Finite Group Databasebeta

← #13392133

Subgroups — #13392133

Order 1792.

1260 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2433C211yesmaximal, characteristic
2C22×#6406C211yesmaximal, characteristic, complemented
3C2×#89805C211yesmaximal, characteristic, complemented
4#11730901C211yesmaximal, characteristic, complemented
5#11729882C211yesmaximal, characteristic, complemented
6#11729516C212yesmaximal, complemented
7#11729515C212yesmaximal, complemented
8#11729435C211yesmaximal, characteristic, complemented
9#11727407C212yesmaximal, complemented
10#11727390C212yesmaximal, complemented
11#11727203C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C2×C7×#127C2212yes—
14C2×C7×#121C2211yescharacteristic
15C2×C7×#120—21no—
16C2×C7×#120—21no—
17C22×C4⋊Dic7C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C2×#6440—21no—
20C7×#392—22no—
21C7×#390C2212yes—
22C4×C7×C22⋊C4C2211yescharacteristic
23C2×#6406—24no—
24C2×#6406C2212yes—
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28C2×#6406C2211yescharacteristic, complemented
29C2×#6406C2211yescharacteristic, complemented
30C2×#6406C2212yescomplemented
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33#89805—24no—
34C2×#6381C2211yescharacteristic, complemented
35C2×#6381C2211yescharacteristic, complemented
36#89715—21no—
37C4×C4⋊Dic7C2212yescomplemented
38#89700C2212yescomplemented
39C2×C4⋊(C7⋊C8)C2212yescomplemented
40#89425—22no—
41#89324C2212yescomplemented
42#89319C2212yescomplemented
43#24476—71nomaximal, Sylow
44C25×C7—21no—
45C22×C7×D4—24no—
46C22×C7×D4—21no—
47C22×C7×D4—22no—
48C22×C7×D4—21no—
49C22×C7×D4—22no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—42no—
53C22×C7×D4C2311yescharacteristic
54C22×C7×D4C2312yes—
55C23×C4×C7C2311yescharacteristic
56C23×Dic7—21no—
57C7×#127—24no—
58C7×#127—42no—
59C7×#121—22no—
60C7×#121—22no—
61C7×#120—44no—
62C7×#120—44no—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4D411yescharacteristic
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C42×C7C2312yes—
70C2×C22⋊Dic7—21no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7D414yescomplemented
74C2×C4⋊Dic7C2312yes—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77C2×Dic7⋊C4—22no—
78C2×C4×Dic7—22no—
79C22×C7⋊C8—22no—
80C22×C7⋊C8—42no—
81C22×C7⋊C8C2311yescharacteristic
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2311yescharacteristic
84#6440—44no—
85C7×#95D412yes—
86#6406—24no—
87#6406—24no—
88#6406—24no—
89#6406—22no—
90#6406—22no—
91#6406—42no—
92#6406—42no—
93C7⋊#95—21no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C22⋊(C7⋊C8)—22no—
97#6381—22no—
98#6381—22no—
99C4⋊(C7⋊C8)—42no—
100C2×#480—71no—
101C22×#102—71no—
102#2433D711yescharacteristic
103#2344—71no—
104#2067—72no—
105#2041—72no—
106#1982—72no—
107#1925—72no—
108#1893—71no—
109#1850—71no—
110#1786—71no—
111C24×C7—22no—
112C24×C7—21no—
113C24×C7—44no—
114C24×C7—42no—
115C24×C7—42no—
116C24×C7—42no—
117C24×C7—82no—
118C24×C7C2×D411yescharacteristic
119C2×C7×D4—28no—
120C2×C7×D4—22no—
121C2×C7×D4—21no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—21no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—48no—
130C2×C7×D4—44no—
131C2×C7×D4—44no—
132C2×C7×D4—44no—
133C2×C7×D4—44no—
134C2×C7×D4—44no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
139C2×C7×D4—44no—
140C2×C7×D4—82no—
141C2×C7×D4—82no—
142C2×C7×D4#4114yes—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7#4112yes—
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—24no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—42no—
157C22×Dic7—42no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—21no—
160C7×C22⋊C4—22no—
161C7×C22⋊C4—44no—
162C7×C22⋊C4—42no—
163C7×C22⋊C4—44no—
164C7×C22⋊C4D814yes—
165C42×C7—22no—
166C42×C7—22no—
167C22⋊Dic7—42no—
168C4⋊Dic7—24no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—22no—
172C4⋊Dic7—22no—
173C4⋊Dic7C2×D412yes—
174C4⋊Dic7C2×D412yes—
175Dic7⋊C4—44no—
176C4×Dic7—44no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—44no—
185C23×D4C2×D711yescharacteristic
186C23×C8—71no—
187C2×#127C2×D712yes—
188C2×#121C2×D711yescharacteristic
189C2×#120—21no—
190C2×#120—21no—
191C2×#117—141no—
192C22×C4⋊C4—71no—
193#480—144no—
194C2×#107—71no—
195C2×#107—71no—
196C2×C4⋊C8—72no—
197C2×#102—71no—
198C2×#102—71no—
199C2×#102—72no—
200C2×#102—72no—
201C2×#102—72no—
202C2×#102—72no—
203C2×#102—72no—
204C2×#102—144no—
205C2×C22⋊C8—71no—
206#392—22no—
207#390C2×D712yes—
208#389—72no—
209#387—142no—
210#381—141no—
211#380—71no—
212C4×C4⋊C4—72no—
213C4×C22⋊C4C2×D711yescharacteristic
214#340—72no—
215#336—72no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—24no—
219C23×C7—48no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—41no—
227C23×C7—42no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—42no—
231C23×C7—88no—
232C23×C7—88no—
233C23×C7—84no—
234C23×C7—84no—
235C23×C7#12112yes—
236C23×C7C2×D812yes—
237C23×C7AGL(1,ℤ/8)12yes—
238C23×C7C22×D411yescharacteristic
239C7×D4—24no—
240C7×D4—24no—
241C7×D4—48no—
242C7×D4—41no—
243C7×D4—42no—
244C7×D4—41no—
245C7×D4—42no—
246C7×D4—42no—
247C7×D4—88no—
248C7×D4—88no—
249C7×D4—84no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7#12112yes—
263C2×C4×C7C2×D814yes—
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4112yes—
266C2×C4×C7C2×#4111yescharacteristic
267C2×C4×C7H(5,𝔽2)12yes—
268C2×Dic7—22no—
269C2×Dic7—22no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7—48no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C7⋊C8—41no—
279C7⋊C8—41no—
280C7⋊C8—42no—
281C25—21no—
282C22×D4—24no—
283C22×D4—21no—
284C22×D4—22no—
285C22×D4—22no—
286C22×D4—21no—
287C22×D4—22no—
288C22×D4—22no—
289C22×D4—42no—
290C22×D4C22×D711yescharacteristic
291C22×D4C22×D712yes—
292C23×C4—141no—
293C23×C4C22×D711yescharacteristic
294C22×C8—72no—
295C22×C8—71no—
296C22×C8—71no—
297C22×C8—142no—
298C22×C8—282no—
299#127—24no—
300#127—42no—
301#121—22no—
302#121—22no—
303#120—44no—
304#120—44no—
305#117—284no—
306C2×C4⋊C4—74no—
307C2×C4⋊C4—71no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—72no—
310C2×C4⋊C4—142no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4C22×D711yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4C22⋊D711yescharacteristic
319C2×C22⋊C4C22⋊D711yescharacteristic
320C2×C42—142no—
321C2×C42C22×D712yes—
322#107—142no—
323#107—142no—
324C4⋊C8—282no—
325#102—144no—
326#102—144no—
327#102—142no—
328#102—142no—
329#102—144no—
330#102—282no—
331#102—282no—
332C22⋊C8—142no—
333#95—142no—
334#95—141no—
335#95—142no—
336#95C22⋊D712yes—
337C22×C7—22no—
338C22×C7—24no—
339C22×C7—22no—
340C22×C7—24no—
341C22×C7—44no—
342C22×C7—44no—
343C22×C7—44no—
344C22×C7—816no—
345C22×C7—82no—
346C22×C7—82no—
347C22×C7—82no—
348C22×C7—82no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—82no—
352C22×C7—81no—
353C22×C7—82no—
354C22×C7#45914yes—
355C22×C7C2×#12111yescharacteristic
356C22×C7#54612yes—
357C22×C7C22×D811yescharacteristic
358C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
359C22×C7#57712yes—
360C4×C7—22no—
361C4×C7—24no—
362C4×C7—42no—
363C4×C7—81no—
364C4×C7#52511yescharacteristic
365C4×C7#53511yescharacteristic
366C4×C7#55112yes—
367Dic7—42no—
368Dic7—42no—
369Dic7—82no—
370C24—22no—
371C24—21no—
372C24—44no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—82no—
377C24C2×C22⋊D711yescharacteristic
378C2×D4—28no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4#115914yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—144no—
406C22×C4—141no—
407C22×C4—141no—
408C22×C4—142no—
409C22×C4—282no—
410C22×C4—282no—
411C22×C4D4×D712yes—
412C22×C4#115912yes—
413C22×C4C2×C22⋊D712yes—
414C22×C4C2×C22⋊D712yes—
415C22×C4C2×C22⋊D711yescharacteristic
416C22×C4C23×D711yescharacteristic
417C2×C8—141no—
418C2×C8—141no—
419C2×C8—142no—
420C2×C8—142no—
421C2×C8—142no—
422C2×C8—142no—
423C2×C8—142no—
424C2×C8—284no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—72no—
428C4⋊C4—72no—
429C4⋊C4—144no—
430C4⋊C4—141no—
431C4⋊C4—141no—
432C4⋊C4—142no—
433C4⋊C4—142no—
434C4⋊C4—284no—
435C22⋊C4—22no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—282no—
440C22⋊C4#114114yes—
441C42—22no—
442C42—22no—
443C42—284no—
444C2×C7—24no—
445C2×C7—84no—
446C2×C7—161no—
447C2×C7#241111yescharacteristic
448C2×C7C2×#45911yescharacteristic
449C2×C7#304311yescharacteristic
450C2×C7#329412yes—
451C2×C7#331012yes—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—48no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—88no—
468C23—88no—
469C23—84no—
470C23—84no—
471C23C2×#114112yes—
472C23#649512yes—
473C23#650112yes—
474C23C22×C22⋊D711yescharacteristic
475D4—24no—
476D4—24no—
477D4—48no—
478D4—41no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—41no—
483D4—88no—
484D4—88no—
485D4—84no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—288no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4C2×#114114yes—
509C2×C4#650112yes—
510C2×C4C2×D4×D711yescharacteristic
511C2×C4C2×#115912yes—
512C2×C4C2×#115911yescharacteristic
513C2×C4#654812yes—
514C8—281no—
515C8—281no—
516C8—282no—
517C7#2447611yescharacteristic, complemented, Sylow
518C22—24no—
519C22—22no—
520C22—22no—
521C22—24no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—816no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—82no—
535C22#8984414yes—
536C22C22×#114111yescharacteristic
537C22C2×#649511yescharacteristic
538C22C2×#650111yescharacteristic
539C22#9049712yes—
540C22#9052712yes—
541C4—22no—
542C4—24no—
543C4—42no—
544C4—81no—
545C4—282no—
546C4—282no—
547C4—562no—
548C4#9027711yescharacteristic
549C4#9027812yes—
550C4#9028311yescharacteristic
551C2—24no—
552C2—84no—
553C2—161no—
554C2#1174002811yescharacteristic
555C2C2×#8984411yescharacteristic
556C2#1174083611yescharacteristic
557C2#1174092712yes—
558C2#1174092912yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2433C2
socleC23×C7C22×D4
hypercenter#2433D7
hypocenterC7#24476
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2433

Complements

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

NH ≅ G/Ntype
2. C22×#6406C2semidirect
3. C2×#89805C2semidirect
4. #11730901C2semidirect
5. #11729882C2semidirect
6. #11729516C2semidirect
7. #11729515C2semidirect
8. #11729435C2semidirect
9. #11727407C2semidirect
10. #11727390C2semidirect
11. #11727203C2semidirect
28. C2×#6406C22semidirect
29. C2×#6406C22semidirect
30. C2×#6406C22semidirect
34. C2×#6381C22semidirect
35. C2×#6381C22semidirect
37. C4×C4⋊Dic7C22semidirect
38. #89700C22semidirect
39. C2×C4⋊(C7⋊C8)C22semidirect
41. #89324C22semidirect
42. #89319C22semidirect
73. C2×C4⋊Dic7D4semidirect
517. C7#24476semidirect

Maximal subgroups

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

subgroupindex
C7×#24332
C22×#64062
C2×#898052
#117309012
#117298822
#117295162
#117295152
#117294352
#117274072
#117273902
#117272032
#244767