Finite Group Databasebeta

← #13410791

Subgroups — #13410791

Order 1792.

1100 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×#89839C211yesmaximal, characteristic
3#11739921C211yesmaximal, characteristic
4C2×#89799C211yesmaximal, characteristic
5#11729546C212yesmaximal, complemented
6#11729497C212yesmaximal, complemented
7#11729492C211yesmaximal, characteristic, complemented
8#11729389C211yesmaximal, characteristic, complemented
9#11729387C212yesmaximal, complemented
10#11729387C212yesmaximal, complemented
11#11729385C211yesmaximal, characteristic, complemented
12C7×#557—24no—
13C7×#557—24no—
14C2×C7×C4⋊Q8—21no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×#125—21no—
18C2×C7×#125C2211yescharacteristic
19C2×C7×#125C2211yescharacteristic
20C2×C4×C7×Q8—21no—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×Q8C2211yescharacteristic
24C2×C4×C7×Q8C2211yescharacteristic
25C2×#6510—21no—
26C22×#1143C2212yes—
27C2×#6437C2211yescharacteristic
28C2×#6409C2212yes—
29#89839—24no—
30#89799—24no—
31#89763—21no—
32#89763—21no—
33#89760—21no—
34C2×#6383C2212yes—
35C2×#6383C2212yes—
36C2×#6383C2212yes—
37#89711C2212yes—
38#89711C2212yes—
39C4×C4⋊Dic7C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89319C2212yes—
46#89319C2212yes—
47#25944—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×Q8C2311yescharacteristic
51C22×D28—21no—
52C22×D28—21no—
53C22×D28C2312yes—
54C7×C4⋊Q8—28no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×#125—28no—
58C7×#125—28no—
59C7×#125—22no—
60C7×#125—22no—
61C7×#125—22no—
62C7×#125D414yes—
63C4×C7×Q8—28no—
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×Q8—22no—
71C4×C7×Q8D414yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4D411yescharacteristic
82C2×C7×C4⋊C4D411yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C42×C7—21no—
87C2×C42×C7—21no—
88C2×C42×C7—21no—
89C2×C42×C7C2311yescharacteristic
90C2×C42×C7C2311yescharacteristic
91C2×C42×C7C2311yescharacteristic
92#6510—44no—
93C2×#1143—24no—
94C2×#1143—24no—
95C2×#1143—24no—
96C2×(C2×D7)⋊C4—22no—
97C2×(C2×D7)⋊C4—22no—
98C2×(C2×D7)⋊C4—22no—
99C2×(C2×D7)⋊C4—22no—
100C2×C4⋊Dic7C2312yes—
101C2×C4×Dic7—21no—
102C22×C7⋊C8C2312yes—
103C22×C7⋊C8C2312yes—
104#6437—22no—
105#6437—22no—
106#6409—24no—
107C7⋊#95—21no—
108#6383—24no—
109#6383—24no—
110#6383—24no—
111C4⋊(C7⋊C8)—22no—
112#6377—22no—
113C2×#557D711yescharacteristic, complemented
114C2×#490—71no—
115C2×#460—71no—
116#2548—71no—
117#2025—72no—
118#2023—72no—
119#1989—71no—
120#1985—71no—
121#1880—72no—
122#1880—72no—
123#1763—71no—
124C23×D7—22no—
125C2×C7×Q8—24no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—41no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8#4114yes—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2411yescharacteristic
146C22×Dic7—22no—
147C2×D28—22no—
148C2×D28—22no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—22no—
153C2×D28—42no—
154C2×D28—42no—
155C2×D28—42no—
156C2×D28—42no—
157C2×D28—82no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4C2×D412yes—
173C7×C4⋊C4C2×D412yes—
174C7×C4⋊C4C2×D412yes—
175C42×C7—24no—
176C42×C7—24no—
177C42×C7—24no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7C2×D412yes—
183C42×C7C2×D412yes—
184#1143—48no—
185(C2×D7)⋊C4—44no—
186(C2×D7)⋊C4—44no—
187(C2×D7)⋊C4—82no—
188(C2×D7)⋊C4—82no—
189C4⋊Dic7—22no—
190C4⋊Dic7—22no—
191C4×Dic7—42no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C22×SD16—72no—
199#557—24no—
200#557—24no—
201C2×C4⋊Q8—21no—
202C2×C4⋊Q8C2×D711yescharacteristic
203C2×#127—71no—
204C2×#125—21no—
205C2×#125—21no—
206C2×#125C2×D711yescharacteristic
207C2×#125C2×D711yescharacteristic
208C2×#124—141no—
209C2×C4×Q8—21no—
210C2×C4×Q8C2×D711yescharacteristic
211C2×C4×Q8C2×D711yescharacteristic
212C2×C4×Q8C2×D711yescharacteristic
213C2×C4×Q8C2×D711yescharacteristic
214#490—144no—
215#460—144no—
216C2×C4⋊C8—71no—
217C2×#103—72no—
218C2×#102—72no—
219C2×#102—72no—
220C2×#102—72no—
221C2×#97—71no—
222#394—141no—
223#394—141no—
224#390—72no—
225#390—72no—
226#390—141no—
227C4×C4⋊C4—71no—
228#340—72no—
229#340—72no—
230#340—72no—
231#336—72no—
232#336—72no—
233C23×C7C22×D411yescharacteristic
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C22×D7—42no—
240C22×D7—42no—
241C7×Q8—22no—
242C7×Q8—24no—
243C7×Q8—44no—
244C7×Q8—48no—
245C7×Q8#12414yescomplemented
246C2×C4×C7—22no—
247C2×C4×C7—21no—
248C2×C4×C7—24no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7#12012yes—
269C2×C4×C7#12012yes—
270C2×C4×C7AGL(1,ℤ/8)12yes—
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4112yes—
274C2×C4×C72−1+412yes—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278D28—42no—
279D28—42no—
280D28—42no—
281D28—42no—
282D28—84no—
283D28—84no—
284C7⋊C8—42no—
285C7⋊C8—42no—
286C22×Q8—21no—
287C22×Q8C22⋊D711yescharacteristic
288C22×Q8C22×D711yescharacteristic
289C22×D4—72no—
290C22×D4—141no—
291C22×D4—141no—
292C2×SD16—144no—
293C2×SD16—144no—
294C2×SD16—144no—
295C22×C8—72no—
296C22×C8—72no—
297C4⋊Q8—28no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300#127—142no—
301#127—142no—
302#125—28no—
303#125—28no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125C22⋊D714yes—
308#124—284no—
309C4×Q8—28no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8C22⋊D714yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—72no—
327C2×C4⋊C4C22⋊D711yescharacteristic
328C2×C4⋊C4C22⋊D711yescharacteristic
329C2×C4⋊C4C22⋊D711yescharacteristic
330C2×C4⋊C4C22×D711yescharacteristic
331C2×C4⋊C4C22×D711yescharacteristic
332C2×C4⋊C4C22×D711yescharacteristic
333C2×C22⋊C4—142no—
334C2×C22⋊C4—142no—
335C2×C22⋊C4—142no—
336C2×C22⋊C4—142no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—141no—
341C2×C42C22×D711yescharacteristic
342C2×C42C22×D711yescharacteristic
343C2×C42C22×D711yescharacteristic
344C4⋊C8—142no—
345#103—144no—
346#102—144no—
347#102—144no—
348#102—144no—
349#97—142no—
350#95—141no—
351C22×C7C2×#12011yescharacteristic
352C22×C7#54812yes—
353C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C7#57511yescharacteristic
355C22×C7#57712yes—
356C2×D7—82no—
357C2×D7—82no—
358C2×D7—82no—
359C2×D7—82no—
360C2×D7—82no—
361C2×D7—82no—
362C2×D7—82no—
363C4×C7—24no—
364C4×C7—22no—
365C4×C7—22no—
366C4×C7—22no—
367C4×C7—41no—
368C4×C7—42no—
369C4×C7—42no—
370C4×C7—41no—
371C4×C7—41no—
372C4×C7—42no—
373C4×C7C2×#12411yescharacteristic
374C4×C7#53611yescharacteristic
375C4×C7#54812yes—
376Dic7—82no—
377C24—142no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—41no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8#116114yes—
387C2×Q8C2×C22⋊D712yes—
388C2×D4—142no—
389C2×D4—142no—
390C2×D4—142no—
391C2×D4—142no—
392C2×D4—142no—
393C2×D4—142no—
394C2×D4—282no—
395C2×D4—282no—
396C2×D4—282no—
397C2×D4—282no—
398C2×D4—562no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—142no—
404C22×C4C2×C22⋊D711yescharacteristic
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C2×C22⋊D711yescharacteristic
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C2×C22⋊D711yescharacteristic
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411SD16—288no—
412C2×C8—142no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—142no—
417C2×C8—142no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—142no—
433C4⋊C4—142no—
434C4⋊C4C2×C22⋊D712yes—
435C4⋊C4C2×C22⋊D712yes—
436C4⋊C4C2×C22⋊D712yes—
437C22⋊C4—284no—
438C22⋊C4—284no—
439C22⋊C4—562no—
440C22⋊C4—562no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—282no—
449C42C2×C22⋊D712yes—
450C42C2×C22⋊D712yes—
451C2×C7#243611yescharacteristic
452C2×C7#300611yescharacteristic
453C2×C7#301011yescharacteristic
454C2×C7#336812yes—
455C2×C7#337112yes—
456D7—162no—
457C23—282no—
458C23—282no—
459C23—282no—
460C23—282no—
461C23—282no—
462C23—282no—
463C23—282no—
464C23C22×C22⋊D711yescharacteristic
465Q8—22no—
466Q8—24no—
467Q8—44no—
468Q8—48no—
469Q8#651014yescomplemented
470D4—282no—
471D4—282no—
472D4—282no—
473D4—282no—
474D4—564no—
475D4—564no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4#651212yes—
502C2×C4#651612yes—
503C2×C4#651612yes—
504C2×C4C2×#116112yes—
505C2×C4#655112yes—
506C2×C4C22×C22⋊D711yescharacteristic
507C2×C4C22×C22⋊D711yescharacteristic
508C8—282no—
509C8—282no—
510C7#2594411yescharacteristic, complemented, Sylow
511C22—562no—
512C22—562no—
513C22—562no—
514C22—562no—
515C22—562no—
516C22—562no—
517C22—562no—
518C22#9050812yes—
519C22C2×#651211yescharacteristic
520C22#9051311yescharacteristic
521C22#9052812yes—
522C22C2×#651611yescharacteristic
523C4—24no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—562no—
534C4C2×#651011yescharacteristic
535C4#9050711yescharacteristic
536C4#9050812yes—
537C2—1122no—
538C2#1174113512yes—
539C2#1174113612yes—
540C2#1174151111yescharacteristic
541C2#1174152711yescharacteristic
542C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#557C2
socleC23×C7C22×D4
prime radicalC2×#6437C22
hypercenterC2×#557D7
hypocenterC7#25944
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
5. #11729546C2semidirect
6. #11729497C2semidirect
7. #11729492C2semidirect
8. #11729389C2semidirect
9. #11729387C2semidirect
10. #11729387C2semidirect
11. #11729385C2semidirect
113. C2×#557D7semidirect
245. C7×Q8#124semidirect
469. Q8#6510semidirect
510. C7#25944semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×#898392
#117399212
C2×#897992
#117295462
#117294972
#117294922
#117293892
#117293872
#117293852
#259447