Finite Group Databasebeta

← #13332532

Subgroups — #13332532

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2#11740667C211yesmaximal, characteristic, complemented
3#11739759C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic, complemented
5#11729683C214yesmaximal, complemented
6#11729682C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic, complemented
8#11729352C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic, complemented
10C7×#555—22no—
11C7×#555—22no—
12C7×#555—22no—
13C7×#555C2214yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#124—22no—
16C2×C7×#121—21no—
17C2×C7×#121—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C2×C7×#117—21no—
21C2×C7×#117—21no—
22#90161C2211yescharacteristic
23#90000—22no—
24#89990—21no—
25#89987—21no—
26#89864C2212yescomplemented
27#89862—22no—
28#89853C2212yescomplemented
29#89791C2212yescomplemented
30C4×#1142—22no—
31#89781—22no—
32#89780C2212yescomplemented
33#89780C2212yescomplemented
34#89779C2212yescomplemented
35#89743—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yescomplemented
41D4⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yescomplemented
43#18773—71nomaximal, Sylow
44C22×C7×Q8D412yes—
45C22×C7×D4—22no—
46C22×C7×D4D411yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8D412yes—
53C7×C4⋊Q8C2312yes—
54C7×#124—28no—
55C7×#124—28no—
56C7×#121—24no—
57C7×#121—24no—
58C7×#121—24no—
59C7×#121—24no—
60C4×C7×D4—21no—
61C4×C7×D4—21no—
62C4×C7×D4—21no—
63C4×C7×D4—21no—
64C4×C7×D4D412yes—
65C4×C7×D4D412yes—
66C4×C7×D4C2312yes—
67C4×C7×D4C2312yes—
68C7×#117—24no—
69C7×#117—24no—
70C7×#117—24no—
71C7×#117—24no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7—22no—
81C2×C42×C7C2311yescharacteristic
82C2×#1144—24no—
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#6406—22no—
95#6406—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—22no—
100#6382—22no—
101#6382—22no—
102#6381—22no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)C2312yes—
106C4⋊(C7⋊C8)C2312yes—
107C4×C7⋊C8—21no—
108C4×C7⋊C8C2312yes—
109C2×#555D711yescharacteristic
110#3209—71no—
111#3101—71no—
112#3077—71no—
113#1646—74no—
114#1634—74no—
115#1548—71no—
116#1486—71no—
117#1478—71no—
118C24×C7—21no—
119C24×C7—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—82no—
122C2×C7×Q8C2×D414yes—
123C2×C7×D4—28no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—42no—
129C2×C7×D4—84no—
130C2×C7×D4C2×D412yes—
131C2×C7×D4C2×D412yes—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—44no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C2×Dic14—21no—
146C2×Dic14—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4C2×D412yes—
154C7×C4⋊C4C2×D412yes—
155C7×C22⋊C4—28no—
156C7×C22⋊C4—28no—
157C7×C22⋊C4—22no—
158C7×C22⋊C4—22no—
159C7×C22⋊C4—22no—
160C7×C22⋊C4—22no—
161C42×C7—28no—
162C42×C7C2×D411yescharacteristic
163C42×C7C2×D411yescharacteristic
164C42×C7C2×D411yescharacteristic
165C42×C7C2411yescharacteristic
166C22⋊Dic7—41no—
167#1144—84no—
168#1142—44no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173Dic7⋊C4—41no—
174Dic7⋊C4—41no—
175C4×Dic7—41no—
176#1136—42no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180#555—22no—
181#555—22no—
182#555—22no—
183#555C2×D714yes—
184#533—71no—
185C2×C4⋊Q8C2×D711yescharacteristic
186C2×#124—22no—
187C2×#121—21no—
188C2×#121—21no—
189C2×C4×D4C2×D711yescharacteristic
190C2×C4×D4C2×D711yescharacteristic
191C2×#117—21no—
192C2×#117—21no—
193#494—72no—
194#487—72no—
195#483—141no—
196#478—142no—
197#476—72no—
198#476—72no—
199#474—72no—
200#462—142no—
201#461—72no—
202#451—142no—
203#450—141no—
204C4×SD16—142no—
205#431—71no—
206#431—71no—
207#428—141no—
208#423—71no—
209#418—141no—
210#333—74no—
211D4⋊C8—74no—
212D4⋊C8—74no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7#12011yescharacteristic
222C7×Q8—44no—
223C7×Q8—82no—
224C7×D4—22no—
225C7×D4—22no—
226C7×D4—22no—
227C7×D4—22no—
228C7×D4—42no—
229C7×D4—42no—
230C7×D4—88no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—24no—
235C2×C4×C7—41no—
236C2×C4×C7—44no—
237C2×C4×C7—42no—
238C2×C4×C7—44no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—44no—
243C2×C4×C7—44no—
244C2×C4×C7#12011yescharacteristic
245C2×C4×C7#12011yescharacteristic
246C2×C4×C7#12011yescharacteristic
247C2×C4×C7#13512yes—
248C2×C4×C7#13712yes—
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C22×D411yescharacteristic
252C2×Dic7—41no—
253C2×Dic7—41no—
254Dic14—42no—
255Dic14—42no—
256C7⋊C8—42no—
257C7⋊C8—42no—
258C7⋊C8—42no—
259C22×Q8C22⋊D712yes—
260C22×D4—22no—
261C22×D4C22⋊D711yescharacteristic
262C22×D4C22⋊D711yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C2×Q16—144no—
266C2×SD16—142no—
267C2×M16—141no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—71no—
271C4⋊Q8—71no—
272C4⋊Q8C22⋊D712yes—
273C4⋊Q8C22×D712yes—
274#125—141no—
275#124—28no—
276#124—28no—
277C22⋊Q8—141no—
278C22⋊Q8—141no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283C4×Q8—141no—
284C4×Q8—141no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C22⋊D712yes—
290C4×D4C22⋊D712yes—
291C4×D4C22×D712yes—
292C4×D4C22×D712yes—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—141no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C22⋊D711yescharacteristic
301C2×C4⋊C4C22⋊D711yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C42—22no—
307C2×C42C22×D711yescharacteristic
308#107—142no—
309#106—142no—
310#106—142no—
311C4⋊C8—72no—
312C4⋊C8—72no—
313C4⋊C8—141no—
314C4⋊C8—141no—
315#103—144no—
316#103—142no—
317#102—142no—
318#102—142no—
319#102—142no—
320C22⋊C8—141no—
321C22⋊C8—141no—
322C4×C8—72no—
323C4×C8—141no—
324C22×C7—21no—
325C22×C7—21no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7—42no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—84no—
333C22×C7—84no—
334C22×C7#45212yes—
335C22×C7C2×#12011yescharacteristic
336C4×C7—21no—
337C4×C7—21no—
338C4×C7—21no—
339C4×C7—42no—
340C4×C7—42no—
341C4×C7—44no—
342C4×C7#45212yes—
343C4×C7C2×#13511yescharacteristic
344C4×C7C2×#13711yescharacteristic
345C4×C7#57512yes—
346Dic7—81no—
347Dic7—81no—
348C24—21no—
349C24—21no—
350C2×Q8—22no—
351C2×Q8—82no—
352C2×Q8—141no—
353C2×Q8—141no—
354C2×Q8C2×C22⋊D714yes—
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—42no—
361C2×D4—84no—
362C2×D4C2×C22⋊D712yes—
363C2×D4C2×C22⋊D712yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C2×C22⋊D711yescharacteristic
375C22×C4C2×C22⋊D711yescharacteristic
376C22×C4C2×C22⋊D711yescharacteristic
377Q16—564no—
378SD16—284no—
379M16—282no—
380C2×C8—142no—
381C2×C8—142no—
382C2×C8—142no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—141no—
390C4⋊C4—141no—
391C4⋊C4—141no—
392C4⋊C4—141no—
393C4⋊C4—281no—
394C4⋊C4—281no—
395C4⋊C4C2×C22⋊D712yes—
396C4⋊C4C2×C22⋊D712yes—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—281no—
404C42—28no—
405C42—281no—
406C42C2×C22⋊D711yescharacteristic
407C42C2×C22⋊D711yescharacteristic
408C42C2×C22⋊D711yescharacteristic
409C42C23×D711yescharacteristic
410C2×C7—22no—
411C2×C7—82no—
412C2×C7—82no—
413C2×C7C2×#45211yescharacteristic
414C2×C7#300511yescharacteristic
415C2×C7#301911yescharacteristic
416C23—22no—
417C23—22no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23#651611yescharacteristic
425Q8—44no—
426Q8—82no—
427Q8—282no—
428Q8—282no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—42no—
434D4—42no—
435D4—88no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—281no—
450C2×C4—281no—
451C2×C4#651312yes—
452C2×C4#651412yes—
453C2×C4#651611yescharacteristic
454C2×C4#651611yescharacteristic
455C2×C4#651611yescharacteristic
456C2×C4C22×C22⋊D711yescharacteristic
457C2×C4C22×C22⋊D711yescharacteristic
458C2×C4C22×C22⋊D711yescharacteristic
459C8—282no—
460C8—282no—
461C8—282no—
462C7#1877311yescharacteristic, complemented, Sylow
463C22—21no—
464C22—21no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—84no—
472C22—84no—
473C22#9001112yes—
474C22C2×#651611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—44no—
481C4—561no—
482C4—561no—
483C4#9001112yes—
484C4C2×#651311yescharacteristic
485C4#9051312yes—
486C4C2×#651411yescharacteristic
487C2—22no—
488C2—82no—
489C2—82no—
490C2#1174152611yescharacteristic
491C2C2×#9001111yescharacteristic
492C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#555C2
socleC22×C7C2×#120
hypercenterC2×#555D7
hypocenterC7#18773
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
2. #11740667C2semidirect
3. #11739759C2semidirect
4. #11739741C2semidirect
5. #11729683C2semidirect
6. #11729682C2semidirect
7. #11729528C2semidirect
8. #11729352C2semidirect
9. #11729351C2semidirect
26. #89864C22semidirect
28. #89853C22semidirect
29. #89791C22semidirect
32. #89780C22semidirect
33. #89780C22semidirect
34. #89779C22semidirect
40. #89341C22semidirect
41. D4⋊(C7⋊C8)C22semidirect
42. D4⋊(C7⋊C8)C22semidirect
462. C7#18773semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5552
#117406672
#117397592
#117397412
#117296832
#117296822
#117295282
#117293522
#117293512
#187737