Finite Group Databasebeta

← #13388571

Subgroups — #13388571

Order 1792.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic
2C2×#89847C211yesmaximal, characteristic, complemented
3#11739927C211yesmaximal, characteristic, complemented
4C2×C4×#1142C211yesmaximal, characteristic, complemented
5#11729522C212yesmaximal, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729516C212yesmaximal, complemented
8#11729388C211yesmaximal, characteristic, complemented
9#11729366C212yesmaximal, complemented
10#11729361C212yesmaximal, complemented
11#11729358C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C7×#550—24no—
14C7×#550—24no—
15C2×C7×#127C2212yes—
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121—21no—
18C2×C7×#121—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C22×#1142C2211yescharacteristic
23C22×#1142C2212yes—
24C2×C4×Dic14C2211yescharacteristic
25#90002—21no—
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31#89847—24no—
32C4×#1142—24no—
33#89756—22no—
34#89745—21no—
35C2×#6384C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37#89707C2211yescharacteristic
38#89701C2212yes—
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24188—71nomaximal, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×D4—22no—
51C22×C7×D4—21no—
52C22×C7×D4—22no—
53C22×C7×D4—21no—
54C22×C7×D4C2312yes—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7—21no—
58C22×Dic14C2311yescharacteristic
59C7×#127—24no—
60C7×#127—24no—
61C7×#127—22no—
62C7×#127D414yes—
63C7×#121—28no—
64C7×#121—216no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×D4—24no—
68C4×C7×D4—24no—
69C4×C7×D4—22no—
70C4×C7×D4D414yes—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D412yes—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7C2312yes—
77C2×C42×C7C2311yescharacteristic
78C2×#1142—22no—
79C2×#1142—22no—
80C2×#1142—24no—
81C2×#1142—24no—
82C2×#1142—22no—
83C2×#1142—42no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2312yes—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—22no—
90C2×C4×Dic7—21no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2311yescharacteristic
94C4×Dic14—22no—
95C4×Dic14—22no—
96#6406—22no—
97#6406—24no—
98#6406—24no—
99#6406—42no—
100#6406—42no—
101C7⋊#95—22no—
102C7⋊#95—22no—
103C7⋊#95—22no—
104#6384—22no—
105#6382—22no—
106C4⋊(C7⋊C8)—22no—
107C4×C7⋊C8—22no—
108C2×#550D711yescharacteristic
109C2×#460—71no—
110C2×C4×SD16—71no—
111#2582—71no—
112#2041—72no—
113#2014—72no—
114#1984—71no—
115#1957—71no—
116#1877—72no—
117#1864—72no—
118#1802—71no—
119C24×C7—22no—
120C24×C7—21no—
121C7×#41—432no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8SD1614yescomplemented
125C2×C7×D4—28no—
126C2×C7×D4—28no—
127C2×C7×D4—24no—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C2×C7×D4—21no—
131C2×C7×D4—22no—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—22no—
137C2×C7×D4—21no—
138C2×C7×D4—22no—
139C2×C7×D4—44no—
140C2×C7×D4—44no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4—81no—
144C2×C7×D4—81no—
145C2×C7×D4—82no—
146C2×C7×D4—82no—
147C2×C7×D4C2×D414yes—
148C2×C7×D4C2×D412yes—
149C22×C4×C7—28no—
150C22×C4×C7—21no—
151C22×C4×C7—24no—
152C22×C4×C7—44no—
153C22×C4×C7—44no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14#4112yescomplemented
170C7×C4⋊C4—24no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4C2×D412yes—
174C7×C22⋊C4—28no—
175C7×C22⋊C4—24no—
176C42×C7—22no—
177C42×C7—22no—
178C42×C7C2×D412yes—
179C42×C7C2×D412yes—
180#1142—28no—
181#1142—48no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—22no—
185C4⋊Dic7—22no—
186C4⋊Dic7—42no—
187C4⋊Dic7—81no—
188Dic7⋊C4—42no—
189Dic7⋊C4—82no—
190C4×Dic7—42no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8#4112yescomplemented
200C22×#41—21no—
201C22×SD16—71no—
202C22×SD16—72no—
203#550—24no—
204#550—24no—
205C2×#127C2×D712yes—
206C2×#127C2×D711yescharacteristic
207C2×#121—21no—
208C2×#121—22no—
209C2×C4×Q8—71no—
210C2×C4×Q8C2×D711yescharacteristic
211C2×C4×D4C2×D712yes—
212C2×C4×D4C2×D711yescharacteristic
213#460—144no—
214C4×SD16—144no—
215C2×#106—71no—
216C2×C4⋊C8—71no—
217C2×#103—71no—
218C2×#102—72no—
219C2×#102—72no—
220C2×#102—71no—
221C2×#102—72no—
222C2×#102—72no—
223C2×C4×C8—71no—
224#398—142no—
225#391—72no—
226#391—141no—
227#389—72no—
228#389—141no—
229#384—71no—
230#340—72no—
231#336—72no—
232#336—72no—
233C23×C7—24no—
234C23×C7—22no—
235C23×C7—41no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—41no—
239C23×C7—42no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—41no—
243C23×C7—42no—
244C23×C7—41no—
245C23×C7—42no—
246C23×C7—41no—
247C23×C7C22×D411yescharacteristic
248C7×Q8—28no—
249C7×D4—22no—
250C7×D4—22no—
251C7×D4—24no—
252C7×D4—24no—
253C7×D4—416no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—41no—
257C7×D4—41no—
258C7×D4—48no—
259C7×D4—84no—
260C7×D4—88no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—416no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—48no—
273C2×C4×C7#12012yes—
274C2×C4×C7#12012yes—
275C2×C4×C7C2×SD1614yes—
276C2×C4×C7C2×SD1612yes—
277C2×C4×C7AGL(1,ℤ/8)12yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—41no—
291C2×Dic7—42no—
292C2×Dic7—41no—
293Dic14—22no—
294Dic14—22no—
295Dic14—41no—
296Dic14—41no—
297C7⋊C8—22no—
298C7⋊C8—41no—
299C7⋊C8—42no—
300C2×#41—28no—
301C2×#41—44no—
302C2×#41—44no—
303C22×Q8—71no—
304C22×Q8C22⋊D711yescharacteristic
305C22×D4—22no—
306C22×D4—21no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4C22×D712yes—
310C22×D4C22×D711yescharacteristic
311C23×C4—22no—
312C23×C4—21no—
313C2×SD16—142no—
314C2×SD16—142no—
315C2×SD16—142no—
316C2×SD16—144no—
317C2×SD16—144no—
318C2×SD16—282no—
319C22×C8—71no—
320C22×C8—72no—
321C22×C8—71no—
322#127—24no—
323#127—24no—
324#127—22no—
325#127C22⋊D714yes—
326#121—28no—
327#121—216no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—142no—
331C4×Q8—142no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—22no—
335C4×D4C22⋊D714yes—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—72no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—142no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22⋊D711yescharacteristic
343C2×C4⋊C4C22⋊D712yes—
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C42—141no—
348C2×C42C22×D712yes—
349C2×C42C22×D711yescharacteristic
350#106—142no—
351C4⋊C8—142no—
352#103—142no—
353#102—144no—
354#102—142no—
355#102—144no—
356#102—282no—
357#102—282no—
358C4×C8—142no—
359#95—142no—
360#95—142no—
361#95—142no—
362C22×C7—42no—
363C22×C7—44no—
364C22×C7—44no—
365C22×C7—42no—
366C22×C7—44no—
367C22×C7—42no—
368C22×C7—81no—
369C22×C7—82no—
370C22×C7—82no—
371C22×C7—81no—
372C22×C7—82no—
373C22×C7—81no—
374C22×C7—82no—
375C22×C7—81no—
376C22×C7C2×#12011yescharacteristic
377C22×C7#54811yescharacteristic
378C22×C7#55011yescharacteristic
379C22×C7C22×SD1611yescharacteristic
380C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C7#57611yescharacteristic
382C22×C7#57711yescharacteristic
383C4×C7—24no—
384C4×C7—22no—
385C4×C7—42no—
386C4×C7—42no—
387C4×C7#45014yes—
388C4×C7#52511yescharacteristic
389C4×C7#53711yescharacteristic
390C4×C7#54611yescharacteristic
391C4×C7#54711yescharacteristic
392Dic7—42no—
393Dic7—82no—
394Dic7—81no—
395C24—22no—
396C24—21no—
397#41—432no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—72no—
401C2×Q8—141no—
402C2×Q8—141no—
403C2×Q8—141no—
404C2×Q8—141no—
405C2×Q8—141no—
406C2×Q8#114214yescomplemented
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4C2×C22⋊D714yes—
430C2×D4C2×C22⋊D712yes—
431C22×C4—28no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—141no—
439C22×C4—142no—
440C22×C4—141no—
441C22×C4C2×C22⋊D712yes—
442C22×C4C2×C22⋊D712yes—
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C23×D711yescharacteristic
446SD16—148no—
447SD16—288no—
448C2×C8—72no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—142no—
452C2×C8—141no—
453C2×C8—142no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—142no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—142no—
461C4⋊C4—142no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4—562no—
467C4⋊C4—561no—
468C4⋊C4C2×C22⋊D712yes—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C42—22no—
472C42—22no—
473C42—282no—
474C42C2×C22⋊D712yes—
475C42C2×C22⋊D712yes—
476C2×C7—82no—
477C2×C7—84no—
478C2×C7#240711yescharacteristic
479C2×C7C2×#45011yescharacteristic
480C2×C7#301211yescharacteristic
481C2×C7#336211yescharacteristic
482C2×C7#336811yescharacteristic
483C2×C7#339211yescharacteristic
484C2×C7#339911yescharacteristic
485C23—24no—
486C23—22no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23C22×C22⋊D711yescharacteristic
500Q8—28no—
501Q8—142no—
502Q8—142no—
503Q8—281no—
504Q8—281no—
505D4—24no—
506D4—22no—
507D4—24no—
508D4—22no—
509D4—416no—
510D4—48no—
511D4—41no—
512D4—42no—
513D4—41no—
514D4—42no—
515D4—84no—
516D4—88no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—416no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—48no—
529C2×C4—142no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4#649512yes—
539C2×C4C2×#114214yes—
540C2×C4C2×#114212yes—
541C2×C4#651612yes—
542C2×C4#651612yes—
543C2×C4D7×#4111yescharacteristic
544C2×C4D7×#4111yescharacteristic
545C2×C4#655311yescharacteristic
546C2×C4#655411yescharacteristic
547C2×C4C22×C22⋊D711yescharacteristic
548C2×C4C22×C22⋊D711yescharacteristic
549C8—142no—
550C8—282no—
551C8—281no—
552C7#2418811yescharacteristic, complemented, Sylow
553C22—44no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—82no—
560C22—81no—
561C22—82no—
562C22—81no—
563C22—82no—
564C22—81no—
565C22—82no—
566C22—81no—
567C22C2×#649511yescharacteristic
568C22C22×#114211yescharacteristic
569C22#9052211yescharacteristic
570C22#9052511yescharacteristic
571C22#9052611yescharacteristic
572C22#9052711yescharacteristic
573C22C2×#651611yescharacteristic
574C4—24no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—282no—
579C4—562no—
580C4—561no—
581C4#8999014yes—
582C4#9051511yescharacteristic
583C4#9051911yescharacteristic
584C4#9052011yescharacteristic
585C4#9052411yescharacteristic
586C2—84no—
587C2—82no—
588C2#1174103111yescharacteristic
589C2#1174103211yescharacteristic
590C2#1174103311yescharacteristic
591C2#1174103411yescharacteristic
592C2C2×#8999011yescharacteristic
593C2#1174154211yescharacteristic
594C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#550C2
socleC23×C7C22×D4
hypercenterC2×#550D7
hypocenterC7#24188
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#89847C2semidirect
3. #11739927C2semidirect
4. C2×C4×#1142C2semidirect
5. #11729522C2semidirect
6. #11729518C2semidirect
7. #11729516C2semidirect
8. #11729388C2semidirect
9. #11729366C2semidirect
10. #11729361C2semidirect
11. #11729358C2semidirect
124. C2×C7×Q8SD16semidirect
169. C2×Dic14#41semidirect
199. C2×C7⋊C8#41semidirect
406. C2×Q8#1142semidirect
552. C7#24188semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5502
C2×#898472
#117399272
C2×C4×#11422
#117295222
#117295182
#117295162
#117293882
#117293662
#117293612
#117293582
#241887