Finite Group Databasebeta

← #13341793

Subgroups — #13341793

Order 1792.

750 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#486C211yesmaximal, characteristic
2#11740668C211yesmaximal, characteristic, complemented
3#11740643C211yesmaximal, characteristic, complemented
4#11740614C211yesmaximal, characteristic
5#11740604C211yesmaximal, characteristic
6#11735920C211yesmaximal, characteristic, complemented
7#11728947C212yesmaximal, complemented
8#11728944C212yesmaximal, complemented
9#11728937C212yesmaximal, complemented
10#11728935C212yesmaximal, complemented
11C8×#1136C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×#127C2211yescharacteristic
14#90161C2211yescharacteristic
15#90161C2211yescharacteristic
16C7×#486—22no—
17C7×#486C2212yes—
18C7×#486C2212yes—
19C2×C7×#102—22no—
20C2×C4×C7×C8C2211yescharacteristic
21#89997—22no—
22#89990—22no—
23#89987—22no—
24#89888—22no—
25#89866—22no—
26#89865C2212yescomplemented
27#89854—22no—
28#89853C2212yescomplemented
29#89853C2212yescomplemented
30#89851C2212yescomplemented
31#89848C2212yes—
32#89847—22no—
33#89843—22no—
34#89842C2212yes—
35#89842C2212yes—
36#89842C2212yes—
37#89743—22no—
38#89697C2211yescharacteristic
39C4×#1136C2211yescharacteristic
40#89471C2211yescharacteristic, complemented
41C4×Dic28—22no—
42#89454C2211yescharacteristic, complemented
43#89453C2211yescharacteristic, complemented
44#89453C2211yescharacteristic, complemented
45#89249C2212yescomplemented
46C8×C7⋊C8C2212yescomplemented
47#19637—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C22×C7×C8D412yes—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8C2312yes—
57C7×#127—21no—
58C7×#127—21no—
59C7×#127—21no—
60C7×#127C2312yes—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C42×C7C2311yescharacteristic
64C2×#1142—24no—
65#6489—21no—
66#6489—21no—
67#6487—21no—
68#6487—21no—
69#6487—21no—
70#6487—21no—
71C2×#1136D412yes—
72C2×Dic28—22no—
73#6433—21no—
74#6433—21no—
75#6432C2311yescharacteristic
76#6432C2311yescharacteristic
77#6432C2311yescharacteristic
78#6432C2311yescharacteristic
79C4×Dic14—21no—
80C4×Dic14—21no—
81C4×Dic14—21no—
82C4×Dic14—21no—
83C7×#102—24no—
84C7×#102—24no—
85C4×C7×C8D412yes—
86C4×C7×C8C2311yescharacteristic
87C4×C7×C8C2311yescharacteristic
88#6409—24no—
89#6409—24no—
90#6406—24no—
91#6406—24no—
92#6406—24no—
93C22⋊(C7⋊C8)D412yes—
94#6392—22no—
95#6390—22no—
96#6390—22no—
97#6384—24no—
98#6382—24no—
99C4⋊(C7⋊C8)D412yes—
100#6377D412yes—
101C4×C7⋊C8C2312yes—
102C4×C7⋊C8C2312yes—
103#3229—71no—
104#3208—71no—
105#3170—71no—
106#3139—71no—
107#3138—71no—
108C2×#486D711yescharacteristic
109#1720—72no—
110#1719—72no—
111#1711—72no—
112#1708—72no—
113C8×M16—71no—
114C24×C7—41no—
115C2×C7×Q8—22no—
116C2×C7×Q8—41no—
117C2×C7×Q8—44no—
118C2×C7×D4—22no—
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4—44no—
122C2×C7×D4—41no—
123C2×C7×D4—41no—
124C2×C7×D4—41no—
125C2×C7×D4—44no—
126C2×C7×D4—82no—
127C22×C4×C7—41no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C2×Dic14—21no—
132C2×Dic14—21no—
133C2×Dic14—21no—
134C2×Dic14—21no—
135C2×C7×C8—42no—
136C2×C7×C8C2×D412yes—
137C2×C7×C8C2×D412yes—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—41no—
141C7×C4⋊C4—41no—
142C42×C7C2×D411yescharacteristic
143C42×C7C2×D411yescharacteristic
144C42×C7C2×D411yescharacteristic
145C42×C7C2411yescharacteristic
146C22⋊Dic7—41no—
147C22⋊Dic7—41no—
148#1142—84no—
149C4⋊Dic7—21no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157Dic7⋊C4—41no—
158Dic7⋊C4—41no—
159Dic7⋊C4—41no—
160Dic7⋊C4—41no—
161C4×Dic7—41no—
162C4×Dic7—41no—
163#1136—24no—
164C2×C7⋊C8C2×D414yes—
165C2×C7⋊C8C2×D414yes—
166Dic28—44no—
167#533—71no—
168#533—71no—
169C2×C4⋊Q8C2×D711yescharacteristic
170C2×#127C2×D711yescharacteristic
171#500—71no—
172#498—72no—
173#494—71no—
174#492—72no—
175#491—142no—
176#490—142no—
177#487—72no—
178#487—72no—
179#487—72no—
180#487—71no—
181#487—71no—
182#486—22no—
183#486—72no—
184#486—72no—
185#486—72no—
186#486C2×D712yes—
187#486C2×D712yes—
188#483—142no—
189#475—142no—
190#467—142no—
191#460—142no—
192#450—142no—
193C4×Q16—142no—
194#431—71no—
195#419—142no—
196#418—142no—
197C2×#102—22no—
198C4×M16—71no—
199C2×C4×C8C2×D711yescharacteristic
200#322—72no—
201C82—72no—
202C23×C7—42no—
203C23×C7—82no—
204C23×C7—82no—
205C23×C7—82no—
206C23×C7#12711yescharacteristic
207C7×Q8—44no—
208C7×Q8—82no—
209C7×D4—44no—
210C7×D4—44no—
211C7×D4—44no—
212C7×D4—82no—
213C7×D4—82no—
214C7×D4—82no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—42no—
218C2×C4×C7—82no—
219C2×C4×C7#12711yescharacteristic
220C2×C4×C7#12711yescharacteristic
221C2×C4×C7#12711yescharacteristic
222C2×C4×C7#13514yes—
223C2×C4×C7C22×D411yescharacteristic
224C2×C4×C7C22×D411yescharacteristic
225C2×C4×C7C22×D411yescharacteristic
226C2×Dic7—41no—
227C2×Dic7—41no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230Dic14—42no—
231Dic14—42no—
232Dic14—42no—
233Dic14—42no—
234C7×C8—22no—
235C7×C8—22no—
236C7⋊C8—24no—
237C7⋊C8—24no—
238C22×Q8—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C2×Q16—142no—
243C2×SD16—144no—
244C2×M16—72no—
245C22×C8C22⋊D712yes—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—71no—
250C4⋊Q8—71no—
251C4⋊Q8—71no—
252C4⋊Q8—71no—
253C4⋊Q8C22×D712yes—
254#127—21no—
255#127—21no—
256#127—21no—
257#127C22×D712yes—
258#125—141no—
259#125—141no—
260C22⋊Q8—141no—
261C22⋊Q8—141no—
262C22⋊Q8—141no—
263C22⋊Q8—141no—
264C4×Q8—141no—
265C4×Q8—141no—
266C4×Q8—141no—
267C4×Q8—141no—
268#117—141no—
269#117—141no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C42C22×D711yescharacteristic
273#107—142no—
274#106—144no—
275C4⋊C8—72no—
276#103—144no—
277#103—144no—
278#103—144no—
279#103—142no—
280#103—142no—
281#102—24no—
282#102—24no—
283#102—144no—
284#102—144no—
285#102—144no—
286C22⋊C8—72no—
287#97—72no—
288C4×C8—72no—
289C4×C8—72no—
290C4×C8C22⋊D712yes—
291C4×C8C22×D711yescharacteristic
292C4×C8C22×D711yescharacteristic
293C22×C7—21no—
294C22×C7—21no—
295C22×C7—82no—
296C22×C7—82no—
297C22×C7—82no—
298C22×C7—164no—
299C22×C7#49512yes—
300C22×C7C2×#12711yescharacteristic
301C4×C7—21no—
302C4×C7—21no—
303C4×C7—82no—
304C4×C7#49512yes—
305C4×C7C2×#13512yes—
306C4×C7#57512yes—
307C4×C7#57712yes—
308Dic7—81no—
309Dic7—81no—
310Dic7—81no—
311Dic7—81no—
312C24—41no—
313C2×Q8—22no—
314C2×Q8—41no—
315C2×Q8—44no—
316C2×Q8—141no—
317C2×Q8—141no—
318C2×Q8—141no—
319C2×Q8—141no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—82no—
329C22×C4—41no—
330C22×C4D4×D711yescharacteristic
331C22×C4D4×D711yescharacteristic
332C22×C4C2×C22⋊D711yescharacteristic
333Q16—284no—
334SD16—564no—
335M16—144no—
336C2×C8—42no—
337C2×C8—74no—
338C2×C8—74no—
339C2×C8C2×C22⋊D712yes—
340C2×C8C2×C22⋊D712yes—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—141no—
346C4⋊C4—141no—
347C4⋊C4—141no—
348C4⋊C4—141no—
349C4⋊C4—141no—
350C4⋊C4—141no—
351C4⋊C4—141no—
352C4⋊C4—141no—
353C4⋊C4—281no—
354C4⋊C4—281no—
355C4⋊C4—281no—
356C4⋊C4—281no—
357C22⋊C4—281no—
358C22⋊C4—281no—
359C42—281no—
360C42—281no—
361C42D4×D711yescharacteristic
362C42D4×D711yescharacteristic
363C42C2×C22⋊D711yescharacteristic
364C42C23×D711yescharacteristic
365C2×C7—22no—
366C2×C7—162no—
367C2×C7C2×#49511yescharacteristic
368C2×C7#315311yescharacteristic
369C2×C7#315411yescharacteristic
370C23—42no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23#650311yescharacteristic
375Q8—44no—
376Q8—82no—
377Q8—282no—
378Q8—282no—
379Q8—282no—
380Q8—282no—
381D4—44no—
382D4—44no—
383D4—44no—
384D4—82no—
385D4—82no—
386D4—82no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—281no—
392C2×C4—281no—
393C2×C4—281no—
394C2×C4—281no—
395C2×C4#647514yes—
396C2×C4#650311yescharacteristic
397C2×C4#650311yescharacteristic
398C2×C4#650311yescharacteristic
399C2×C4C2×D4×D711yescharacteristic
400C2×C4C2×D4×D711yescharacteristic
401C2×C4C22×C22⋊D711yescharacteristic
402C8—22no—
403C8—22no—
404C8—144no—
405C8—144no—
406C7#1963711yescharacteristic, complemented, Sylow
407C22—21no—
408C22—21no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—164no—
413C22#8992712yes—
414C22C2×#650311yescharacteristic
415C4—21no—
416C4—21no—
417C4—82no—
418C4—561no—
419C4—561no—
420C4—561no—
421C4—561no—
422C4#8992712yes—
423C4C2×#647512yes—
424C4#9045312yes—
425C4#9051312yes—
426C2—22no—
427C2—162no—
428C2#1174038811yescharacteristic
429C2C2×#8992711yescharacteristic
430C2#1174129911yescharacteristic
subgroupG/H
centerC22C2×#6503
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#486C2
socleC22×C7C2×#127
prime radicalC2×C7×#127C22
hypercenterC2×#486D7
hypocenterC7#19637
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
2. #11740668C2semidirect
3. #11740643C2semidirect
6. #11735920C2semidirect
7. #11728947C2semidirect
8. #11728944C2semidirect
9. #11728937C2semidirect
10. #11728935C2semidirect
11. C8×#1136C2semidirect
26. #89865C22semidirect
28. #89853C22semidirect
29. #89853C22semidirect
30. #89851C22semidirect
40. #89471C22semidirect
42. #89454C22semidirect
43. #89453C22semidirect
44. #89453C22semidirect
45. #89249C22semidirect
46. C8×C7⋊C8C22semidirect
406. C7#19637semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4862
#117406682
#117406432
#117406142
#117406042
#117359202
#117289472
#117289442
#117289372
#117289352
C8×#11362
#196377