Finite Group Databasebeta

← #13380958

Subgroups — #13380958

Order 1792.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#124C211yesmaximal, characteristic, complemented
2C22×#6505C211yesmaximal, characteristic, complemented
3C22×#6495C211yesmaximal, characteristic
4C2×#89997C211yesmaximal, characteristic, complemented
5C2×#89987C211yesmaximal, characteristic, complemented
6#11739515C211yesmaximal, characteristic, complemented
7#11729483C214yesmaximal, complemented
8#11729482C214yesmaximal, complemented
9#11729281C211yesmaximal, characteristic, complemented
10C23×C7×Q8C2211yescharacteristic
11C23×C7×D4C2211yescharacteristic
12C22×#1157C2211yescharacteristic
13C2×C7×#124—24no—
14C2×C7×#124—24no—
15C2×C7×#124—22no—
16C2×C7×#124C2214yes—
17C22×C7×C22⋊C4—22no—
18C22×C42×C7C2211yescharacteristic
19C22×#1144C2212yescomplemented
20C2×#6505—24no—
21C2×#6505—24no—
22C2×#6505—24no—
23C22×#1143C2212yescomplemented
24C22×#1142C2212yes—
25C2×#6495—24no—
26C2×#6495—24no—
27C2×#6495—24no—
28C22×#1141C2212yes—
29C2×#6489C2211yescharacteristic
30C22×#1136C2211yescharacteristic
31C22×#1136C2211yescharacteristic
32#89997—24no—
33C2×#6409C2214yescomplemented
34#89987—24no—
35C2×#6406C2214yes—
36#89985—22no—
37#89711C2212yescomplemented
38C4×(C2×D7)⋊C4—22no—
39#89701C2212yescomplemented
40#89319C2214yescomplemented
41#23431—71nomaximal, Sylow
42C25×C7—21no—
43C22×C7×Q8—22no—
44C22×C7×Q8—21no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8—22no—
48C22×C7×Q8—22no—
49C22×C7×Q8D414yes—
50C22×C7×Q8C2312yes—
51C22×C7×D4—22no—
52C22×C7×D4—22no—
53C22×C7×D4—22no—
54C22×C7×D4—22no—
55C22×C7×D4—21no—
56C22×C7×D4—22no—
57C22×C7×D4D414yes—
58C22×C7×D4C2312yes—
59C23×C4×C7—21no—
60C23×C4×C7D412yes—
61C23×C4×C7C2311yescharacteristic
62C22×C22⋊D7—21no—
63C2×#1157—22no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157—22no—
67C2×#1157—22no—
68C2×#1157D414yes—
69C22×D28C2311yescharacteristic
70C22×C4×D7—21no—
71C22×Dic14C2311yescharacteristic
72C7×#124—216no—
73C7×#124—28no—
74C7×#124—28no—
75C2×C7×C22⋊C4—216no—
76C2×C7×C22⋊C4—28no—
77C2×C42×C7—22no—
78C2×C42×C7—22no—
79C2×C42×C7D412yes—
80C2×C42×C7C2312yes—
81C2×C22⋊Dic7—21no—
82C2×#1144—24no—
83C2×#1144—24no—
84C2×#1144—24no—
85#6505—232no—
86C2×#1143—24no—
87C2×#1143—24no—
88C2×#1143—24no—
89C2×#1142—24no—
90C2×#1142—24no—
91C2×#1142—24no—
92#6495—232no—
93C2×#1141—24no—
94C2×#1141—24no—
95C2×#1141—24no—
96C2×(C2×D7)⋊C4—24no—
97#6489—22no—
98#6489—22no—
99C2×C4⋊Dic7C2312yes—
100C2×C4×Dic7—21no—
101C2×#1136—22no—
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136—22no—
105C2×#1136D414yes—
106C2×#1136D414yes—
107C22×C7⋊C8C2312yes—
108C22×C7⋊C8C2312yes—
109#6409—28no—
110#6406—28no—
111C7⋊#95—24no—
112C22×#137—71no—
113C22×AGL(1,ℤ/8)—71no—
114C22×#124D711yescharacteristic, complemented
115C2×#419—71no—
116C2×#418—71no—
117#2297—71no—
118#2015—74no—
119#2012—74no—
120#1728—71no—
121C24×C7—24no—
122C24×C7—22no—
123C24×C7—48no—
124C24×C7—42no—
125C24×C7—42no—
126C24×C7C2×D411yescharacteristic
127C23×D7—21no—
128C2×C7×Q8—24no—
129C2×C7×Q8—22no—
130C2×C7×Q8—28no—
131C2×C7×Q8—28no—
132C2×C7×Q8—22no—
133C2×C7×Q8—22no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8—44no—
137C2×C7×Q8—42no—
138C2×C7×Q8—44no—
139C2×C7×Q8—42no—
140C2×C7×Q8C2×D414yes—
141C2×C7×D4—28no—
142C2×C7×D4—24no—
143C2×C7×D4—28no—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—22no—
149C2×C7×D4—44no—
150C2×C7×D4—42no—
151C2×C7×D4—44no—
152C2×C7×D4—42no—
153C2×C7×D4C2×D414yes—
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—44no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7C2×D414yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7#4112yes—
168C22×C4×C7C2411yescharacteristic
169C2×C22⋊D7—24no—
170C2×C22⋊D7—42no—
171C2×C22⋊D7—42no—
172C22×Dic7—21no—
173C22×Dic7—22no—
174#1157—28no—
175#1157—28no—
176#1157—44no—
177#1157—44no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28C2×D412yes—
184C2×C4×D7—24no—
185C2×C4×D7—42no—
186C2×C4×D7—42no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—21no—
191C2×Dic14—21no—
192C2×Dic14C2×D412yes—
193C7×C22⋊C4—232no—
194C42×C7—24no—
195C42×C7—22no—
196C42×C7—22no—
197C22⋊Dic7—42no—
198#1144—216no—
199#1143—216no—
200#1142—216no—
201#1141—216no—
202(C2×D7)⋊C4—48no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4×Dic7—42no—
206#1136—28no—
207#1136—28no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8C2×D414yes—
213C2×C7⋊C8C2×D414yes—
214C22×#41—71no—
215C23×Q8C2×D711yescharacteristic
216C23×D4C2×D711yescharacteristic
217C2×#137—144no—
218C2×#137—144no—
219C2×#137—144no—
220C2×AGL(1,ℤ/8)—144no—
221C2×AGL(1,ℤ/8)—144no—
222C2×AGL(1,ℤ/8)—144no—
223C22×Q16—72no—
224C22×SD16—72no—
225C22×SD16—72no—
226C22×D8—72no—
227C22×M16—71no—
228C22×M16—71no—
229C2×#124—24no—
230C2×#124—24no—
231C2×#124—22no—
232C2×#124C2×D714yes—
233C2×#117—71no—
234C22×C22⋊C4—22no—
235C22×C42C2×D711yescharacteristic
236#419—144no—
237#418—144no—
238C2×#103—74no—
239C2×#102—74no—
240#391—72no—
241#390—72no—
242#386—142no—
243C4×C22⋊C4—142no—
244#336—74no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—416no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—48no—
251C23×C7—48no—
252C23×C7—88no—
253C23×C7—88no—
254C23×C7#12012yes—
255C23×C7#12112yes—
256C23×C7#12712yes—
257C23×C7C22×D411yescharacteristic
258C22×D7—22no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—44no—
262C7×Q8—24no—
263C7×Q8—24no—
264C7×Q8—42no—
265C7×Q8—42no—
266C7×Q8—48no—
267C7×D4—24no—
268C7×D4—24no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—48no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—48no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—48no—
280C2×C4×C7#12014yes—
281C2×C4×C7#12014yes—
282C2×C4×C7#12012yes—
283C2×C4×C7#12112yes—
284C2×C4×C7#12114yes—
285C2×C4×C7#12114yes—
286C2×C4×C7#12712yes—
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C22⋊D7—48no—
294C2×Dic7—22no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—42no—
298C2×Dic7—44no—
299D28—22no—
300D28—22no—
301D28—41no—
302D28—41no—
303C4×D7—48no—
304Dic14—22no—
305Dic14—22no—
306Dic14—41no—
307Dic14—41no—
308C7⋊C8—24no—
309C7⋊C8—24no—
310C25—21no—
311C2×#41—74no—
312C2×#41—142no—
313C2×#41—142no—
314C2×#41—142no—
315C2×#41—142no—
316C2×#41—142no—
317C22×Q8—22no—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—71no—
324C22×Q8C22⋊D714yes—
325C22×Q8C22×D712yes—
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—21no—
332C22×D4—71no—
333C22×D4—141no—
334C22×D4C22⋊D714yes—
335C22×D4C22×D712yes—
336C23×C4—21no—
337C23×C4—141no—
338C23×C4C22⋊D712yes—
339C23×C4C22×D711yescharacteristic
340#137—1432no—
341AGL(1,ℤ/8)—1432no—
342C2×Q16—144no—
343C2×Q16—144no—
344C2×Q16—144no—
345C2×SD16—144no—
346C2×SD16—144no—
347C2×SD16—144no—
348C2×SD16—144no—
349C2×SD16—144no—
350C2×SD16—144no—
351C2×D8—144no—
352C2×D8—144no—
353C2×D8—144no—
354C2×M16—74no—
355C2×M16—74no—
356C2×M16—142no—
357C2×M16—142no—
358C2×M16—142no—
359C2×M16—142no—
360C22×C8—72no—
361C22×C8—72no—
362#124—216no—
363#124—28no—
364#124—28no—
365#117—142no—
366#117—142no—
367C2×C4⋊C4—72no—
368C2×C22⋊C4—216no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4—144no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—141no—
375C2×C42C22⋊D712yes—
376C2×C42C22×D712yes—
377#103—148no—
378#102—148no—
379#95—144no—
380C22×C7—24no—
381C22×C7—24no—
382C22×C7—24no—
383C22×C7—44no—
384C22×C7—48no—
385C22×C7—816no—
386C22×C7—84no—
387C22×C7—84no—
388C22×C7#39214yes—
389C22×C7C2×#12011yescharacteristic
390C22×C7C2×#12111yescharacteristic
391C22×C7C2×#12711yescharacteristic
392C22×C7#57514yes—
393C2×D7—42no—
394C2×D7—44no—
395C2×D7—82no—
396C2×D7—82no—
397C4×C7—44no—
398C4×C7—44no—
399C4×C7#39214yes—
400C4×C7C2×#12011yescharacteristic
401C4×C7C2×#12011yescharacteristic
402C4×C7C2×#12111yescharacteristic
403C4×C7C2×#12111yescharacteristic
404Dic7—42no—
405Dic7—82no—
406C24—24no—
407C24—22no—
408C24—48no—
409C24—42no—
410C24—42no—
411C24—141no—
412C24C2×C22⋊D711yescharacteristic
413#41—148no—
414#41—148no—
415#41—284no—
416#41—284no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—72no—
430C2×Q8—141no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8C2×C22⋊D714yes—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—72no—
449C2×D4—144no—
450C2×D4—141no—
451C2×D4—141no—
452C2×D4—141no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4—282no—
456C2×D4—282no—
457C2×D4C2×C22⋊D714yes—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—141no—
465C22×C4—144no—
466C22×C4—142no—
467C22×C4—282no—
468C22×C4—282no—
469C22×C4D4×D712yes—
470C22×C4D4×D712yes—
471C22×C4D4×D712yes—
472C22×C4#115912yes—
473C22×C4C2×C22⋊D714yes—
474C22×C4C2×C22⋊D712yes—
475C22×C4C2×C22⋊D712yes—
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C23×D711yescharacteristic
478Q16—1416no—
479SD16—1416no—
480SD16—1416no—
481D8—1416no—
482M16—148no—
483M16—148no—
484C2×C8—74no—
485C2×C8—74no—
486C2×C8—142no—
487C2×C8—142no—
488C2×C8—142no—
489C2×C8—142no—
490C4⋊C4—142no—
491C4⋊C4—142no—
492C22⋊C4—232no—
493C22⋊C4—288no—
494C22⋊C4—282no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—282no—
499C2×C7—24no—
500C2×C7—84no—
501C2×C7C2×#39211yescharacteristic
502C2×C7#300512yes—
503C2×C7#304012yes—
504C2×C7#315312yes—
505D7—82no—
506C23—24no—
507C23—24no—
508C23—416no—
509C23—42no—
510C23—42no—
511C23—48no—
512C23—48no—
513C23—88no—
514C23—88no—
515C23—142no—
516C23—281no—
517C23—284no—
518C23—281no—
519C23#650112yes—
520C23#650312yes—
521C23#651612yes—
522C23C22×C22⋊D711yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—42no—
526Q8—42no—
527Q8—48no—
528Q8—142no—
529Q8—142no—
530Q8—281no—
531Q8—281no—
532D4—24no—
533D4—24no—
534D4—42no—
535D4—42no—
536D4—48no—
537D4—142no—
538D4—142no—
539D4—288no—
540D4—281no—
541D4—281no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—142no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—288no—
554C2×C4—282no—
555C2×C4—284no—
556C2×C4#650014yes—
557C2×C4#650014yes—
558C2×C4#650112yes—
559C2×C4#650214yes—
560C2×C4#650214yes—
561C2×C4#650312yes—
562C2×C4#651612yes—
563C2×C4C2×D4×D711yescharacteristic
564C2×C4C2×D4×D711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×#115911yescharacteristic
567C2×C4C22×C22⋊D711yescharacteristic
568C2×C4C22×C22⋊D711yescharacteristic
569C8—144no—
570C8—144no—
571C7#2343111yescharacteristic, complemented, Sylow
572C22—24no—
573C22—24no—
574C22—24no—
575C22—44no—
576C22—48no—
577C22—816no—
578C22—84no—
579C22—84no—
580C22—282no—
581C22—284no—
582C22—562no—
583C22—562no—
584C22#8999514yes—
585C22C2×#650111yescharacteristic
586C22C2×#650311yescharacteristic
587C22#9051314yes—
588C22C2×#651611yescharacteristic
589C4—44no—
590C4—44no—
591C4—282no—
592C4—562no—
593C4#8999514yes—
594C4C2×#650011yescharacteristic
595C4C2×#650011yescharacteristic
596C4C2×#650211yescharacteristic
597C4C2×#650211yescharacteristic
598C2—24no—
599C2—84no—
600C2—562no—
601C2#1174038812yes—
602C2#1174039012yes—
603C2C2×#8999511yescharacteristic
604C2#1174152612yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#124C2
socleC23×C7C22×D4
prime radicalC22×#6495C2
hypercenterC22×#124D7
hypocenterC7#23431
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. C22×C7×#124C2semidirect
2. C22×#6505C2semidirect
4. C2×#89997C2semidirect
5. C2×#89987C2semidirect
6. #11739515C2semidirect
7. #11729483C2semidirect
8. #11729482C2semidirect
9. #11729281C2semidirect
19. C22×#1144C22semidirect
23. C22×#1143C22semidirect
33. C2×#6409C22semidirect
37. #89711C22semidirect
39. #89701C22semidirect
40. #89319C22semidirect
114. C22×#124D7semidirect
571. C7#23431semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1242
C22×#65052
C22×#64952
C2×#899972
C2×#899872
#117395152
#117294832
#117294822
#117292812
#234317