Finite Group Databasebeta

← #13332319

Subgroups — #13332319

Order 1792.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2#11740658C211yesmaximal, characteristic, complemented
3#11739886C211yesmaximal, characteristic, complemented
4#11739753C211yesmaximal, characteristic
5#11729680C214yesmaximal, complemented
6#11729677C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic, complemented
8#11729375C211yesmaximal, characteristic, complemented
9#11729352C211yesmaximal, characteristic, complemented
10C7×#554—22no—
11C7×#554—22no—
12C7×#554—22no—
13C7×#554C2214yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22C2×#6505—21no—
23#90168C2211yescharacteristic
24#89999—22no—
25#89989—21no—
26#89857C2212yescomplemented
27#89855—22no—
28#89853C2212yescomplemented
29#89803C2212yescomplemented
30#89800—22no—
31#89799C2212yescomplemented
32#89787C2212yes—
33C4×#1141—22no—
34#89779C2212yes—
35#89739—21no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18752—71nomaximal, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8D412yes—
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—21no—
49C2×#1157—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8D412yes—
53C7×C4⋊Q8C2312yes—
54C7×#125—24no—
55C7×#125—24no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—28no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—28no—
60C4×C7×Q8—21no—
61C4×C7×Q8—21no—
62C4×C7×Q8D412yes—
63C4×C7×Q8C2312yes—
64C4×C7×D4—24no—
65C4×C7×D4—24no—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C4×C7×D4D412yes—
69C4×C7×D4C2312yes—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—24no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—24no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×#1144—22no—
84#6505—44no—
85C2×#1143—22no—
86C2×#1143—24no—
87C2×#1141—22no—
88#6493—21no—
89#6493—21no—
90C2×#1136—21no—
91#6438—21no—
92#6437C2311yescharacteristic
93C4×D28—21no—
94C4×D28—21no—
95#6432C2311yescharacteristic
96#6409—24no—
97#6406—22no—
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)—21no—
100#6384—22no—
101#6383—22no—
102#6383—22no—
103#6381—22no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)C2312yes—
107C4⋊(C7⋊C8)C2312yes—
108C4×C7⋊C8—21no—
109C4×C7⋊C8C2312yes—
110C2×#554D711yescharacteristic, complemented
111#3206—71no—
112#3100—71no—
113#3037—71no—
114#1644—74no—
115#1622—74no—
116#1548—71no—
117#1486—71no—
118#1479—71no—
119C24×C7—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—82no—
125C2×C7×Q8C2×D412yes—
126C2×C7×Q8C2×D414yes—
127C2×C7×D4—24no—
128C2×C7×D4—21no—
129C2×C7×D4C2×D412yes—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—28no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7—84no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C2×C22⋊D7—41no—
144#1157—42no—
145#1157—42no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28—21no—
149C2×C4×D7—41no—
150C2×Dic14—21no—
151C7×C4⋊C4—28no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4C2×D412yes—
162C7×C4⋊C4C2×D412yes—
163C7×C22⋊C4—28no—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C42×C7—22no—
167C42×C7—22no—
168C42×C7C2×D411yescharacteristic
169C42×C7C2×D411yescharacteristic
170C42×C7C2×D411yescharacteristic
171C42×C7C2411yescharacteristic
172#1144—44no—
173#1143—44no—
174#1143—84no—
175#1141—44no—
176(C2×D7)⋊C4—41no—
177(C2×D7)⋊C4—41no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180#1136—42no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×#137—141no—
185#554—22no—
186#554—22no—
187#554—22no—
188#554C2×D714yes—
189#532—71no—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×#125—21no—
192C2×C22⋊Q8—21no—
193C2×C22⋊Q8—22no—
194C2×C4×Q8C2×D711yescharacteristic
195C2×C4×D4—21no—
196C2×C4×D4C2×D711yescharacteristic
197C22×C4⋊C4—21no—
198#492—72no—
199#487—72no—
200#480—141no—
201#474—72no—
202#464—142no—
203#462—72no—
204#461—142no—
205#460—72no—
206#459—72no—
207#448—142no—
208#447—141no—
209C4×D8—142no—
210#431—71no—
211#431—71no—
212#423—71no—
213#417—141no—
214#333—74no—
215Q8⋊C8—74no—
216D4⋊C8—74no—
217C23×C7—24no—
218C23×C7—22no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7#12011yescharacteristic
222C22×D7—41no—
223C7×Q8—22no—
224C7×Q8—22no—
225C7×Q8—44no—
226C7×Q8—42no—
227C7×Q8—82no—
228C7×D4—24no—
229C7×D4—22no—
230C7×D4#13714yescomplemented
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—24no—
236C2×C4×C7—41no—
237C2×C4×C7—42no—
238C2×C4×C7—44no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—44no—
244C2×C4×C7—88no—
245C2×C4×C7#12011yescharacteristic
246C2×C4×C7#12011yescharacteristic
247C2×C4×C7#12011yescharacteristic
248C2×C4×C7#13512yes—
249C2×C4×C7AGL(1,ℤ/8)12yes—
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C22⋊D7—82no—
254C2×Dic7—41no—
255D28—42no—
256D28—41no—
257D28—41no—
258D28—42no—
259C4×D7—82no—
260Dic14—41no—
261Dic14—41no—
262C7⋊C8—42no—
263C7⋊C8—42no—
264C7⋊C8—42no—
265C2×#41—141no—
266C22×Q8C22⋊D711yescharacteristic
267C22×Q8C22⋊D712yes—
268C22×D4C22⋊D711yescharacteristic
269C23×C4—22no—
270C23×C4—21no—
271#137—284no—
272C2×Q16—142no—
273C2×SD16—142no—
274C2×SD16—144no—
275C2×D8—142no—
276C2×M16—141no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—71no—
280C4⋊Q8C22⋊D712yes—
281C4⋊Q8C22×D712yes—
282#127—71no—
283#125—24no—
284#125—24no—
285#124—141no—
286C22⋊Q8—24no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8—28no—
290#121—141no—
291#121—141no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8C22⋊D712yes—
295C4×Q8C22×D712yes—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—141no—
301C4×D4—141no—
302C4×D4C22⋊D712yes—
303C4×D4C22×D712yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4C22⋊D711yescharacteristic
312C2×C4⋊C4C22⋊D711yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42C22×D711yescharacteristic
317#107—142no—
318C4⋊C8—72no—
319C4⋊C8—72no—
320C4⋊C8—141no—
321C4⋊C8—141no—
322#103—144no—
323#103—142no—
324#102—142no—
325#102—142no—
326#102—142no—
327C22⋊C8—141no—
328C22⋊C8—141no—
329C4×C8—72no—
330C4×C8—141no—
331C22×C7—21no—
332C22×C7—21no—
333C22×C7—24no—
334C22×C7—48no—
335C22×C7—42no—
336C22×C7—42no—
337C22×C7#44912yes—
338C22×C7C2×#12011yescharacteristic
339C2×D7—81no—
340C2×D7—81no—
341C2×D7—81no—
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—21no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—42no—
348C4×C7—44no—
349C4×C7#44912yes—
350C4×C7C2×#13511yescharacteristic
351C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
352C4×C7C2×#13712yes—
353Dic7—81no—
354C24—21no—
355#41—282no—
356#41—282no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—82no—
362C2×Q8—141no—
363C2×Q8C2×C22⋊D712yes—
364C2×Q8C2×C22⋊D714yes—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—141no—
368C2×D4—141no—
369C2×D4—141no—
370C2×D4—281no—
371C2×D4C2×C22⋊D712yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—84no—
382C22×C4—281no—
383C22×C4C2×C22⋊D711yescharacteristic
384C22×C4C2×C22⋊D711yescharacteristic
385C22×C4C2×C22⋊D711yescharacteristic
386Q16—284no—
387SD16—284no—
388SD16—564no—
389D8—284no—
390M16—282no—
391C2×C8—142no—
392C2×C8—142no—
393C2×C8—142no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—141no—
405C4⋊C4—141no—
406C4⋊C4C2×C22⋊D712yes—
407C4⋊C4C2×C22⋊D712yes—
408C22⋊C4—28no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—281no—
412C22⋊C4—281no—
413C42—22no—
414C42—22no—
415C42C2×C22⋊D711yescharacteristic
416C42C2×C22⋊D711yescharacteristic
417C42C2×C22⋊D711yescharacteristic
418C42C23×D711yescharacteristic
419C2×C7—22no—
420C2×C7—44no—
421C2×C7C2×#44911yescharacteristic
422C2×C7#300411yescharacteristic
423C2×C7#301911yescharacteristic
424D7—161no—
425C23—24no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—281no—
430C23#651611yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—44no—
434Q8—42no—
435Q8—82no—
436Q8—281no—
437Q8—281no—
438D4—24no—
439D4—22no—
440D4—282no—
441D4—282no—
442D4—281no—
443D4—281no—
444D4—562no—
445D4#650514yescomplemented
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—88no—
460C2×C4—281no—
461C2×C4—562no—
462C2×C4#651212yes—
463C2×C4#651312yes—
464C2×C4#651611yescharacteristic
465C2×C4#651611yescharacteristic
466C2×C4#651611yescharacteristic
467C2×C4C22×C22⋊D711yescharacteristic
468C2×C4C22×C22⋊D711yescharacteristic
469C2×C4C22×C22⋊D711yescharacteristic
470C8—282no—
471C8—282no—
472C8—282no—
473C7#1875211yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—24no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—561no—
481C22—561no—
482C22—561no—
483C22#9001012yes—
484C22C2×#651611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—561no—
493C4#9001012yes—
494C4C2×#650512yes—
495C4C2×#651211yescharacteristic
496C4C2×#651311yescharacteristic
497C2—22no—
498C2—44no—
499C2—1121no—
500C2#1174150411yescharacteristic
501C2C2×#9001011yescharacteristic
502C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#554C2
socleC22×C7C2×#120
prime radical#11739753C2
hypercenterC2×#554D7
hypocenterC7#18752
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C7×#554C2semidirect
2. #11740658C2semidirect
3. #11739886C2semidirect
5. #11729680C2semidirect
6. #11729677C2semidirect
7. #11729528C2semidirect
8. #11729375C2semidirect
9. #11729352C2semidirect
26. #89857C22semidirect
28. #89853C22semidirect
29. #89803C22semidirect
31. #89799C22semidirect
40. #89341C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
110. C2×#554D7semidirect
230. C7×D4#137semidirect
445. D4#6505semidirect
473. C7#18752semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
#117406582
#117398862
#117397532
#117296802
#117296772
#117295282
#117293752
#117293522
#187527