Finite Group Databasebeta

← #13380896

Subgroups — #13380896

Order 1792.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic, complemented
2C22×#6512C211yesmaximal, characteristic
3C23×#1143C211yesmaximal, characteristic, complemented
4C2×#90005C211yesmaximal, characteristic, complemented
5C2×#89997C211yesmaximal, characteristic, complemented
6#11739714C211yesmaximal, characteristic, complemented
7#11729545C214yesmaximal, complemented
8#11729482C214yesmaximal, complemented
9#11729333C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C23×D28C2211yescharacteristic
13C7×#536—24no—
14C7×#536—24no—
15C2×C7×C4⋊Q8C2212yescomplemented
16C2×C7×#124C2212yescomplemented
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8—22no—
19C2×C7×#117C2211yescharacteristic
20C2×#6512—24no—
21C2×#6512—24no—
22C2×#6512—24no—
23C22×#1143—24no—
24C22×#1143—24no—
25C22×#1143—24no—
26C22×#1143C2214yes—
27C22×#1143C2212yes—
28C22×#1141C2212yes—
29C2×#6489C2211yescharacteristic
30C22×#1136C2211yescharacteristic
31C23×C7⋊C8C2211yescharacteristic
32#90005—24no—
33#89997—24no—
34C2×#6409C2214yescomplemented
35C2×#6409C2212yescomplemented
36C2×#6406C2212yescomplemented
37#89760—22no—
38#89727—22no—
39#89711C2214yes—
40#89319C2214yescomplemented
41#23424—71nomaximal, Sylow
42C24×D7—21no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41D414yes—
49C22×C7×Q8—21no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8—22no—
55C22×C7×Q8D414yes—
56C22×C7×Q8C2312yes—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4—21no—
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C22×D28—22no—
64C22×D28—22no—
65C22×D28—22no—
66C22×D28—22no—
67C22×D28—21no—
68C22×D28—22no—
69C22×D28D414yes—
70C22×D28C2312yes—
71C7×C4⋊Q8—24no—
72C7×C4⋊Q8—24no—
73C7×#124—24no—
74C7×#124—24no—
75C7×#123—216no—
76C7×C22⋊Q8—216no—
77C7×#117—22no—
78C7×#117—22no—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4D412yes—
81C2×C7×C22⋊C4—24no—
82C2×C7×C22⋊C4D412yes—
83C2×C42×C7C2312yes—
84C2×C22⋊Dic7—21no—
85#6512—232no—
86C2×#1143—232no—
87C2×#1143—28no—
88C2×#1143—24no—
89C2×#1143—28no—
90C2×#1143—24no—
91C2×#1143—28no—
92C2×#1143—24no—
93C2×#1141—24no—
94C2×#1141—24no—
95C2×#1141—24no—
96C2×(C2×D7)⋊C4—28no—
97#6489—22no—
98#6489—22no—
99C2×C4⋊Dic7C2312yes—
100C2×C4×Dic7—21no—
101C2×#1136—22no—
102C2×#1136—22no—
103C2×#1136D414yes—
104C22×C7⋊C8—22no—
105C22×C7⋊C8—22no—
106C22×C7⋊C8D414yes—
107C22×C7⋊C8C2312yes—
108C22×C7⋊C8C2312yes—
109#6409—28no—
110#6409—24no—
111#6406—24no—
112C22×AGL(1,ℤ/8)—71no—
113C23×SD16—71no—
114C2×#536D711yescharacteristic, complemented
115C2×#419—71no—
116C2×#416—71no—
117#2309—71no—
118#2013—74no—
119#2012—74no—
120#1729—71no—
121C24×C7—21no—
122C24×C7C2×D411yescharacteristic
123C23×D7—22no—
124C23×D7—24no—
125C23×D7—42no—
126C23×D7—42no—
127C23×D7—84no—
128C7×#41—28no—
129C7×#41—28no—
130C7×#41—44no—
131C7×#41—44no—
132C2×C7×Q8—24no—
133C2×C7×Q8—22no—
134C2×C7×Q8—21no—
135C2×C7×Q8—28no—
136C2×C7×Q8—28no—
137C2×C7×Q8—22no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—22no—
144C2×C7×Q8—21no—
145C2×C7×Q8—44no—
146C2×C7×Q8—42no—
147C2×C7×Q8—44no—
148C2×C7×Q8—42no—
149C2×C7×Q8C2×D414yes—
150C2×C7×Q8C2×D412yes—
151C2×C7×D4—24no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4—21no—
157C2×C7×D4—42no—
158C2×C7×D4—42no—
159C2×C7×D4C2×D412yes—
160C22×C4×C7—24no—
161C22×C4×C7—22no—
162C22×C4×C7—21no—
163C22×C4×C7—24no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—42no—
167C22×C4×C7—42no—
168C22×C4×C7C2×D414yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D412yes—
175C22×C4×C7#4112yes—
176C22×C4×C7C2411yescharacteristic
177C22×Dic7—22no—
178C2×D28—28no—
179C2×D28—28no—
180C2×D28—22no—
181C2×D28—22no—
182C2×D28—22no—
183C2×D28—22no—
184C2×D28—24no—
185C2×D28—22no—
186C2×D28—42no—
187C2×D28—44no—
188C2×D28—44no—
189C2×D28—42no—
190C2×D28C2×D414yes—
191C7×C4⋊C4—216no—
192C7×C4⋊C4—24no—
193C7×C22⋊C4—216no—
194C7×C22⋊C4—24no—
195C42×C7—22no—
196C42×C7—22no—
197C22⋊Dic7—42no—
198#1143—232no—
199#1143—216no—
200#1141—216no—
201(C2×D7)⋊C4—88no—
202C4⋊Dic7—22no—
203C4⋊Dic7—22no—
204C4×Dic7—42no—
205#1136—28no—
206C2×C7⋊C8—28no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8C2×D414yes—
212C2×C7⋊C8C2×D414yes—
213C22×#41C2×D711yescharacteristic
214C23×Q8C2×D711yescharacteristic
215C23×D4—71no—
216C2×AGL(1,ℤ/8)—144no—
217C2×AGL(1,ℤ/8)—144no—
218C2×AGL(1,ℤ/8)—144no—
219C22×SD16—74no—
220C22×SD16—72no—
221C22×SD16—144no—
222C22×SD16—144no—
223C22×SD16—144no—
224C22×D8—72no—
225C22×M16—71no—
226C23×C8—71no—
227#536—24no—
228#536—24no—
229C2×C4⋊Q8C2×D712yescomplemented
230C2×#124C2×D712yescomplemented
231C2×#123—22no—
232C2×C22⋊Q8—22no—
233C2×#117—71no—
234C2×#117C2×D711yescharacteristic
235#419—144no—
236#416—144no—
237C2×#103—72no—
238C2×#103—74no—
239C2×#102—72no—
240#390—74no—
241#390—142no—
242#379—142no—
243#336—74no—
244C23×C7—24no—
245C23×C7—22no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—44no—
249C23×C7#12012yes—
250C23×C7#12112yes—
251C23×C7#12712yes—
252C23×C7C22×D411yescharacteristic
253C22×D7—24no—
254C22×D7—42no—
255C22×D7—42no—
256C22×D7—48no—
257C22×D7—48no—
258C22×D7—88no—
259C22×D7—88no—
260C22×D7—88no—
261C7×Q8—24no—
262C7×Q8—22no—
263C7×Q8—24no—
264C7×Q8—22no—
265C7×Q8—42no—
266C7×Q8—41no—
267C7×Q8—42no—
268C7×Q8—41no—
269C7×Q8—48no—
270C7×D4—22no—
271C7×D4—22no—
272C7×D4—48no—
273C7×D4—41no—
274C7×D4—41no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—28no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—48no—
284C2×C4×C7—48no—
285C2×C4×C7#12014yes—
286C2×C4×C7#12014yes—
287C2×C4×C7#12012yes—
288C2×C4×C7#12112yes—
289C2×C4×C7#12114yes—
290C2×C4×C7#12114yes—
291C2×C4×C7#12712yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×Dic7—42no—
299C2×Dic7—44no—
300D28—24no—
301D28—24no—
302D28—42no—
303D28—42no—
304D28—48no—
305C7⋊C8—24no—
306C7⋊C8—24no—
307C25—141no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41C22⋊D714yes—
314C22×Q8—21no—
315C22×Q8—22no—
316C22×Q8—22no—
317C22×Q8—22no—
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8C22⋊D714yes—
321C22×Q8C22×D712yes—
322C22×Q8C22×D711yescharacteristic
323C22×D4—21no—
324C22×D4—74no—
325C22×D4—72no—
326C22×D4—142no—
327C22×D4—142no—
328C22×D4—142no—
329C22×D4—142no—
330C22×D4—142no—
331C22×D4—141no—
332C22×D4C22×D711yescharacteristic
333C23×C4—21no—
334C23×C4—21no—
335C23×C4C22×D711yescharacteristic
336AGL(1,ℤ/8)—1432no—
337C2×SD16—148no—
338C2×SD16—144no—
339C2×SD16—148no—
340C2×SD16—144no—
341C2×SD16—1432no—
342C2×SD16—148no—
343C2×SD16—144no—
344C2×D8—144no—
345C2×D8—144no—
346C2×D8—144no—
347C2×M16—74no—
348C2×M16—142no—
349C2×M16—142no—
350C22×C8—74no—
351C22×C8—72no—
352C22×C8—72no—
353C22×C8—142no—
354C22×C8—142no—
355C4⋊Q8—24no—
356C4⋊Q8—24no—
357#124—24no—
358#124—24no—
359#123—216no—
360C22⋊Q8—216no—
361#117—22no—
362#117—22no—
363#117—142no—
364#117—142no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—72no—
367C2×C4⋊C4C22⋊D712yes—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—148no—
371C2×C22⋊C4C22⋊D712yes—
372C2×C42—141no—
373C2×C42C22×D712yes—
374#103—144no—
375#103—148no—
376#102—144no—
377C22×C7—24no—
378C22×C7—24no—
379C22×C7—24no—
380C22×C7—42no—
381C22×C7—44no—
382C22×C7—82no—
383C22×C7—82no—
384C22×C7#39214yes—
385C22×C7C2×#12011yescharacteristic
386C22×C7C2×#12111yescharacteristic
387C22×C7C2×#12711yescharacteristic
388C22×C7#57714yes—
389C2×D7—44no—
390C2×D7—48no—
391C2×D7—816no—
392C2×D7—84no—
393C2×D7—84no—
394C4×C7—42no—
395C4×C7—44no—
396C4×C7—44no—
397C4×C7#39214yes—
398C4×C7C2×#12011yescharacteristic
399C4×C7C2×#12011yescharacteristic
400C4×C7C2×#12111yescharacteristic
401C4×C7C2×#12111yescharacteristic
402Dic7—82no—
403C24—21no—
404C24—142no—
405C24—144no—
406C24—282no—
407C24—282no—
408C24—564no—
409C24C2×C22⋊D711yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—42no—
431C2×Q8C2×C22⋊D714yes—
432C2×Q8C2×C22⋊D712yes—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—74no—
442C2×D4—148no—
443C2×D4—148no—
444C2×D4—144no—
445C2×D4—142no—
446C2×D4—142no—
447C2×D4—142no—
448C2×D4—142no—
449C2×D4—142no—
450C2×D4—284no—
451C2×D4—284no—
452C2×D4—282no—
453C2×D4—282no—
454C2×D4C2×C22⋊D712yes—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—142no—
464C22×C4D4×D712yes—
465C22×C4D4×D712yes—
466C22×C4D4×D712yes—
467C22×C4#115912yes—
468C22×C4C2×C22⋊D714yes—
469C22×C4C2×C22⋊D712yes—
470C22×C4C2×C22⋊D712yes—
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C23×D711yescharacteristic
473SD16—1432no—
474SD16—1416no—
475D8—1416no—
476M16—148no—
477C2×C8—74no—
478C2×C8—74no—
479C2×C8—148no—
480C2×C8—142no—
481C2×C8—142no—
482C2×C8—142no—
483C2×C8—142no—
484C4⋊C4—216no—
485C4⋊C4—24no—
486C4⋊C4—142no—
487C4⋊C4—142no—
488C22⋊C4—216no—
489C22⋊C4—24no—
490C22⋊C4—282no—
491C22⋊C4—568no—
492C42—22no—
493C42—22no—
494C42—282no—
495C2×C7—24no—
496C2×C7—82no—
497C2×C7C2×#39211yescharacteristic
498C2×C7#301012yes—
499C2×C7#304212yes—
500C2×C7#315412yes—
501D7—84no—
502C23—24no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—44no—
507C23—144no—
508C23—282no—
509C23—288no—
510C23—288no—
511C23—282no—
512C23—568no—
513C23—568no—
514C23—568no—
515C23#650112yes—
516C23#650312yes—
517C23#651612yes—
518C23C22×C22⋊D711yescharacteristic
519Q8—24no—
520Q8—22no—
521Q8—24no—
522Q8—22no—
523Q8—42no—
524Q8—41no—
525Q8—42no—
526Q8—41no—
527Q8—48no—
528D4—22no—
529D4—22no—
530D4—48no—
531D4—41no—
532D4—41no—
533D4—144no—
534D4—144no—
535D4—282no—
536D4—282no—
537D4—288no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—282no—
549C2×C4—284no—
550C2×C4#650014yes—
551C2×C4#650014yes—
552C2×C4#650112yes—
553C2×C4#650214yes—
554C2×C4#650214yes—
555C2×C4#650312yes—
556C2×C4#651612yes—
557C2×C4C2×D4×D711yescharacteristic
558C2×C4C2×D4×D711yescharacteristic
559C2×C4C2×D4×D711yescharacteristic
560C2×C4C2×#115911yescharacteristic
561C2×C4C22×C22⋊D711yescharacteristic
562C2×C4C22×C22⋊D711yescharacteristic
563C8—144no—
564C8—144no—
565C7#2342411yescharacteristic, complemented, Sylow
566C22—24no—
567C22—24no—
568C22—24no—
569C22—42no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—284no—
574C22—288no—
575C22—5616no—
576C22—564no—
577C22—564no—
578C22#8999514yes—
579C22C2×#650111yescharacteristic
580C22C2×#650311yescharacteristic
581C22#9052814yes—
582C22C2×#651611yescharacteristic
583C4—42no—
584C4—44no—
585C4—44no—
586C4—562no—
587C4#8999514yes—
588C4C2×#650011yescharacteristic
589C4C2×#650011yescharacteristic
590C4C2×#650211yescharacteristic
591C4C2×#650211yescharacteristic
592C2—24no—
593C2—82no—
594C2—564no—
595C2#1174084712yes—
596C2#1174085012yes—
597C2C2×#8999511yescharacteristic
598C2#1174155812yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#536C2
socleC23×C7C22×D4
prime radicalC22×#6512C2
hypercenterC2×#536D7
hypocenterC7#23424
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C7×#536C2semidirect
3. C23×#1143C2semidirect
4. C2×#90005C2semidirect
5. C2×#89997C2semidirect
6. #11739714C2semidirect
7. #11729545C2semidirect
8. #11729482C2semidirect
9. #11729333C2semidirect
15. C2×C7×C4⋊Q8C22semidirect
16. C2×C7×#124C22semidirect
34. C2×#6409C22semidirect
35. C2×#6409C22semidirect
36. C2×#6406C22semidirect
40. #89319C22semidirect
114. C2×#536D7semidirect
229. C2×C4⋊Q8C2×D7semidirect
230. C2×#124C2×D7semidirect
565. C7#23424semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C22×#65122
C23×#11432
C2×#900052
C2×#899972
#117397142
#117295452
#117294822
#117293332
#234247