Finite Group Databasebeta

← #13330938

Subgroups — #13330938

Order 1792.

1066 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3503C211yesmaximal, characteristic, complemented
2#11740800C211yesmaximal, characteristic, complemented
3C2×#89857C211yesmaximal, characteristic, complemented
4#11739971C211yesmaximal, characteristic
5#11729693C214yesmaximal, complemented
6#11729677C214yesmaximal, complemented
7#11729580C211yesmaximal, characteristic, complemented
8#11729528C211yesmaximal, characteristic, complemented
9#11729392C211yesmaximal, characteristic, complemented
10C7×Q82C2212yescomplemented
11C7×#557—21no—
12C7×#554—22no—
13C7×#554C2212yescomplemented
14C7×D4×Q8—22no—
15C7×D4×Q8—21no—
16C7×#544—22no—
17C7×#544—21no—
18C7×#533—21no—
19C7×#533C2211yescharacteristic
20C2×C7×C4⋊Q8—21no—
21C2×C7×C4⋊Q8C2211yescharacteristic
22C7×#520—21no—
23C4×C7×#41C2211yescharacteristic
24C2×#6512—21no—
25C22×#1143—22no—
26C2×#6437C2211yescharacteristic
27#89860—21no—
28#89857—21no—
29#89857—22no—
30#89857C2214yes—
31#89818—22no—
32#89799—21no—
33#89799C2214yes—
34#89799C2212yes—
35#89787—21no—
36#89787C2212yes—
37#89773—21no—
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89341C2214yescomplemented
42Q8⋊(C7⋊C8)C2214yescomplemented
43Q8⋊(C7⋊C8)C2212yescomplemented
44D4⋊(C7⋊C8)C2212yescomplemented
45#18622—71nomaximal, Sylow
46C2×C7×#41D411yescharacteristic
47C22×C7×Q8—22no—
48C22×C7×Q8D412yes—
49C22×D28—21no—
50C22×D28—21no—
51C22×D28—21no—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8D412yes—
60C7×C4⋊Q8D411yescharacteristic
61C7×C4⋊Q8D411yescharacteristic
62C7×C4⋊Q8C2312yes—
63C7×#125—22no—
64C7×#125—24no—
65C7×#125—22no—
66C7×#125—21no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—24no—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8—24no—
72C7×C22⋊Q8—21no—
73C7×C22⋊Q8D412yes—
74C4×C7×Q8—22no—
75C4×C7×Q8—22no—
76C4×C7×Q8—22no—
77C4×C7×Q8—21no—
78C4×C7×Q8D411yescharacteristic
79C4×C7×Q8C2311yescharacteristic
80C4×C7×Q8C2312yes—
81C4×C7×D4—24no—
82C4×C7×D4—22no—
83C4×C7×D4—21no—
84C4×C7×D4—21no—
85C4×C7×D4D411yescharacteristic
86C4×C7×D4C2311yescharacteristic
87C7×#117—22no—
88C7×#117—21no—
89C7×#117—21no—
90C7×#117D411yescharacteristic
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—22no—
95C2×C42×C7—21no—
96C2×C42×C7C2311yescharacteristic
97#6512—44no—
98C2×#1143—28no—
99C2×#1143—22no—
100C2×#1143—24no—
101C2×#1143—48no—
102C2×#1141—22no—
103C2×#1136—21no—
104C22×C7⋊C8—21no—
105#6437—21no—
106#6437—21no—
107#6437—21no—
108#6437C2312yes—
109C22⋊(C7⋊C8)—21no—
110#6383—24no—
111#6383—22no—
112#6383—22no—
113C4⋊(C7⋊C8)—21no—
114C4⋊(C7⋊C8)C2312yes—
115C4⋊(C7⋊C8)C2312yes—
116C4×C7⋊C8—21no—
117C4×C7⋊C8—21no—
118C4×C7⋊C8C2312yes—
119#3503D711yescharacteristic, complemented
120#3208—71no—
121C2×#492—71no—
122#3038—71no—
123#1623—74no—
124#1622—74no—
125#1548—71no—
126#1493—71no—
127#1476—71no—
128C23×D7—41no—
129C7×#41—22no—
130C7×#41SD1614yescomplemented
131C2×C7×Q8—24no—
132C2×C7×Q8—22no—
133C2×C7×Q8—21no—
134C2×C7×Q8—44no—
135C2×C7×Q8—84no—
136C2×C7×Q8C2×D412yes—
137C2×C7×Q8C2×D411yescharacteristic
138C2×C7×Q8C2×D414yes—
139C2×C7×D4—21no—
140C2×C7×D4C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C2×D28—22no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—44no—
153C2×D28—42no—
154C2×D28—42no—
155C2×D28—41no—
156C2×D28—41no—
157C2×D28—41no—
158C2×D28—82no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4C2×D412yes—
175C7×C4⋊C4C2×D411yescharacteristic
176C7×C4⋊C4C2×D411yescharacteristic
177C7×C22⋊C4—24no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—21no—
180C7×C22⋊C4—21no—
181C7×C22⋊C4—21no—
182C42×C7—24no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—22no—
187C42×C7C2×D411yescharacteristic
188C42×C7C2×D411yescharacteristic
189C42×C7C2×D411yescharacteristic
190C42×C7C2411yescharacteristic
191#1143—416no—
192#1143—44no—
193#1143—84no—
194#1141—44no—
195#1136—42no—
196C2×C7⋊C8—24no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×AGL(1,ℤ/8)—141no—
201C22×SD16—142no—
202Q82C2×D712yescomplemented
203#557—21no—
204#554—22no—
205#554C2×D712yescomplemented
206D4×Q8—22no—
207D4×Q8—21no—
208#544—22no—
209#544—21no—
210#533—21no—
211#533C2×D711yescharacteristic
212C2×C4⋊Q8—21no—
213C2×C4⋊Q8C2×D711yescharacteristic
214C2×#127—71no—
215#520—21no—
216C4×#41C2×D711yescharacteristic
217#492—74no—
218#492—142no—
219#492—141no—
220#486—141no—
221#460—74no—
222#460—72no—
223#460—141no—
224#459—72no—
225#459—141no—
226#450—142no—
227#431—71no—
228#423—71no—
229#418—141no—
230C2×C4×C8—71no—
231#333—74no—
232Q8⋊C8—74no—
233Q8⋊C8—72no—
234D4⋊C8—72no—
235C23×C7—21no—
236C23×C7#12011yescharacteristic
237C22×D7—42no—
238C22×D7—82no—
239C22×D7—82no—
240C22×D7—82no—
241C7×Q8—21no—
242C7×Q8—24no—
243C7×Q8—44no—
244C7×Q8—44no—
245C7×Q8—42no—
246C7×Q8—84no—
247C7×Q8C2×SD1612yescomplemented
248C7×D4—24no—
249C7×D4—21no—
250C7×D4C2×SD1612yescomplemented
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—24no—
259C2×C4×C7—22no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—82no—
263C2×C4×C7—84no—
264C2×C4×C7#12011yescharacteristic
265C2×C4×C7#12011yescharacteristic
266C2×C4×C7#12011yescharacteristic
267C2×C4×C7C2×SD1612yes—
268C2×C4×C7#13712yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272D28—44no—
273D28—42no—
274D28—42no—
275D28—44no—
276D28—82no—
277D28—82no—
278D28—82no—
279C7⋊C8—24no—
280C7⋊C8—42no—
281C7⋊C8—42no—
282C2×#41C22⋊D711yescharacteristic
283C22×Q8—22no—
284C22×Q8C22⋊D712yes—
285C22×D4—141no—
286C22×D4—141no—
287C22×D4—141no—
288AGL(1,ℤ/8)—284no—
289C2×SD16—148no—
290C2×SD16—142no—
291C2×SD16—144no—
292C2×SD16—288no—
293C2×D8—142no—
294C2×M16—141no—
295C22×C8—141no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—24no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8C22⋊D712yes—
304C4⋊Q8C22⋊D711yescharacteristic
305C4⋊Q8C22⋊D711yescharacteristic
306C4⋊Q8C22×D712yes—
307#127—72no—
308#127—141no—
309#127—141no—
310#127—141no—
311#125—22no—
312#125—24no—
313#125—22no—
314#125—21no—
315C22⋊Q8—22no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—24no—
320C22⋊Q8—21no—
321C22⋊Q8C22⋊D712yes—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—21no—
326C4×Q8C22⋊D711yescharacteristic
327C4×Q8C22×D711yescharacteristic
328C4×Q8C22×D712yes—
329C4×D4—24no—
330C4×D4—22no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4C22⋊D711yescharacteristic
334C4×D4C22×D711yescharacteristic
335#117—22no—
336#117—21no—
337#117—21no—
338#117C22⋊D711yescharacteristic
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C42—21no—
344C2×C42C22×D711yescharacteristic
345C4⋊C8—72no—
346C4⋊C8—72no—
347C4⋊C8—141no—
348#102—142no—
349#102—142no—
350#102—144no—
351C22⋊C8—141no—
352C4×C8—72no—
353C4×C8—141no—
354C4×C8—141no—
355C22×C7—21no—
356C22×C7—21no—
357C22×C7—22no—
358C22×C7—41no—
359C22×C7—41no—
360C22×C7#44812yes—
361C22×C7C2×#12011yescharacteristic
362C2×D7—82no—
363C2×D7—82no—
364C2×D7—82no—
365C2×D7—164no—
366C4×C7—22no—
367C4×C7—21no—
368C4×C7—21no—
369C4×C7—41no—
370C4×C7—42no—
371C4×C7—41no—
372C4×C7—44no—
373C4×C7—42no—
374C4×C7#44812yes—
375C4×C7#45312yes—
376C4×C7C22×SD1611yescharacteristic
377C4×C7C2×#13711yescharacteristic
378C4×C7#57712yes—
379C24—281no—
380#41—22no—
381#41#114314yescomplemented
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—44no—
386C2×Q8—84no—
387C2×Q8C2×C22⋊D712yes—
388C2×Q8C2×C22⋊D711yescharacteristic
389C2×Q8C2×C22⋊D714yes—
390C2×D4—21no—
391C2×D4—142no—
392C2×D4—142no—
393C2×D4—142no—
394C2×D4—284no—
395C2×D4—282no—
396C2×D4—282no—
397C2×D4—281no—
398C2×D4—281no—
399C2×D4—281no—
400C2×D4—562no—
401C2×D4C2×C22⋊D711yescharacteristic
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C2×C22⋊D711yescharacteristic
409C22×C4C2×C22⋊D711yescharacteristic
410SD16—2816no—
411SD16—284no—
412SD16—564no—
413D8—284no—
414M16—282no—
415C2×C8—144no—
416C2×C8—142no—
417C2×C8—142no—
418C2×C8—142no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4C2×C22⋊D712yes—
435C4⋊C4C2×C22⋊D711yescharacteristic
436C4⋊C4C2×C22⋊D711yescharacteristic
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C42—24no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42C2×C22⋊D711yescharacteristic
448C42C2×C22⋊D711yescharacteristic
449C42C2×C22⋊D711yescharacteristic
450C42C23×D711yescharacteristic
451C2×C7—22no—
452C2×C7—42no—
453C2×C7C2×#44811yescharacteristic
454C2×C7#301011yescharacteristic
455C2×C7C2×#45311yescharacteristic
456D7—162no—
457C23—21no—
458C23—282no—
459C23—562no—
460C23—562no—
461C23—562no—
462C23#651611yescharacteristic
463Q8—21no—
464Q8—24no—
465Q8—44no—
466Q8—44no—
467Q8—42no—
468Q8—84no—
469Q8C2×#114312yescomplemented
470D4—24no—
471D4—21no—
472D4—284no—
473D4—282no—
474D4—282no—
475D4—284no—
476D4—562no—
477D4—562no—
478D4—562no—
479D4C2×#114312yescomplemented
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4C2×#114312yes—
494C2×C4#650512yes—
495C2×C4#651611yescharacteristic
496C2×C4#651611yescharacteristic
497C2×C4#651611yescharacteristic
498C2×C4C22×C22⋊D711yescharacteristic
499C2×C4C22×C22⋊D711yescharacteristic
500C2×C4C22×C22⋊D711yescharacteristic
501C8—144no—
502C8—282no—
503C8—282no—
504C7#1862211yescharacteristic, complemented, Sylow
505C22—21no—
506C22—21no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—562no—
511C22—562no—
512C22—562no—
513C22—1124no—
514C22#8999912yes—
515C22C2×#651611yescharacteristic
516C4—22no—
517C4—21no—
518C4—21no—
519C4—41no—
520C4—44no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4#8999912yes—
525C4#9001312yes—
526C4C22×#114311yescharacteristic
527C4C2×#650511yescharacteristic
528C4#9052812yes—
529C2—22no—
530C2—42no—
531C2—1122no—
532C2C2×#8999911yescharacteristic
533C2C2×#9001311yescharacteristic
534C2#1174155811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3503C2
socleC22×C7C2×#120
prime radical#11739971C2
hypercenter#3503D7
hypocenterC7#18622
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. C7×#3503C2semidirect
2. #11740800C2semidirect
3. C2×#89857C2semidirect
5. #11729693C2semidirect
6. #11729677C2semidirect
7. #11729580C2semidirect
8. #11729528C2semidirect
9. #11729392C2semidirect
10. C7×Q82C22semidirect
13. C7×#554C22semidirect
41. #89341C22semidirect
42. Q8⋊(C7⋊C8)C22semidirect
43. Q8⋊(C7⋊C8)C22semidirect
44. D4⋊(C7⋊C8)C22semidirect
119. #3503D7semidirect
130. C7×#41SD16semidirect
202. Q82C2×D7semidirect
205. #554C2×D7semidirect
247. C7×Q8C2×SD16semidirect
250. C7×D4C2×SD16semidirect
381. #41#1143semidirect
469. Q8C2×#1143semidirect
479. D4C2×#1143semidirect
504. C7#18622semidirect

Maximal subgroups

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

subgroupindex
C7×#35032
#117408002
C2×#898572
#117399712
#117296932
#117296772
#117295802
#117295282
#117293922
#186227