×C71C2411yescharacteristic
177C22×C4×C71C2413yes—
178C22×C4×C71C2413yes—
179C4⋊C4×C71—26no—
180C4⋊C4×C71—26no—
181C4⋊C4×C71C22×C4112yes—
182C42×C71—26no—
183C42×C71—26no—
184C42×C71—46no—
185C42×C71C22×C4112yes—
186C23×C71C2511yescharacteristic
187C22×D71—21no—
188C22×D71—21no—
189C22×D71—21no—
190C22×D71—21no—
191C22×D71—21no—
192C22×D71—21no—
193C22×D71C23×C412yes—
194Q8×C71#117116yes—
195C2×C4×C71—21no—
196C2×C4×C71—23no—
197C2×C4×C71—23no—
198C2×C4×C71—21no—
199C2×C4×C71—21no—
200C2×C4×C71—21no—
201C2×C4×C71—23no—
202C2×C4×C71—23no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—23no—
206C2×C4×C71—21no—
207C2×C4×C71—23no—
208C2×C4×C71—23no—
209C2×C4×C71—23no—
210C2×C4×C71—21no—
211C2×C4×C71C23×C412yes—
212C2×C4×C71C23×C416yes—
213C2×C4×C71C23×C416yes—
214C2×C4×C71C2×#4116yes—
215C2×C4×C71C2×#4116yes—
216C2×Dic71—21no—
217C2×Dic71—21no—
218C2×Dic71—23no—
219C2×Dic71—23no—
220C2×Dic71—21no—
221C2×Dic71—23no—
222C2×Dic71—21no—
223C2×Dic71—21no—
224C2×Dic71—21no—
225C2×Dic71—23no—
226C2×Dic71—23no—
227C2×Dic71—23no—
228C2×Dic71—23no—
229C2×Dic71—23no—
230C2×Dic71—23no—
231C2×Dic71—23no—
232C2×Dic71C23×C412yes—
233C2×Dic71C23×C416yes—
234C2×Dic71C23×C416yes—
235C2×Dic712−1+412yes—
236C2×Dic712−1+412yes—
237C4×D71—224no—
238Dic142—224no—
239C22×C71C24×C411yescharacteristic
240C22×C71C22×#4111yescharacteristic
241C22×C71C22×#4111yescharacteristic
242C22×C71C2×2−1+411yescharacteristic
243C22×C71C2×2−1+411yescharacteristic
244C22×C71#58511yescharacteristic
245C22×C71#58511yescharacteristic
246C2×D71—22no—
247C2×D71—22no—
248C2×D71—22no—
249C2×D71—41no—
250C2×D71—41no—
251C2×D71—41no—
252C2×D71—41no—
253C4×C71—22no—
254C4×C71—26no—
255C4×C71—43no—
256C4×C71—41no—
257C4×C71—41no—
258C4×C71—43no—
259C4×C71C2×#117112yes—
260Dic71—26no—
261Dic71—26no—
262Dic71—22no—
263Dic71—22no—
264Dic71—22no—
265Dic71—43no—
266Dic71—43no—
267#34262—711nomaximal, Sylow
268C2×C71C22×#11711yescharacteristic
269C2×C71C4×2−1+411yescharacteristic
270C2×C71C4×2−1+411yescharacteristic
271C2×C71#351611yescharacteristic
272C2×C71#351811yescharacteristic
273C2×C71#352311yescharacteristic
274C2×C71#352411yescharacteristic
275D71—42no—
276C2×#519—711no—
277#2356—713no—
278#2355—713no—
279#2351—711no—
280Q8×C4⋊C4—711no—
281#2347—713no—
282#2346—713no—
283Q8×C22⋊C4—711no—
284Q8×C22⋊C4—711no—
285#2320—713no—
286#2314—713no—
287#2298—713no—
288#2295—713no—
289#2276D7111yescharacteristic, complemented
290C42×Q8—711no—
291C71#3426211yescharacteristic, complemented, Sylow
292C23×Q8—711no—
293#519—7116no—
294C2×C4×Q8—711no—
295C2×C4×Q8—711no—
296C2×C4×Q8—711no—
297C2×C4×Q8—713no—
298C2×C4×Q8—713no—
299C2×C4×Q8C2×D7111yescharacteristic
300C2×C4×Q8C2×D7111yescharacteristic
301C2×C4×Q8C2×D7111yescharacteristic
302C2×#117—713no—
303C2×#117—713no—
304#391—713no—
305#391—713no—
306#391—713no—
307#391—713no—
308#389—713no—
309#389—713no—
310#389—713no—
311#389—713no—
312#388—716no—
313#388—713no—
314#388—713no—
315#388—713no—
316#388—711no—
317#387—716no—
318#387—716no—
319#387—713no—
320#387—713no—
321#387—713no—
322#387—713no—
323#384—713no—
324#383—716no—
325#382—713no—
326#382—716no—
327#380—713no—
328#380—713no—
329C4×C4⋊C4—713no—
330C4×C4⋊C4—713no—
331C4×C4⋊C4—713no—
332C4×C4⋊C4—716no—
333C4×C4⋊C4—713no—
334C4×C4⋊C4—713no—
335C4×C4⋊C4C2×D7113yes—
336C4×C4⋊C4C2×D7113yes—
337C4×C4⋊C4C2×D7113yes—
338C4×C22⋊C4—713no—
339C4×C22⋊C4—713no—
340#376—716no—
341#376C2×D7113yes—
342C43—713no—
343C22×Q8—718no—
344C22×Q8—713no—
345C22×Q8—1424no—
346C22×Q8—1424no—
347C22×Q8C22×D7111yescharacteristic
348C23×C4—713no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—718no—
352C4×Q8—7124no—
353C4×Q8—7124no—
354C4×Q8—1424no—
355C4×Q8—1424no—
356C4×Q8C4×D7118yes—
357#117—7124no—
358#117—7124no—
359C2×C4⋊C4—713no—
360C2×C4⋊C4—716no—
361C2×C4⋊C4—713no—
362C2×C4⋊C4—713no—
363C2×C4⋊C4—713no—
364C2×C4⋊C4—713no—
365C2×C4⋊C4—713no—
366C2×C4⋊C4—713no—
367C2×C4⋊C4—711no—
368C2×C4⋊C4—713no—
369C2×C4⋊C4C22×D7113yes—
370C2×C4⋊C4C22×D7113yes—
371C2×C4⋊C4C22×D7113yes—
372C2×C22⋊C4—713no—
373C2×C22⋊C4—713no—
374C2×C22⋊C4—713no—
375C2×C22⋊C4—711no—
376C2×C22⋊C4—711no—
377C2×C22⋊C4—711no—
378C2×C42—713no—
379C2×C42—713no—
380C2×C42—716no—
381C2×C42—713no—
382C2×C42—713no—
383C2×C42—713no—
384C2×C42—711no—
385C2×C42—713no—
386C2×C42C22×D7113yes—
387C2×C42C22×D7113yes—
388C2×C42C22×D7116yes—
389C2×C42C22×D7113yes—
390#95—713no—
391#95—713no—
392#95—716no—
393#95—716no—
394#95—713no—
395#95—713no—
396#95—713no—
397#95—716no—
398#95—711no—
399#95—716no—
400#95—713no—
401#95—713no—
402#95C22×D7111yescharacteristic
403#95C22×D7113yes—
404#95C22×D7113yes—
405#95C22×D7113yes—
406C24—711no—
407C2×Q8—7112no—
408C2×Q8—14232no—
409C2×Q8—1426no—
410C2×Q8—1426no—
411C2×Q8C2×C4×D7114yes—
412C2×Q8#1176586514yes—
413C2×Q8#1176586514yes—
414C22×C4—713no—
415C22×C4—711no—
416C22×C4—711no—
417C22×C4—711no—
418C22×C4—713no—
419C22×C4—7112no—
420C22×C4—713no—
421C22×C4—713no—
422C22×C4—1426no—
423C22×C4—1426no—
424C22×C4C23×D7111yescharacteristic
425C22×C4C23×D7113yes—
426C22×C4C23×D7111yescharacteristic
427C22×C4C23×D7113yes—
428C22×C4C23×D7113yes—
429C22×C4C23×D7111yescharacteristic
430C22×C4C23×D7113yes—
431C4⋊C4—26no—
432C4⋊C4—26no—
433C4⋊C4—2843no—
434C4⋊C4—2846no—
435C4⋊C4—2843no—
436C4⋊C4—7112no—
437C4⋊C4—7112no—
438C4⋊C4—7112no—
439C4⋊C4—1422no—
440C4⋊C4—1426no—
441C4⋊C4—1426no—
442C4⋊C4—1426no—
443C4⋊C4C2×C4×D71112yes—
444C22⋊C4—2843no—
445C22⋊C4—2843no—
446C22⋊C4—7112no—
447C22⋊C4—714no—
448C22⋊C4—1422no—
449C22⋊C4—1422no—
450C42—26no—
451C42—26no—
452C42—46no—
453C42—2843no—
454C42—2843no—
455C42—7112no—
456C42—7124no—
457C42—1426no—
458C42—1422no—
459C42—1426no—
460C42—1426no—
461C42C2×C4×D71112yes—
462C23—712no—
463C23—1421no—
464C23—1421no—
465C23—1421no—
466C23—1421no—
467C23—1421no—
468C23—1421no—
469C23C24×D7111yescharacteristic
470Q8—14224no—
471Q8#14556085116yes—
472C2×C4—21no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—23no—
483C2×C4—21no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—23no—
487C2×C4—21no—
488C2×C4—716no—
489C2×C4—712no—
490C2×C4—716no—
491C2×C4—712no—
492C2×C4—712no—
493C2×C4—1423no—
494C2×C4—1423no—
495C2×C4—1423no—
496C2×C4—1423no—
497C2×C4—1423no—
498C2×C4—1423no—
499C2×C4—1421no—
500C2×C4—1421no—
501C2×C4—1423no—
502C2×C4—1423no—
503C2×C4—1421no—
504C2×C4—1423no—
505C2×C4—1421no—
506C2×C4—1421no—
507C2×C4—1421no—
508C2×C4—1423no—
509C2×C4—14224no—
510C2×C4C22×C4×D7116yes—
511C2×C4C22×C4×D7112yes—
512C2×C4C22×C4×D7116yes—
513C2×C4C2×#1176586516yes—
514C2×C4C2×#1176586516yes—
515C22—2841no—
516C22—2841no—
517C22—2841no—
518C22—2841no—
519C22—1422no—
520C22—1422no—
521C22—1422no—
522C22C23×C4×D7111yescharacteristic
523C22C22×#1176586511yescharacteristic
524C22C22×#1176586511yescharacteristic
525C222−1+4×D7111yescharacteristic
526C222−1+4×D7111yescharacteristic
527C22#1798683411yescharacteristic
528C22#1798683611yescharacteristic
529C4—22no—
530C4—26no—
531C4—43no—
532C4—41no—
533C4—41no—
534C4—43no—
535C4—2843no—
536C4—2843no—
537C4—1426no—
538C4—1426no—
539C4—1422no—
540C4—1422no—
541C4—1422no—
542C4C2×#14556085112yes—
543C2—2842no—
544C2C22×#1455608511yescharacteristic
545C2#2260154911yescharacteristic
546C2#2260155111yescharacteristic
547C2#2260232511yescharacteristic
548C2#2260235111yescharacteristic
549C2#2260243011yescharacteristic
550C2#2260244111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2276C2
socle, cosocleC23×C71C25
prime radicalC23×D71C24
hypercenter#2276D71
hypocenterC71#34262
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. #22591301C2semidirect
3. #22591293C2semidirect
4. #22591258C2semidirect
5. Q8×Dic71⋊C4C2semidirect
10. #22590992C2semidirect
13. #22590814C2semidirect
14. C4×Q8×Dic71C2semidirect
15. C71×#2276C2semidirect
17. #17986475C4semidirect
289. #2276D71semidirect
291. C71#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#179864752
#225913012
#225912932
#225912582
Q8×Dic71⋊C42
#225912292
#225912212
Q8×(C2×D71)⋊C42
#225909922
#225909452
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