Finite Group Databasebeta

← #13395101

Subgroups — #13395101

Order 1792.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2857C211yesmaximal, characteristic, complemented
2C2×#89859C211yesmaximal, characteristic, complemented
3C4×#6493C211yesmaximal, characteristic
4C2×#89698C211yesmaximal, characteristic, complemented
5#11729534C211yesmaximal, characteristic, complemented
6#11727375C212yesmaximal, complemented
7#11727371C212yesmaximal, complemented
8#11727338C211yesmaximal, characteristic, complemented
9#11727322C211yesmaximal, characteristic, complemented
10#11727278C212yesmaximal
11#11727273C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×C22⋊Q8—21no—
14C22×C42×C7C2211yescharacteristic
15C2×#6493C2211yescharacteristic
16C2×C4×C22⋊D7—21no—
17C2×C4×D28C2211yescharacteristic
18C7×#391—21no—
19C7×#389—21no—
20C7×#389C2212yes—
21C7×#384C2211yescharacteristic
22C7×#380—21no—
23C7×#380C2212yes—
24#89985—21no—
25#89984C2211yescharacteristic
26C2×C22⋊(C7⋊C8)C2212yes—
27#89859—24no—
28C2×#6383C2212yes—
29C2×#6383C2212yes—
30C2×#6382C2212yes—
31C2×#6381C2212yes—
32#89712—21no—
33#89711C2211yescharacteristic
34C4×(C2×D7)⋊C4—21no—
35C4×C4⋊Dic7C2211yescharacteristic
36#89700C2211yescharacteristic
37#89698—24no—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×#6377C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24709—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C22×C22⋊D7—21no—
51C22×D28C2311yescharacteristic
52C22×C4×D7—21no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8—22no—
55C7×C22⋊Q8—44no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C22⋊C4—22no—
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C42×C7D411yescharacteristic
66C2×C42×C7D411yescharacteristic
67C2×C42×C7C2311yescharacteristic
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—21no—
70#6493—22no—
71#6493—22no—
72C4×C22⋊D7—44no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×C4×Dic7—21no—
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2312yes—
84C4×D28—22no—
85C4×D28—22no—
86C7×#95—22no—
87C7×#95—22no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C22⋊(C7⋊C8)—24no—
92#6383—24no—
93#6383—24no—
94#6382—24no—
95#6381—24no—
96C4⋊(C7⋊C8)—22no—
97C4⋊(C7⋊C8)D414yescomplemented
98#6377—24no—
99C2×#474—71no—
100C2×#432—71no—
101#2857D711yescharacteristic, complemented
102C4×#121—71no—
103#2076—72no—
104#2034—72no—
105#1988—71no—
106#1948—71no—
107#1944—72no—
108#1936—72no—
109#1802—71no—
110C24×C7C2×D411yescharacteristic
111C23×D7—21no—
112C2×C7×Q8—42no—
113C2×C7×Q8—42no—
114C2×C7×Q8—42no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—41no—
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2411yescharacteristic
135C2×C22⋊D7—24no—
136C2×C22⋊D7—42no—
137C2×C22⋊D7—42no—
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C2×D28—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28#4112yes—
147C2×C4×D7—42no—
148C2×C4×D7—42no—
149C2×C4×D7—42no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—42no—
155C7×C4⋊C4—44no—
156C7×C4⋊C4—42no—
157C7×C4⋊C4—42no—
158C7×C22⋊C4—44no—
159C42×C7—22no—
160C42×C7—22no—
161C42×C7—21no—
162C42×C7—21no—
163C42×C7C2×D412yes—
164C42×C7#4112yes—
165C22⋊Dic7—42no—
166(C2×D7)⋊C4—24no—
167(C2×D7)⋊C4—42no—
168(C2×D7)⋊C4—42no—
169(C2×D7)⋊C4—42no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173C4⋊Dic7—21no—
174C4⋊Dic7—21no—
175C4⋊Dic7#4112yes—
176Dic7⋊C4—42no—
177C4×Dic7—42no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8C2×D414yes—
184C2×C4⋊Q8C2×D711yescharacteristic
185C2×C22⋊Q8—21no—
186C2×#121—71no—
187C2×C4×D4—71no—
188C2×C4×D4—141no—
189C22×C42C2×D711yescharacteristic
190#474—144no—
191#432—144no—
192C2×#107—72no—
193C2×#106—72no—
194C2×C4⋊C8—71no—
195C2×C4⋊C8—71no—
196C2×#102—72no—
197C2×#102—72no—
198C2×C22⋊C8—72no—
199C2×#97—72no—
200#391—21no—
201#390—71no—
202#389—21no—
203#389—71no—
204#389C2×D712yes—
205#388—141no—
206#386—141no—
207#384C2×D711yescharacteristic
208#380—21no—
209#380—71no—
210#380C2×D712yes—
211C4×C4⋊C4—71no—
212C4×C22⋊C4—141no—
213#340—72no—
214#340—72no—
215#340—72no—
216#340—72no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7#13712yes—
224C23×C7C22×D411yescharacteristic
225C22×D7—22no—
226C22×D7—41no—
227C22×D7—41no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C7×Q8—84no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—44no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—44no—
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#12112yes—
257C2×C4×C7#12112yes—
258C2×C4×C7AGL(1,ℤ/8)12yes—
259C2×C4×C7AGL(1,ℤ/8)12yes—
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C2×#4111yescharacteristic
262C2×C4×C7C2×#4111yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C72−1+411yescharacteristic
265C2×C4×C72−1+411yescharacteristic
266C22⋊D7—48no—
267C2×Dic7—22no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C2×Dic7—41no—
276D28—22no—
277D28—22no—
278D28—41no—
279D28—41no—
280C4×D7—84no—
281C7⋊C8—24no—
282C7⋊C8—42no—
283C22×Q8—21no—
284C22×D4—71no—
285C22×D4—141no—
286C23×C4—141no—
287C23×C4C22⋊D711yescharacteristic
288C23×C4C22⋊D711yescharacteristic
289C23×C4C22×D711yescharacteristic
290C22×C8—72no—
291C22×C8—72no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C22⋊Q8—44no—
295#121—142no—
296#121—142no—
297C4×D4—142no—
298C4×D4—142no—
299C4×D4—284no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—71no—
305C2×C4⋊C4—71no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—141no—
308C2×C4⋊C4C22×D712yes—
309C2×C4⋊C4C22×D712yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—141no—
312C2×C22⋊C4—141no—
313C2×C22⋊C4—141no—
314C2×C22⋊C4—141no—
315C2×C22⋊C4—141no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—141no—
319C2×C42C22⋊D711yescharacteristic
320C2×C42C22⋊D711yescharacteristic
321C2×C42C22×D711yescharacteristic
322C2×C42C22×D711yescharacteristic
323#107—144no—
324#106—144no—
325C4⋊C8—74no—
326C4⋊C8—142no—
327#102—144no—
328#102—144no—
329C22⋊C8—144no—
330#97—144no—
331#95—22no—
332#95—22no—
333#95—141no—
334#95—141no—
335#95—141no—
336C22×C7—22no—
337C22×C7—22no—
338C22×C7—22no—
339C22×C7—41no—
340C22×C7—41no—
341C22×C7—41no—
342C22×C7—41no—
343C22×C7C2×#12111yescharacteristic
344C22×C7#52511yescharacteristic
345C22×C7#53611yescharacteristic
346C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C7C2×#13711yescharacteristic
349C22×C7#57511yescharacteristic
350C2×D7—42no—
351C2×D7—42no—
352C2×D7—42no—
353C2×D7—81no—
354C2×D7—81no—
355C2×D7—81no—
356C2×D7—81no—
357C4×C7—22no—
358C4×C7—41no—
359C4×C7—41no—
360C4×C7—41no—
361C4×C7—82no—
362C4×C7—82no—
363C4×C7#46914yes—
364C4×C7#54711yescharacteristic
365C4×C7#54811yescharacteristic
366C4×C7#55211yescharacteristic
367C4×C7#55611yescharacteristic
368Dic7—42no—
369Dic7—81no—
370Dic7—81no—
371C24—141no—
372C24C2×C22⋊D711yescharacteristic
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×D4—72no—
377C2×D4—144no—
378C2×D4—141no—
379C2×D4—141no—
380C2×D4—141no—
381C2×D4—141no—
382C2×D4—141no—
383C2×D4—282no—
384C2×D4—282no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—282no—
403C22×C4—282no—
404C22×C4—282no—
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C2×C22⋊D711yescharacteristic
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C2×C22⋊D711yescharacteristic
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411C2×C8—74no—
412C2×C8—142no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—142no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—72no—
426C4⋊C4—141no—
427C4⋊C4—141no—
428C4⋊C4—141no—
429C4⋊C4—141no—
430C4⋊C4—141no—
431C4⋊C4—282no—
432C22⋊C4—44no—
433C22⋊C4—144no—
434C22⋊C4—282no—
435C22⋊C4—282no—
436C22⋊C4—282no—
437C22⋊C4—282no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—282no—
443C42D4×D712yes—
444C42#115912yes—
445C2×C7—42no—
446C2×C7#241511yescharacteristic
447C2×C7C2×#46911yescharacteristic
448C2×C7#305611yescharacteristic
449C2×C7#311011yescharacteristic
450C2×C7#311511yescharacteristic
451C2×C7#323511yescharacteristic
452C2×C7#323911yescharacteristic
453D7—82no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—142no—
461C23—281no—
462C23—281no—
463C23—281no—
464C23—281no—
465C23—281no—
466C23—281no—
467C23#650512yes—
468C23C22×C22⋊D711yescharacteristic
469Q8—84no—
470D4—142no—
471D4—142no—
472D4—288no—
473D4—281no—
474D4—281no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—142no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—564no—
508C2×C4#649512yes—
509C2×C4#650212yes—
510C2×C4#650212yes—
511C2×C4#651212yes—
512C2×C4C2×D4×D711yescharacteristic
513C2×C4C2×#115911yescharacteristic
514C2×C4#655111yescharacteristic
515C2×C4#655111yescharacteristic
516C2×C4D7×#4111yescharacteristic
517C2×C4D7×#4111yescharacteristic
518C8—144no—
519C8—282no—
520C7#2470911yescharacteristic, complemented, Sylow
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—282no—
529C22—282no—
530C22—282no—
531C22—561no—
532C22—561no—
533C22—561no—
534C22—561no—
535C22C2×#649511yescharacteristic
536C22C2×#650211yescharacteristic
537C22C2×#650511yescharacteristic
538C22#9050711yescharacteristic
539C22C2×#651211yescharacteristic
540C22#9051311yescharacteristic
541C22#9051911yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4—282no—
549C4—561no—
550C4—561no—
551C4#8981114yes—
552C4#9032211yescharacteristic
553C4#9032411yescharacteristic
554C4#9033011yescharacteristic
555C4#9033211yescharacteristic
556C2—42no—
557C2—562no—
558C2C2×#8981111yescharacteristic
559C2#1174017711yescharacteristic
560C2#1174025911yescharacteristic
561C2#1174066311yescharacteristic
562C2#1174074611yescharacteristic
563C2#1174079811yescharacteristic
564C2#1174080611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2857C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2857D7
hypocenterC7#24709
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C7×#2857C2semidirect
2. C2×#89859C2semidirect
4. C2×#89698C2semidirect
5. #11729534C2semidirect
6. #11727375C2semidirect
7. #11727371C2semidirect
8. #11727338C2semidirect
9. #11727322C2semidirect
11. #11727273C2semidirect
97. C4⋊(C7⋊C8)D4semidirect
101. #2857D7semidirect
520. C7#24709semidirect

Maximal subgroups

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

subgroupindex
C7×#28572
C2×#898592
C4×#64932
C2×#896982
#117295342
#117273752
#117273712
#117273382
#117273222
#117272782
#117272732
#247097