Finite Group Databasebeta

← #13332478

Subgroups — #13332478

Order 1792.

1234 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic
2#11740614C211yesmaximal, characteristic, complemented
3#11739876C211yesmaximal, characteristic, complemented
4#11739759C211yesmaximal, characteristic, complemented
5#11729675C214yesmaximal, complemented
6#11729673C214yesmaximal, complemented
7#11729513C211yesmaximal, characteristic, complemented
8#11729375C211yesmaximal, characteristic, complemented
9#11729352C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#550—22no—
12C7×#550—22no—
13C7×#550—22no—
14C7×#550C2214yes—
15C2×C7×#127—21no—
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121—22no—
18C2×C7×#121—21no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22#90161C2211yescharacteristic
23#89997—21no—
24#89990—21no—
25#89990—22no—
26#89848C2212yescomplemented
27#89847—22no—
28#89842C2212yescomplemented
29#89794—22no—
30#89793C2212yescomplemented
31#89792C2212yescomplemented
32#89791C2212yescomplemented
33C4×#1142—22no—
34#89780C2212yescomplemented
35#89743—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89339C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yescomplemented
43#18767—71nomaximal, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
48C22×C7×Q8D411yescharacteristic
49C22×C7×D4—22no—
50C22×C7×D4—22no—
51C22×C7×D4—21no—
52C22×C7×D4—21no—
53C22×C7×D4D411yescharacteristic
54C22×C7×D4D412yes—
55C23×C4×C7—22no—
56C23×C4×C7—21no—
57C7×#127—24no—
58C7×#127—24no—
59C7×#127—21no—
60C7×#127—21no—
61C7×#127D412yes—
62C7×#127C2312yes—
63C7×#121—24no—
64C7×#121—28no—
65C7×#121—28no—
66C7×#121—24no—
67C4×C7×Q8—21no—
68C4×C7×Q8—21no—
69C4×C7×Q8D412yes—
70C4×C7×Q8C2312yes—
71C4×C7×D4—24no—
72C4×C7×D4—24no—
73C4×C7×D4—21no—
74C4×C7×D4—21no—
75C4×C7×D4D412yes—
76C4×C7×D4C2312yes—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×#1142—22no—
85C2×#1142—24no—
86#6489—21no—
87#6487—21no—
88#6487—21no—
89C2×#1136—21no—
90#6433—21no—
91#6432C2311yescharacteristic
92#6432C2311yescharacteristic
93C4×Dic14—21no—
94C4×Dic14—21no—
95#6409—22no—
96#6409—22no—
97#6406—22no—
98#6406—24no—
99C22⋊(C7⋊C8)—21no—
100C22⋊(C7⋊C8)—21no—
101#6384—22no—
102#6382—22no—
103#6382—22no—
104#6381—22no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2312yes—
109C4×C7⋊C8—21no—
110C4×C7⋊C8C2312yes—
111C2×#550D711yescharacteristic
112#3208—71no—
113#3101—71no—
114#3078—71no—
115#1642—74no—
116#1631—74no—
117#1545—71no—
118#1486—71no—
119#1479—71no—
120C24×C7—22no—
121C24×C7—21no—
122C7×#41—48no—
123C7×#41—416no—
124C7×#41—48no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8SD1614yescomplemented
128C2×C7×Q8C2×D412yes—
129C2×C7×D4—28no—
130C2×C7×D4—28no—
131C2×C7×D4—21no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—42no—
138C2×C7×D4—44no—
139C2×C7×D4—48no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4—82no—
143C2×C7×D4—84no—
144C2×C7×D4C2×D412yes—
145C2×C7×D4C2×D414yes—
146C22×C4×C7—21no—
147C22×C4×C7—28no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—48no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C2×Dic14—21no—
159C2×Dic14—21no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4C2×D412yes—
165C7×C4⋊C4C2×D412yes—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—22no—
168C7×C22⋊C4—22no—
169C42×C7—22no—
170C42×C7—22no—
171C42×C7C2×D411yescharacteristic
172C42×C7C2×D411yescharacteristic
173C42×C7C2×D411yescharacteristic
174C42×C7C2411yescharacteristic
175C22⋊Dic7—41no—
176#1142—44no—
177#1142—84no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182Dic7⋊C4—41no—
183Dic7⋊C4—41no—
184C4×Dic7—41no—
185#1136—42no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C22×#41—21no—
190#550—22no—
191#550—22no—
192#550—22no—
193#550C2×D714yes—
194#533—71no—
195C2×#127—21no—
196C2×#127C2×D711yescharacteristic
197C2×#121—22no—
198C2×#121—21no—
199C2×C4×Q8C2×D711yescharacteristic
200C2×C4×D4—21no—
201C2×C4×D4C2×D711yescharacteristic
202#492—72no—
203#486—72no—
204#483—141no—
205#479—142no—
206#477—72no—
207#476—72no—
208#475—72no—
209#461—72no—
210#460—142no—
211#450—141no—
212#450—142no—
213C4×SD16—142no—
214#431—71no—
215#431—71no—
216#428—141no—
217#423—71no—
218#419—141no—
219#331—74no—
220Q8⋊C8—74no—
221D4⋊C8—74no—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—42no—
225C23×C7—44no—
226C23×C7—44no—
227C23×C7—42no—
228C23×C7—44no—
229C23×C7—42no—
230C23×C7#12011yescharacteristic
231C7×Q8—22no—
232C7×Q8—24no—
233C7×Q8C2×SD1614yes—
234C7×D4—22no—
235C7×D4—22no—
236C7×D4—44no—
237C7×D4—42no—
238C7×D4—44no—
239C7×D4—416no—
240C7×D4—42no—
241C7×D4—42no—
242C7×D4—44no—
243C7×D4—44no—
244C7×D4—82no—
245C7×D4—88no—
246C2×C4×C7—22no—
247C2×C4×C7—24no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—41no—
252C2×C4×C7—416no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—44no—
258C2×C4×C7#12011yescharacteristic
259C2×C4×C7#12011yescharacteristic
260C2×C4×C7#12011yescharacteristic
261C2×C4×C7C2×SD1612yes—
262C2×C4×C7AGL(1,ℤ/8)12yes—
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C2×Dic7—41no—
267C2×Dic7—41no—
268Dic14—42no—
269Dic14—42no—
270C7⋊C8—42no—
271C7⋊C8—42no—
272C7⋊C8—42no—
273C2×#41—24no—
274C2×#41—24no—
275C2×#41—44no—
276C2×#41—44no—
277C22×Q8C22⋊D711yescharacteristic
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22⋊D711yescharacteristic
283C22×D4C22⋊D712yes—
284C23×C4—22no—
285C23×C4—21no—
286C2×SD16—142no—
287C2×SD16—144no—
288C2×M16—141no—
289C4⋊Q8—71no—
290C4⋊Q8—71no—
291#127—24no—
292#127—24no—
293#127—21no—
294#127—21no—
295#127C22⋊D712yes—
296#127C22×D712yes—
297#125—141no—
298C22⋊Q8—141no—
299C22⋊Q8—141no—
300#121—24no—
301#121—28no—
302#121—28no—
303#121—24no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—141no—
307C4×Q8—141no—
308C4×Q8C22⋊D712yes—
309C4×Q8C22×D712yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4C22⋊D712yes—
315C4×D4C22×D712yes—
316#117—141no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C22⋊D711yescharacteristic
319C2×C4⋊C4C22⋊D711yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—21no—
323C2×C42C22×D711yescharacteristic
324#107—142no—
325#106—142no—
326#106—142no—
327C4⋊C8—72no—
328C4⋊C8—72no—
329C4⋊C8—141no—
330C4⋊C8—141no—
331#103—142no—
332#103—142no—
333#103—142no—
334#102—142no—
335#102—144no—
336C22⋊C8—141no—
337C22⋊C8—141no—
338C4×C8—72no—
339C4×C8—141no—
340C22×C7—21no—
341C22×C7—21no—
342C22×C7—44no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—44no—
346C22×C7—44no—
347C22×C7—42no—
348C22×C7—84no—
349C22×C7—88no—
350C22×C7#45012yes—
351C22×C7C2×#12011yescharacteristic
352C4×C7—24no—
353C4×C7—21no—
354C4×C7—21no—
355C4×C7—21no—
356C4×C7—42no—
357C4×C7—42no—
358C4×C7#45012yes—
359C4×C7C22×SD1611yescharacteristic
360C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
361C4×C7#57512yes—
362Dic7—81no—
363Dic7—81no—
364C24—22no—
365C24—21no—
366#41—48no—
367#41—416no—
368#41—48no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—141no—
372C2×Q8—141no—
373C2×Q8#114214yescomplemented
374C2×Q8C2×C22⋊D712yes—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—42no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—82no—
389C2×D4—84no—
390C2×D4C2×C22⋊D712yes—
391C2×D4C2×C22⋊D714yes—
392C22×C4—21no—
393C22×C4—28no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—48no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C2×C22⋊D711yescharacteristic
402C22×C4C2×C22⋊D711yescharacteristic
403C22×C4C2×C22⋊D711yescharacteristic
404SD16—284no—
405SD16—564no—
406M16—282no—
407C2×C8—142no—
408C2×C8—142no—
409C2×C8—142no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—141no—
415C4⋊C4—141no—
416C4⋊C4—141no—
417C4⋊C4—141no—
418C4⋊C4—281no—
419C4⋊C4—281no—
420C4⋊C4C2×C22⋊D712yes—
421C4⋊C4C2×C22⋊D712yes—
422C22⋊C4—28no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—281no—
426C42—22no—
427C42—22no—
428C42—281no—
429C42C2×C22⋊D711yescharacteristic
430C42C2×C22⋊D711yescharacteristic
431C42C2×C22⋊D711yescharacteristic
432C42C23×D711yescharacteristic
433C2×C7—22no—
434C2×C7—82no—
435C2×C7—84no—
436C2×C7C2×#45011yescharacteristic
437C2×C7#300511yescharacteristic
438C2×C7#301911yescharacteristic
439C23—22no—
440C23—24no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—44no—
446C23—42no—
447C23#651611yescharacteristic
448Q8—22no—
449Q8—24no—
450Q8—282no—
451Q8—282no—
452Q8C2×#114214yes—
453D4—22no—
454D4—22no—
455D4—44no—
456D4—42no—
457D4—44no—
458D4—416no—
459D4—42no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—82no—
464D4—88no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—416no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4#649512yes—
480C2×C4C2×#114212yes—
481C2×C4#651611yescharacteristic
482C2×C4#651611yescharacteristic
483C2×C4#651611yescharacteristic
484C2×C4C22×C22⋊D711yescharacteristic
485C2×C4C22×C22⋊D711yescharacteristic
486C2×C4C22×C22⋊D711yescharacteristic
487C8—282no—
488C8—282no—
489C8—282no—
490C7#1876711yescharacteristic, complemented, Sylow
491C22—21no—
492C22—21no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—84no—
500C22—88no—
501C22#8999012yes—
502C22C2×#651611yescharacteristic
503C4—24no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—42no—
508C4—42no—
509C4—561no—
510C4—561no—
511C4#8999012yes—
512C4C2×#649511yescharacteristic
513C4C22×#114211yescharacteristic
514C4#9051312yes—
515C2—22no—
516C2—82no—
517C2—84no—
518C2C2×#8999011yescharacteristic
519C2#1174152611yescharacteristic
520C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#550C2
socleC22×C7C2×#120
hypercenterC2×#550D7
hypocenterC7#18767
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. #11740614C2semidirect
3. #11739876C2semidirect
4. #11739759C2semidirect
5. #11729675C2semidirect
6. #11729673C2semidirect
7. #11729513C2semidirect
8. #11729375C2semidirect
9. #11729352C2semidirect
26. #89848C22semidirect
28. #89842C22semidirect
30. #89793C22semidirect
31. #89792C22semidirect
32. #89791C22semidirect
34. #89780C22semidirect
40. #89339C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
42. D4⋊(C7⋊C8)C22semidirect
127. C2×C7×Q8SD16semidirect
373. C2×Q8#1142semidirect
490. C7#18767semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5502
#117406142
#117398762
#117397592
#117296752
#117296732
#117295132
#117293752
#117293522
#187677