Finite Group Databasebeta

← #13332514

Subgroups — #13332514

Order 1792.

1234 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic
2#11740614C211yesmaximal, characteristic, complemented
3#11739892C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic, complemented
5#11729674C214yesmaximal, complemented
6#11729673C214yesmaximal, complemented
7#11729513C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#550—22no—
12C7×#550—22no—
13C7×#550—22no—
14C7×#550C2214yes—
15C2×C7×#127—21no—
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121—22no—
18C2×C7×#121—21no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22#90161C2211yescharacteristic
23#90000—21no—
24#89990—22no—
25#89987—21no—
26#89848C2212yescomplemented
27#89847—22no—
28#89842C2212yescomplemented
29#89803C2212yescomplemented
30C4×#1144—22no—
31#89793C2212yescomplemented
32#89781—22no—
33#89780C2212yescomplemented
34#89779C2212yescomplemented
35#89742—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89339C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yescomplemented
43#18771—71nomaximal, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—48no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C22×C7×D4D411yescharacteristic
53C22×C7×D4D412yes—
54C23×C4×C7—22no—
55C23×C4×C7—21no—
56C7×#127—24no—
57C7×#127—24no—
58C7×#127—21no—
59C7×#127—21no—
60C7×#127D412yes—
61C7×#127C2312yes—
62C7×#121—24no—
63C7×#121—28no—
64C7×#121—28no—
65C7×#121—24no—
66C4×C7×Q8—21no—
67C4×C7×Q8—21no—
68C4×C7×Q8D412yes—
69C4×C7×Q8C2312yes—
70C4×C7×D4—24no—
71C4×C7×D4—24no—
72C4×C7×D4—21no—
73C4×C7×D4—21no—
74C4×C7×D4D412yes—
75C4×C7×D4C2312yes—
76C2×C7×C4⋊C4—21no—
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—
84C2×#1142—24no—
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#6409—22no—
95#6406—22no—
96#6406—22no—
97#6406—24no—
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)—21no—
100#6384—22no—
101#6382—22no—
102#6381—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×#550D711yescharacteristic
111#3208—71no—
112#3104—71no—
113#3077—71no—
114#1641—74no—
115#1631—74no—
116#1545—71no—
117#1487—71no—
118#1478—71no—
119C24×C7—21no—
120C24×C7—22no—
121C7×#41—416no—
122C7×#41—48no—
123C7×#41—48no—
124C2×C7×Q8—24no—
125C2×C7×Q8—21no—
126C2×C7×Q8C2×D412yes—
127C2×C7×D4—28no—
128C2×C7×D4—28no—
129C2×C7×D4—21no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—48no—
138C2×C7×D4—44no—
139C2×C7×D4—42no—
140C2×C7×D4—42no—
141C2×C7×D4—82no—
142C2×C7×D4—84no—
143C2×C7×D4C2×D412yes—
144C2×C7×D4C2×D414yes—
145C22×C4×C7—21no—
146C22×C4×C7—28no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—48no—
152C22×C4×C7—44no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C2×Dic14—21no—
157C2×Dic14—21no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4C2×D412yes—
163C7×C4⋊C4C2×D412yes—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—22no—
166C7×C22⋊C4—22no—
167C42×C7—22no—
168C42×C7—22no—
169C42×C7C2×D411yescharacteristic
170C42×C7C2×D411yescharacteristic
171C42×C7C2×D411yescharacteristic
172C42×C7C2411yescharacteristic
173C22⋊Dic7—41no—
174#1144—44no—
175#1142—84no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180Dic7⋊C4—41no—
181Dic7⋊C4—41no—
182C4×Dic7—41no—
183#1136—42no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C22×#41—21no—
188#550—22no—
189#550—22no—
190#550—22no—
191#550C2×D714yes—
192#533—71no—
193C2×#127—21no—
194C2×#127C2×D711yescharacteristic
195C2×#121—22no—
196C2×#121—21no—
197C2×C4×Q8C2×D711yescharacteristic
198C2×C4×D4—21no—
199C2×C4×D4C2×D711yescharacteristic
200#492—72no—
201#486—72no—
202#484—141no—
203#478—142no—
204#477—72no—
205#476—72no—
206#474—72no—
207#462—72no—
208#460—142no—
209#451—141no—
210#450—142no—
211C4×Q16—142no—
212#431—71no—
213#431—71no—
214#428—141no—
215#423—71no—
216#418—141no—
217#331—74no—
218Q8⋊C8—74no—
219D4⋊C8—74no—
220C23×C7—22no—
221C23×C7—24no—
222C23×C7—44no—
223C23×C7—42no—
224C23×C7—44no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—44no—
228C23×C7#12011yescharacteristic
229C7×Q8—24no—
230C7×Q8—22no—
231C7×Q8AGL(1,ℤ/8)14yes—
232C7×D4—22no—
233C7×D4—22no—
234C7×D4—44no—
235C7×D4—416no—
236C7×D4—44no—
237C7×D4—42no—
238C7×D4—44no—
239C7×D4—42no—
240C7×D4—42no—
241C7×D4—44no—
242C7×D4—82no—
243C7×D4—88no—
244C2×C4×C7—24no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—41no—
250C2×C4×C7—416no—
251C2×C4×C7—44no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—44no—
255C2×C4×C7#12011yescharacteristic
256C2×C4×C7#12011yescharacteristic
257C2×C4×C7#12011yescharacteristic
258C2×C4×C7#13512yes—
259C2×C4×C7#13712yes—
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×Dic7—41no—
264C2×Dic7—41no—
265Dic14—42no—
266Dic14—42no—
267C7⋊C8—42no—
268C7⋊C8—42no—
269C7⋊C8—42no—
270C2×#41—24no—
271C2×#41—24no—
272C2×#41—48no—
273C22×Q8C22⋊D711yescharacteristic
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C22⋊D711yescharacteristic
279C22×D4C22⋊D712yes—
280C23×C4—22no—
281C23×C4—21no—
282C2×Q16—142no—
283C2×SD16—144no—
284C2×M16—141no—
285C4⋊Q8—71no—
286C4⋊Q8—71no—
287#127—24no—
288#127—24no—
289#127—21no—
290#127—21no—
291#127C22⋊D712yes—
292#127C22×D712yes—
293#125—141no—
294C22⋊Q8—141no—
295C22⋊Q8—141no—
296#121—24no—
297#121—28no—
298#121—28no—
299#121—24no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—141no—
303C4×Q8—141no—
304C4×Q8C22⋊D712yes—
305C4×Q8C22×D712yes—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4C22⋊D712yes—
311C4×D4C22×D712yes—
312#117—141no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C22⋊D711yescharacteristic
315C2×C4⋊C4C22⋊D711yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C42—21no—
319C2×C42C22×D711yescharacteristic
320#107—142no—
321#107—142no—
322#106—142no—
323C4⋊C8—72no—
324C4⋊C8—72no—
325C4⋊C8—141no—
326C4⋊C8—141no—
327#103—142no—
328#103—142no—
329#102—142no—
330#102—142no—
331#102—144no—
332C22⋊C8—141no—
333C22⋊C8—141no—
334C4×C8—72no—
335C4×C8—141no—
336C22×C7—21no—
337C22×C7—21no—
338C22×C7—42no—
339C22×C7—44no—
340C22×C7—44no—
341C22×C7—42no—
342C22×C7—44no—
343C22×C7—42no—
344C22×C7—84no—
345C22×C7—88no—
346C22×C7#45212yes—
347C22×C7C2×#12011yescharacteristic
348C4×C7—24no—
349C4×C7—21no—
350C4×C7—21no—
351C4×C7—21no—
352C4×C7—42no—
353C4×C7—42no—
354C4×C7#45212yes—
355C4×C7C2×#13511yescharacteristic
356C4×C7C2×AGL(1,ℤ/8)12yes—
357C4×C7C2×#13711yescharacteristic
358Dic7—81no—
359Dic7—81no—
360C24—21no—
361C24—22no—
362#41—416no—
363#41—48no—
364#41—48no—
365C2×Q8—24no—
366C2×Q8—21no—
367C2×Q8—141no—
368C2×Q8—141no—
369C2×Q8C2×C22⋊D712yes—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—82no—
385C2×D4—84no—
386C2×D4C2×C22⋊D712yes—
387C2×D4C2×C22⋊D714yes—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4C2×C22⋊D711yescharacteristic
397C22×C4C2×C22⋊D711yescharacteristic
398C22×C4C2×C22⋊D711yescharacteristic
399Q16—284no—
400SD16—564no—
401M16—282no—
402C2×C8—142no—
403C2×C8—142no—
404C2×C8—142no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—141no—
410C4⋊C4—141no—
411C4⋊C4—141no—
412C4⋊C4—141no—
413C4⋊C4—281no—
414C4⋊C4—281no—
415C4⋊C4C2×C22⋊D712yes—
416C4⋊C4C2×C22⋊D712yes—
417C22⋊C4—28no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—281no—
421C42—22no—
422C42—22no—
423C42—281no—
424C42C2×C22⋊D711yescharacteristic
425C42C2×C22⋊D711yescharacteristic
426C42C2×C22⋊D711yescharacteristic
427C42C23×D711yescharacteristic
428C2×C7—22no—
429C2×C7—82no—
430C2×C7—84no—
431C2×C7C2×#45211yescharacteristic
432C2×C7#300311yescharacteristic
433C2×C7#301911yescharacteristic
434C23—22no—
435C23—24no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23#651611yescharacteristic
443Q8—24no—
444Q8—22no—
445Q8—282no—
446Q8—282no—
447Q8#649514yes—
448D4—22no—
449D4—22no—
450D4—44no—
451D4—416no—
452D4—44no—
453D4—44no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—44no—
458D4—82no—
459D4—88no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—416no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4#651312yes—
474C2×C4#651412yes—
475C2×C4#651611yescharacteristic
476C2×C4#651611yescharacteristic
477C2×C4#651611yescharacteristic
478C2×C4C22×C22⋊D711yescharacteristic
479C2×C4C22×C22⋊D711yescharacteristic
480C2×C4C22×C22⋊D711yescharacteristic
481C8—282no—
482C8—282no—
483C8—282no—
484C7#1877111yescharacteristic, complemented, Sylow
485C22—21no—
486C22—21no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—84no—
494C22—88no—
495C22#9001112yes—
496C22C2×#651611yescharacteristic
497C4—24no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—561no—
504C4—561no—
505C4#9001112yes—
506C4C2×#649512yes—
507C4C2×#651311yescharacteristic
508C4C2×#651411yescharacteristic
509C2—22no—
510C2—82no—
511C2—84no—
512C2#1174148311yescharacteristic
513C2C2×#9001111yescharacteristic
514C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#550C2
socleC22×C7C2×#120
hypercenterC2×#550D7
hypocenterC7#18771
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. #11740614C2semidirect
3. #11739892C2semidirect
4. #11739741C2semidirect
5. #11729674C2semidirect
6. #11729673C2semidirect
7. #11729513C2semidirect
8. #11729376C2semidirect
9. #11729351C2semidirect
26. #89848C22semidirect
28. #89842C22semidirect
29. #89803C22semidirect
31. #89793C22semidirect
33. #89780C22semidirect
34. #89779C22semidirect
40. #89339C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
42. D4⋊(C7⋊C8)C22semidirect
484. C7#18771semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5502
#117406142
#117398922
#117397412
#117296742
#117296732
#117295132
#117293762
#117293512
#187717