Finite Group Databasebeta

← #13404531

Subgroups — #13404531

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×#2542C211yesmaximal, characteristic, complemented
2C2×#89855C211yesmaximal, characteristic, complemented
3C2×#89820C212yesmaximal, complemented
4#11739862C211yesmaximal, characteristic
5C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
6#11729539C211yesmaximal, characteristic, complemented
7#11729453C212yesmaximal, complemented
8#11729450C212yesmaximal, complemented
9#11729448C212yesmaximal, complemented
10C4×#6383C211yesmaximal, characteristic, complemented
11#11727221C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×C22⋊Q8C2212yes—
14C2×C7×C22⋊Q8C2212yes—
15C2×C4×C7×D4C2211yescharacteristic
16C22×#1143C2212yes—
17C2×#6501—21no—
18C2×#6493C2212yes—
19C23×C7⋊C8C2212yes—
20C2×C4×D28C2211yescharacteristic
21C7×#399—22no—
22C7×#390—21no—
23C7×#384—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30#89855—24no—
31#89820—28no—
32D4×C7⋊C8—24no—
33#89759—21no—
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6382C2212yes—
37#89733—22no—
38#89707C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89324C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#25385—71nomaximal, Sylow
45C22×C7×Q8C2312yes—
46C22×C7×D4D411yescharacteristic
47C23×C4×C7C2312yes—
48C22×C22⋊D7—22no—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8—22no—
53C7×C22⋊Q8—24no—
54C7×C22⋊Q8—24no—
55C7×C22⋊Q8—42no—
56C7×C22⋊Q8—42no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
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×C22⋊Dic7—22no—
72C2×#1143—24no—
73C2×#1143—24no—
74C2×#1143—42no—
75#6501—44no—
76#6493—24no—
77#6493—42no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C22×C7⋊C8—24no—
85C22×C7⋊C8—24no—
86C22×C7⋊C8—42no—
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2312yes—
90C4×D28—22no—
91C4×D28—22no—
92C7×#95—21no—
93#6409—24no—
94#6409—24no—
95#6409—42no—
96C7⋊#95—22no—
97C22⋊(C7⋊C8)—24no—
98#6383—22no—
99#6383D414yescomplemented
100#6382—24no—
101C4⋊(C7⋊C8)D414yescomplemented
102C4×C7⋊C8—22no—
103C2×#465—72no—
104C2×#461—71no—
105C2×C8×D4—71no—
106#2542D711yescharacteristic, complemented
107#2537—71no—
108#2035—72no—
109#2000—72no—
110#1956—71no—
111#1948—71no—
112#1892—72no—
113C4×#102—71no—
114C24×C7C2×D412yes—
115C23×D7—21no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4SD1614yescomplemented
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
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×C7C2×D412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2411yescharacteristic
138C2×C22⋊D7—44no—
139C2×C22⋊D7—44no—
140C2×C22⋊D7—82no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28C2×D412yes—
149C2×C4×D7—42no—
150C2×C4×D7—42no—
151C2×C4×D7—42no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—44no—
156C7×C4⋊C4—44no—
157C7×C4⋊C4—42no—
158C7×C4⋊C4—42no—
159C7×C4⋊C4C2×D412yes—
160C7×C4⋊C4#4112yes—
161C7×C22⋊C4—24no—
162C7×C22⋊C4—44no—
163C7×C22⋊C4—44no—
164C42×C7—42no—
165C42×C7C2×D412yes—
166C42×C7#4112yes—
167C22⋊Dic7—44no—
168#1143—48no—
169(C2×D7)⋊C4—44no—
170(C2×D7)⋊C4—42no—
171C4⋊Dic7—21no—
172C4⋊Dic7—22no—
173C4⋊Dic7—21no—
174C4⋊Dic7—22no—
175C4⋊Dic7—42no—
176C4⋊Dic7—81no—
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⋊Q8C2×D711yescharacteristic
190C2×C22⋊Q8C2×D712yes—
191C2×C22⋊Q8C2×D712yes—
192C2×#121—72no—
193C2×#121—141no—
194C2×C4×D4—71no—
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—22no—
209#394—142no—
210#390—21no—
211#384—21no—
212#384—71no—
213#380—72no—
214#380—141no—
215C4×C4⋊C4C2×D711yescharacteristic
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×C7C2×SD1614yes—
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—42no—
235C7×Q8—42no—
236C7×Q8—42no—
237C7×Q8—42no—
238C7×D4—28no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7#12112yes—
258C2×C4×C7#12112yes—
259C2×C4×C7C2×SD1612yes—
260C2×C4×C7#13512yes—
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7C2×#4111yescharacteristic
265C2×C4×C7H(5,𝔽2)11yescharacteristic
266C2×C4×C7H(5,𝔽2)11yescharacteristic
267C22⋊D7—88no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—42no—
274D28—22no—
275D28—22no—
276D28—41no—
277D28—41no—
278C4×D7—84no—
279C7⋊C8—22no—
280C7⋊C8—24no—
281C7⋊C8—41no—
282C22×Q8C22×D712yes—
283C22×D4—71no—
284C22×D4—142no—
285C22×D4C22⋊D711yescharacteristic
286C23×C4—141no—
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—22no—
298C4⋊Q8—22no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303#121—144no—
304#121—282no—
305#121—284no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—142no—
309C4×D4—142no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—72no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4C22⋊D711yescharacteristic
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C4⋊C4C22×D711yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—141no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4C22⋊D712yes—
327C2×C42—21no—
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—21no—
339#95—142no—
340C22×C7—24no—
341C22×C7—24no—
342C22×C7—24no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7C2×#12111yescharacteristic
348C22×C7D4211yescharacteristic
349C22×C7#54611yescharacteristic
350C22×C7C22×SD1611yescharacteristic
351C22×C7C2×#13511yescharacteristic
352C22×C7#57711yescharacteristic
353C22×C7#57811yescharacteristic
354C2×D7—42no—
355C2×D7—42no—
356C2×D7—42no—
357C2×D7—81no—
358C2×D7—81no—
359C2×D7—81no—
360C2×D7—81no—
361C4×C7—22no—
362C4×C7—42no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—82no—
366C4×C7#46514yescomplemented
367C4×C7#53211yescharacteristic
368C4×C7#54011yescharacteristic
369C4×C7D4211yescharacteristic
370C4×C7#55011yescharacteristic
371Dic7—82no—
372Dic7—81no—
373C24—141no—
374C24C2×C22⋊D712yes—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—72no—
384C2×D4—141no—
385C2×D4—141no—
386C2×D4—141no—
387C2×D4—141no—
388C2×D4—141no—
389C2×D4—284no—
390C2×D4—284no—
391C2×D4—562no—
392C2×D4#114314yescomplemented
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—141no—
403C22×C4—142no—
404C22×C4—282no—
405C22×C4—282no—
406C22×C4—282no—
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—21no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—141no—
430C4⋊C4—142no—
431C4⋊C4—141no—
432C4⋊C4—142no—
433C4⋊C4—282no—
434C4⋊C4—561no—
435C4⋊C4D4×D712yes—
436C4⋊C4#115912yes—
437C22⋊C4—24no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—284no—
441C22⋊C4—284no—
442C22⋊C4—282no—
443C42—42no—
444C42D4×D712yes—
445C42#115912yes—
446C2×C7—44no—
447C2×C7#242811yescharacteristic
448C2×C7C2×#46511yescharacteristic
449C2×C7#306311yescharacteristic
450C2×C7D4×SD1611yescharacteristic
451C2×C7#328311yescharacteristic
452C2×C7#329811yescharacteristic
453C2×C7#330311yescharacteristic
454D7—82no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—142no—
462C23—281no—
463C23—281no—
464C23—281no—
465C23—281no—
466C23—281no—
467C23—281no—
468C23C2×#114314yes—
469C23C22×C22⋊D711yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—42no—
473Q8—42no—
474D4—28no—
475D4—142no—
476D4—142no—
477D4—281no—
478D4—281no—
479D4—568no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—282no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—282no—
502C2×C4—281no—
503C2×C4—282no—
504C2×C4—564no—
505C2×C4#650212yes—
506C2×C4#650212yes—
507C2×C4C2×#114312yes—
508C2×C4#651312yes—
509C2×C4C2×D4×D711yescharacteristic
510C2×C4C2×D4×D711yescharacteristic
511C2×C4C2×#115911yescharacteristic
512C2×C4C2×#115911yescharacteristic
513C2×C4#654811yescharacteristic
514C2×C4#654811yescharacteristic
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—282no—
527C22—282no—
528C22—282no—
529C22—561no—
530C22—561no—
531C22—561no—
532C22—561no—
533C22D4×C22⋊D711yescharacteristic
534C22C2×#650211yescharacteristic
535C22#9049711yescharacteristic
536C22C22×#114311yescharacteristic
537C22C2×#651311yescharacteristic
538C22#9052811yescharacteristic
539C22#9052911yescharacteristic
540C4—22no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—561no—
546C4—562no—
547C4#8982014yescomplemented
548C4#9040711yescharacteristic
549C4#9040811yescharacteristic
550C4#9040911yescharacteristic
551C4#9041011yescharacteristic
552C2—44no—
553C2—562no—
554C2C2×#8982011yescharacteristic
555C2#1174063111yescharacteristic
556C2#1174084511yescharacteristic
557C2#1174095611yescharacteristic
558C2#1174096911yescharacteristic
559C2#1174099511yescharacteristic
560C2D4×#114311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2542C2
socleC23×C7C22×D4
prime radical#11739862C2
hypercenter#2542D7
hypocenterC7#25385
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C7×#2542C2semidirect
2. C2×#89855C2semidirect
3. C2×#89820C2semidirect
5. C2×D4×C7⋊C8C2semidirect
6. #11729539C2semidirect
7. #11729453C2semidirect
8. #11729450C2semidirect
9. #11729448C2semidirect
10. C4×#6383C2semidirect
11. #11727221C2semidirect
99. #6383D4semidirect
101. C4⋊(C7⋊C8)D4semidirect
106. #2542D7semidirect
124. C2×C7×D4SD16semidirect
366. C4×C7#465semidirect
392. C2×D4#1143semidirect
518. C7#25385semidirect
547. C4#89820semidirect

Maximal subgroups

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

subgroupindex
C7×#25422
C2×#898552
C2×#898202
#117398622
C2×D4×C7⋊C82
#117295392
#117294532
#117294502
#117294482
C4×#63832
#117272212
#253857