Finite Group Databasebeta

← #13404530

Subgroups — #13404530

Order 1792.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2537C211yesmaximal, characteristic
2C2×#89813C212yesmaximal, complemented
3#11739824C211yesmaximal, characteristic, complemented
4C2×#89791C211yesmaximal, characteristic, complemented
5C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
6#11729439C212yesmaximal, complemented
7C4×#6406C211yesmaximal, characteristic, complemented
8#11727333C211yesmaximal, characteristic, complemented
9#11727322C211yesmaximal, characteristic, complemented
10#11727280C212yesmaximal, complemented
11#11727268C212yesmaximal, complemented
12C2×C7×#121—21no—
13C2×C7×#121C2212yes—
14C2×C4×C7×D4C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C22×#1142C2212yes—
17C2×#6487C2212yes—
18C2×#6487C2212yes—
19C23×C7⋊C8C2212yes—
20C2×#6432C2211yescharacteristic
21C7×#394—22no—
22C7×#384C2211yescharacteristic
23C7×#380—21no—
24C7×#380C2212yes—
25#89993—21no—
26C2×#6406C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×C22⋊(C7⋊C8)C2212yes—
29#89813—28no—
30#89791—24no—
31D4×C7⋊C8—24no—
32#89748—21no—
33C2×#6384C2212yes—
34C2×#6384C2212yes—
35C2×#6384C2212yes—
36C2×#6382C2212yes—
37#89721—22no—
38C4×C4⋊Dic7C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89324C2212yes—
42#89324C2211yescharacteristic
43#89324C2211yescharacteristic
44#25385—71nomaximal, Sylow
45C22×C7×D4—22no—
46C22×C7×D4D411yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7C2312yes—
50C22×Dic14C2312yes—
51C7×#121—24no—
52C7×#121—42no—
53C7×#121—44no—
54C4×C7×D4—22no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4D412yes—
67C2×C42×C7C2311yescharacteristic
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—22no—
70C2×#1142—24no—
71C2×#1142—24no—
72C2×#1142—42no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77#6487—24no—
78#6487—24no—
79#6487—42no—
80#6487—42no—
81C2×Dic7⋊C4—22no—
82C2×Dic7⋊C4—22no—
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#6432—22no—
91#6432—22no—
92C7×#95—22no—
93#6406—22no—
94#6406D414yescomplemented
95C7⋊#95—21no—
96C22⋊(C7⋊C8)—24no—
97#6384—24no—
98#6384—24no—
99#6384—42no—
100#6382—24no—
101C4⋊(C7⋊C8)D414yescomplemented
102C4×C7⋊C8—22no—
103C2×#465—72no—
104C2×#461—71no—
105C2×C8×D4—71no—
106#2542—71no—
107#2537D711yescharacteristic
108#2035—72no—
109#2000—72no—
110#1956—71no—
111#1948—71no—
112#1892—72no—
113C4×#102—71no—
114C24×C7—21no—
115C24×C7C2×D412yes—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4—21no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
125C2×C7×D4—82no—
126C2×C7×D4SD1614yescomplemented
127C2×C7×D4C2×D412yes—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—22no—
146C2×Dic14—22no—
147C2×Dic14—22no—
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—22no—
151C2×Dic14—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—42no—
158C7×C4⋊C4—81no—
159C7×C22⋊C4—24no—
160C7×C22⋊C4—44no—
161C7×C22⋊C4—44no—
162C7×C22⋊C4—42no—
163C42×C7C2×D412yes—
164C42×C7#4112yes—
165C22⋊Dic7—44no—
166C22⋊Dic7—44no—
167#1142—48no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—42no—
171C4⋊Dic7—42no—
172C4⋊Dic7C2×D412yes—
173C4⋊Dic7#4112yes—
174Dic7⋊C4—44no—
175Dic7⋊C4—44no—
176C4×Dic7—42no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—48no—
185C2×C7⋊C8C2×D414yes—
186C2×C7⋊C8C2×D412yes—
187C22×SD16—72no—
188C23×C8—72no—
189C2×C4⋊Q8—71no—
190C2×C22⋊Q8—72no—
191C2×C22⋊Q8—72no—
192C2×#121—21no—
193C2×#121C2×D712yes—
194C2×C4×D4C2×D711yescharacteristic
195C2×C4×D4C2×D711yescharacteristic
196#465—148no—
197#461—144no—
198C8×D4—144no—
199C2×#106—72no—
200C2×C4⋊C8—71no—
201C2×#103—72no—
202C2×#103—72no—
203C2×#103—72no—
204C2×#102—71no—
205C2×#102—71no—
206C2×C22⋊C8—72no—
207C2×C4×C8—71no—
208#399—142no—
209#394—22no—
210#390—141no—
211#384—141no—
212#384C2×D711yescharacteristic
213#380—21no—
214#380C2×D712yes—
215C4×C4⋊C4—71no—
216#340—71no—
217#340—72no—
218#340—71no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7C2×SD1614yes—
233C23×C7C22×D411yescharacteristic
234C7×D4—22no—
235C7×D4—22no—
236C7×D4—28no—
237C7×D4—41no—
238C7×D4—41no—
239C7×D4—88no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—84no—
256C2×C4×C7#12112yes—
257C2×C4×C7#12112yes—
258C2×C4×C7C2×SD1612yes—
259C2×C4×C7#13512yes—
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C2×#4111yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7H(5,𝔽2)11yescharacteristic
265C2×C4×C7H(5,𝔽2)11yescharacteristic
266C2×Dic7—22no—
267C2×Dic7—22no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275Dic14—42no—
276Dic14—42no—
277Dic14—42no—
278Dic14—42no—
279C7⋊C8—22no—
280C7⋊C8—24no—
281C7⋊C8—41no—
282C22×Q8—72no—
283C22×D4—22no—
284C22×D4C22⋊D711yescharacteristic
285C22×D4C22×D711yescharacteristic
286C23×C4—21no—
287C23×C4C22×D712yes—
288C2×SD16—144no—
289C2×SD16—144no—
290C2×SD16—282no—
291C22×C8—71no—
292C22×C8—72no—
293C22×C8—71no—
294C22×C8—144no—
295C22×C8—144no—
296C22×C8—282no—
297C4⋊Q8—142no—
298C4⋊Q8—142no—
299C22⋊Q8—144no—
300C22⋊Q8—144no—
301C22⋊Q8—282no—
302C22⋊Q8—282no—
303#121—24no—
304#121—42no—
305#121—44no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—71no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4C22⋊D711yescharacteristic
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C4⋊C4C22×D712yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4C22⋊D712yes—
327C2×C42—141no—
328C2×C42C22×D711yescharacteristic
329#106—144no—
330C4⋊C8—74no—
331#103—144no—
332#103—144no—
333#103—282no—
334#102—74no—
335#102—142no—
336C22⋊C8—144no—
337C4×C8—142no—
338#95—22no—
339#95—141no—
340C22×C7—24no—
341C22×C7—24no—
342C22×C7—24no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7—81no—
354C22×C7C2×#12111yescharacteristic
355C22×C7D4211yescharacteristic
356C22×C7#54611yescharacteristic
357C22×C7C22×SD1611yescharacteristic
358C22×C7C2×#13511yescharacteristic
359C22×C7#57711yescharacteristic
360C22×C7#57811yescharacteristic
361C4×C7—22no—
362C4×C7—42no—
363C4×C7—82no—
364C4×C7—81no—
365C4×C7#46514yescomplemented
366C4×C7#53211yescharacteristic
367C4×C7#54011yescharacteristic
368C4×C7D4211yescharacteristic
369C4×C7#55011yescharacteristic
370Dic7—42no—
371Dic7—42no—
372Dic7—82no—
373C24—21no—
374C24C2×C22⋊D712yes—
375C2×Q8—142no—
376C2×Q8—142no—
377C2×Q8—142no—
378C2×Q8—142no—
379C2×Q8—142no—
380C2×Q8—142no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—82no—
391C2×D4#114214yescomplemented
392C2×D4D4×D712yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—141no—
405C22×C4—141no—
406C22×C4—142no—
407C22×C4C2×C22⋊D712yes—
408C22×C4C2×C22⋊D711yescharacteristic
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411SD16—288no—
412C2×C8—74no—
413C2×C8—72no—
414C2×C8—141no—
415C2×C8—142no—
416C2×C8—141no—
417C2×C8—142no—
418C2×C8—141no—
419C2×C8—141no—
420C2×C8—141no—
421C2×C8—288no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—42no—
428C4⋊C4—72no—
429C4⋊C4—72no—
430C4⋊C4—81no—
431C4⋊C4—141no—
432C4⋊C4—141no—
433C4⋊C4—284no—
434C4⋊C4—284no—
435C4⋊C4—282no—
436C4⋊C4—282no—
437C22⋊C4—24no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—284no—
442C22⋊C4—284no—
443C42—282no—
444C42D4×D712yes—
445C42#115912yes—
446C2×C7—44no—
447C2×C7—82no—
448C2×C7#242811yescharacteristic
449C2×C7C2×#46511yescharacteristic
450C2×C7#306311yescharacteristic
451C2×C7D4×SD1611yescharacteristic
452C2×C7#328311yescharacteristic
453C2×C7#329811yescharacteristic
454C2×C7#330311yescharacteristic
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C2×#114214yes—
469C23C22×C22⋊D711yescharacteristic
470Q8—282no—
471Q8—282no—
472Q8—282no—
473Q8—282no—
474D4—22no—
475D4—22no—
476D4—28no—
477D4—41no—
478D4—41no—
479D4—88no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—142no—
497C2×C4—142no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4C2×#114212yes—
506C2×C4#650212yes—
507C2×C4#650212yes—
508C2×C4#651312yes—
509C2×C4C2×D4×D711yescharacteristic
510C2×C4C2×D4×D711yescharacteristic
511C2×C4C2×#115911yescharacteristic
512C2×C4#654811yescharacteristic
513C2×C4D7×#4111yescharacteristic
514C2×C4#655311yescharacteristic
515C8—142no—
516C8—144no—
517C8—281no—
518C7#2538511yescharacteristic, complemented, Sylow
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22C22×#114211yescharacteristic
534C22D4×C22⋊D711yescharacteristic
535C22C2×#650211yescharacteristic
536C22C2×#651311yescharacteristic
537C22#9052011yescharacteristic
538C22#9052711yescharacteristic
539C22#9052911yescharacteristic
540C4—22no—
541C4—42no—
542C4—82no—
543C4—81no—
544C4—282no—
545C4—282no—
546C4—562no—
547C4#8981314yescomplemented
548C4#9028711yescharacteristic
549C4#9029011yescharacteristic
550C4#9030011yescharacteristic
551C4#9031011yescharacteristic
552C2—44no—
553C2—82no—
554C2C2×#8981311yescharacteristic
555C2#1174009411yescharacteristic
556C2#1174086811yescharacteristic
557C2D4×#114211yescharacteristic
558C2#1174096911yescharacteristic
559C2#1174097611yescharacteristic
560C2#1174109711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2537C2
socleC23×C7C22×D4
hypercenter#2537D7
hypocenterC7#25385
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. C2×#89813C2semidirect
3. #11739824C2semidirect
4. C2×#89791C2semidirect
5. C2×D4×C7⋊C8C2semidirect
6. #11729439C2semidirect
7. C4×#6406C2semidirect
8. #11727333C2semidirect
9. #11727322C2semidirect
10. #11727280C2semidirect
11. #11727268C2semidirect
94. #6406D4semidirect
101. C4⋊(C7⋊C8)D4semidirect
126. C2×C7×D4SD16semidirect
365. C4×C7#465semidirect
391. C2×D4#1142semidirect
518. C7#25385semidirect
547. C4#89813semidirect

Maximal subgroups

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

subgroupindex
C7×#25372
C2×#898132
#117398242
C2×#897912
C2×D4×C7⋊C82
#117294392
C4×#64062
#117273332
#117273222
#117272802
#117272682
#253857