Finite Group Databasebeta

← #13367605

Subgroups — #13367605

Order 1792.

1074 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×C7×#496C211yesmaximal, characteristic
2Dic7×#127C211yesmaximal, characteristic, complemented
3Dic7×#124C211yesmaximal, characteristic, complemented
4#11729802C212yesmaximal, complemented
5#11729801C212yesmaximal, complemented
6#11729791C212yesmaximal, complemented
7#11729791C212yesmaximal, complemented
8#11729519C211yesmaximal, characteristic, complemented
9#11729480C211yesmaximal, characteristic, complemented
10#11727569C211yesmaximal, characteristic, complemented
11#11727559C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2212yes—
13C22×C7×D8C2212yes—
14C2×C7×#127C2211yescharacteristic
15C2×C7×#124C2211yescharacteristic
16C2×Q8×Dic7C2211yescharacteristic
17C2×D4×Dic7—21no—
18C2×D4×Dic7—21no—
19C2×D4×Dic7C2211yescharacteristic
20C2×D4×Dic7C2212yes—
21C7×#496C418yescomplemented
22C2×C7×#97C2211yescharacteristic
23#90002C2211yescharacteristic
24C2×#6409C2212yes—
25#89993—21no—
26#89993—21no—
27#89993C2211yescharacteristic
28#89993C2212yes—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32#89723—21no—
33#89723—21no—
34C22⋊C4×Dic7—21no—
35C22⋊C4×Dic7—21no—
36#89708C2211yescharacteristic
37#89707C2211yescharacteristic
38C42×Dic7C2211yescharacteristic
39C2×#6377C2211yescharacteristic
40#89346C2212yes—
41#89346C2212yes—
42#89345C2212yes—
43#89345C2212yes—
44#22272—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4C2311yescharacteristic
49C22×C7×D4C2312yes—
50C23×Dic7—21no—
51C23×Dic7—22no—
52C2×C7×SD16—42no—
53C2×C7×SD16—42no—
54C2×C7×SD16C2×C418yescomplemented
55C2×C7×D8—42no—
56C2×C7×D8—42no—
57C2×C7×D8C2×C418yescomplemented
58C22×C7×C8C2312yes—
59C7×#127—22no—
60C7×#127C2×C414yescomplemented
61C7×#124—22no—
62C7×#124C2×C414yescomplemented
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C42×C7C2311yescharacteristic
66C2×C22⋊Dic7—21no—
67C2×C22⋊Dic7—21no—
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—21no—
71C2×C22⋊Dic7—22no—
72Q8×Dic7—42no—
73D4×Dic7—42no—
74D4×Dic7—42no—
75D4×Dic7—42no—
76D4×Dic7—44no—
77D4×Dic7—44no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2312yes—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4×Dic7—21no—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7D411yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2312yes—
92C7×#97C2×C414yescomplemented
93#6409—42no—
94#6406—42no—
95#6406—42no—
96#6406—42no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101#6377—22no—
102C2×#496D711yescharacteristic
103C4×#127—71no—
104C4×#124—71no—
105#1961—71no—
106#1959—71no—
107#1880—72no—
108#1879—72no—
109#1879—72no—
110#1877—72no—
111#1836—71no—
112#1745—71no—
113C24×C7—21no—
114C24×C7—22no—
115C2×C7×Q8—21no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8C22×C412yes—
119C2×C7×D4—24no—
120C2×C7×D4—24no—
121C2×C7×D4—21no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4—42no—
132C2×C7×D4—42no—
133C2×C7×D4C22×C412yes—
134C2×C7×D4C22×C414yes—
135C22×C4×C7—21no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2411yescharacteristic
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—22no—
142C22×Dic7—42no—
143C22×Dic7—44no—
144C22×Dic7—44no—
145C22×Dic7—42no—
146C22×Dic7—44no—
147C22×Dic7—42no—
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C7×SD16—48no—
153C7×D8—48no—
154C2×C7×C8—22no—
155C2×C7×C8—22no—
156C2×C7×C8C22×C414yes—
157C7×C22⋊C4—24no—
158C7×C22⋊C4—24no—
159C42×C7C22×C412yes—
160C42×C7#4112yes—
161C22⋊Dic7—42no—
162C22⋊Dic7—44no—
163C22⋊Dic7—44no—
164C22⋊Dic7—42no—
165C22⋊Dic7—44no—
166C22⋊Dic7—42no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7—22no—
170C4⋊Dic7—22no—
171C4⋊Dic7—42no—
172C4⋊Dic7—42no—
173C4⋊Dic7—42no—
174C4×Dic7—22no—
175C4×Dic7—22no—
176C4×Dic7—22no—
177C4×Dic7—22no—
178C4×Dic7—41no—
179C4×Dic7—41no—
180C4×Dic7—42no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C22×SD16C2×D712yes—
185C22×D8C2×D712yes—
186C2×#127C2×D711yescharacteristic
187C2×#124C2×D711yescharacteristic
188C2×C4×Q8—71no—
189C2×C4×D4—72no—
190C2×C4×D4—71no—
191C2×C4×D4—141no—
192C2×C4×D4—141no—
193#496Dic718yescomplemented
194C2×#103—72no—
195C2×#102—72no—
196C2×#102—72no—
197C2×#102—72no—
198C2×#97—71no—
199C2×#97C2×D711yescharacteristic
200#391—71no—
201#390—72no—
202#390—71no—
203#390—141no—
204#390—141no—
205#388—141no—
206#388—141no—
207#384—71no—
208#382—71no—
209C4×C22⋊C4—141no—
210C4×C22⋊C4—141no—
211C43—71no—
212#340—72no—
213#340—72no—
214#336—72no—
215#336—72no—
216C23×C7—22no—
217C23×C7—24no—
218C23×C7—41no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—41no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7C22×D411yescharacteristic
231C7×Q8—24no—
232C7×Q8—42no—
233C7×D4—24no—
234C7×D4—24no—
235C7×D4—24no—
236C7×D4—48no—
237C7×D4—48no—
238C7×D4—42no—
239C7×D4—42no—
240C7×D4—42no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—22no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7C4×D412yes—
249C2×C4×C7C4×D412yes—
250C2×C4×C7C23×C411yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7H(5,𝔽2)12yes—
253C2×C4×C7H(5,𝔽2)12yes—
254C2×Dic7—21no—
255C2×Dic7—21no—
256C2×Dic7—21no—
257C2×Dic7—21no—
258C2×Dic7—21no—
259C2×Dic7—21no—
260C2×Dic7—21no—
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—21no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—84no—
274C2×Dic7—88no—
275C2×Dic7AGL(1,ℤ/8)12yes—
276C2×Dic7AGL(1,ℤ/8)12yes—
277C7×C8—24no—
278C7⋊C8—42no—
279C22×Q8C22×D711yescharacteristic
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22×D711yescharacteristic
283C22×D4C22×D712yes—
284C23×C4—142no—
285C23×C4—141no—
286C2×SD16—42no—
287C2×SD16—42no—
288C2×SD16C2×Dic718yescomplemented
289C2×D8—42no—
290C2×D8—42no—
291C2×D8C2×Dic718yescomplemented
292C22×C8—72no—
293C22×C8C22×D712yes—
294#127—22no—
295#127C2×Dic714yescomplemented
296#124—22no—
297#124C2×Dic714yescomplemented
298C4×Q8—282no—
299C4×D4—282no—
300C4×D4—282no—
301C4×D4—282no—
302C4×D4—284no—
303C4×D4—284no—
304C2×C4⋊C4—71no—
305C2×C4⋊C4—72no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—141no—
308C2×C4⋊C4—141no—
309C2×C4⋊C4—141no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—141no—
313C2×C22⋊C4—142no—
314C2×C22⋊C4—142no—
315C2×C22⋊C4—142no—
316C2×C22⋊C4—141no—
317C2×C22⋊C4—141no—
318C2×C42—71no—
319C2×C42—71no—
320C2×C42—71no—
321C2×C42—71no—
322C2×C42—71no—
323C2×C42—71no—
324C2×C42—141no—
325C2×C42C22×D711yescharacteristic
326#103—282no—
327#102—282no—
328#102—282no—
329#102—282no—
330#97—142no—
331#97C2×Dic714yescomplemented
332#95—141no—
333#95—141no—
334#95—141no—
335#95—141no—
336C22×C7—42no—
337C22×C7—44no—
338C22×C7—44no—
339C22×C7—44no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—81no—
343C22×C7—82no—
344C22×C7—81no—
345C22×C7—82no—
346C22×C7—82no—
347C22×C7—81no—
348C22×C7—82no—
349C22×C7—81no—
350C22×C7C2×C4×D411yescharacteristic
351C22×C7#53511yescharacteristic
352C22×C7#53511yescharacteristic
353C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C7#57511yescharacteristic
356C22×C7#57511yescharacteristic
357C4×C7—22no—
358C4×C7—22no—
359C4×C7—42no—
360C4×C7#51812yes—
361C4×C7#54012yes—
362Dic7—22no—
363Dic7—22no—
364Dic7—41no—
365Dic7—41no—
366Dic7—81no—
367Dic7—81no—
368Dic7—82no—
369C24—21no—
370C24—22no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C22×Dic712yes—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C22×Dic712yes—
390C2×D4C22×Dic714yes—
391C22×C4—21no—
392C22×C4—71no—
393C22×C4—71no—
394C22×C4—71no—
395C22×C4—71no—
396C22×C4—141no—
397C22×C4—141no—
398C22×C4—142no—
399C22×C4—284no—
400C22×C4—282no—
401C22×C4—282no—
402C22×C4—284no—
403C22×C4—284no—
404C22×C4—282no—
405C22×C4D4×D711yescharacteristic
406C22×C4D4×D711yescharacteristic
407C22×C4C23×D711yescharacteristic
408SD16—48no—
409D8—48no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—142no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8C22×Dic714yes—
416C4⋊C4—142no—
417C4⋊C4—142no—
418C4⋊C4—142no—
419C4⋊C4—142no—
420C4⋊C4—282no—
421C4⋊C4—282no—
422C4⋊C4—282no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—284no—
426C22⋊C4—282no—
427C22⋊C4—284no—
428C22⋊C4—282no—
429C22⋊C4—284no—
430C22⋊C4—282no—
431C42—142no—
432C42—142no—
433C42—142no—
434C42—142no—
435C42—281no—
436C42—281no—
437C42—282no—
438C42#115912yes—
439C42C22×Dic712yes—
440C2×C7—84no—
441C2×C7—82no—
442C2×C7#232811yescharacteristic
443C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
444C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
445C2×C7#321311yescharacteristic
446C2×C7#321411yescharacteristic
447C2×C7#321711yescharacteristic
448C2×C7#321911yescharacteristic
449C23—22no—
450C23—24no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23C2×D4×D711yescharacteristic
464Q8—24no—
465Q8—42no—
466D4—24no—
467D4—24no—
468D4—24no—
469D4—48no—
470D4—48no—
471D4—42no—
472D4—42no—
473D4—42no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—72no—
482C2×C4—72no—
483C2×C4—141no—
484C2×C4—141no—
485C2×C4—141no—
486C2×C4—141no—
487C2×C4—141no—
488C2×C4—141no—
489C2×C4—141no—
490C2×C4—141no—
491C2×C4—141no—
492C2×C4—141no—
493C2×C4—281no—
494C2×C4—281no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—564no—
503C2×C4—568no—
504C2×C4D4×Dic712yes—
505C2×C4D4×Dic712yes—
506C2×C4C2×#115911yescharacteristic
507C2×C4#654812yes—
508C2×C4#654812yes—
509C2×C4C23×Dic711yescharacteristic
510C8—24no—
511C8—282no—
512C7#2227211yescharacteristic, complemented, Sylow
513C22—44no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—81no—
520C22—82no—
521C22—81no—
522C22—82no—
523C22—82no—
524C22—81no—
525C22—82no—
526C22—81no—
527C22#9040611yescharacteristic
528C22#9040611yescharacteristic
529C22D7×AGL(1,ℤ/8)11yescharacteristic
530C22D7×AGL(1,ℤ/8)11yescharacteristic
531C22#9046411yescharacteristic
532C22#9046411yescharacteristic
533C22C2×D4×Dic711yescharacteristic
534C4—22no—
535C4—22no—
536C4—42no—
537C4—142no—
538C4—142no—
539C4—281no—
540C4—281no—
541C4—561no—
542C4—561no—
543C4—562no—
544C4#9041012yes—
545C4#9048912yes—
546C2—84no—
547C2—82no—
548C2#1173929411yescharacteristic
549C2#1173929511yescharacteristic
550C2#1173929611yescharacteristic
551C2#1173929711yescharacteristic
552C2#1174061711yescharacteristic
553C2Dic7×AGL(1,ℤ/8)11yescharacteristic
554C2Dic7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#496C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#496D7
hypocenterC7#22272
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
2. Dic7×#127C2semidirect
3. Dic7×#124C2semidirect
4. #11729802C2semidirect
5. #11729801C2semidirect
6. #11729791C2semidirect
7. #11729791C2semidirect
8. #11729519C2semidirect
9. #11729480C2semidirect
10. #11727569C2semidirect
11. #11727559C2semidirect
21. C7×#496C4semidirect
54. C2×C7×SD16C2×C4semidirect
57. C2×C7×D8C2×C4semidirect
60. C7×#127C2×C4semidirect
62. C7×#124C2×C4semidirect
92. C7×#97C2×C4semidirect
193. #496Dic7semidirect
288. C2×SD16C2×Dic7semidirect
291. C2×D8C2×Dic7semidirect
295. #127C2×Dic7semidirect
297. #124C2×Dic7semidirect
331. #97C2×Dic7semidirect
512. C7#22272semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4962
Dic7×#1272
Dic7×#1242
#117298022
#117298012
#117297912
#117295192
#117294802
#117275692
#117275592
#222727