Finite Group Databasebeta

← #13329908

Subgroups — #13329908

Order 1792.

962 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2#11740643C211yesmaximal, characteristic, complemented
3#11739893C211yesmaximal, characteristic, complemented
4#11739760C211yesmaximal, characteristic
5#11729680C214yesmaximal, complemented
6#11729679C214yesmaximal, complemented
7#11729531C211yesmaximal, characteristic, complemented
8#11729377C211yesmaximal, characteristic, complemented
9#11729353C211yesmaximal, characteristic
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—
22#90161C2211yescharacteristic
23#90000—21no—
24#89997—22no—
25#89990—21no—
26#89854—22no—
27#89853C2212yescomplemented
28#89853C2212yescomplemented
29#89803C2212yescomplemented
30#89802—22no—
31#89792C2212yescomplemented
32#89791C2212yes—
33#89790—22no—
34#89779C2212yes—
35#89743—21no—
36#89742—21no—
37#89698C2211yescharacteristic
38#89698C2211yescharacteristic
39C4×#1136C2211yescharacteristic
40#89341C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18530—71nomaximal, Sylow
44C22×C7×Q8D412yes—
45C22×C7×Q8D411yescharacteristic
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8D412yes—
52C7×C4⋊Q8C2312yes—
53C7×#125—24no—
54C7×#125—24no—
55C7×C22⋊Q8—24no—
56C7×C22⋊Q8—28no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—28no—
59C4×C7×Q8—21no—
60C4×C7×Q8—21no—
61C4×C7×Q8D412yes—
62C4×C7×Q8C2312yes—
63C4×C7×D4—24no—
64C4×C7×D4—24no—
65C4×C7×D4—21no—
66C4×C7×D4—21no—
67C4×C7×D4D412yes—
68C4×C7×D4C2312yes—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×#1144—22no—
83C2×#1142—22no—
84#6489—21no—
85#6487—21no—
86#6487—21no—
87C2×#1136—21no—
88#6433—21no—
89#6432C2311yescharacteristic
90#6432C2311yescharacteristic
91C4×Dic14—21no—
92C4×Dic14—21no—
93#6409—24no—
94#6409—24no—
95#6409—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—22no—
100#6384—22no—
101#6382—22no—
102#6381—22no—
103C4⋊(C7⋊C8)C2312yes—
104C4⋊(C7⋊C8)C2312yes—
105#6377—21no—
106#6377—21no—
107#6377—21no—
108C4×C7⋊C8C2312yes—
109C2×#554D711yescharacteristic
110#3139—71no—
111#3113—71no—
112#3110—71no—
113#1644—74no—
114#1643—74no—
115#1558—71no—
116#1491—71no—
117#1490—71no—
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—42no—
122C2×C7×Q8—44no—
123C2×C7×Q8C2×D412yes—
124C2×C7×Q8C2×D414yes—
125C2×C7×D4—24no—
126C2×C7×D4—21no—
127C2×C7×D4C2×D412yes—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—28no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—44no—
136C22×C4×C7—84no—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C2×Dic14—21no—
141C2×Dic14—21no—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—28no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4C2×D412yes—
153C7×C4⋊C4C2×D412yes—
154C7×C22⋊C4—28no—
155C7×C22⋊C4—22no—
156C7×C22⋊C4—22no—
157C42×C7—22no—
158C42×C7—22no—
159C42×C7C2×D411yescharacteristic
160C42×C7C2×D411yescharacteristic
161C42×C7C2×D411yescharacteristic
162C42×C7C2411yescharacteristic
163C22⋊Dic7—41no—
164#1144—44no—
165#1142—44no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170Dic7⋊C4—41no—
171Dic7⋊C4—41no—
172C4×Dic7—41no—
173#1136—42no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177#554—22no—
178#554—22no—
179#554—22no—
180#554C2×D714yes—
181#533—71no—
182C2×C4⋊Q8C2×D711yescharacteristic
183C2×#125—21no—
184C2×C22⋊Q8—21no—
185C2×C22⋊Q8—22no—
186C2×C4×Q8C2×D711yescharacteristic
187C2×C4×D4—21no—
188C2×C4×D4C2×D711yescharacteristic
189C22×C4⋊C4—21no—
190#491—142no—
191#487—72no—
192#487—72no—
193#484—141no—
194#483—141no—
195#475—72no—
196#474—72no—
197#462—72no—
198#461—72no—
199#451—141no—
200#450—141no—
201#441—142no—
202#440—142no—
203#432—71no—
204#432—71no—
205#419—142no—
206C4×M16—71no—
207#333—74no—
208Q8⋊C8—74no—
209D4⋊C8—74no—
210C23×C7—24no—
211C23×C7—22no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7#12011yescharacteristic
215C7×Q8—22no—
216C7×Q8—22no—
217C7×Q8—44no—
218C7×Q8—42no—
219C7×Q8—82no—
220C7×D4—24no—
221C7×D4—22no—
222C7×D4#13714yes—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—44no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—44no—
235C2×C4×C7—44no—
236C2×C4×C7—88no—
237C2×C4×C7#12011yescharacteristic
238C2×C4×C7#12011yescharacteristic
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7C22×D411yescharacteristic
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C2×Dic7—41no—
244C2×Dic7—41no—
245Dic14—42no—
246Dic14—42no—
247C7⋊C8—42no—
248C7⋊C8—42no—
249C7⋊C8—42no—
250C22×Q8C22⋊D711yescharacteristic
251C22×Q8C22⋊D712yes—
252C22×D4C22⋊D711yescharacteristic
253C23×C4—22no—
254C23×C4—21no—
255C2×Q16—142no—
256C2×SD16—142no—
257C2×M16—141no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—71no—
261C4⋊Q8—71no—
262C4⋊Q8C22⋊D712yes—
263C4⋊Q8C22×D712yes—
264#125—24no—
265#125—24no—
266#125—141no—
267C22⋊Q8—24no—
268C22⋊Q8—28no—
269C22⋊Q8—24no—
270C22⋊Q8—28no—
271C22⋊Q8—141no—
272C22⋊Q8—141no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—141no—
276C4×Q8—141no—
277C4×Q8C22⋊D712yes—
278C4×Q8C22×D712yes—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C22⋊D712yes—
284C4×D4C22×D712yes—
285#117—141no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4C22⋊D711yescharacteristic
294C2×C4⋊C4C22⋊D711yescharacteristic
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C42—21no—
298C2×C42C22×D711yescharacteristic
299#107—142no—
300#106—142no—
301C4⋊C8—72no—
302C4⋊C8—72no—
303#103—144no—
304#103—144no—
305#103—142no—
306#103—142no—
307#103—142no—
308#102—142no—
309C22⋊C8—141no—
310C22⋊C8—141no—
311#97—141no—
312#97—141no—
313#97—141no—
314C4×C8—72no—
315C22×C7—21no—
316C22×C7—21no—
317C22×C7—24no—
318C22×C7—48no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7#45512yes—
322C22×C7C2×#12011yescharacteristic
323C4×C7—21no—
324C4×C7—21no—
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—42no—
328C4×C7—42no—
329C4×C7—42no—
330C4×C7—44no—
331C4×C7C2×#13712yes—
332C4×C7#57512yes—
333Dic7—81no—
334Dic7—81no—
335C24—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—42no—
339C2×Q8—44no—
340C2×Q8—141no—
341C2×Q8—141no—
342C2×Q8C2×C22⋊D712yes—
343C2×Q8C2×C22⋊D714yes—
344C2×D4—24no—
345C2×D4—21no—
346C2×D4C2×C22⋊D712yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—44no—
355C22×C4—84no—
356C22×C4C2×C22⋊D711yescharacteristic
357C22×C4C2×C22⋊D711yescharacteristic
358C22×C4C2×C22⋊D711yescharacteristic
359Q16—284no—
360SD16—284no—
361M16—282no—
362C2×C8—142no—
363C2×C8—142no—
364C2×C8—142no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—141no—
376C4⋊C4—141no—
377C4⋊C4—141no—
378C4⋊C4—141no—
379C4⋊C4—281no—
380C4⋊C4—281no—
381C4⋊C4C2×C22⋊D712yes—
382C4⋊C4C2×C22⋊D712yes—
383C22⋊C4—28no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—281no—
387C42—22no—
388C42—22no—
389C42—281no—
390C42C2×C22⋊D711yescharacteristic
391C42C2×C22⋊D711yescharacteristic
392C42C2×C22⋊D711yescharacteristic
393C42C23×D711yescharacteristic
394C2×C7—22no—
395C2×C7—44no—
396C2×C7#300411yescharacteristic
397C2×C7#300511yescharacteristic
398C2×C7C2×#45511yescharacteristic
399C23—24no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23#651611yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—44no—
407Q8—42no—
408Q8—82no—
409Q8—282no—
410Q8—282no—
411D4—24no—
412D4—22no—
413D4#650514yes—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—88no—
428C2×C4—281no—
429C2×C4—281no—
430C2×C4#651611yescharacteristic
431C2×C4#651611yescharacteristic
432C2×C4#651611yescharacteristic
433C2×C4C22×C22⋊D711yescharacteristic
434C2×C4C22×C22⋊D711yescharacteristic
435C2×C4C22×C22⋊D711yescharacteristic
436C8—282no—
437C8—282no—
438C8—282no—
439C7#1853011yescharacteristic, complemented, Sylow
440C22—21no—
441C22—21no—
442C22—24no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22#9001412yes—
447C22C2×#651611yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—44no—
456C4—561no—
457C4—561no—
458C4C2×#650512yes—
459C4#9051312yes—
460C2—22no—
461C2—44no—
462C2#1174150411yescharacteristic
463C2#1174152611yescharacteristic
464C2C2×#9001411yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#554C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7#18530
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
2. #11740643C2semidirect
3. #11739893C2semidirect
5. #11729680C2semidirect
6. #11729679C2semidirect
7. #11729531C2semidirect
8. #11729377C2semidirect
27. #89853C22semidirect
28. #89853C22semidirect
29. #89803C22semidirect
31. #89792C22semidirect
40. #89341C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
439. C7#18530semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
#117406432
#117398932
#117397602
#117296802
#117296792
#117295312
#117293772
#117293532
#185307