Finite Group Databasebeta

← #13395338

Subgroups — #13395338

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×D8C211yesmaximal, characteristic
2C2×#89925C212yesmaximal
3#11739827C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic7C211yesmaximal, characteristic, complemented
5C2×#89791C211yesmaximal, characteristic, complemented
6#11729358C211yesmaximal, characteristic, complemented
7#11728397C212yesmaximal, complemented
8#11727932C212yesmaximal, complemented
9#11727925C212yesmaximal, complemented
10#11727540C211yesmaximal, characteristic, complemented
11#11727525C211yesmaximal, characteristic, complemented
12C22×C7×D8C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×#6499C2212yes—
16C2×D4×Dic7C2212yes—
17C22×#1142C2212yes—
18C22×C4⋊Dic7—21no—
19C2×#6432C2211yescharacteristic
20C4×C7×D8—24no—
21C2×C7×#107C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C2×#6406C2212yes—
26#89984—21no—
27#89984—21no—
28#89984—21no—
29#89925—28no—
30C2×#6390C2212yes—
31C2×#6390C2212yes—
32C2×Dic7⋊C8C2212yes—
33#89791—24no—
34#89757C2212yes—
35#89746C2212yes—
36C2×#6384C2212yes—
37#89725—22no—
38#89722—22no—
39#89707C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89345C2212yes—
43#89324C2212yes—
44#24727—71nomaximal, Sylow
45C22×C7×D4C2311yescharacteristic
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×C4×C7—21no—
49C23×Dic7—22no—
50C22×Dic14C2312yes—
51C2×C7×D8—22no—
52C2×C7×D8—22no—
53C2×C7×D8D414yes—
54C22×C7×C8C2311yescharacteristic
55C22×C7×C8C2311yescharacteristic
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C4×C7×D4D414yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C42×C7C2311yescharacteristic
65C2×C22⋊Dic7—22no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—22no—
68#6499—24no—
69#6499—42no—
70D4×Dic7—24no—
71D4×Dic7—42no—
72C2×#1142—24no—
73C2×#1142—24no—
74C2×#1142—24no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—44no—
81C2×C4⋊Dic7D411yescharacteristic
82C2×C4⋊Dic7D411yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×C4×Dic7C2312yes—
85C22×C7⋊C8C2312yes—
86#6432—22no—
87#6432—22no—
88C7×#107—22no—
89C7×#102—22no—
90C7×#102—22no—
91C4×C7×C8—22no—
92#6406—24no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95D412yes—
97#6390—24no—
98#6390—42no—
99Dic7⋊C8—24no—
100#6384—24no—
101C4⋊(C7⋊C8)—22no—
102C2×#463—72no—
103C2×#461—71no—
104C2×C4×D8D711yescharacteristic
105#2547—71no—
106D4×C4⋊C4—71no—
107#2068—72no—
108#2042—72no—
109#1956—71no—
110#1939—71no—
111#1877—72no—
112#1802—71no—
113C24×C7—21no—
114C24×C7—21no—
115C2×C7×D4—21no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4C2×D412yes—
124C2×C7×D4C2×D412yes—
125C2×C7×D4C2×D412yes—
126C2×C7×D4#4112yes—
127C22×C4×C7—21no—
128C22×C4×C7—24no—
129C22×C4×C7—21no—
130C22×C4×C7—42no—
131C22×C4×C7—42no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7—42no—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—44no—
141C22×Dic7—44no—
142C22×Dic7—82no—
143C22×Dic7C2×D412yes—
144C22×Dic7C2×D412yes—
145C2×Dic14—22no—
146C2×Dic14—22no—
147C2×Dic14—22no—
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—22no—
151C7×D8—28no—
152C2×C7×C8—21no—
153C2×C7×C8—21no—
154C2×C7×C8—21no—
155C2×C7×C8—21no—
156C2×C7×C8—21no—
157C2×C7×C8C2×D412yes—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4C2×D412yes—
162C7×C22⋊C4—24no—
163C7×C22⋊C4—42no—
164C42×C7C2×D412yes—
165C42×C7#4112yes—
166C22⋊Dic7—44no—
167C22⋊Dic7—44no—
168C22⋊Dic7—44no—
169#1142—48no—
170C4⋊Dic7—22no—
171C4⋊Dic7—22no—
172C4⋊Dic7—22no—
173C4⋊Dic7—22no—
174C4⋊Dic7—44no—
175C4⋊Dic7—42no—
176C4⋊Dic7—42no—
177C4⋊Dic7—42no—
178C4⋊Dic7—81no—
179C4×Dic7—22no—
180C4×Dic7—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C22×SD16—72no—
185C22×D8C2×D711yescharacteristic
186C2×C4⋊Q8—71no—
187C2×#124—72no—
188C2×C4×D4—72no—
189C2×C4×D4C2×D711yescharacteristic
190C2×C4×D4C2×D711yescharacteristic
191C22×C4⋊C4—141no—
192#463—148no—
193#461—144no—
194C4×D8—24no—
195C2×#107C2×D711yescharacteristic
196C2×C4⋊C8—72no—
197C2×C4⋊C8—71no—
198C2×#103—72no—
199C2×#103—72no—
200C2×#103—72no—
201C2×#102—72no—
202C2×#102C2×D711yescharacteristic
203C2×#102C2×D711yescharacteristic
204C2×C4×C8C2×D711yescharacteristic
205#397—142no—
206#391—72no—
207#389—72no—
208#385—142no—
209#384—71no—
210#380—141no—
211#380—141no—
212#380—141no—
213C4×C4⋊C4—71no—
214#340—72no—
215#336—72no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7C22×D411yescharacteristic
231C7×D4—22no—
232C7×D4—22no—
233C7×D4—22no—
234C7×D4—22no—
235C7×D4—22no—
236C7×D4—41no—
237C7×D4—41no—
238C7×D4#12114yes—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—22no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—48no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—84no—
249C2×C4×C7#12112yes—
250C2×C4×C7#12112yes—
251C2×C4×C7#13512yes—
252C2×C4×C7AGL(1,ℤ/8)12yes—
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C2×#4111yescharacteristic
256C2×C4×C7C2×#4111yescharacteristic
257C2×C4×C7H(5,𝔽2)11yescharacteristic
258C2×C4×C72−1+411yescharacteristic
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—88no—
272Dic14—42no—
273Dic14—42no—
274Dic14—42no—
275Dic14—42no—
276C7×C8—22no—
277C7×C8—41no—
278C7⋊C8—42no—
279C22×Q8—72no—
280C22×D4C22×D711yescharacteristic
281C22×D4C22×D711yescharacteristic
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—142no—
285C2×SD16—144no—
286C2×SD16—144no—
287C2×SD16—144no—
288C2×D8—22no—
289C2×D8—22no—
290C2×D8D2814yes—
291C22×C8—72no—
292C22×C8C22×D711yescharacteristic
293C22×C8C22×D711yescharacteristic
294C4⋊Q8—142no—
295C4⋊Q8—142no—
296#124—144no—
297#124—282no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—144no—
302C4×D4—282no—
303C4×D4C22⋊D714yes—
304C2×C4⋊C4—71no—
305C2×C4⋊C4—71no—
306C2×C4⋊C4—72no—
307C2×C4⋊C4—142no—
308C2×C4⋊C4—141no—
309C2×C4⋊C4—141no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—284no—
313C2×C4⋊C4C22×D711yescharacteristic
314C2×C4⋊C4C22×D711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—142no—
318C2×C22⋊C4—142no—
319C2×C22⋊C4—142no—
320C2×C42—72no—
321C2×C42C22×D711yescharacteristic
322#107—22no—
323C4⋊C8—144no—
324C4⋊C8—142no—
325#103—144no—
326#103—144no—
327#103—282no—
328#102—22no—
329#102—22no—
330#102—144no—
331C4×C8—22no—
332#95—72no—
333#95—142no—
334#95—142no—
335#95—142no—
336C22×C7—24no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—81no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7C2×#12111yescharacteristic
351C22×C7D4211yescharacteristic
352C22×C7#54811yescharacteristic
353C22×C7C2×#13511yescharacteristic
354C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C7#57711yescharacteristic
356C22×C7#57811yescharacteristic
357C4×C7—22no—
358C4×C7—42no—
359C4×C7—81no—
360C4×C7#46314yescomplemented
361C4×C7C2×#12111yescharacteristic
362C4×C7#53711yescharacteristic
363C4×C7#54711yescharacteristic
364C4×C7#55311yescharacteristic
365Dic7—42no—
366Dic7—42no—
367Dic7—82no—
368Dic7—82no—
369C24—21no—
370C24—21no—
371C2×Q8—142no—
372C2×Q8—142no—
373C2×Q8—142no—
374C2×Q8—142no—
375C2×Q8—142no—
376C2×Q8—142no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4C2×D2812yes—
386C2×D4C2×D2812yes—
387C2×D4#115712yes—
388C2×D4C2×C22⋊D712yes—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—72no—
398C22×C4—72no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—284no—
402C22×C4—284no—
403C22×C4—562no—
404C22×C4D4×D711yescharacteristic
405C22×C4D4×D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407SD16—288no—
408D8—28no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—142no—
417C2×C8C2×D2812yes—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—142no—
422C4⋊C4—142no—
423C4⋊C4—142no—
424C4⋊C4—142no—
425C4⋊C4—284no—
426C4⋊C4—282no—
427C4⋊C4—282no—
428C4⋊C4—282no—
429C4⋊C4—561no—
430C4⋊C4C2×C22⋊D712yes—
431C22⋊C4—24no—
432C22⋊C4—42no—
433C22⋊C4—284no—
434C22⋊C4—284no—
435C22⋊C4—284no—
436C42—142no—
437C42—142no—
438C42#115912yes—
439C42C2×C22⋊D712yes—
440C2×C7—44no—
441C2×C7—82no—
442C2×C7#241611yescharacteristic
443C2×C7C2×#46311yescharacteristic
444C2×C7#304611yescharacteristic
445C2×C7#327211yescharacteristic
446C2×C7#328311yescharacteristic
447C2×C7#336011yescharacteristic
448C2×C7#336711yescharacteristic
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C2×D4×D711yescharacteristic
464Q8—282no—
465Q8—282no—
466Q8—282no—
467Q8—282no—
468D4—22no—
469D4—22no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—41no—
474D4—41no—
475D4#649314yes—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—48no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—84no—
486C2×C4—142no—
487C2×C4—142no—
488C2×C4—142no—
489C2×C4—142no—
490C2×C4—142no—
491C2×C4—142no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—568no—
499C2×C4#647412yes—
500C2×C4#647512yes—
501C2×C4#650212yes—
502C2×C4#650212yes—
503C2×C4C22×D2811yescharacteristic
504C2×C4C2×#115711yescharacteristic
505C2×C4C2×#115911yescharacteristic
506C2×C4#655311yescharacteristic
507C2×C4#655411yescharacteristic
508C2×C4C22×C22⋊D711yescharacteristic
509C8—22no—
510C8—41no—
511C8—282no—
512C7#2472711yescharacteristic, complemented, Sylow
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22D4×D2811yescharacteristic
528C22#9024311yescharacteristic
529C22C2×#647411yescharacteristic
530C22C2×#647511yescharacteristic
531C22#9046111yescharacteristic
532C22#9046211yescharacteristic
533C22C2×#650211yescharacteristic
534C4—22no—
535C4—42no—
536C4—81no—
537C4—282no—
538C4—282no—
539C4—562no—
540C4—562no—
541C4#8992514yescomplemented
542C4#9024611yescharacteristic
543C4#9025611yescharacteristic
544C4C2×#649311yescharacteristic
545C4#9052411yescharacteristic
546C2—44no—
547C2—82no—
548C2#1173701311yescharacteristic
549C2#1173701611yescharacteristic
550C2#1173704811yescharacteristic
551C2#1173705211yescharacteristic
552C2#1173979711yescharacteristic
553C2C2×#8992511yescharacteristic
554C2#1174134711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×D8C2
socleC23×C7C22×D4
prime radicalC22×C7×D8C22
hypercenterC2×C4×D8D7
hypocenterC7#24727
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
3. #11739827C2semidirect
4. D4×C4⋊Dic7C2semidirect
5. C2×#89791C2semidirect
6. #11729358C2semidirect
7. #11728397C2semidirect
8. #11727932C2semidirect
9. #11727925C2semidirect
10. #11727540C2semidirect
11. #11727525C2semidirect
360. C4×C7#463semidirect
512. C7#24727semidirect
541. C4#89925semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D82
C2×#899252
#117398272
D4×C4⋊Dic72
C2×#897912
#117293582
#117283972
#117279322
#117279252
#117275402
#117275252
#247277