Finite Group Databasebeta

← #13395323

Subgroups — #13395323

Order 1792.

1222 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×#89844C212yesmaximal, complemented
3C2×#89844C211yesmaximal, characteristic, complemented
4C2×C4×#1141C211yesmaximal, characteristic, complemented
5C4⋊C4×D28C211yesmaximal, characteristic, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729516C212yesmaximal, complemented
8#11729358C211yesmaximal, characteristic, complemented
9#11727411C212yesmaximal, complemented
10#11727407C212yesmaximal, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C7×#127C2211yescharacteristic
13C2×C7×#127C2212yes—
14C2×C7×#121—21no—
15C2×C7×#121—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C22×#1141C2212yes—
18C22×#1141C2211yescharacteristic
19C2×D7×C4⋊C4—21no—
20C2×C4×D28C2212yes—
21C2×C4×D28C2211yescharacteristic
22C7×#392—22no—
23C7×#390C2212yes—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2211yescharacteristic
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30#89844—28no—
31#89844—24no—
32C4×#1141—24no—
33#89759—21no—
34C2×#6383C2211yescharacteristic
35C2×#6381C2211yescharacteristic
36#89707C2211yescharacteristic
37#89700C2212yes—
38C2×C4⋊(C7⋊C8)C2212yes—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89440—22no—
42#89324C2212yes—
43#89319C2212yes—
44#24726—71nomaximal, Sylow
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4—21no—
49C22×C7×D4C2312yes—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C22×D28C2311yescharacteristic
53C22×C4×D7—21no—
54C22×C4×D7—22no—
55C7×#127—24no—
56C7×#127—22no—
57C7×#127—22no—
58C7×#127—42no—
59C7×#121—44no—
60C7×#121—44no—
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C42×C7C2312yes—
70C2×C42×C7C2311yescharacteristic
71C2×#1141—22no—
72C2×#1141—22no—
73C2×#1141—24no—
74C2×#1141—24no—
75C2×#1141—24no—
76C2×#1141D414yescomplemented
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—22no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2312yes—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×Dic7⋊C4—22no—
84C22×C7⋊C8C2311yescharacteristic
85C22×C7⋊C8C2312yes—
86C22×C7⋊C8C2311yescharacteristic
87D7×C4⋊C4—44no—
88C4×D28—22no—
89C4×D28—24no—
90C4×D28—42no—
91C4×D28D414yescomplemented
92C7×#95D412yes—
93#6406—24no—
94#6406—22no—
95#6406—24no—
96#6406—24no—
97#6406—42no—
98C7⋊#95—22no—
99#6383—22no—
100#6381—22no—
101C4⋊(C7⋊C8)—22no—
102C4⋊(C7⋊C8)—24no—
103C4×C7⋊C8—22no—
104C2×#459—71no—
105C2×#459—72no—
106C2×C4×D8—71no—
107#2541D711yescharacteristic, complemented
108D4×C4⋊C4—71no—
109#2067—72no—
110#2041—72no—
111#1957—71no—
112#1939—71no—
113#1879—72no—
114#1802—71no—
115C24×C7—21no—
116C24×C7—22no—
117C23×D7—21no—
118C2×C7×D4—21no—
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—21no—
126C2×C7×D4—22no—
127C2×C7×D4—21no—
128C2×C7×D4—22no—
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—44no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C2×C7×D4—44no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4—81no—
138C2×C7×D4—82no—
139C2×C7×D4—81no—
140C2×C7×D4—82no—
141C2×C7×D4C2×D412yes—
142C22×C4×C7—21no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—21no—
152C22×Dic7—22no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28C2×D412yes—
157C2×D28C2×D412yes—
158C2×D28#4112yes—
159C2×C4×D7—24no—
160C2×C4×D7—42no—
161C2×C4×D7—42no—
162C2×C4×D7—44no—
163C2×C4×D7—44no—
164C2×C4×D7—82no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4D814yescomplemented
169C7×C22⋊C4—44no—
170C7×C22⋊C4—42no—
171C7×C22⋊C4—44no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7C2×D412yes—
175C42×C7#4112yes—
176#1141—28no—
177#1141—48no—
178(C2×D7)⋊C4—24no—
179(C2×D7)⋊C4—44no—
180C4⋊Dic7—21no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7—42no—
184C4⋊Dic7—81no—
185C4⋊Dic7C2×D412yes—
186Dic7⋊C4—44no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8C2×D412yes—
196C22×D8—71no—
197C22×D8—72no—
198C2×#127C2×D711yescharacteristic
199C2×#127C2×D712yes—
200C2×#121—21no—
201C2×#121—21no—
202C2×C4×D4—71no—
203C2×C4×D4—72no—
204C2×C4×D4C2×D711yescharacteristic
205C22×C4⋊C4—141no—
206#459—144no—
207#459—148no—
208C4×D8—144no—
209C2×#107—71no—
210C2×C4⋊C8—71no—
211C2×C4⋊C8—72no—
212C2×#102—72no—
213C2×#102—72no—
214C2×#102—72no—
215C2×#102—71no—
216C2×#102—71no—
217C2×#102—72no—
218C2×C4×C8—71no—
219#392—22no—
220#390C2×D712yes—
221#389—72no—
222#385—142no—
223#384—71no—
224#380—141no—
225C4×C4⋊C4C2×D711yescharacteristic
226#340—72no—
227#336—72no—
228C23×C7—22no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7C22×D411yescharacteristic
243C22×D7—22no—
244C22×D7—22no—
245C22×D7—22no—
246C22×D7—41no—
247C22×D7—41no—
248C22×D7—41no—
249C22×D7—41no—
250C7×D4—22no—
251C7×D4—22no—
252C7×D4—42no—
253C7×D4—41no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—41no—
257C7×D4—42no—
258C7×D4—88no—
259C7×D4—84no—
260C7×D4—88no—
261C7×D4—84no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—84no—
273C2×C4×C7#12112yes—
274C2×C4×C7#12112yes—
275C2×C4×C7C2×D814yes—
276C2×C4×C7C2×D812yes—
277C2×C4×C7AGL(1,ℤ/8)12yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290D28—22no—
291D28—22no—
292D28—22no—
293D28#12114yescomplemented
294C4×D7—48no—
295C4×D7—88no—
296C7⋊C8—22no—
297C7⋊C8—42no—
298C7⋊C8—41no—
299C22×D4—21no—
300C22×D4—22no—
301C22×D4—22no—
302C22×D4—21no—
303C22×D4—71no—
304C22×D4C22×D712yes—
305C22×D4C22×D711yescharacteristic
306C23×C4—21no—
307C23×C4—141no—
308C23×C4—142no—
309C2×D8—74no—
310C2×D8—142no—
311C2×D8—142no—
312C2×D8—144no—
313C2×D8—144no—
314C2×D8—144no—
315C22×C8—71no—
316C22×C8—72no—
317C22×C8—71no—
318#127—24no—
319#127—22no—
320#127—22no—
321#127—42no—
322#121—44no—
323#121—44no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—74no—
327C4×D4—144no—
328C4×D4—142no—
329C4×D4—282no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—72no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—284no—
336C2×C4⋊C4C22⋊D711yescharacteristic
337C2×C4⋊C4C22⋊D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—142no—
344C2×C42C22×D712yes—
345C2×C42C22×D711yescharacteristic
346#107—142no—
347C4⋊C8—144no—
348C4⋊C8—142no—
349#102—142no—
350#102—142no—
351#102—144no—
352#102—144no—
353#102—144no—
354#102—282no—
355C4×C8—142no—
356#95—142no—
357#95C22⋊D712yes—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—81no—
364C22×C7—81no—
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#54811yescharacteristic
375C22×C7C22×D811yescharacteristic
376C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
377C22×C7#57611yescharacteristic
378C22×C7#57711yescharacteristic
379C2×D7—24no—
380C2×D7—42no—
381C2×D7—42no—
382C2×D7—42no—
383C2×D7—42no—
384C2×D7—42no—
385C2×D7—42no—
386C4×C7—24no—
387C4×C7—22no—
388C4×C7—42no—
389C4×C7—81no—
390C4×C7#45914yescomplemented
391C4×C7C2×#12111yescharacteristic
392C4×C7#53711yescharacteristic
393C4×C7#54711yescharacteristic
394C4×C7#55311yescharacteristic
395Dic7—42no—
396Dic7—82no—
397C24—21no—
398C24—22no—
399C24—141no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—72no—
420C2×D4—72no—
421C2×D4—72no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—81no—
425C2×D4—82no—
426C2×D4—141no—
427C2×D4—141no—
428C2×D4—141no—
429C2×D4D4×D712yes—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—141no—
435C22×C4—144no—
436C22×C4—142no—
437C22×C4—284no—
438C22×C4—282no—
439C22×C4—282no—
440C22×C4—284no—
441C22×C4—562no—
442C22×C4C2×C22⋊D712yes—
443C22×C4C2×C22⋊D712yes—
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447D8—148no—
448D8—288no—
449C2×C8—72no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—142no—
454C2×C8—142no—
455C2×C8—141no—
456C2×C8—142no—
457C2×C8—141no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—72no—
462C4⋊C4—141no—
463C4⋊C4—142no—
464C4⋊C4—142no—
465C4⋊C4—284no—
466C4⋊C4—282no—
467C4⋊C4—561no—
468C4⋊C4#114114yescomplemented
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—42no—
472C22⋊C4—144no—
473C22⋊C4—284no—
474C42—22no—
475C42—22no—
476C42D4×D712yes—
477C42#115912yes—
478C2×C7—82no—
479C2×C7—162no—
480C2×C7#241611yescharacteristic
481C2×C7C2×#45911yescharacteristic
482C2×C7#304511yescharacteristic
483C2×C7#327211yescharacteristic
484C2×C7D4×D811yescharacteristic
485C2×C7#335811yescharacteristic
486C2×C7#336611yescharacteristic
487D7—44no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—142no—
503C23—142no—
504C23—142no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23—281no—
509C23C22×C22⋊D711yescharacteristic
510D4—22no—
511D4—22no—
512D4—42no—
513D4—41no—
514D4—42no—
515D4—42no—
516D4—41no—
517D4—42no—
518D4—74no—
519D4—88no—
520D4—84no—
521D4—88no—
522D4—84no—
523D4—142no—
524D4—142no—
525D4—142no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
537C2×C4—142no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—288no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—568no—
545C2×C4C2×#114114yes—
546C2×C4C2×#114112yes—
547C2×C4#649512yes—
548C2×C4#650112yes—
549C2×C4#650112yes—
550C2×C4C2×D4×D711yescharacteristic
551C2×C4C2×D4×D711yescharacteristic
552C2×C4C2×#115911yescharacteristic
553C2×C4#654811yescharacteristic
554C2×C4D7×#4111yescharacteristic
555C2×C4#655411yescharacteristic
556C8—142no—
557C8—282no—
558C8—281no—
559C7#2472611yescharacteristic, complemented, Sylow
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—144no—
575C22—282no—
576C22—282no—
577C22—282no—
578C22—282no—
579C22—282no—
580C22—282no—
581C22C22×#114111yescharacteristic
582C22C2×#649511yescharacteristic
583C22C2×#650111yescharacteristic
584C22D4×C22⋊D711yescharacteristic
585C22#9052211yescharacteristic
586C22#9052611yescharacteristic
587C22#9052711yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—81no—
592C4—282no—
593C4—562no—
594C4#8984414yescomplemented
595C4D7×#12111yescharacteristic
596C4#9029911yescharacteristic
597C4#9030211yescharacteristic
598C4#9030511yescharacteristic
599C2—82no—
600C2—162no—
601C2—284no—
602C2#1174008611yescharacteristic
603C2C2×#8984411yescharacteristic
604C2#1174083011yescharacteristic
605C2#1174093411yescharacteristic
606C2D4×#114111yescharacteristic
607C2#1174102511yescharacteristic
608C2#1174102611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2541C2
socleC23×C7C22×D4
hypercenter#2541D7
hypocenterC7#24726
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C7×#2541C2semidirect
2. C2×#89844C2semidirect
3. C2×#89844C2semidirect
4. C2×C4×#1141C2semidirect
5. C4⋊C4×D28C2semidirect
6. #11729518C2semidirect
7. #11729516C2semidirect
8. #11729358C2semidirect
9. #11727411C2semidirect
10. #11727407C2semidirect
11. #11727238C2semidirect
76. C2×#1141D4semidirect
91. C4×D28D4semidirect
107. #2541D7semidirect
168. C7×C4⋊C4D8semidirect
293. D28#121semidirect
390. C4×C7#459semidirect
468. C4⋊C4#1141semidirect
559. C7#24726semidirect
594. C4#89844semidirect

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×C4×#11412
C4⋊C4×D282
#117295182
#117295162
#117293582
#117274112
#117274072
#117272382
#247267