Finite Group Databasebeta

← #13332634

Subgroups — #13332634

Order 1792.

1170 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×#547C211yesmaximal, characteristic
2#11740669C211yesmaximal, characteristic, complemented
3#11739759C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic, complemented
5#11729666C214yesmaximal, complemented
6#11729665C214yesmaximal, complemented
7#11729529C211yesmaximal, characteristic, complemented
8#11729352C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#547—22no—
12C7×#547—22no—
13C7×#547—22no—
14C7×#547C2214yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×#121—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22#90161C2211yescharacteristic
23#89990—21no—
24#89987—21no—
25#89865C2212yescomplemented
26#89863—22no—
27#89850C2212yescomplemented
28#89791C2212yescomplemented
29C4×#1142—22no—
30#89781—22no—
31#89780C2212yescomplemented
32#89780C2212yescomplemented
33#89779C2212yescomplemented
34#89743—21no—
35#89742—22no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41D4⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yescomplemented
43#18783—71nomaximal, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—48no—
47C22×C7×Q8—21no—
48C22×C7×D4—22no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4D411yescharacteristic
51C23×C4×C7—22no—
52C23×C4×C7—21no—
53C23×C4×C7—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8D412yes—
57C7×C4⋊Q8C2312yes—
58C7×#123—28no—
59C7×#123—28no—
60C7×C22⋊Q8—24no—
61C7×C22⋊Q8—24no—
62C7×#121—24no—
63C7×#121—24no—
64C4×C7×D4—21no—
65C4×C7×D4—21no—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C4×C7×D4D412yes—
69C4×C7×D4D412yes—
70C4×C7×D4C2312yes—
71C4×C7×D4C2312yes—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—24no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D412yes—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C42×C7C2311yescharacteristic
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#6406—22no—
94#6406—22no—
95#6406—22no—
96C22⋊(C7⋊C8)—21no—
97C22⋊(C7⋊C8)—21no—
98#6384—24no—
99#6384—22no—
100#6382—22no—
101#6382—22no—
102#6381—24no—
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×#547D711yescharacteristic
111#3230—71no—
112#3101—71no—
113#3077—71no—
114#1647—74no—
115#1635—74no—
116#1549—71no—
117#1486—71no—
118#1478—71no—
119C24×C7—21no—
120C24×C7—21no—
121C7×#41—416no—
122C7×#41—48no—
123C7×#41—48no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—44no—
127C2×C7×D4—24no—
128C2×C7×D4—28no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—48no—
134C2×C7×D4C2×D412yes—
135C2×C7×D4C2×D412yes—
136C22×C4×C7—28no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—44no—
145C22×C4×C7—48no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C2×Dic14—21no—
152C2×Dic14—21no—
153C7×C4⋊C4—28no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4C2×D412yes—
160C7×C4⋊C4C2×D412yes—
161C7×C4⋊C4C2×D414yes—
162C7×C22⋊C4—28no—
163C7×C22⋊C4—22no—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C7×C22⋊C4—22no—
167C42×C7C2×D411yescharacteristic
168C42×C7C2×D411yescharacteristic
169C42×C7C2×D411yescharacteristic
170C42×C7C2411yescharacteristic
171C22⋊Dic7—41no—
172#1142—44no—
173C4⋊Dic7—21no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177Dic7⋊C4—41no—
178Dic7⋊C4—41no—
179C4×Dic7—41no—
180#1136—42no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C22×#41—21no—
185#547—22no—
186#547—22no—
187#547—22no—
188#547C2×D714yes—
189#533—71no—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×#123—22no—
192C2×C22⋊Q8—21no—
193C2×#121—21no—
194C2×C4×D4C2×D711yescharacteristic
195C2×C4×D4C2×D711yescharacteristic
196C22×C4⋊C4—21no—
197#500—72no—
198#487—72no—
199#484—142no—
200#483—141no—
201#478—142no—
202#477—142no—
203#476—72no—
204#476—72no—
205#474—72no—
206#461—72no—
207#450—141no—
208C4×SD16—142no—
209#431—71no—
210#431—71no—
211#428—141no—
212#423—71no—
213#418—141no—
214#332—74no—
215D4⋊C8—74no—
216D4⋊C8—74no—
217C23×C7—24no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7#12011yescharacteristic
226C7×Q8—44no—
227C7×Q8—44no—
228C7×D4—24no—
229C7×D4—22no—
230C7×D4—22no—
231C7×D4—22no—
232C7×D4—416no—
233C7×D4—42no—
234C7×D4AGL(1,ℤ/8)14yes—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—24no—
239C2×C4×C7—41no—
240C2×C4×C7—44no—
241C2×C4×C7—42no—
242C2×C4×C7—44no—
243C2×C4×C7—416no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—44no—
247C2×C4×C7—42no—
248C2×C4×C7—44no—
249C2×C4×C7#12011yescharacteristic
250C2×C4×C7#12011yescharacteristic
251C2×C4×C7#12011yescharacteristic
252C2×C4×C7#13512yes—
253C2×C4×C7#13712yes—
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×Dic7—41no—
258C2×Dic7—41no—
259Dic14—42no—
260Dic14—42no—
261C7⋊C8—42no—
262C7⋊C8—42no—
263C7⋊C8—42no—
264C2×#41—24no—
265C2×#41—24no—
266C2×#41—48no—
267C22×Q8—21no—
268C22×D4—22no—
269C22×D4C22⋊D711yescharacteristic
270C22×D4C22⋊D711yescharacteristic
271C23×C4—22no—
272C23×C4—21no—
273C23×C4—21no—
274C2×SD16—142no—
275C2×M16—141no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—71no—
279C4⋊Q8—71no—
280C4⋊Q8C22⋊D712yes—
281C4⋊Q8C22×D712yes—
282#125—141no—
283#123—28no—
284#123—28no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—141no—
288C22⋊Q8—141no—
289#121—24no—
290#121—24no—
291C4×Q8—141no—
292C4×Q8—141no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4C22⋊D712yes—
298C4×D4C22⋊D712yes—
299C4×D4C22×D712yes—
300C4×D4C22×D712yes—
301#117—141no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4C22⋊D711yescharacteristic
307C2×C4⋊C4C22⋊D711yescharacteristic
308C2×C4⋊C4C22⋊D712yes—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42C22×D711yescharacteristic
313#107—142no—
314#107—144no—
315#106—142no—
316#106—142no—
317C4⋊C8—72no—
318C4⋊C8—72no—
319C4⋊C8—141no—
320C4⋊C8—141no—
321#103—142no—
322#103—144no—
323#102—142no—
324#102—142no—
325#102—142no—
326C22⋊C8—141no—
327C22⋊C8—141no—
328C4×C8—72no—
329C4×C8—141no—
330C22×C7—21no—
331C22×C7—21no—
332C22×C7—24no—
333C22×C7—48no—
334C22×C7—42no—
335C22×C7—42no—
336C22×C7—42no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—84no—
340C22×C7#45212yes—
341C22×C7C2×#12011yescharacteristic
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—21no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—44no—
348C4×C7#45212yes—
349C4×C7C2×#13511yescharacteristic
350C4×C7C2×AGL(1,ℤ/8)12yes—
351C4×C7C2×#13711yescharacteristic
352Dic7—81no—
353Dic7—81no—
354C24—21no—
355C24—21no—
356#41—416no—
357#41—48no—
358#41—48no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—44no—
362C2×Q8—141no—
363C2×Q8—141no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—48no—
371C2×D4C2×C22⋊D712yes—
372C2×D4C2×C22⋊D712yes—
373C22×C4—28no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4C2×C22⋊D711yescharacteristic
386C22×C4C2×C22⋊D711yescharacteristic
387C22×C4C2×C22⋊D711yescharacteristic
388SD16—284no—
389M16—282no—
390C2×C8—142no—
391C2×C8—142no—
392C2×C8—142no—
393C4⋊C4—28no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—141no—
400C4⋊C4—141no—
401C4⋊C4—141no—
402C4⋊C4—141no—
403C4⋊C4—281no—
404C4⋊C4—281no—
405C4⋊C4C2×C22⋊D712yes—
406C4⋊C4C2×C22⋊D714yes—
407C4⋊C4C2×C22⋊D712yes—
408C22⋊C4—28no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—281no—
414C42—281no—
415C42C2×C22⋊D711yescharacteristic
416C42C2×C22⋊D711yescharacteristic
417C42C2×C22⋊D711yescharacteristic
418C42C23×D711yescharacteristic
419C2×C7—22no—
420C2×C7—44no—
421C2×C7—82no—
422C2×C7C2×#45211yescharacteristic
423C2×C7#300311yescharacteristic
424C2×C7#302011yescharacteristic
425C23—24no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23#651611yescharacteristic
434Q8—44no—
435Q8—44no—
436Q8—282no—
437Q8—282no—
438D4—24no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—416no—
443D4—42no—
444D4#649514yes—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—416no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—281no—
460C2×C4—281no—
461C2×C4#651312yes—
462C2×C4#651412yes—
463C2×C4#651611yescharacteristic
464C2×C4#651611yescharacteristic
465C2×C4#651611yescharacteristic
466C2×C4C22×C22⋊D711yescharacteristic
467C2×C4C22×C22⋊D711yescharacteristic
468C2×C4C22×C22⋊D711yescharacteristic
469C8—282no—
470C8—282no—
471C8—282no—
472C7#1878311yescharacteristic, complemented, Sylow
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22#9001112yes—
484C22C2×#651611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—561no—
492C4—561no—
493C4#9001112yes—
494C4C2×#649512yes—
495C4C2×#651311yescharacteristic
496C4C2×#651411yescharacteristic
497C2—22no—
498C2—44no—
499C2—82no—
500C2#1174148311yescharacteristic
501C2C2×#9001111yescharacteristic
502C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#547C2
socleC22×C7C2×#120
hypercenterC2×#547D7
hypocenterC7#18783
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. #11740669C2semidirect
3. #11739759C2semidirect
4. #11739741C2semidirect
5. #11729666C2semidirect
6. #11729665C2semidirect
7. #11729529C2semidirect
8. #11729352C2semidirect
9. #11729351C2semidirect
25. #89865C22semidirect
27. #89850C22semidirect
28. #89791C22semidirect
31. #89780C22semidirect
32. #89780C22semidirect
33. #89779C22semidirect
40. #89342C22semidirect
41. D4⋊(C7⋊C8)C22semidirect
42. D4⋊(C7⋊C8)C22semidirect
472. C7#18783semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
#117406692
#117397592
#117397412
#117296662
#117296652
#117295292
#117293522
#117293512
#187837