Finite Group Databasebeta

← #13332367

Subgroups — #13332367

Order 1792.

1224 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2#11740660C211yesmaximal, characteristic, complemented
3#11739753C211yesmaximal, characteristic, complemented
4#11739752C211yesmaximal, characteristic, complemented
5#11729666C214yesmaximal, complemented
6#11729664C214yesmaximal, complemented
7#11729527C211yesmaximal, 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—
22C2×#6495—21no—
23#90168C2211yescharacteristic
24#89989—21no—
25#89860C2212yescomplemented
26#89859—22no—
27#89850C2212yescomplemented
28#89791C2212yescomplemented
29#89788—22no—
30#89787C2212yescomplemented
31#89787C2212yescomplemented
32C4×#1141—22no—
33#89779C2212yescomplemented
34#89739—21no—
35#89739—22no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41D4⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yescomplemented
43#18756—71nomaximal, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
48C22×C7×Q8—21no—
49C22×C7×D4—22no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4D411yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C2×#1157—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8D412yes—
59C7×C4⋊Q8C2312yes—
60C7×#123—28no—
61C7×#123—28no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—24no—
64C7×#121—24no—
65C7×#121—24no—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C4×C7×D4—21no—
69C4×C7×D4—21no—
70C4×C7×D4D412yes—
71C4×C7×D4D412yes—
72C4×C7×D4C2312yes—
73C4×C7×D4C2312yes—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—24no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4D412yes—
81C2×C7×C22⋊C4—22no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C42×C7C2311yescharacteristic
85C2×#1142—22no—
86#6495—44no—
87C2×#1141—22no—
88C2×#1141—22no—
89#6493—21no—
90#6493—21no—
91C2×#1136—21no—
92#6438—21no—
93#6437C2311yescharacteristic
94C4×D28—21no—
95C4×D28—21no—
96#6432C2311yescharacteristic
97#6406—22no—
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)—21no—
100#6384—22no—
101#6383—24no—
102#6383—22no—
103#6383—22no—
104#6381—24no—
105#6381—22no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)—21no—
108C4⋊(C7⋊C8)C2312yes—
109C4⋊(C7⋊C8)C2312yes—
110C4×C7⋊C8—21no—
111C4×C7⋊C8C2312yes—
112C2×#547D711yescharacteristic, complemented
113#3226—71no—
114#3100—71no—
115#3036—71no—
116#1647—74no—
117#1625—74no—
118#1546—71no—
119#1486—71no—
120#1478—71no—
121C24×C7—21no—
122C24×C7—21no—
123C7×#41—48no—
124C7×#41—416no—
125C7×#41—48no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—42no—
129C2×C7×Q8—42no—
130C2×C7×D4—28no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—42no—
135C2×C7×D4—48no—
136C2×C7×D4D814yescomplemented
137C2×C7×D4C2×D412yes—
138C2×C7×D4C2×D412yes—
139C22×C4×C7—28no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—44no—
148C22×C4×C7—48no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C2×C22⋊D7—41no—
155#1157—42no—
156#1157—42no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×C4×D7—41no—
161C2×Dic14—21no—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4C2×D412yes—
169C7×C4⋊C4C2×D412yes—
170C7×C4⋊C4C2×D414yes—
171C7×C22⋊C4—28no—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—22no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C42×C7C2×D411yescharacteristic
177C42×C7C2×D411yescharacteristic
178C42×C7C2×D411yescharacteristic
179C42×C7C2411yescharacteristic
180#1142—44no—
181#1141—44no—
182#1141—44no—
183(C2×D7)⋊C4—41no—
184(C2×D7)⋊C4—41no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187#1136—42no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C22×#41—21no—
192C2×AGL(1,ℤ/8)—141no—
193#547—22no—
194#547—22no—
195#547—22no—
196#547C2×D714yes—
197#532—71no—
198C2×C4⋊Q8C2×D711yescharacteristic
199C2×#123—22no—
200C2×C22⋊Q8—21no—
201C2×#121—21no—
202C2×C4×D4C2×D711yescharacteristic
203C2×C4×D4C2×D711yescharacteristic
204C22×C4⋊C4—21no—
205#500—72no—
206#486—72no—
207#480—141no—
208#480—142no—
209#474—72no—
210#474—142no—
211#463—142no—
212#461—72no—
213#459—72no—
214#459—72no—
215#447—141no—
216C4×D8—142no—
217#431—71no—
218#431—71no—
219#423—71no—
220#417—141no—
221#332—74no—
222D4⋊C8—74no—
223D4⋊C8—74no—
224C23×C7—22no—
225C23×C7—24no—
226C23×C7—22no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7#12011yescharacteristic
233C22×D7—41no—
234C7×Q8—42no—
235C7×Q8—42no—
236C7×Q8—44no—
237C7×D4—22no—
238C7×D4—22no—
239C7×D4—22no—
240C7×D4—24no—
241C7×D4—42no—
242C7×D4—416no—
243C7×D4C2×D814yes—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—24no—
248C2×C4×C7—41no—
249C2×C4×C7—44no—
250C2×C4×C7—42no—
251C2×C4×C7—416no—
252C2×C4×C7—42no—
253C2×C4×C7—44no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—44no—
257C2×C4×C7—44no—
258C2×C4×C7#12011yescharacteristic
259C2×C4×C7#12011yescharacteristic
260C2×C4×C7#12011yescharacteristic
261C2×C4×C7C2×D812yes—
262C2×C4×C7AGL(1,ℤ/8)12yes—
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C22⋊D7—82no—
267C2×Dic7—41no—
268D28—42no—
269D28—41no—
270D28—41no—
271D28—42no—
272C4×D7—82no—
273Dic14—41no—
274Dic14—41no—
275C7⋊C8—42no—
276C7⋊C8—42no—
277C7⋊C8—42no—
278C2×#41—24no—
279C2×#41—24no—
280C2×#41—44no—
281C2×#41—44no—
282C2×#41—141no—
283C22×Q8—21no—
284C22×D4—22no—
285C22×D4C22⋊D711yescharacteristic
286C22×D4C22⋊D711yescharacteristic
287C23×C4—22no—
288C23×C4—21no—
289C23×C4—21no—
290AGL(1,ℤ/8)—284no—
291C2×SD16—142no—
292C2×D8—142no—
293C2×D8—142no—
294C2×M16—141no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—71no—
298C4⋊Q8C22⋊D712yes—
299C4⋊Q8C22×D712yes—
300#127—71no—
301#124—141no—
302#123—28no—
303#123—28no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306#121—24no—
307#121—24no—
308#121—141no—
309#121—141no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—141no—
315C4×D4—141no—
316C4×D4C22⋊D712yes—
317C4×D4C22⋊D712yes—
318C4×D4C22×D712yes—
319C4×D4C22×D712yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4C22⋊D711yescharacteristic
325C2×C4⋊C4C22⋊D711yescharacteristic
326C2×C4⋊C4C22⋊D712yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C42C22×D711yescharacteristic
331#107—142no—
332#107—144no—
333C4⋊C8—72no—
334C4⋊C8—72no—
335C4⋊C8—141no—
336C4⋊C8—141no—
337#103—142no—
338#102—142no—
339#102—142no—
340#102—142no—
341#102—144no—
342C22⋊C8—141no—
343C22⋊C8—141no—
344C4×C8—72no—
345C4×C8—141no—
346C22×C7—21no—
347C22×C7—21no—
348C22×C7—24no—
349C22×C7—48no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—84no—
356C22×C7#44712yes—
357C22×C7C2×#12011yescharacteristic
358C2×D7—81no—
359C2×D7—81no—
360C2×D7—81no—
361C4×C7—21no—
362C4×C7—21no—
363C4×C7—21no—
364C4×C7—42no—
365C4×C7—44no—
366C4×C7—42no—
367C4×C7#44712yes—
368C4×C7C22×D811yescharacteristic
369C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
370C4×C7#57512yes—
371Dic7—81no—
372C24—21no—
373C24—21no—
374#41—48no—
375#41—416no—
376#41—48no—
377#41—282no—
378#41—282no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—141no—
384C2×D4—28no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—48no—
390C2×D4—141no—
391C2×D4—141no—
392C2×D4—141no—
393C2×D4—281no—
394C2×D4#114114yescomplemented
395C2×D4C2×C22⋊D712yes—
396C2×D4C2×C22⋊D712yes—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—281no—
410C22×C4C2×C22⋊D711yescharacteristic
411C22×C4C2×C22⋊D711yescharacteristic
412C22×C4C2×C22⋊D711yescharacteristic
413SD16—284no—
414D8—284no—
415D8—284no—
416M16—282no—
417C2×C8—142no—
418C2×C8—142no—
419C2×C8—142no—
420C4⋊C4—28no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—141no—
427C4⋊C4—141no—
428C4⋊C4C2×C22⋊D712yes—
429C4⋊C4C2×C22⋊D712yes—
430C4⋊C4C2×C22⋊D714yes—
431C22⋊C4—28no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—281no—
437C22⋊C4—281no—
438C42C2×C22⋊D711yescharacteristic
439C42C2×C22⋊D711yescharacteristic
440C42C2×C22⋊D711yescharacteristic
441C42C23×D711yescharacteristic
442C2×C7—22no—
443C2×C7—44no—
444C2×C7—82no—
445C2×C7C2×#44711yescharacteristic
446C2×C7#300511yescharacteristic
447C2×C7#302011yescharacteristic
448D7—161no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—281no—
458C23#651611yescharacteristic
459Q8—42no—
460Q8—42no—
461Q8—44no—
462Q8—281no—
463Q8—281no—
464D4—22no—
465D4—24no—
466D4—22no—
467D4—22no—
468D4—42no—
469D4—416no—
470D4—282no—
471D4—281no—
472D4—281no—
473D4—282no—
474D4—562no—
475D4C2×#114114yes—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—416no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—281no—
491C2×C4—562no—
492C2×C4C2×#114112yes—
493C2×C4#649512yes—
494C2×C4#651611yescharacteristic
495C2×C4#651611yescharacteristic
496C2×C4#651611yescharacteristic
497C2×C4C22×C22⋊D711yescharacteristic
498C2×C4C22×C22⋊D711yescharacteristic
499C2×C4C22×C22⋊D711yescharacteristic
500C8—282no—
501C8—282no—
502C8—282no—
503C7#1875611yescharacteristic, complemented, Sylow
504C22—21no—
505C22—21no—
506C22—24no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22—561no—
515C22—561no—
516C22—561no—
517C22#8998912yes—
518C22C2×#651611yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—44no—
524C4—42no—
525C4—561no—
526C4#8998912yes—
527C4C22×#114111yescharacteristic
528C4C2×#649511yescharacteristic
529C4#9051312yes—
530C2—22no—
531C2—44no—
532C2—82no—
533C2—1121no—
534C2C2×#8998911yescharacteristic
535C2#1174152611yescharacteristic
536C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#547C2
socleC22×C7C2×#120
hypercenterC2×#547D7
hypocenterC7#18756
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
1. C2×C7×#547C2semidirect
2. #11740660C2semidirect
3. #11739753C2semidirect
4. #11739752C2semidirect
5. #11729666C2semidirect
6. #11729664C2semidirect
7. #11729527C2semidirect
8. #11729352C2semidirect
9. #11729351C2semidirect
25. #89860C22semidirect
27. #89850C22semidirect
28. #89791C22semidirect
30. #89787C22semidirect
31. #89787C22semidirect
33. #89779C22semidirect
40. #89342C22semidirect
41. D4⋊(C7⋊C8)C22semidirect
42. D4⋊(C7⋊C8)C22semidirect
112. C2×#547D7semidirect
136. C2×C7×D4D8semidirect
394. C2×D4#1141semidirect
503. C7#18756semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
#117406602
#117397532
#117397522
#117296662
#117296642
#117295272
#117293522
#117293512
#187567