Finite Group Databasebeta

← #13332376

Subgroups — #13332376

Order 1792.

1170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2#11740668C211yesmaximal, characteristic, complemented
3#11739759C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic, complemented
5#11729667C214yesmaximal, complemented
6#11729665C214yesmaximal, complemented
7#11729527C211yesmaximal, characteristic, complemented
8#11729352C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#547—22no—
12C7×#547—22no—
13C7×#547—22no—
14C7×#547C2214yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×#121—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22#90161C2211yescharacteristic
23#89990—21no—
24#89987—21no—
25#89866—22no—
26#89865C2212yescomplemented
27#89851C2212yescomplemented
28#89791C2212yescomplemented
29C4×#1142—22no—
30#89781—22no—
31#89780C2212yescomplemented
32#89780C2212yescomplemented
33#89779C2212yescomplemented
34#89743—21no—
35#89743—22no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41D4⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yescomplemented
43#18757—71nomaximal, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
48C22×C7×Q8—21no—
49C22×C7×D4—22no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4D411yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8D412yes—
58C7×C4⋊Q8C2312yes—
59C7×#123—28no—
60C7×#123—28no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—24no—
63C7×#121—24no—
64C7×#121—24no—
65C4×C7×D4—21no—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C4×C7×D4—21no—
69C4×C7×D4D412yes—
70C4×C7×D4D412yes—
71C4×C7×D4C2312yes—
72C4×C7×D4C2312yes—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—24no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—24no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D412yes—
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7C2311yescharacteristic
84C2×#1142—22no—
85#6489—21no—
86#6487—21no—
87#6487—21no—
88C2×#1136—21no—
89#6433—21no—
90#6432C2311yescharacteristic
91#6432C2311yescharacteristic
92C4×Dic14—21no—
93C4×Dic14—21no—
94#6406—22no—
95#6406—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—24no—
100#6384—22no—
101#6382—24no—
102#6382—22no—
103#6382—22no—
104#6381—22no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2312yes—
109C4×C7⋊C8—21no—
110C4×C7⋊C8C2312yes—
111C2×#547D711yescharacteristic
112#3229—71no—
113#3101—71no—
114#3077—71no—
115#1654—74no—
116#1635—74no—
117#1546—71no—
118#1486—71no—
119#1478—71no—
120C24×C7—21no—
121C24×C7—21no—
122C7×#41—48no—
123C7×#41—416no—
124C7×#41—48no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C2×C7×D4—28no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—22no—
133C2×C7×D4—42no—
134C2×C7×D4—48no—
135C2×C7×D4SD1614yescomplemented
136C2×C7×D4C2×D412yes—
137C2×C7×D4C2×D412yes—
138C22×C4×C7—28no—
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—44no—
147C22×C4×C7—48no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C2×Dic14—21no—
154C2×Dic14—21no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4C2×D412yes—
162C7×C4⋊C4C2×D412yes—
163C7×C4⋊C4C2×D414yes—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—22no—
166C7×C22⋊C4—22no—
167C7×C22⋊C4—22no—
168C7×C22⋊C4—22no—
169C42×C7C2×D411yescharacteristic
170C42×C7C2×D411yescharacteristic
171C42×C7C2×D411yescharacteristic
172C42×C7C2411yescharacteristic
173C22⋊Dic7—41no—
174#1142—44no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179Dic7⋊C4—41no—
180Dic7⋊C4—41no—
181C4×Dic7—41no—
182#1136—42no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C22×#41—21no—
187#547—22no—
188#547—22no—
189#547—22no—
190#547C2×D714yes—
191#533—71no—
192C2×C4⋊Q8C2×D711yescharacteristic
193C2×#123—22no—
194C2×C22⋊Q8—21no—
195C2×#121—21no—
196C2×C4×D4C2×D711yescharacteristic
197C2×C4×D4C2×D711yescharacteristic
198C22×C4⋊C4—21no—
199#498—72no—
200#487—72no—
201#483—141no—
202#483—142no—
203#478—142no—
204#476—72no—
205#476—72no—
206#475—142no—
207#474—72no—
208#461—72no—
209#450—141no—
210C4×SD16—142no—
211#431—71no—
212#431—71no—
213#428—141no—
214#423—71no—
215#418—141no—
216#332—74no—
217D4⋊C8—74no—
218D4⋊C8—74no—
219C23×C7—22no—
220C23×C7—24no—
221C23×C7—22no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7#12011yescharacteristic
228C7×Q8—42no—
229C7×Q8—42no—
230C7×Q8—44no—
231C7×D4—22no—
232C7×D4—22no—
233C7×D4—22no—
234C7×D4—24no—
235C7×D4—42no—
236C7×D4—416no—
237C7×D4C2×SD1614yes—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—24no—
242C2×C4×C7—41no—
243C2×C4×C7—44no—
244C2×C4×C7—42no—
245C2×C4×C7—416no—
246C2×C4×C7—42no—
247C2×C4×C7—44no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—44no—
251C2×C4×C7—44no—
252C2×C4×C7#12011yescharacteristic
253C2×C4×C7#12011yescharacteristic
254C2×C4×C7#12011yescharacteristic
255C2×C4×C7C2×SD1612yes—
256C2×C4×C7AGL(1,ℤ/8)12yes—
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C22×D411yescharacteristic
260C2×Dic7—41no—
261C2×Dic7—41no—
262Dic14—42no—
263Dic14—42no—
264C7⋊C8—42no—
265C7⋊C8—42no—
266C7⋊C8—42no—
267C2×#41—24no—
268C2×#41—24no—
269C2×#41—44no—
270C2×#41—44no—
271C22×Q8—21no—
272C22×D4—22no—
273C22×D4C22⋊D711yescharacteristic
274C22×D4C22⋊D711yescharacteristic
275C23×C4—22no—
276C23×C4—21no—
277C23×C4—21no—
278C2×SD16—142no—
279C2×M16—141no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—71no—
283C4⋊Q8—71no—
284C4⋊Q8C22⋊D712yes—
285C4⋊Q8C22×D712yes—
286#125—141no—
287#123—28no—
288#123—28no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—141no—
292C22⋊Q8—141no—
293#121—24no—
294#121—24no—
295C4×Q8—141no—
296C4×Q8—141no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4C22⋊D712yes—
302C4×D4C22⋊D712yes—
303C4×D4C22×D712yes—
304C4×D4C22×D712yes—
305#117—141no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4C22⋊D711yescharacteristic
311C2×C4⋊C4C22⋊D711yescharacteristic
312C2×C4⋊C4C22⋊D712yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C42C22×D711yescharacteristic
317#107—142no—
318#106—142no—
319#106—142no—
320#106—144no—
321C4⋊C8—72no—
322C4⋊C8—72no—
323C4⋊C8—141no—
324C4⋊C8—141no—
325#103—142no—
326#103—144no—
327#102—142no—
328#102—142no—
329#102—142no—
330C22⋊C8—141no—
331C22⋊C8—141no—
332C4×C8—72no—
333C4×C8—141no—
334C22×C7—21no—
335C22×C7—21no—
336C22×C7—24no—
337C22×C7—48no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—84no—
344C22×C7#45012yes—
345C22×C7C2×#12011yescharacteristic
346C4×C7—21no—
347C4×C7—21no—
348C4×C7—21no—
349C4×C7—42no—
350C4×C7—44no—
351C4×C7—42no—
352C4×C7#45012yes—
353C4×C7C22×SD1611yescharacteristic
354C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C4×C7#57512yes—
356Dic7—81no—
357Dic7—81no—
358C24—21no—
359C24—21no—
360#41—48no—
361#41—416no—
362#41—48no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—141no—
368C2×Q8—141no—
369C2×D4—28no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—42no—
374C2×D4—48no—
375C2×D4#114214yescomplemented
376C2×D4C2×C22⋊D712yes—
377C2×D4C2×C22⋊D712yes—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C22⋊D711yescharacteristic
391C22×C4C2×C22⋊D711yescharacteristic
392C22×C4C2×C22⋊D711yescharacteristic
393SD16—284no—
394M16—282no—
395C2×C8—142no—
396C2×C8—142no—
397C2×C8—142no—
398C4⋊C4—28no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—141no—
405C4⋊C4—141no—
406C4⋊C4—141no—
407C4⋊C4—141no—
408C4⋊C4—281no—
409C4⋊C4—281no—
410C4⋊C4C2×C22⋊D712yes—
411C4⋊C4C2×C22⋊D714yes—
412C4⋊C4C2×C22⋊D712yes—
413C22⋊C4—28no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—281no—
419C42—281no—
420C42C2×C22⋊D711yescharacteristic
421C42C2×C22⋊D711yescharacteristic
422C42C2×C22⋊D711yescharacteristic
423C42C23×D711yescharacteristic
424C2×C7—22no—
425C2×C7—44no—
426C2×C7—82no—
427C2×C7C2×#45011yescharacteristic
428C2×C7#300511yescharacteristic
429C2×C7#302011yescharacteristic
430C23—24no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23#651611yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—44no—
442Q8—282no—
443Q8—282no—
444D4—22no—
445D4—24no—
446D4—22no—
447D4—22no—
448D4—42no—
449D4—416no—
450D4C2×#114214yes—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—416no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4#649512yes—
468C2×C4C2×#114212yes—
469C2×C4#651611yescharacteristic
470C2×C4#651611yescharacteristic
471C2×C4#651611yescharacteristic
472C2×C4C22×C22⋊D711yescharacteristic
473C2×C4C22×C22⋊D711yescharacteristic
474C2×C4C22×C22⋊D711yescharacteristic
475C8—282no—
476C8—282no—
477C8—282no—
478C7#1875711yescharacteristic, complemented, Sylow
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22#8999012yes—
490C22C2×#651611yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—44no—
496C4—42no—
497C4—561no—
498C4—561no—
499C4#8999012yes—
500C4C2×#649511yescharacteristic
501C4C22×#114211yescharacteristic
502C4#9051312yes—
503C2—22no—
504C2—44no—
505C2—82no—
506C2C2×#8999011yescharacteristic
507C2#1174152611yescharacteristic
508C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#547C2
socleC22×C7C2×#120
hypercenterC2×#547D7
hypocenterC7#18757
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. #11740668C2semidirect
3. #11739759C2semidirect
4. #11739741C2semidirect
5. #11729667C2semidirect
6. #11729665C2semidirect
7. #11729527C2semidirect
8. #11729352C2semidirect
9. #11729351C2semidirect
26. #89865C22semidirect
27. #89851C22semidirect
28. #89791C22semidirect
31. #89780C22semidirect
32. #89780C22semidirect
33. #89779C22semidirect
40. #89342C22semidirect
41. D4⋊(C7⋊C8)C22semidirect
42. D4⋊(C7⋊C8)C22semidirect
135. C2×C7×D4SD16semidirect
375. C2×D4#1142semidirect
478. C7#18757semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
#117406682
#117397592
#117397412
#117296672
#117296652
#117295272
#117293522
#117293512
#187577