Finite Group Databasebeta

← #13353868

Subgroups — #13353868

Order 1792.

1338 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1161C211yesmaximal, characteristic
2C2×#89477C211yesmaximal, characteristic, complemented
3#11731197C211yesmaximal, characteristic, complemented
4C7×#1763C211yesmaximal, characteristic, complemented
5#11729385C211yesmaximal, characteristic, complemented
6#11728021C212yesmaximal
7#11727976C212yesmaximal, complemented
8#11727566C211yesmaximal, characteristic, complemented
9#11727228C211yesmaximal, characteristic, complemented
10#11726647C212yesmaximal, complemented
11#11726627C212yesmaximal, complemented
12C22×#1161C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C4×#1161—24no—
16C4×#1161C418yescomplemented
17C2×#6456—21no—
18C2×#6456C2211yescharacteristic
19C2×#6455—21no—
20C2×#6455C2211yescharacteristic
21C2×C4×D28—21no—
22C2×C4×D28C2211yescharacteristic
23C2×C42×D7—21no—
24C2×C42×D7C2211yescharacteristic
25C2×C7×#103C2212yes—
26C2×C7×#97C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C2×#6409C2212yes—
29C2×#6395C2212yes—
30C2×(C2×D7)⋊C8C2212yes—
31C2×Dic7⋊C8C2212yes—
32#89759C2212yes—
33#89749C2212yes—
34C2×#6383C2212yes—
35C4×(C2×D7)⋊C4—21no—
36C4×C4⋊Dic7C2211yescharacteristic
37#89704—21no—
38C2×#6377C2211yescharacteristic
39#89477—24no—
40#89436—21no—
41#89423—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#20742—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C2×#1161—24no—
49C2×#1161—24no—
50C2×#1161—22no—
51C2×#1161C2×C414yes—
52C22×D28—21no—
53C22×D28C2311yescharacteristic
54C22×C4×D7—21no—
55C22×C4×D7—21no—
56C22×C4×D7—21no—
57C22×C4×D7D411yescharacteristic
58C22×C4×D7D411yescharacteristic
59C22×C4×D7C2311yescharacteristic
60C22×C7×C8C2312yes—
61C4×C7×Q8—22no—
62C4×C7×Q8C2×C414yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2312yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C42×C7—21no—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69Q8×Dic7—22no—
70Q8×Dic7C2×C414yes—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7C2312yes—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4×Dic7—21no—
79C2×C4×Dic7—21no—
80C2×C4×Dic7—21no—
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7D411yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85#6456—28no—
86#6456—22no—
87#6456C2×C414yes—
88#6455—22no—
89#6455—28no—
90#6455C2×C414yes—
91C4×D28—28no—
92C4×D28—22no—
93C4×D28C2×C414yes—
94C42×D7—28no—
95C42×D7C2×C414yes—
96C42×D7D414yes—
97C7×#103—24no—
98C7×#97—22no—
99C7×#95—21no—
100#6409—24no—
101C7⋊#95—21no—
102C7⋊#95—22no—
103C7⋊#95—22no—
104C7⋊#95—22no—
105#6395—24no—
106(C2×D7)⋊C8—24no—
107Dic7⋊C8—24no—
108#6383—24no—
109#6377—22no—
110C2×C4×#41—71no—
111C2×#432—71no—
112#2283—71no—
113#1825—72no—
114#1805—72no—
115#1804—72no—
116#1788—72no—
117#1763—71no—
118#1763D711yescharacteristic, complemented
119#1762—71no—
120#1762—71no—
121C23×D7—21no—
122C23×D7C2×D411yescharacteristic
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22×C412yes—
126C2×C7×Q8#4114yes—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2411yescharacteristic
133C22×Dic7—21no—
134C22×Dic7—22no—
135C22×Dic7C2×D411yescharacteristic
136C22×Dic7C2×D411yescharacteristic
137C22×Dic7C2×D411yescharacteristic
138#1161—216no—
139#1161—28no—
140#1161—28no—
141C2×D28—28no—
142C2×D28—24no—
143C2×D28—21no—
144C2×D28—22no—
145C2×D28C22×C412yes—
146C2×D28#4114yes—
147C2×C4×D7—28no—
148C2×C4×D7—22no—
149C2×C4×D7—22no—
150C2×C4×D7—24no—
151C2×C4×D7—24no—
152C2×C4×D7—24no—
153C2×C4×D7—41no—
154C2×C4×D7—41no—
155C2×C4×D7—42no—
156C2×C4×D7—42no—
157C2×C4×D7—42no—
158C2×C4×D7—44no—
159C2×C4×D7C22⋊C414yes—
160C2×C4×D7C22⋊C414yes—
161C2×C4×D7C22×C412yes—
162C2×C4×D7C2×D412yes—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4C22×C412yes—
171C42×C7—24no—
172C42×C7—42no—
173C42×C7C22×C412yes—
174C42×C7C2×D412yes—
175(C2×D7)⋊C4—24no—
176(C2×D7)⋊C4—24no—
177(C2×D7)⋊C4—24no—
178(C2×D7)⋊C4—44no—
179C4⋊Dic7—21no—
180C4⋊Dic7—24no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7C22×C412yes—
184C4×Dic7—24no—
185C4×Dic7—24no—
186C4×Dic7—42no—
187C4×Dic7C22⋊C414yes—
188C4×Dic7C22⋊C414yes—
189C4×Dic7C22×C412yes—
190C4×Dic7C2×D412yes—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C22×#41—71no—
195C4×#41—78no—
196C4×#41—144no—
197C2×C4×Q8—71no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×D4—71no—
200C2×C4×D4—71no—
201C2×C4×D4—141no—
202C2×C4×D4—141no—
203C2×#117—71no—
204C2×#117—141no—
205C22×C42—71no—
206C22×C42—141no—
207#432—144no—
208C2×C4⋊C8—72no—
209C2×#103—72no—
210C2×#103C2×D712yes—
211C2×#102—72no—
212C2×#102—72no—
213C2×C22⋊C8—72no—
214C2×#97—71no—
215C2×#97C2×D711yescharacteristic
216#383—141no—
217#382—72no—
218#381—141no—
219#380—72no—
220C4×C4⋊C4—71no—
221C4×C4⋊C4C2×D711yescharacteristic
222C4×C22⋊C4—141no—
223#376—141no—
224#340—72no—
225#340—72no—
226#340—72no—
227#340C2×D712yes—
228C23×C7C22×D411yescharacteristic
229C22×D7—22no—
230C22×D7—22no—
231C22×D7—24no—
232C22×D7—22no—
233C22×D7—21no—
234C22×D7—21no—
235C22×D7—41no—
236C22×D7—41no—
237C22×D7—42no—
238C22×D7C2×C22⋊C412yes—
239C7×Q8—22no—
240C7×Q8—24no—
241C7×Q8#11714yes—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7C2×C22⋊C412yes—
254C2×C4×C7C2×C22⋊C412yes—
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C2×#4112yes—
258C2×C4×C7C2×#4112yes—
259C2×Dic7—22no—
260C2×Dic7—21no—
261C2×Dic7—21no—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7C2×C22⋊C412yes—
269C2×Dic7C2×C22⋊C412yes—
270C2×Dic7C2×C22⋊C412yes—
271D28—216no—
272D28—22no—
273D28—24no—
274D28#11714yes—
275C4×D7—216no—
276C4×D7—22no—
277C4×D7—22no—
278C4×D7—24no—
279C4×D7—44no—
280C4×D7—44no—
281C4×D7—48no—
282C4×D7—48no—
283C7×C8—42no—
284C7⋊C8—42no—
285C2×#41—74no—
286C2×#41—142no—
287C2×#41—144no—
288C2×#41—144no—
289C22×Q8C22×D711yescharacteristic
290C22×D4—71no—
291C22×D4—141no—
292C23×C4—71no—
293C23×C4—71no—
294C23×C4—71no—
295C23×C4—141no—
296C23×C4—141no—
297C23×C4—141no—
298C22×C8—72no—
299C22×C8C22×D712yes—
300C4×Q8—22no—
301C4×Q8—74no—
302C4×Q8—142no—
303C4×Q8C4×D714yes—
304C4×D4—74no—
305C4×D4—74no—
306C4×D4—148no—
307C4×D4—142no—
308C4×D4—148no—
309C4×D4—142no—
310#117—74no—
311#117—148no—
312#117—142no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—72no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4C22×D712yes—
318C2×C4⋊C4C22×D711yescharacteristic
319C2×C22⋊C4—141no—
320C2×C22⋊C4—141no—
321C2×C22⋊C4—141no—
322C2×C22⋊C4—142no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42—71no—
328C2×C42—74no—
329C2×C42—74no—
330C2×C42—148no—
331C2×C42—141no—
332C2×C42—141no—
333C2×C42—141no—
334C2×C42C22×D711yescharacteristic
335C4⋊C8—144no—
336#103—24no—
337#103—144no—
338#102—144no—
339#102—144no—
340C22⋊C8—144no—
341#97—22no—
342#97—142no—
343#95—21no—
344#95—142no—
345#95—142no—
346#95—142no—
347#95—141no—
348C22×C7C22×C22⋊C411yescharacteristic
349C22×C7#52512yes—
350C22×C7#57511yescharacteristic
351C22×C7#57511yescharacteristic
352C22×C7#57512yes—
353C2×D7—22no—
354C2×D7—22no—
355C2×D7—22no—
356C2×D7—24no—
357C2×D7—42no—
358C2×D7—42no—
359C2×D7—44no—
360C2×D7—44no—
361C2×D7—42no—
362C2×D7—42no—
363C4×C7—22no—
364C4×C7—22no—
365C4×C7—24no—
366C4×C7—42no—
367C4×C7—82no—
368C4×C7C2×#11711yescharacteristic
369C4×C7C2×#11711yescharacteristic
370C4×C7#52512yes—
371Dic7—22no—
372Dic7—22no—
373Dic7—22no—
374Dic7—42no—
375Dic7—82no—
376C24—71no—
377C24—141no—
378#41—148no—
379#41—1416no—
380#41—148no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8C2×C4×D712yes—
384C2×Q8#115714yes—
385C2×D4—72no—
386C2×D4—74no—
387C2×D4—144no—
388C2×D4—148no—
389C2×D4—142no—
390C2×D4—141no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—71no—
395C22×C4—71no—
396C22×C4—71no—
397C22×C4—72no—
398C22×C4—72no—
399C22×C4—74no—
400C22×C4—74no—
401C22×C4—144no—
402C22×C4—144no—
403C22×C4—148no—
404C22×C4—142no—
405C22×C4—142no—
406C22×C4—141no—
407C22×C4—144no—
408C22×C4—142no—
409C22×C4—281no—
410C22×C4—281no—
411C22×C4—282no—
412C22×C4—284no—
413C22×C4—282no—
414C22×C4—282no—
415C22×C4D4×D711yescharacteristic
416C22×C4D4×D711yescharacteristic
417C22×C4C23×D711yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—142no—
422C2×C8—142no—
423C2×C8—142no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—72no—
429C4⋊C4—144no—
430C4⋊C4—141no—
431C4⋊C4—142no—
432C4⋊C4—142no—
433C4⋊C4C2×C4×D712yes—
434C22⋊C4—144no—
435C22⋊C4—144no—
436C22⋊C4—144no—
437C22⋊C4—284no—
438C42—24no—
439C42—42no—
440C42—74no—
441C42—74no—
442C42—72no—
443C42—72no—
444C42—144no—
445C42—144no—
446C42—282no—
447C42C2×C4×D712yes—
448C42D4×D712yes—
449C2×C7#228011yescharacteristic
450C2×C7#289611yescharacteristic
451C2×C7#289611yescharacteristic
452C2×C7#311112yes—
453C2×C7#311112yes—
454D7—42no—
455D7—44no—
456C23—72no—
457C23—142no—
458C23—141no—
459C23—142no—
460C23—141no—
461C23—144no—
462C23—142no—
463C23—281no—
464C23—281no—
465C23—282no—
466C23C2×D4×D711yescharacteristic
467Q8—22no—
468Q8—24no—
469Q8#643514yes—
470D4—74no—
471D4—1416no—
472D4—142no—
473D4—144no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—72no—
486C2×C4—72no—
487C2×C4—72no—
488C2×C4—1416no—
489C2×C4—141no—
490C2×C4—142no—
491C2×C4—141no—
492C2×C4—142no—
493C2×C4—142no—
494C2×C4—142no—
495C2×C4—144no—
496C2×C4—142no—
497C2×C4—284no—
498C2×C4—284no—
499C2×C4—288no—
500C2×C4—288no—
501C2×C4—282no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4D7×C22⋊C412yes—
506C2×C4D7×C22⋊C412yes—
507C2×C4C22×C4×D711yescharacteristic
508C2×C4C2×#115712yes—
509C2×C4C2×D4×D711yescharacteristic
510C2×C4D7×#4112yes—
511C8—42no—
512C8—282no—
513C7#2074211yescharacteristic, complemented, Sylow
514C22—142no—
515C22—144no—
516C22—142no—
517C22—142no—
518C22—282no—
519C22—282no—
520C22—282no—
521C22—284no—
522C22—282no—
523C22—284no—
524C22C2×D7×C22⋊C411yescharacteristic
525C22#9023812yes—
526C22#9046411yescharacteristic
527C22#9046811yescharacteristic
528C22#9046812yes—
529C4—22no—
530C4—24no—
531C4—22no—
532C4—42no—
533C4—82no—
534C4—142no—
535C4—142no—
536C4—142no—
537C4—282no—
538C4—562no—
539C4C2×#643511yescharacteristic
540C4D7×#11711yescharacteristic
541C4#9023812yes—
542C2—282no—
543C2—284no—
544C2#1173100611yescharacteristic
545C2#1173632611yescharacteristic
546C2#1173640111yescharacteristic
547C2#1173725512yes—
548C2#1173729212yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1763C2
socleC23×C7C22×D4
prime radicalC22×#1161C22
cosocleC22×C4×C7C24
hypercenter#1763D7
hypocenterC7#20742
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#89477C2semidirect
3. #11731197C2semidirect
4. C7×#1763C2semidirect
5. #11729385C2semidirect
7. #11727976C2semidirect
8. #11727566C2semidirect
9. #11727228C2semidirect
10. #11726647C2semidirect
11. #11726627C2semidirect
16. C4×#1161C4semidirect
118. #1763D7semidirect
513. C7#20742semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11612
C2×#894772
#117311972
C7×#17632
#117293852
#117280212
#117279762
#117275662
#117272282
#117266472
#117266272
#207427