Finite Group Databasebeta

← #13405976

Subgroups — #13405976

Order 1792.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#532C211yesmaximal, characteristic
2C2×#89813C212yesmaximal, complemented
3#11739520C211yesmaximal, characteristic, complemented
4#11729548C211yesmaximal, characteristic, complemented
5#11729537C211yesmaximal, characteristic, complemented
6#11729522C211yesmaximal, characteristic, complemented
7#11729516C211yesmaximal, characteristic, complemented
8#11729435C212yesmaximal, complemented
9#11729366C212yesmaximal, complemented
10#11729361C212yesmaximal, complemented
11C2×#89319C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C7×#532—28no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#127C2211yescharacteristic
16C2×C7×#124—21no—
17C2×C7×#121—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C22×#1142C2212yes—
23C2×#6487C2211yescharacteristic
24C23×C7⋊C8C2212yes—
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28#89984C2211yescharacteristic
29#89983—21no—
30#89982—21no—
31#89813—28no—
32C2×#6384C2212yes—
33C2×#6382C2212yes—
34#89705—21no—
35#89705—21no—
36#89701C2211yescharacteristic
37#89701C2211yescharacteristic
38#89700C2211yescharacteristic
39#89700C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89319C2211yescharacteristic
43#89319C2211yescharacteristic
44#89319C2212yes—
45#25495—71nomaximal, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8—21no—
50C22×C7×D4—22no—
51C22×C7×D4—21no—
52C22×C7×D4—21no—
53C22×C7×D4C2312yes—
54C23×C4×C7—22no—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C22×Dic14C2311yescharacteristic
59C7×C4⋊Q8—24no—
60C7×#127—24no—
61C7×#124—28no—
62C7×#121—28no—
63C7×#121—24no—
64C7×#121D418yes—
65C4×C7×D4—28no—
66C4×C7×D4—24no—
67C4×C7×D4—24no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C42×C7—24no—
73C2×C42×C7D411yescharacteristic
74C2×C42×C7D411yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×#1142—24no—
80C2×#1142—24no—
81C2×#1142—42no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85#6487—24no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C22×C7⋊C8—24no—
90C22×C7⋊C8—24no—
91C22×C7⋊C8—42no—
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2312yes—
94#6406—24no—
95#6406—24no—
96#6406—42no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104#6384—24no—
105#6382—24no—
106C2×#532D711yescharacteristic
107C2×#465—72no—
108#2855—71no—
109#2058—71no—
110#2041—71no—
111#2024—71no—
112#2014—71no—
113#1893—72no—
114#1877—72no—
115#1864—72no—
116C2×#336—71no—
117C24×C7—22no—
118C24×C7C2×D411yescharacteristic
119C7×#41—432no—
120C2×C7×Q8—24no—
121C2×C7×Q8—42no—
122C2×C7×Q8—42no—
123C2×C7×D4—24no—
124C2×C7×D4—24no—
125C2×C7×D4—28no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—42no—
132C2×C7×D4—42no—
133C2×C7×D4—44no—
134C2×C7×D4—44no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4C2×D414yes—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—28no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—41no—
148C22×C4×C7—44no—
149C22×C4×C7—44no—
150C22×C4×C7SD1614yescomplemented
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D414yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4C2×D414yes—
169C7×C22⋊C4—28no—
170C7×C22⋊C4—28no—
171C42×C7—22no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7—22no—
175C22⋊Dic7—42no—
176C22⋊Dic7—42no—
177#1142—48no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182Dic7⋊C4—42no—
183Dic7⋊C4—42no—
184Dic7⋊C4—42no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—48no—
191C2×C7⋊C8#4114yescomplemented
192C22×#41—21no—
193C22×SD16—72no—
194C23×C8—72no—
195#532—28no—
196C2×C4⋊Q8C2×D711yescharacteristic
197C2×#127C2×D711yescharacteristic
198C2×#124—21no—
199C2×C22⋊Q8—71no—
200C2×#121—21no—
201C2×#121C2×D712yes—
202C2×C4×D4—21no—
203C2×C4×D4C2×D712yes—
204C22×C42C2×D711yescharacteristic
205#465—148no—
206C2×#106—72no—
207C2×#103—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×#102—72no—
211#391—71no—
212#391—71no—
213#389—71no—
214#389—71no—
215#387—141no—
216#387—141no—
217#385—141no—
218#381—141no—
219#380—71no—
220#340—72no—
221#340—72no—
222#336—71no—
223#336—72no—
224#336—71no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—24no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7#12012yes—
237C23×C7C2×SD1612yes—
238C23×C7AGL(1,ℤ/8)12yes—
239C23×C7C22×D411yescharacteristic
240C7×Q8—48no—
241C7×D4—24no—
242C7×D4—24no—
243C7×D4—48no—
244C7×D4—48no—
245C7×D4—416no—
246C7×D4—42no—
247C7×D4—42no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—28no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—21no—
260C2×C4×C7—24no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—44no—
264C2×C4×C7—416no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7#12012yes—
268C2×C4×C7C2×SD1612yes—
269C2×C4×C7C2×SD1612yes—
270C2×C4×C7#13512yes—
271C2×C4×C7C22×D412yes—
272C2×C4×C7C2×#4112yes—
273C2×C4×C7H(5,𝔽2)11yescharacteristic
274C2×C4×C72−1+411yescharacteristic
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287Dic14—42no—
288Dic14—42no—
289C7⋊C8—24no—
290C7⋊C8—42no—
291C2×#41—28no—
292C2×#41—44no—
293C2×#41—44no—
294C22×Q8—21no—
295C22×Q8—71no—
296C22×D4—22no—
297C22×D4—21no—
298C22×D4—21no—
299C22×D4C22×D712yes—
300C23×C4—22no—
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22×D711yescharacteristic
304C2×SD16—144no—
305C2×SD16—144no—
306C2×SD16—282no—
307C22×C8—72no—
308C22×C8—72no—
309C22×C8—144no—
310C22×C8—144no—
311C22×C8—282no—
312C4⋊Q8—24no—
313#127—24no—
314#124—28no—
315C22⋊Q8—144no—
316#121—28no—
317#121—24no—
318#121C22⋊D718yes—
319C4×D4—28no—
320C4×D4—24no—
321C4×D4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4C22×D712yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—141no—
333C2×C22⋊C4—141no—
334C2×C42—24no—
335C2×C42C22⋊D711yescharacteristic
336C2×C42C22⋊D711yescharacteristic
337C2×C42C22×D711yescharacteristic
338C2×C42C22×D711yescharacteristic
339#106—144no—
340#103—144no—
341#102—144no—
342#102—144no—
343#102—282no—
344#95—141no—
345#95—141no—
346#95—141no—
347#95—141no—
348#95—141no—
349#95—141no—
350#95—141no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7—44no—
358C22×C7—44no—
359C22×C7—44no—
360C22×C7—82no—
361C22×C7—82no—
362C22×C7—82no—
363C22×C7—82no—
364C22×C7#45014yes—
365C22×C7C2×#12011yescharacteristic
366C22×C7#53211yescharacteristic
367C22×C7#53711yescharacteristic
368C22×C7C22×SD1611yescharacteristic
369C22×C7C2×#13511yescharacteristic
370C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C7#57511yescharacteristic
372C4×C7—22no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—22no—
376C4×C7—44no—
377C4×C7—41no—
378C4×C7#54612yes—
379C4×C7#54712yes—
380Dic7—81no—
381Dic7—81no—
382Dic7—81no—
383Dic7—81no—
384C24—22no—
385C24C2×C22⋊D711yescharacteristic
386#41—432no—
387C2×Q8—24no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—142no—
391C2×Q8—141no—
392C2×Q8—141no—
393C2×Q8—142no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C2×C22⋊D714yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4#114214yescomplemented
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D714yes—
432C22×C4C23×D711yescharacteristic
433SD16—288no—
434C2×C8—74no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C2×C8—142no—
439C2×C8—142no—
440C2×C8—288no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—142no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—142no—
447C4⋊C4—282no—
448C4⋊C4—282no—
449C4⋊C4—282no—
450C4⋊C4C2×C22⋊D714yes—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—282no—
454C22⋊C4—282no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C2×C7—24no—
460C2×C7—84no—
461C2×C7#243011yescharacteristic
462C2×C7C2×#45011yescharacteristic
463C2×C7#300211yescharacteristic
464C2×C7#316711yescharacteristic
465C2×C7#318011yescharacteristic
466C2×C7#324611yescharacteristic
467C2×C7#324711yescharacteristic
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23#649512yes—
480C23C2×#114212yes—
481C23#651612yes—
482C23C22×C22⋊D711yescharacteristic
483Q8—48no—
484Q8—282no—
485Q8—282no—
486D4—24no—
487D4—24no—
488D4—48no—
489D4—48no—
490D4—416no—
491D4—42no—
492D4—42no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—44no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—281no—
513C2×C4—281no—
514C2×C4—281no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4C2×#114212yes—
525C2×C4C2×#114212yes—
526C2×C4#651312yes—
527C2×C4#651612yes—
528C2×C4D7×#4112yes—
529C2×C4#655311yescharacteristic
530C2×C4#655411yescharacteristic
531C2×C4C22×C22⋊D712yes—
532C8—144no—
533C8—282no—
534C7#2549511yescharacteristic, complemented, Sylow
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22#8999014yes—
549C22C2×#649511yescharacteristic
550C22C22×#114211yescharacteristic
551C22C2×#651311yescharacteristic
552C22#9051311yescharacteristic
553C22#9052311yescharacteristic
554C22#9052411yescharacteristic
555C22C2×#651611yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—41no—
562C4—561no—
563C4—561no—
564C4—561no—
565C4—561no—
566C4#9051512yes—
567C4#9052012yes—
568C2—24no—
569C2—84no—
570C2#1174073511yescharacteristic
571C2#1174073811yescharacteristic
572C2#1174075611yescharacteristic
573C2#1174076511yescharacteristic
574C2C2×#8999011yescharacteristic
575C2#1174152511yescharacteristic
576C2#1174153611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#532C2
socleC23×C7C22×D4
hypercenterC2×#532D7
hypocenterC7#25495
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#89813C2semidirect
3. #11739520C2semidirect
4. #11729548C2semidirect
5. #11729537C2semidirect
6. #11729522C2semidirect
7. #11729516C2semidirect
8. #11729435C2semidirect
9. #11729366C2semidirect
10. #11729361C2semidirect
11. C2×#89319C2semidirect
150. C22×C4×C7SD16semidirect
191. C2×C7⋊C8#41semidirect
425. C22×C4#1142semidirect
534. C7#25495semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5322
C2×#898132
#117395202
#117295482
#117295372
#117295222
#117295162
#117294352
#117293662
#117293612
C2×#893192
#254957