Finite Group Databasebeta

← #13395349

Subgroups — #13395349

Order 1792.

1118 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
2C2×#89935C212yesmaximal, complemented
3Q8×C4⋊Dic7C211yesmaximal, characteristic, complemented
4C2×#89799C211yesmaximal, characteristic, complemented
5#11739823C211yesmaximal, characteristic
6#11729358C211yesmaximal, characteristic, complemented
7#11728339C212yesmaximal, complemented
8#11727994C212yesmaximal, complemented
9#11727931C212yesmaximal, complemented
10#11727540C211yesmaximal, characteristic, complemented
11#11727526C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×Q8×Dic7C2212yes—
16C22×#1143C2212yes—
17C2×#6503C2212yes—
18C2×#6493—21no—
19C2×#6493—21no—
20C2×#6437C2211yescharacteristic
21C4×C7×SD16—24no—
22C2×C7×#106C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C2×C4×C7×C8C2211yescharacteristic
26C2×#6406C2212yes—
27#89935—28no—
28C2×#6395C2212yes—
29C2×#6395C2212yes—
30C2×Dic7⋊C8C2212yes—
31#89799—24no—
32#89760C2212yes—
33#89757—22no—
34#89757C2212yes—
35C2×#6383C2212yes—
36#89732—22no—
37#89707—21no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7—21no—
40C4×C4⋊Dic7C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89345C2212yes—
43#89324C2212yes—
44#24728—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C22×C22⋊D7—22no—
49C22×C22⋊D7—22no—
50C22×D28—21no—
51C22×D28—21no—
52C22×D28C2312yes—
53C2×C7×SD16—22no—
54C2×C7×SD16—22no—
55C2×C7×SD16D414yescomplemented
56C22×C7×C8C2311yescharacteristic
57C22×C7×C8C2311yescharacteristic
58C4×C7×Q8—22no—
59C4×C7×Q8D414yescomplemented
60C4×C7×D4—22no—
61C4×C7×D4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C42×C7—21no—
67C2×C42×C7C2311yescharacteristic
68Q8×Dic7—24no—
69Q8×Dic7—42no—
70C2×#1143—24no—
71C2×#1143—24no—
72C2×#1143—24no—
73#6503—24no—
74#6503—42no—
75#6493—44no—
76#6493—44no—
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—22no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7D411yescharacteristic
84C2×C4⋊Dic7D411yescharacteristic
85C2×C4⋊Dic7C2312yes—
86C2×C4×Dic7—22no—
87C2×C4×Dic7C2312yes—
88C22×C7⋊C8C2312yes—
89#6437—22no—
90#6437—22no—
91C7×#106—22no—
92C7×#103—22no—
93C7×#102—22no—
94C4×C7×C8—22no—
95#6406—24no—
96C7⋊#95—22no—
97C7⋊#95D412yes—
98#6395—24no—
99#6395—42no—
100Dic7⋊C8—24no—
101#6383—24no—
102C4⋊(C7⋊C8)—22no—
103C2×#460—72no—
104C2×#460—71no—
105C2×C4×SD16D711yescharacteristic, complemented
106#2541—71no—
107Q8×C4⋊C4—71no—
108#2065—72no—
109#2041—72no—
110#1957—71no—
111#1939—71no—
112#1880—72no—
113#1802—71no—
114C24×C7—21no—
115C23×D7—22no—
116C2×C7×Q8—21no—
117C2×C7×Q8—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8C2×D412yes—
120C2×C7×Q8C2×D412yes—
121C2×C7×Q8#4112yes—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4C2×D412yes—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—42no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—44no—
138C2×C22⋊D7—44no—
139C2×C22⋊D7—44no—
140C2×C22⋊D7—44no—
141C2×C22⋊D7—82no—
142C2×C22⋊D7—82no—
143C22×Dic7—22no—
144C22×Dic7C2×D412yes—
145C22×Dic7C2×D412yes—
146C2×D28—22no—
147C2×D28—22no—
148C2×D28—22no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—42no—
153C2×D28—42no—
154C2×D28—42no—
155C2×D28—42no—
156C2×D28—81no—
157C2×D28—81no—
158C7×SD16—28no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8C2×D412yes—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4C2×D412yes—
170C7×C22⋊C4—42no—
171C42×C7—24no—
172C42×C7C2×D412yes—
173C42×C7#4112yes—
174#1143—48no—
175(C2×D7)⋊C4—44no—
176(C2×D7)⋊C4—44no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—44no—
181C4⋊Dic7—42no—
182C4⋊Dic7—82no—
183C4⋊Dic7—81no—
184C4⋊Dic7SD1614yescomplemented
185C4×Dic7—22no—
186C4×Dic7—22no—
187C4×Dic7—44no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C22×SD16—72no—
192C22×SD16C2×D711yescharacteristic
193C2×#127—72no—
194C2×#127—71no—
195C2×#121—141no—
196C2×#121—141no—
197C2×C4×Q8—72no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×D4C2×D711yescharacteristic
200#460—148no—
201#460—144no—
202C4×SD16—24no—
203C2×#106C2×D711yescharacteristic
204C2×C4⋊C8—72no—
205C2×C4⋊C8—71no—
206C2×#103C2×D711yescharacteristic
207C2×#102—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×#102—72no—
211C2×#102C2×D711yescharacteristic
212C2×C4×C8C2×D711yescharacteristic
213#392—142no—
214#390—72no—
215#389—72no—
216#389—142no—
217#384—71no—
218#384—141no—
219C4×C4⋊C4—71no—
220C4×C4⋊C4—141no—
221#340—72no—
222#336—72no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7C22×D411yescharacteristic
231C22×D7—42no—
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—42no—
238C7×Q8—22no—
239C7×Q8—22no—
240C7×Q8—22no—
241C7×Q8#12114yescomplemented
242C7×D4—22no—
243C7×D4—22no—
244C7×D4—41no—
245C7×D4—41no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—84no—
258C2×C4×C7#12112yes—
259C2×C4×C7#12112yes—
260C2×C4×C7C2×SD1612yes—
261C2×C4×C7AGL(1,ℤ/8)12yes—
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C2×#4111yescharacteristic
265C2×C4×C7C2×#4111yescharacteristic
266C2×C4×C7H(5,𝔽2)11yescharacteristic
267C2×C4×C72−1+411yescharacteristic
268C22⋊D7—88no—
269C22⋊D7—88no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7C2×SD1614yes—
279D28—42no—
280D28—42no—
281D28—42no—
282D28—42no—
283D28—84no—
284D28—84no—
285C7×C8—22no—
286C7×C8—41no—
287C7⋊C8—42no—
288C22×Q8C22×D711yescharacteristic
289C22×D4—72no—
290C22×D4—142no—
291C22×D4—142no—
292C22×D4—141no—
293C22×D4—141no—
294C22×D4C22×D711yescharacteristic
295C23×C4—21no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16—144no—
299C2×SD16—144no—
300C2×SD16—144no—
301C2×SD16D2814yescomplemented
302C22×C8—72no—
303C22×C8C22×D711yescharacteristic
304C22×C8C22×D711yescharacteristic
305#127—144no—
306#127—142no—
307#127—142no—
308#127—282no—
309#121—284no—
310#121—284no—
311C4×Q8—22no—
312C4×Q8—144no—
313C4×Q8—282no—
314C4×Q8C22⋊D714yescomplemented
315C4×D4—22no—
316C4×D4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4C22×D711yescharacteristic
326C2×C4⋊C4C22×D711yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—142no—
329C2×C22⋊C4—142no—
330C2×C42—21no—
331C2×C42—72no—
332C2×C42—142no—
333C2×C42C22×D711yescharacteristic
334#106—22no—
335C4⋊C8—144no—
336C4⋊C8—142no—
337#103—22no—
338#102—22no—
339#102—144no—
340#102—144no—
341#102—144no—
342#102—282no—
343C4×C8—22no—
344#95—72no—
345#95—142no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7C2×#12111yescharacteristic
354C22×C7D4211yescharacteristic
355C22×C7#54811yescharacteristic
356C22×C7C22×SD1611yescharacteristic
357C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C7#57611yescharacteristic
359C22×C7#57711yescharacteristic
360C2×D7—82no—
361C2×D7—82no—
362C2×D7—82no—
363C2×D7—82no—
364C2×D7—82no—
365C2×D7—82no—
366C2×D7—82no—
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—42no—
370C4×C7—81no—
371C4×C7#46014yescomplemented
372C4×C7C2×#12111yescharacteristic
373C4×C7#53711yescharacteristic
374C4×C7#54711yescharacteristic
375C4×C7#55311yescharacteristic
376Dic7—24no—
377Dic7—42no—
378Dic7—82no—
379C24—21no—
380C24—142no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8C2×D2812yes—
385C2×Q8#115712yes—
386C2×Q8C2×C22⋊D712yes—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—142no—
393C2×D4—142no—
394C2×D4—142no—
395C2×D4—142no—
396C2×D4—142no—
397C2×D4—142no—
398C2×D4—284no—
399C2×D4—284no—
400C2×D4—284no—
401C2×D4—284no—
402C2×D4—282no—
403C2×D4—282no—
404C2×D4—282no—
405C2×D4—282no—
406C2×D4—561no—
407C2×D4—561no—
408C2×D4—562no—
409C2×D4—562no—
410C2×D4C2×D2812yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—142no—
420C22×C4D4×D711yescharacteristic
421C22×C4D4×D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423SD16—28no—
424SD16—288no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—142no—
431C2×C8—142no—
432C2×C8—142no—
433C2×C8C2×D2812yes—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—74no—
439C4⋊C4—142no—
440C4⋊C4—142no—
441C4⋊C4—142no—
442C4⋊C4—284no—
443C4⋊C4—282no—
444C4⋊C4—561no—
445C4⋊C4—562no—
446C4⋊C4C2×C22⋊D712yes—
447C22⋊C4—42no—
448C22⋊C4—284no—
449C22⋊C4—284no—
450C42—24no—
451C42—142no—
452C42—142no—
453C42—284no—
454C42#115912yes—
455C42C2×C22⋊D712yes—
456C2×C7—82no—
457C2×C7#241611yescharacteristic
458C2×C7C2×#46011yescharacteristic
459C2×C7#304511yescharacteristic
460C2×C7#327411yescharacteristic
461C2×C7D4×SD1611yescharacteristic
462C2×C7#336211yescharacteristic
463C2×C7#336811yescharacteristic
464D7—162no—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—282no—
473C23—282no—
474C23—282no—
475C23—282no—
476C23—282no—
477C23—282no—
478C23—282no—
479C23C2×D4×D711yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—22no—
483Q8#649314yescomplemented
484D4—22no—
485D4—22no—
486D4—41no—
487D4—41no—
488D4—282no—
489D4—282no—
490D4—282no—
491D4—282no—
492D4—564no—
493D4—568no—
494D4—568no—
495D4—564no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—74no—
508C2×C4—84no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—282no—
517C2×C4D7×SD1612yes—
518C2×C4#647712yes—
519C2×C4#650212yes—
520C2×C4#650212yes—
521C2×C4C22×D2811yescharacteristic
522C2×C4C2×#115711yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4#655311yescharacteristic
525C2×C4#655411yescharacteristic
526C2×C4C22×C22⋊D711yescharacteristic
527C8—22no—
528C8—41no—
529C8—282no—
530C7#2472811yescharacteristic, complemented, Sylow
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—562no—
539C22—562no—
540C22—562no—
541C22—562no—
542C22—562no—
543C22—562no—
544C22—562no—
545C22D4×D2811yescharacteristic
546C22#9024311yescharacteristic
547C22C2×D7×SD1611yescharacteristic
548C22C2×#647711yescharacteristic
549C22#9046011yescharacteristic
550C22#9046111yescharacteristic
551C22C2×#650211yescharacteristic
552C4—24no—
553C4—22no—
554C4—42no—
555C4—81no—
556C4—144no—
557C4—282no—
558C4—562no—
559C4#8993514yescomplemented
560C4#9024611yescharacteristic
561C4#9025611yescharacteristic
562C4C2×#649311yescharacteristic
563C4#9052411yescharacteristic
564C2—82no—
565C2—1122no—
566C2#1173706711yescharacteristic
567C2#1173707011yescharacteristic
568C2#1173711511yescharacteristic
569C2SD16×D2811yescharacteristic
570C2#1173979711yescharacteristic
571C2C2×#8993511yescharacteristic
572C2#1174134611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radical#11739823C2
hypercenterC2×C4×SD16D7
hypocenterC7#24728
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×SD16C2semidirect
2. C2×#89935C2semidirect
3. Q8×C4⋊Dic7C2semidirect
4. C2×#89799C2semidirect
6. #11729358C2semidirect
7. #11728339C2semidirect
8. #11727994C2semidirect
9. #11727931C2semidirect
10. #11727540C2semidirect
11. #11727526C2semidirect
55. C2×C7×SD16D4semidirect
59. C4×C7×Q8D4semidirect
105. C2×C4×SD16D7semidirect
184. C4⋊Dic7SD16semidirect
241. C7×Q8#121semidirect
301. C2×SD16D28semidirect
314. C4×Q8C22⋊D7semidirect
371. C4×C7#460semidirect
483. Q8#6493semidirect
530. C7#24728semidirect
559. C4#89935semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×SD162
C2×#899352
Q8×C4⋊Dic72
C2×#897992
#117398232
#117293582
#117283392
#117279942
#117279312
#117275402
#117275262
#247287