Finite Group Databasebeta

← #13404520

Subgroups — #13404520

Order 1792.

1238 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2541C211yesmaximal, characteristic, complemented
2C2×#89844C211yesmaximal, characteristic, complemented
3C2×#89810C212yesmaximal, complemented
4#11739862C211yesmaximal, characteristic, complemented
5C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729438C212yesmaximal, complemented
8#11729435C212yesmaximal, complemented
9#11729433C212yesmaximal, complemented
10C4×#6383C211yesmaximal, characteristic, complemented
11#11727220C211yesmaximal, characteristic, complemented
12C2×C7×#127—21no—
13C2×C7×#127C2211yescharacteristic
14C2×C7×#121C2212yes—
15C2×C7×#121C2212yes—
16C2×C4×C7×D4C2211yescharacteristic
17C2×#6501—21no—
18C22×#1141C2212yes—
19C2×#6493C2212yes—
20C23×C7⋊C8C2212yes—
21C2×C4×D28C2211yescharacteristic
22C7×#392—22no—
23C7×#390—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30#89844—24no—
31#89810—28no—
32D4×C7⋊C8—24no—
33#89759—21no—
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6381C2212yes—
37#89733—22no—
38#89707C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89324C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#25384—71nomaximal, Sylow
45C22×C7×D4—22no—
46C22×C7×D4—22no—
47C22×C7×D4—21no—
48C22×C7×D4—21no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4C2312yes—
51C23×C4×C7C2312yes—
52C22×C22⋊D7—22no—
53C22×D28C2311yescharacteristic
54C22×C4×D7—21no—
55C7×#127—22no—
56C7×#127—22no—
57C7×#127—44no—
58C7×#121—24no—
59C7×#121—24no—
60C7×#121—42no—
61C7×#121—42no—
62C4×C7×D4—22no—
63C4×C7×D4—22no—
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4D412yes—
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72C2×C22⋊Dic7—22no—
73#6501—44no—
74C2×#1141—24no—
75C2×#1141—24no—
76C2×#1141—42no—
77#6493—24no—
78#6493—42no—
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2312yes—
85C22×C7⋊C8—24no—
86C22×C7⋊C8—24no—
87C22×C7⋊C8—42no—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C4×D28—22no—
92C4×D28—22no—
93C7×#95—21no—
94#6406—24no—
95#6406—24no—
96#6406—42no—
97C7⋊#95—22no—
98C22⋊(C7⋊C8)—24no—
99#6383—22no—
100#6383D414yescomplemented
101#6381—24no—
102C4⋊(C7⋊C8)D414yescomplemented
103C4×C7⋊C8—22no—
104C2×#466—72no—
105C2×#459—71no—
106C2×C8×D4—71no—
107#2541D711yescharacteristic, complemented
108#2537—71no—
109#2033—72no—
110#1999—72no—
111#1957—71no—
112#1947—71no—
113#1893—72no—
114C4×#102—71no—
115C24×C7—22no—
116C24×C7C2×D412yes—
117C23×D7—21no—
118C2×C7×D4—22no—
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—44no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4—42no—
131C2×C7×D4—42no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C2×C7×D4—81no—
135C2×C7×D4—81no—
136C2×C7×D4—82no—
137C2×C7×D4—82no—
138C2×C7×D4D814yescomplemented
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2411yescharacteristic
151C2×C22⋊D7—44no—
152C2×C22⋊D7—44no—
153C2×C22⋊D7—82no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C2×D28—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28C2×D412yes—
162C2×C4×D7—42no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4C2×D412yes—
169C7×C4⋊C4#4112yes—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—44no—
172C7×C22⋊C4—44no—
173C42×C7—42no—
174C42×C7C2×D412yes—
175C42×C7#4112yes—
176C22⋊Dic7—44no—
177#1141—48no—
178(C2×D7)⋊C4—44no—
179(C2×D7)⋊C4—42no—
180C4⋊Dic7—21no—
181C4⋊Dic7—22no—
182C4⋊Dic7—21no—
183C4⋊Dic7—22no—
184C4⋊Dic7—42no—
185C4⋊Dic7—81no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8—48no—
194C2×C7⋊C8C2×D414yes—
195C2×C7⋊C8C2×D412yes—
196C22×D8—72no—
197C23×C8—72no—
198C2×#127—21no—
199C2×#127C2×D711yescharacteristic
200C2×#121—72no—
201C2×#121—141no—
202C2×#121C2×D712yes—
203C2×#121C2×D712yes—
204C2×C4×D4—71no—
205C2×C4×D4C2×D711yescharacteristic
206#466—148no—
207#459—144no—
208C8×D4—144no—
209C2×#107—72no—
210C2×C4⋊C8—71no—
211C2×#102—71no—
212C2×#102—72no—
213C2×#102—72no—
214C2×#102—71no—
215C2×#102—72no—
216C2×C22⋊C8—72no—
217C2×C4×C8—71no—
218#394—142no—
219#392—22no—
220#390—21no—
221#384—71no—
222#380—72no—
223#380—141no—
224C4×C4⋊C4C2×D711yescharacteristic
225#340—71no—
226#340—72no—
227#340—71no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7C2×D814yes—
242C23×C7C22×D411yescharacteristic
243C22×D7—22no—
244C22×D7—41no—
245C22×D7—41no—
246C22×D7—41no—
247C22×D7—41no—
248C22×D7—41no—
249C22×D7—41no—
250C7×D4—28no—
251C7×D4—42no—
252C7×D4—42no—
253C7×D4—42no—
254C7×D4—42no—
255C7×D4—84no—
256C7×D4—88no—
257C7×D4—88no—
258C7×D4—84no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7#12112yes—
275C2×C4×C7#12112yes—
276C2×C4×C7C2×D812yes—
277C2×C4×C7#13512yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C7H(5,𝔽2)11yescharacteristic
284C22⋊D7—88no—
285C2×Dic7—41no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—42no—
291D28—22no—
292D28—22no—
293D28—41no—
294D28—41no—
295C4×D7—84no—
296C7⋊C8—22no—
297C7⋊C8—24no—
298C7⋊C8—41no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—71no—
304C22×D4—142no—
305C22×D4C22⋊D711yescharacteristic
306C22×D4C22×D712yes—
307C23×C4—141no—
308C23×C4C22×D712yes—
309C2×D8—144no—
310C2×D8—144no—
311C2×D8—282no—
312C22×C8—71no—
313C22×C8—72no—
314C22×C8—71no—
315C22×C8—144no—
316C22×C8—144no—
317C22×C8—282no—
318#127—22no—
319#127—22no—
320#127—44no—
321#121—24no—
322#121—24no—
323#121—42no—
324#121—42no—
325#121—144no—
326#121—282no—
327#121—284no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—142no—
331C4×D4—142no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22⋊D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4C22⋊D712yes—
345C2×C42—21no—
346C2×C42C22×D711yescharacteristic
347#107—144no—
348C4⋊C8—74no—
349#102—74no—
350#102—144no—
351#102—144no—
352#102—142no—
353#102—282no—
354C22⋊C8—144no—
355C4×C8—142no—
356#95—21no—
357#95—142no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—24no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—82no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7—82no—
369C22×C7—82no—
370C22×C7—82no—
371C22×C7—82no—
372C22×C7C2×#12111yescharacteristic
373C22×C7D4211yescharacteristic
374C22×C7#54611yescharacteristic
375C22×C7C22×D811yescharacteristic
376C22×C7C2×#13511yescharacteristic
377C22×C7#57611yescharacteristic
378C22×C7#57711yescharacteristic
379C2×D7—42no—
380C2×D7—42no—
381C2×D7—42no—
382C2×D7—81no—
383C2×D7—81no—
384C2×D7—81no—
385C2×D7—81no—
386C4×C7—22no—
387C4×C7—42no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7#46614yescomplemented
391C4×C7#53211yescharacteristic
392C4×C7#54011yescharacteristic
393C4×C7D4211yescharacteristic
394C4×C7#55011yescharacteristic
395Dic7—82no—
396Dic7—81no—
397C24—22no—
398C24—141no—
399C24C2×C22⋊D712yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—72no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—141no—
422C2×D4—141no—
423C2×D4—141no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—284no—
427C2×D4—284no—
428C2×D4—562no—
429C2×D4#114114yescomplemented
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—141no—
439C22×C4—142no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4—282no—
443C22×C4C2×C22⋊D712yes—
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447D8—288no—
448C2×C8—74no—
449C2×C8—72no—
450C2×C8—141no—
451C2×C8—142no—
452C2×C8—141no—
453C2×C8—142no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—288no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—141no—
462C4⋊C4—142no—
463C4⋊C4—141no—
464C4⋊C4—142no—
465C4⋊C4—282no—
466C4⋊C4—561no—
467C4⋊C4D4×D712yes—
468C4⋊C4#115912yes—
469C22⋊C4—24no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—284no—
473C22⋊C4—284no—
474C22⋊C4—282no—
475C42—42no—
476C42D4×D712yes—
477C42#115912yes—
478C2×C7—44no—
479C2×C7—162no—
480C2×C7#242811yescharacteristic
481C2×C7C2×#46611yescharacteristic
482C2×C7#306211yescharacteristic
483C2×C7D4×D811yescharacteristic
484C2×C7#328411yescharacteristic
485C2×C7#329411yescharacteristic
486C2×C7#330511yescharacteristic
487D7—82no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—142no—
502C23—281no—
503C23—281no—
504C23—281no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23C2×#114114yes—
509C23C22×C22⋊D711yescharacteristic
510D4—28no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—84no—
516D4—88no—
517D4—88no—
518D4—84no—
519D4—142no—
520D4—142no—
521D4—281no—
522D4—281no—
523D4—568no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—281no—
540C2×C4—282no—
541C2×C4—282no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—282no—
545C2×C4—564no—
546C2×C4C2×#114112yes—
547C2×C4#650212yes—
548C2×C4#650212yes—
549C2×C4#651312yes—
550C2×C4C2×D4×D711yescharacteristic
551C2×C4C2×D4×D711yescharacteristic
552C2×C4C2×#115911yescharacteristic
553C2×C4C2×#115911yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4#654811yescharacteristic
556C8—142no—
557C8—144no—
558C8—281no—
559C7#2538411yescharacteristic, complemented, Sylow
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—282no—
575C22—282no—
576C22—282no—
577C22—561no—
578C22—561no—
579C22—561no—
580C22—561no—
581C22C22×#114111yescharacteristic
582C22D4×C22⋊D711yescharacteristic
583C22C2×#650211yescharacteristic
584C22#9049711yescharacteristic
585C22C2×#651311yescharacteristic
586C22#9052611yescharacteristic
587C22#9052711yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—562no—
593C4—561no—
594C4#8981014yescomplemented
595C4#9040711yescharacteristic
596C4#9040811yescharacteristic
597C4#9040911yescharacteristic
598C4#9041011yescharacteristic
599C2—44no—
600C2—162no—
601C2—562no—
602C2C2×#8981011yescharacteristic
603C2#1174063111yescharacteristic
604C2#1174082411yescharacteristic
605C2#1174092711yescharacteristic
606C2D4×#114111yescharacteristic
607C2#1174101611yescharacteristic
608C2#1174102411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2541C2
socleC23×C7C22×D4
hypercenter#2541D7
hypocenterC7#25384
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C7×#2541C2semidirect
2. C2×#89844C2semidirect
3. C2×#89810C2semidirect
4. #11739862C2semidirect
5. C2×D4×C7⋊C8C2semidirect
6. #11729518C2semidirect
7. #11729438C2semidirect
8. #11729435C2semidirect
9. #11729433C2semidirect
10. C4×#6383C2semidirect
11. #11727220C2semidirect
100. #6383D4semidirect
102. C4⋊(C7⋊C8)D4semidirect
107. #2541D7semidirect
138. C2×C7×D4D8semidirect
390. C4×C7#466semidirect
429. C2×D4#1141semidirect
559. C7#25384semidirect
594. C4#89810semidirect

Maximal subgroups

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

subgroupindex
C7×#25412
C2×#898442
C2×#898102
#117398622
C2×D4×C7⋊C82
#117295182
#117294382
#117294352
#117294332
C4×#63832
#117272202
#253847