Finite Group Databasebeta

← #13404552

Subgroups — #13404552

Order 1792.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2538C211yesmaximal, characteristic, complemented
2C2×#89820C212yesmaximal, complemented
3C2×#89799C211yesmaximal, characteristic, complemented
4#11739823C211yesmaximal, characteristic
5C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
6#11729453C212yesmaximal, complemented
7C4×#6409C211yesmaximal, characteristic, complemented
8#11727334C211yesmaximal, characteristic, complemented
9#11727322C211yesmaximal, characteristic, complemented
10#11727278C212yesmaximal, complemented
11#11727269C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2212yes—
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C22×#1143C2212yes—
16C2×#6503—21no—
17C2×#6493C2212yes—
18C2×#6493C2212yes—
19C23×C7⋊C8C2212yes—
20C2×#6437C2211yescharacteristic
21C7×#396—22no—
22C7×#390—21no—
23C7×#389—21no—
24C7×#384C2211yescharacteristic
25C7×#380C2212yes—
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×C22⋊(C7⋊C8)C2212yes—
29#89820—28no—
30#89799—24no—
31D4×C7⋊C8—24no—
32#89760—21no—
33C2×#6383C2212yes—
34C2×#6383C2212yes—
35C2×#6383C2212yes—
36C2×#6382C2212yes—
37#89732—22no—
38C4×C4⋊Dic7C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89324C2212yes—
42#89324C2211yescharacteristic
43#89324C2211yescharacteristic
44#25387—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4D411yescharacteristic
47C23×C4×C7C2312yes—
48C22×C22⋊D7—22no—
49C22×C22⋊D7—22no—
50C22×D28—21no—
51C22×D28—21no—
52C22×D28C2312yes—
53C7×C22⋊Q8—24no—
54C7×C22⋊Q8—42no—
55C4×C7×Q8—22no—
56C4×C7×Q8—22no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4C2312yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4D412yes—
69C2×C42×C7—21no—
70C2×C42×C7C2311yescharacteristic
71C2×#1143—24no—
72C2×#1143—24no—
73C2×#1143—42no—
74#6503—44no—
75#6493—24no—
76#6493—24no—
77#6493—42no—
78#6493—42no—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—22no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4×Dic7—21no—
84C22×C7⋊C8—24no—
85C22×C7⋊C8—24no—
86C22×C7⋊C8—42no—
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2312yes—
90#6437—22no—
91#6437—22no—
92C7×#95—22no—
93#6409—22no—
94#6409D414yescomplemented
95C7⋊#95—21no—
96C22⋊(C7⋊C8)—24no—
97#6383—24no—
98#6383—24no—
99#6383—42no—
100#6382—24no—
101C4⋊(C7⋊C8)D414yescomplemented
102C4×C7⋊C8—22no—
103C2×#465—72no—
104C2×#460—71no—
105C2×C8×D4—71no—
106#2541—71no—
107#2538D711yescharacteristic, complemented
108#2034—72no—
109#2000—72no—
110#1957—71no—
111#1948—71no—
112#1893—72no—
113C4×#103—71no—
114C24×C7C2×D412yes—
115C23×D7—22no—
116C2×C7×Q8—21no—
117C2×C7×Q8—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8C2×D412yes—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4SD1614yescomplemented
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2411yescharacteristic
138C2×C22⋊D7—44no—
139C2×C22⋊D7—44no—
140C2×C22⋊D7—44no—
141C2×C22⋊D7—44no—
142C2×C22⋊D7—82no—
143C2×C22⋊D7—82no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C2×D28—22no—
147C2×D28—22no—
148C2×D28—22no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—42no—
153C2×D28—42no—
154C2×D28—42no—
155C2×D28—42no—
156C2×D28—81no—
157C2×D28—81no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—44no—
164C7×C4⋊C4—42no—
165C7×C4⋊C4—42no—
166C7×C4⋊C4—81no—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—44no—
169C7×C22⋊C4—82no—
170C42×C7—42no—
171C42×C7C2×D412yes—
172C42×C7#4112yes—
173#1143—48no—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—44no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7C2×D412yes—
179C4⋊Dic7#4112yes—
180C4×Dic7—42no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—48no—
189C2×C7⋊C8C2×D414yes—
190C2×C7⋊C8C2×D412yes—
191C22×SD16—72no—
192C23×C8—72no—
193C2×#127—71no—
194C2×#127—141no—
195C2×C22⋊Q8C2×D712yes—
196C2×#121—72no—
197C2×#121—72no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×D4C2×D711yescharacteristic
200#465—148no—
201#460—144no—
202C8×D4—144no—
203C2×#106—72no—
204C2×C4⋊C8—71no—
205C2×#103—71no—
206C2×#103—71no—
207C2×#102—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×C22⋊C8—72no—
211C2×C4×C8—71no—
212#396—22no—
213#392—142no—
214#390—21no—
215#390—141no—
216#389—21no—
217#384C2×D711yescharacteristic
218#380C2×D712yes—
219C4×C4⋊C4—71no—
220#340—71no—
221#340—72no—
222#340—71no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7C2×SD1614yes—
230C23×C7C22×D411yescharacteristic
231C22×D7—42no—
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—42no—
238C7×Q8—22no—
239C7×Q8—22no—
240C7×Q8—41no—
241C7×Q8—41no—
242C7×D4—28no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7#12112yes—
262C2×C4×C7#12112yes—
263C2×C4×C7C2×SD1612yes—
264C2×C4×C7#13512yes—
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C2×#4111yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7H(5,𝔽2)11yescharacteristic
270C2×C4×C7H(5,𝔽2)11yescharacteristic
271C22⋊D7—88no—
272C22⋊D7—88no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279D28—42no—
280D28—42no—
281D28—42no—
282D28—42no—
283D28—84no—
284D28—84no—
285C7⋊C8—22no—
286C7⋊C8—24no—
287C7⋊C8—41no—
288C22×Q8C22×D711yescharacteristic
289C22×D4—72no—
290C22×D4—142no—
291C22×D4—142no—
292C22×D4—141no—
293C22×D4—141no—
294C22×D4C22⋊D711yescharacteristic
295C23×C4C22×D712yes—
296C2×SD16—144no—
297C2×SD16—144no—
298C2×SD16—282no—
299C22×C8—71no—
300C22×C8—72no—
301C22×C8—71no—
302C22×C8—144no—
303C22×C8—144no—
304C22×C8—282no—
305#127—142no—
306#127—142no—
307#127—284no—
308C22⋊Q8—24no—
309C22⋊Q8—42no—
310#121—144no—
311#121—144no—
312#121—282no—
313#121—282no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×D4—22no—
317C4×D4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4C22⋊D711yescharacteristic
325C2×C4⋊C4C22×D711yescharacteristic
326C2×C4⋊C4C22×D712yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—142no—
330C2×C22⋊C4—142no—
331C2×C22⋊C4C22⋊D712yes—
332C2×C42—21no—
333C2×C42—141no—
334C2×C42C22×D711yescharacteristic
335#106—144no—
336C4⋊C8—74no—
337#103—74no—
338#103—142no—
339#102—144no—
340#102—144no—
341#102—282no—
342C22⋊C8—144no—
343C4×C8—142no—
344#95—22no—
345#95—141no—
346C22×C7—24no—
347C22×C7—24no—
348C22×C7—24no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7C2×#12111yescharacteristic
354C22×C7D4211yescharacteristic
355C22×C7#54611yescharacteristic
356C22×C7C22×SD1611yescharacteristic
357C22×C7C2×#13511yescharacteristic
358C22×C7#57611yescharacteristic
359C22×C7#57711yescharacteristic
360C2×D7—82no—
361C2×D7—82no—
362C2×D7—82no—
363C2×D7—82no—
364C2×D7—82no—
365C2×D7—82no—
366C2×D7—82no—
367C4×C7—22no—
368C4×C7—42no—
369C4×C7—42no—
370C4×C7—82no—
371C4×C7—81no—
372C4×C7#46514yescomplemented
373C4×C7#53211yescharacteristic
374C4×C7#54011yescharacteristic
375C4×C7D4211yescharacteristic
376C4×C7#55011yescharacteristic
377Dic7—42no—
378Dic7—42no—
379C24—142no—
380C24C2×C22⋊D712yes—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8D4×D712yes—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—142no—
390C2×D4—142no—
391C2×D4—142no—
392C2×D4—142no—
393C2×D4—142no—
394C2×D4—142no—
395C2×D4—284no—
396C2×D4—284no—
397C2×D4—284no—
398C2×D4—284no—
399C2×D4—282no—
400C2×D4—282no—
401C2×D4—282no—
402C2×D4—282no—
403C2×D4—561no—
404C2×D4—561no—
405C2×D4—562no—
406C2×D4—562no—
407C2×D4#114314yescomplemented
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4C2×C22⋊D712yes—
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423SD16—288no—
424C2×C8—74no—
425C2×C8—72no—
426C2×C8—141no—
427C2×C8—142no—
428C2×C8—141no—
429C2×C8—142no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—288no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—72no—
443C4⋊C4—72no—
444C4⋊C4—81no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—82no—
450C22⋊C4—284no—
451C22⋊C4—284no—
452C42—42no—
453C42—282no—
454C42D4×D712yes—
455C42#115912yes—
456C2×C7—44no—
457C2×C7#242811yescharacteristic
458C2×C7C2×#46511yescharacteristic
459C2×C7#306211yescharacteristic
460C2×C7D4×SD1611yescharacteristic
461C2×C7#328711yescharacteristic
462C2×C7#329511yescharacteristic
463C2×C7#330411yescharacteristic
464D7—162no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—282no—
472C23—282no—
473C23—282no—
474C23—282no—
475C23—282no—
476C23—282no—
477C23—282no—
478C23C2×#114314yes—
479C23C22×C22⋊D711yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—41no—
483Q8—41no—
484D4—28no—
485D4—282no—
486D4—282no—
487D4—282no—
488D4—282no—
489D4—564no—
490D4—568no—
491D4—568no—
492D4—564no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—281no—
514C2×C4—281no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4#650212yes—
518C2×C4#650212yes—
519C2×C4C2×#114312yes—
520C2×C4#651312yes—
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×D4×D711yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4#654811yescharacteristic
525C2×C4D7×#4111yescharacteristic
526C2×C4#655311yescharacteristic
527C8—142no—
528C8—144no—
529C8—281no—
530C7#2538711yescharacteristic, complemented, Sylow
531C22—24no—
532C22—24no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—562no—
539C22—562no—
540C22—562no—
541C22—562no—
542C22—562no—
543C22—562no—
544C22—562no—
545C22D4×C22⋊D711yescharacteristic
546C22C2×#650211yescharacteristic
547C22C22×#114311yescharacteristic
548C22C2×#651311yescharacteristic
549C22#9052011yescharacteristic
550C22#9052611yescharacteristic
551C22#9052811yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—81no—
557C4—282no—
558C4—282no—
559C4#8982014yescomplemented
560C4#9028711yescharacteristic
561C4#9029011yescharacteristic
562C4#9030011yescharacteristic
563C4#9031011yescharacteristic
564C2—44no—
565C2—1122no—
566C2#1174009411yescharacteristic
567C2C2×#8982011yescharacteristic
568C2#1174086711yescharacteristic
569C2D4×#114311yescharacteristic
570C2#1174103011yescharacteristic
571C2#1174107911yescharacteristic
572C2#1174111811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2538C2
socleC23×C7C22×D4
prime radical#11739823C2
hypercenter#2538D7
hypocenterC7#25387
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C7×#2538C2semidirect
2. C2×#89820C2semidirect
3. C2×#89799C2semidirect
5. C2×D4×C7⋊C8C2semidirect
6. #11729453C2semidirect
7. C4×#6409C2semidirect
8. #11727334C2semidirect
9. #11727322C2semidirect
10. #11727278C2semidirect
11. #11727269C2semidirect
94. #6409D4semidirect
101. C4⋊(C7⋊C8)D4semidirect
107. #2538D7semidirect
124. C2×C7×D4SD16semidirect
372. C4×C7#465semidirect
407. C2×D4#1143semidirect
530. C7#25387semidirect
559. C4#89820semidirect

Maximal subgroups

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

subgroupindex
C7×#25382
C2×#898202
C2×#897992
#117398232
C2×D4×C7⋊C82
#117294532
C4×#64092
#117273342
#117273222
#117272782
#117272692
#253877