Finite Group Databasebeta

← #13388538

Subgroups — #13388538

Order 1792.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic, complemented
2C2×#89855C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic, complemented
4C2×C4×#1141C211yesmaximal, characteristic
5#11729539C211yesmaximal, characteristic, complemented
6#11729482C212yesmaximal
7#11729476C212yesmaximal, complemented
8#11729388C211yesmaximal, characteristic, complemented
9#11729387C212yesmaximal, complemented
10#11729382C212yesmaximal, complemented
11#11729358C211yesmaximal, characteristic, complemented
12C7×#556—24no—
13C7×#556—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—22no—
16C2×C7×#124C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C4×C7×Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22C2×#6509—21no—
23C22×#1143C2212yes—
24C22×#1141C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6406C2211yescharacteristic
30C2×#6406C2212yes—
31#89855—24no—
32C4×#1141—24no—
33#89766—22no—
34#89759—21no—
35C2×#6383C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89711C2212yes—
38#89707C2211yescharacteristic
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24185—71nomaximal, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2312yes—
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C22×D28C2311yescharacteristic
51C22×C4×D7—21no—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8D414yes—
54C7×#126—216no—
55C7×#124—24no—
56C7×#124—24no—
57C7×C22⋊Q8—28no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—22no—
61C4×C7×Q8—22no—
62C4×C7×D4—22no—
63C4×C7×D4D414yes—
64C7×#117—28no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4D412yes—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—22no—
74C2×C42×C7—22no—
75C2×C42×C7C2312yes—
76C2×C42×C7C2311yescharacteristic
77#6509—44no—
78C2×#1143—24no—
79C2×#1143—24no—
80C2×#1143—42no—
81C2×#1141—22no—
82C2×#1141—22no—
83C2×#1141—22no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—22no—
86C2×(C2×D7)⋊C4—22no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2312yes—
91C2×Dic7⋊C4—22no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2312yes—
94C22×C7⋊C8C2311yescharacteristic
95C4×D28—22no—
96C4×D28—22no—
97#6409—24no—
98#6409—24no—
99#6409—42no—
100#6406—22no—
101#6406—42no—
102C7⋊#95—22no—
103#6383—22no—
104#6381—22no—
105C4⋊(C7⋊C8)—22no—
106C4×C7⋊C8—22no—
107C2×#556D711yescharacteristic, complemented
108C2×#461—71no—
109C2×C4×D8—71no—
110#2538—71no—
111#2042—72no—
112#2012—72no—
113#1984—71no—
114#1956—71no—
115#1880—72no—
116#1863—72no—
117#1802—71no—
118C24×C7—21no—
119C23×D7—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8C2×D414yes—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4C2×D412yes—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2411yescharacteristic
145C22×Dic7—21no—
146C22×Dic7—22no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28#4112yes—
153C2×C4×D7—42no—
154C2×C4×D7—42no—
155C2×C4×D7—42no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4C2×D412yes—
163C7×C22⋊C4—28no—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—24no—
166C42×C7—28no—
167C42×C7—22no—
168C42×C7—22no—
169C42×C7C2×D412yes—
170C42×C7C2×D412yes—
171#1143—48no—
172#1141—28no—
173(C2×D7)⋊C4—44no—
174(C2×D7)⋊C4—42no—
175(C2×D7)⋊C4—82no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—42no—
181C4⋊Dic7—81no—
182Dic7⋊C4—44no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8#4112yes—
192C22×SD16—72no—
193C22×D8—71no—
194#556—24no—
195#556—24no—
196C2×C4⋊Q8C2×D711yescharacteristic
197C2×#126—22no—
198C2×#124C2×D712yes—
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8—141no—
201C2×C4×Q8C2×D712yes—
202C2×C4×Q8C2×D711yescharacteristic
203C2×C4×D4—71no—
204C2×C4×D4C2×D711yescharacteristic
205C2×#117—21no—
206#461—144no—
207C4×D8—144no—
208C2×#107—71no—
209C2×C4⋊C8—71no—
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#102—71no—
214C2×#102—71no—
215C2×#102—72no—
216C2×C4×C8—71no—
217#396—142no—
218#390—72no—
219#389—72no—
220#384—71no—
221#380—141no—
222#340—72no—
223#336—72no—
224#336—72no—
225C23×C7—22no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7C22×D411yescharacteristic
233C22×D7—22no—
234C22×D7—41no—
235C22×D7—41no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—41no—
239C22×D7—41no—
240C7×Q8—24no—
241C7×Q8—24no—
242C7×Q8—42no—
243C7×Q8—42no—
244C7×Q8—44no—
245C7×D4—22no—
246C7×D4—22no—
247C7×D4—41no—
248C7×D4—41no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—48no—
264C2×C4×C7#12012yes—
265C2×C4×C7#12012yes—
266C2×C4×C7#13512yes—
267C2×C4×C7AGL(1,ℤ/8)12yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7H(5,𝔽2)11yescharacteristic
273C2×C4×C72−1+411yescharacteristic
274C2×Dic7—41no—
275C2×Dic7—42no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280D28—22no—
281D28—22no—
282D28—41no—
283D28—41no—
284C4×D7—84no—
285C7⋊C8—22no—
286C7⋊C8—41no—
287C7⋊C8—42no—
288C22×Q8C22⋊D711yescharacteristic
289C22×Q8C22×D712yes—
290C22×D4—71no—
291C22×D4C22×D711yescharacteristic
292C23×C4—21no—
293C23×C4—141no—
294C2×SD16—144no—
295C2×SD16—144no—
296C2×SD16—282no—
297C2×D8—142no—
298C2×D8—142no—
299C2×D8—142no—
300C22×C8—71no—
301C22×C8—72no—
302C22×C8—71no—
303C4⋊Q8—22no—
304C4⋊Q8C22⋊D714yes—
305#126—216no—
306#124—24no—
307#124—24no—
308C22⋊Q8—28no—
309C22⋊Q8—284no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×D4—22no—
315C4×D4—142no—
316C4×D4—142no—
317C4×D4C22⋊D714yes—
318#117—28no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—72no—
324C2×C4⋊C4—142no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4C22⋊D711yescharacteristic
328C2×C4⋊C4C22⋊D712yes—
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—141no—
334C2×C22⋊C4—142no—
335C2×C22⋊C4—142no—
336C2×C42—22no—
337C2×C42C22×D712yes—
338C2×C42C22×D711yescharacteristic
339#107—142no—
340C4⋊C8—142no—
341#103—144no—
342#103—144no—
343#103—282no—
344#102—142no—
345#102—142no—
346#102—282no—
347C4×C8—142no—
348#95—142no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—81no—
353C22×C7—81no—
354C22×C7—81no—
355C22×C7—81no—
356C22×C7C2×#12011yescharacteristic
357C22×C7#54811yescharacteristic
358C22×C7#55011yescharacteristic
359C22×C7C2×#13511yescharacteristic
360C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
361C22×C7#57711yescharacteristic
362C22×C7#57811yescharacteristic
363C2×D7—42no—
364C2×D7—42no—
365C2×D7—42no—
366C2×D7—81no—
367C2×D7—81no—
368C2×D7—81no—
369C2×D7—81no—
370C4×C7—22no—
371C4×C7—44no—
372C4×C7—42no—
373C4×C7—42no—
374C4×C7—42no—
375C4×C7#44914yes—
376C4×C7#52511yescharacteristic
377C4×C7#53711yescharacteristic
378C4×C7#54611yescharacteristic
379C4×C7#54711yescharacteristic
380Dic7—82no—
381Dic7—81no—
382C24—21no—
383C24—141no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8C2×C22⋊D714yes—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—72no—
399C2×D4—141no—
400C2×D4—141no—
401C2×D4—141no—
402C2×D4—141no—
403C2×D4—141no—
404C2×D4C2×C22⋊D712yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—141no—
411C22×C4—142no—
412C22×C4—282no—
413C22×C4—282no—
414C22×C4—282no—
415C22×C4C2×C22⋊D712yes—
416C22×C4C2×C22⋊D712yes—
417C22×C4C2×C22⋊D711yescharacteristic
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C23×D711yescharacteristic
420SD16—288no—
421D8—148no—
422C2×C8—72no—
423C2×C8—141no—
424C2×C8—141no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—142no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—284no—
442C4⋊C4—282no—
443C4⋊C4—561no—
444C4⋊C4C2×C22⋊D712yes—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—284no—
449C22⋊C4—282no—
450C22⋊C4—562no—
451C42—28no—
452C42—22no—
453C42—22no—
454C42C2×C22⋊D712yes—
455C42C2×C22⋊D712yes—
456C2×C7—82no—
457C2×C7#240711yescharacteristic
458C2×C7C2×#44911yescharacteristic
459C2×C7#301311yescharacteristic
460C2×C7#336011yescharacteristic
461C2×C7#336711yescharacteristic
462C2×C7#339311yescharacteristic
463C2×C7#339711yescharacteristic
464D7—82no—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—142no—
473C23—281no—
474C23—281no—
475C23—281no—
476C23—281no—
477C23—281no—
478C23—281no—
479C23C22×C22⋊D711yescharacteristic
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8—44no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—142no—
490D4—142no—
491D4—281no—
492D4—281no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—282no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—281no—
514C2×C4—564no—
515C2×C4#651212yes—
516C2×C4#651312yes—
517C2×C4#651612yes—
518C2×C4#651612yes—
519C2×C4D7×#4111yescharacteristic
520C2×C4D7×#4111yescharacteristic
521C2×C4#655311yescharacteristic
522C2×C4#655411yescharacteristic
523C2×C4C22×C22⋊D711yescharacteristic
524C2×C4C22×C22⋊D711yescharacteristic
525C8—142no—
526C8—282no—
527C8—281no—
528C7#2418511yescharacteristic, complemented, Sylow
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—282no—
537C22—282no—
538C22—282no—
539C22—561no—
540C22—561no—
541C22—561no—
542C22—561no—
543C22C2×#651211yescharacteristic
544C22C2×#651311yescharacteristic
545C22#9052211yescharacteristic
546C22#9052511yescharacteristic
547C22#9052811yescharacteristic
548C22#9052911yescharacteristic
549C22C2×#651611yescharacteristic
550C4—22no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—561no—
556C4—562no—
557C4#9001014yes—
558C4#9051511yescharacteristic
559C4#9051911yescharacteristic
560C4#9052011yescharacteristic
561C4#9052411yescharacteristic
562C2—82no—
563C2—562no—
564C2#1174111211yescharacteristic
565C2#1174111311yescharacteristic
566C2#1174111511yescharacteristic
567C2#1174111611yescharacteristic
568C2C2×#9001011yescharacteristic
569C2#1174154211yescharacteristic
570C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenterC2×#556D7
hypocenterC7#24185
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×C7×#556C2semidirect
2. C2×#89855C2semidirect
3. #11739945C2semidirect
5. #11729539C2semidirect
7. #11729476C2semidirect
8. #11729388C2semidirect
9. #11729387C2semidirect
10. #11729382C2semidirect
11. #11729358C2semidirect
107. C2×#556D7semidirect
528. C7#24185semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#898552
#117399452
C2×C4×#11412
#117295392
#117294822
#117294762
#117293882
#117293872
#117293822
#117293582
#241857