Finite Group Databasebeta

← #13364938

Subgroups — #13364938

Order 1792.

1092 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#391C211yesmaximal, characteristic, complemented
2C2×#89999C211yesmaximal, characteristic
3C4×#6493C211yesmaximal, characteristic
4#11729886C211yesmaximal, characteristic, complemented
5#11729314C211yesmaximal, characteristic
6#11729310C211yesmaximal, characteristic, complemented
7C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8#11727441C212yesmaximal, complemented
9#11727430C212yesmaximal, complemented
10#11727428C212yesmaximal, complemented
11#11727420C212yesmaximal, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×#1143C2212yes—
16C2×#6493C2211yescharacteristic
17C2×C4×C22⋊D7—21no—
18C2×C4×D28C2211yescharacteristic
19C7×#391—21no—
20C7×#391—21no—
21C7×#391C2212yes—
22C7×#391C2212yes—
23C2×C7×#95—21no—
24C2×C7×#95—21no—
25#89999C418yescomplemented
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29#89985—21no—
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6383C2212yes—
34C2×#6382C2212yes—
35#89712—21no—
36#89711C2211yescharacteristic
37C4×(C2×D7)⋊C4—21no—
38C4×C4⋊Dic7C2211yescharacteristic
39#89700C2211yescharacteristic
40C2×C4×C7⋊C8C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#22041—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8—42no—
49C22×C7×Q8C2×C414yescomplemented
50C22×C7×Q8C2312yes—
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7D411yescharacteristic
54C23×C4×C7D411yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C22×C22⋊D7—21no—
57C22×D28C2311yescharacteristic
58C22×C4×D7—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—42no—
61C2×C7×C4⋊C4C2312yes—
62C2×C42×C7—42no—
63C2×C42×C7D411yescharacteristic
64C2×C42×C7D411yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C42×C7C2311yescharacteristic
67C2×C22⋊Dic7—21no—
68C2×#1143—42no—
69C2×#1143—42no—
70C2×#1143C2×C418yescomplemented
71#6493—22no—
72#6493C2×C414yes—
73C4×C22⋊D7—44no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×Dic7⋊C4—21no—
82C2×C4×Dic7—21no—
83C22×C7⋊C8C2312yes—
84C22×C7⋊C8C2312yes—
85C4×D28—42no—
86C7×#95—22no—
87C7×#95—22no—
88C7×#95—22no—
89C7×#95—22no—
90#6409—42no—
91#6409—42no—
92#6409C2×C418yescomplemented
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C22⋊(C7⋊C8)—22no—
97C22⋊(C7⋊C8)C2×C414yescomplemented
98#6383—42no—
99#6382—42no—
100C4×C7⋊C8—42no—
101C2×#448—71no—
102C4×#121—71no—
103C2×#391D711yescharacteristic, complemented
104#1968—72no—
105#1933—72no—
106#1927—72no—
107#1877—72no—
108#1874—71no—
109#1853—71no—
110#1787—71no—
111C4×C22⋊C8—71no—
112C24×C7C2×D411yescharacteristic
113C23×D7—21no—
114C2×C7×Q8—24no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—48no—
121C2×C7×Q8—42no—
122C2×C7×Q8—42no—
123C2×C7×Q8—48no—
124C2×C7×Q8C22×C414yes—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—24no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—41no—
133C22×C4×C7—41no—
134C22×C4×C7—41no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7SD1614yescomplemented
141C22×C4×C7C22×C412yes—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7#4112yes—
148C22×C4×C7C2411yescharacteristic
149C2×C22⋊D7—24no—
150C2×C22⋊D7—42no—
151C2×C22⋊D7—42no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C2×D28—22no—
156C2×D28—22no—
157C2×D28—21no—
158C2×D28C22×C412yes—
159C2×C4×D7—42no—
160C2×C4×D7—42no—
161C2×C4×D7—42no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—42no—
165C42×C7—22no—
166C42×C7—22no—
167C42×C7—41no—
168C42×C7—41no—
169C22⋊Dic7—42no—
170#1143—48no—
171(C2×D7)⋊C4—24no—
172(C2×D7)⋊C4—42no—
173(C2×D7)⋊C4—42no—
174(C2×D7)⋊C4—42no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7C22×C412yes—
179Dic7⋊C4—42no—
180C4×Dic7—42no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8C22×C414yes—
187C23×Q8C2×D711yescharacteristic
188C22×SD16—72no—
189C2×#121—71no—
190C2×C4×D4—71no—
191C2×C4×D4—141no—
192C22×C4⋊C4C2×D711yescharacteristic
193C22×C42C2×D711yescharacteristic
194#448—78no—
195C2×#106—72no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103—72no—
199C2×#102—72no—
200C2×C22⋊C8—71no—
201C2×C22⋊C8—71no—
202C2×C4×C8—72no—
203#391—21no—
204#391—21no—
205#391C2×D712yes—
206#391C2×D712yes—
207#390—71no—
208#389—71no—
209#388—141no—
210#386—141no—
211#380—71no—
212C4×C4⋊C4—71no—
213C4×C22⋊C4—141no—
214C2×#95—21no—
215C2×#95—21no—
216#340—72no—
217#340—72no—
218#336—72no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7C4×D412yes—
224C23×C7C2×SD1612yes—
225C23×C7#13512yes—
226C23×C7C22×D411yescharacteristic
227C22×D7—22no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C7×Q8—24no—
235C7×Q8—24no—
236C7×Q8—48no—
237C7×Q8—42no—
238C7×Q8—42no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—44no—
254C2×C4×C7—44no—
255C2×C4×C7—48no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—84no—
262C2×C4×C7C4×D412yes—
263C2×C4×C7C2×SD1612yes—
264C2×C4×C7C2×SD1612yes—
265C2×C4×C7#13712yes—
266C2×C4×C7C23×C411yescharacteristic
267C2×C4×C7C2×#4111yescharacteristic
268C2×C4×C7H(5,𝔽2)12yes—
269C2×C4×C7H(5,𝔽2)12yes—
270C22⋊D7—48no—
271C2×Dic7—22no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280D28—24no—
281D28—42no—
282C4×D7—84no—
283C7⋊C8—24no—
284C7⋊C8—42no—
285C22×Q8—22no—
286C22×Q8—21no—
287C22×Q8—42no—
288C22×Q8—42no—
289C22×Q8C4×D714yescomplemented
290C22×Q8C22×D712yes—
291C22×D4—71no—
292C22×D4—141no—
293C23×C4—21no—
294C23×C4—21no—
295C23×C4—141no—
296C23×C4C22⋊D711yescharacteristic
297C23×C4C22⋊D711yescharacteristic
298C23×C4C22×D711yescharacteristic
299C2×SD16—78no—
300C2×SD16—282no—
301C2×SD16—282no—
302C22×C8—72no—
303C22×C8—72no—
304#121—74no—
305#121—142no—
306C4×D4—282no—
307C4×D4—284no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4C22×D712yes—
315C2×C22⋊C4—141no—
316C2×C22⋊C4—141no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—141no—
319C2×C22⋊C4—141no—
320C2×C42—42no—
321C2×C42—141no—
322C2×C42C22⋊D711yescharacteristic
323C2×C42C22⋊D711yescharacteristic
324C2×C42C22×D711yescharacteristic
325C2×C42C22×D711yescharacteristic
326#106—282no—
327#103—78no—
328#103—282no—
329#103—282no—
330#102—282no—
331C22⋊C8—74no—
332C22⋊C8—142no—
333C4×C8—282no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—141no—
339#95—141no—
340#95—141no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7C4×SD1614yes—
348C22×C7C2×C4×D411yescharacteristic
349C22×C7#53411yescharacteristic
350C22×C7#53511yescharacteristic
351C22×C7C22×SD1611yescharacteristic
352C22×C7C2×#13511yescharacteristic
353C22×C7C2×#13711yescharacteristic
354C22×C7#57511yescharacteristic
355C2×D7—42no—
356C2×D7—42no—
357C2×D7—42no—
358C2×D7—81no—
359C2×D7—81no—
360C2×D7—81no—
361C2×D7—81no—
362C4×C7—22no—
363C4×C7—22no—
364C4×C7—22no—
365C4×C7—22no—
366C4×C7—44no—
367C4×C7—41no—
368C4×C7—82no—
369C4×C7#51812yes—
370C4×C7#53812yes—
371Dic7—42no—
372Dic7—81no—
373Dic7—81no—
374C24—141no—
375C24C2×C22⋊D711yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8C2×C4×D714yes—
387C2×D4—72no—
388C2×D4—144no—
389C2×D4—141no—
390C2×D4—142no—
391C2×D4—142no—
392C2×D4—282no—
393C2×D4—282no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—141no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—282no—
413C22×C4—282no—
414C22×C4—282no—
415C22×C4#114314yescomplemented
416C22×C4C2×C4×D712yes—
417C22×C4#115712yes—
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C2×C22⋊D711yescharacteristic
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C23×D711yescharacteristic
424SD16—288no—
425C2×C8—74no—
426C2×C8—142no—
427C2×C8—142no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8—142no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—72no—
435C4⋊C4—141no—
436C4⋊C4—142no—
437C4⋊C4—142no—
438C4⋊C4—282no—
439C22⋊C4—144no—
440C22⋊C4—282no—
441C22⋊C4—282no—
442C22⋊C4—282no—
443C22⋊C4—282no—
444C42—22no—
445C42—22no—
446C42—41no—
447C42—41no—
448C42—282no—
449C2×C7—24no—
450C2×C7#232111yescharacteristic
451C2×C7C2×C4×SD1611yescharacteristic
452C2×C7C4×#13711yescharacteristic
453C2×C7#318711yescharacteristic
454C2×C7#319011yescharacteristic
455C2×C7#320611yescharacteristic
456C2×C7#321111yescharacteristic
457D7—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—142no—
463C23—281no—
464C23—281no—
465C23—281no—
466C23—281no—
467C23—281no—
468C23—281no—
469C23C4×C22⋊D712yes—
470C23C2×#114312yes—
471C23#651312yes—
472C23C22×C22⋊D711yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—144no—
479D4—288no—
480D4—282no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—142no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—281no—
513C2×C4—564no—
514C2×C4C4×C22⋊D712yes—
515C2×C4C2×#114312yes—
516C2×C4C2×#114312yes—
517C2×C4#650512yes—
518C2×C4C22×C4×D711yescharacteristic
519C2×C4C2×#115711yescharacteristic
520C2×C4#654812yes—
521C2×C4#654812yes—
522C8—144no—
523C8—282no—
524C7#2204111yescharacteristic, complemented, Sylow
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—282no—
532C22—282no—
533C22—282no—
534C22—561no—
535C22—561no—
536C22—561no—
537C22—561no—
538C22C4×#114314yes—
539C22C2×C4×C22⋊D711yescharacteristic
540C22#9049511yescharacteristic
541C22#9049611yescharacteristic
542C22C22×#114311yescharacteristic
543C22C2×#650511yescharacteristic
544C22C2×#651311yescharacteristic
545C22#9051311yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—44no—
552C4—82no—
553C4—282no—
554C4—561no—
555C4—561no—
556C4#9017712yes—
557C4#9018512yes—
558C2—24no—
559C2—562no—
560C2#1173954111yescharacteristic
561C2C2×C4×#114311yescharacteristic
562C2C4×#650511yescharacteristic
563C2#1174018111yescharacteristic
564C2#1174018211yescharacteristic
565C2#1174039111yescharacteristic
566C2#1174065811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#391C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
cosocleC22×C4×C7C24
hypercenterC2×#391D7
hypocenterC7#22041
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
1. C2×C7×#391C2semidirect
4. #11729886C2semidirect
6. #11729310C2semidirect
7. C4×C22⋊(C7⋊C8)C2semidirect
8. #11727441C2semidirect
9. #11727430C2semidirect
10. #11727428C2semidirect
11. #11727420C2semidirect
25. #89999C4semidirect
49. C22×C7×Q8C2×C4semidirect
70. C2×#1143C2×C4semidirect
92. #6409C2×C4semidirect
97. C22⋊(C7⋊C8)C2×C4semidirect
103. C2×#391D7semidirect
140. C22×C4×C7SD16semidirect
289. C22×Q8C4×D7semidirect
415. C22×C4#1143semidirect
524. C7#22041semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3912
C2×#899992
C4×#64932
#117298862
#117293142
#117293102
C4×C22⋊(C7⋊C8)2
#117274412
#117274302
#117274282
#117274202
#220417