Finite Group Databasebeta

← #13403912

Subgroups — #13403912

Order 1792.

958 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C8×D4C211yesmaximal, characteristic, complemented
2#11739862C211yesmaximal, characteristic
3#11739829C211yesmaximal, characteristic, complemented
4C2×#89614C211yesmaximal, characteristic, complemented
5#11729727C212yesmaximal, complemented
6#11729724C212yesmaximal, complemented
7#11728208C211yesmaximal, characteristic, complemented
8#11727694C212yesmaximal, complemented
9#11727689C212yesmaximal, complemented
10#11727531C211yesmaximal, characteristic, complemented
11C4×#6395C211yesmaximal, characteristic, complemented
12C23×C7×C8C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6501—21no—
15C2×#6493C2212yes—
16C2×C4×D28C2211yescharacteristic
17C2×#6432C2211yescharacteristic
18C7×C8×D4—24no—
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C22⋊C8C2212yes—
21C2×C4×C7×C8C2211yescharacteristic
22#89993—21no—
23#89984C2212yes—
24#89984C2212yes—
25#89984C2212yes—
26C2×#6395C2211yescharacteristic
27C2×#6395C2212yes—
28C2×#6395C2211yescharacteristic
29C2×#6391C2212yes—
30C2×#6391C2212yes—
31#89759—21no—
32#89749—21no—
33#89733—22no—
34#89719—22no—
35#89707C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89614—24no—
38#89346C2211yescharacteristic
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2211yescharacteristic
43#25339—71nomaximal, Sylow
44C22×C7×D4D411yescharacteristic
45C23×C4×C7C2312yes—
46C22×C22⋊D7—22no—
47C22×D28C2311yescharacteristic
48C22×C4×D7—21no—
49C22×Dic14—21no—
50C22×C7×C8—24no—
51C22×C7×C8—24no—
52C22×C7×C8—42no—
53C22×C7×C8C2311yescharacteristic
54C22×C7×C8C2312yes—
55C22×C7×C8C2311yescharacteristic
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4D412yes—
60C2×C42×C7C2311yescharacteristic
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64#6501—44no—
65#6493—24no—
66#6493—42no—
67C2×(C2×D7)⋊C4—21no—
68C2×(C2×D7)⋊C4—22no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7C2312yes—
73C2×C4⋊Dic7C2311yescharacteristic
74C2×C4⋊Dic7C2311yescharacteristic
75C2×C4⋊Dic7C2312yes—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4×Dic7—21no—
78C4×D28—22no—
79C4×D28—22no—
80#6432—22no—
81#6432—22no—
82C7×C4⋊C8D414yescomplemented
83C7×C22⋊C8—24no—
84C4×C7×C8—22no—
85C7⋊#95—21no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89#6395—22no—
90#6395—24no—
91#6395—22no—
92#6391—24no—
93#6391—42no—
94C2×#476—71no—
95C2×C8×D4D711yescharacteristic, complemented
96#2563—71no—
97#2537—71no—
98#2077—72no—
99#2034—72no—
100#1990—71no—
101#1948—71no—
102#1893—72no—
103#1807—72no—
104C4×#102—71no—
105C24×C7C2×D412yes—
106C23×D7—21no—
107C2×C7×D4—22no—
108C2×C7×D4—22no—
109C2×C7×D4SD1614yes—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7C2×D412yes—
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2411yescharacteristic
120C2×C22⋊D7—44no—
121C2×C22⋊D7—44no—
122C2×C22⋊D7—82no—
123C22×Dic7—21no—
124C22×Dic7—21no—
125C22×Dic7—22no—
126C22×Dic7—21no—
127C22×Dic7—22no—
128C2×D28—21no—
129C2×D28—21no—
130C2×D28—21no—
131C2×D28—21no—
132C2×D28—21no—
133C2×D28#4112yes—
134C2×C4×D7—42no—
135C2×C4×D7—42no—
136C2×C4×D7—42no—
137C2×Dic14—42no—
138C2×Dic14—81no—
139C2×Dic14—81no—
140C2×C7×C8—21no—
141C2×C7×C8—22no—
142C2×C7×C8—21no—
143C2×C7×C8—22no—
144C2×C7×C8—21no—
145C2×C7×C8—21no—
146C2×C7×C8—21no—
147C2×C7×C8—48no—
148C2×C7×C8C2×D414yes—
149C2×C7×C8#4112yes—
150C7×C4⋊C4—22no—
151C7×C22⋊C4—24no—
152C42×C7C2×D412yes—
153C42×C7#4112yes—
154C22⋊Dic7—44no—
155C22⋊Dic7—82no—
156C22⋊Dic7—82no—
157(C2×D7)⋊C4—44no—
158(C2×D7)⋊C4—42no—
159C4⋊Dic7—22no—
160C4⋊Dic7—21no—
161C4⋊Dic7—22no—
162C4⋊Dic7—21no—
163C4⋊Dic7—22no—
164C4⋊Dic7—21no—
165C4⋊Dic7—21no—
166C4⋊Dic7—22no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—42no—
170C4⋊Dic7—42no—
171C4⋊Dic7—81no—
172C4×Dic7—42no—
173C23×C8C2×D712yes—
174C2×C4⋊Q8—71no—
175C2×#121—72no—
176C2×#121—141no—
177C2×C4×D4—71no—
178C2×C4×D4C2×D711yescharacteristic
179#476—144no—
180C8×D4—24no—
181C2×#106—72no—
182C2×#106—72no—
183C2×C4⋊C8C2×D711yescharacteristic
184C2×#102—71no—
185C2×#102—72no—
186C2×#102—71no—
187C2×C22⋊C8C2×D712yes—
188C2×C4×C8C2×D711yescharacteristic
189#394—142no—
190#393—142no—
191#390—141no—
192#384—71no—
193#382—141no—
194#380—72no—
195#380—72no—
196#380—72no—
197#380—141no—
198C4×C4⋊C4—71no—
199#340—71no—
200#340—72no—
201#340—71no—
202#340—72no—
203#340—72no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7C2×SD1614yes—
211C23×C7C22×D411yescharacteristic
212C22×D7—22no—
213C22×D7—41no—
214C22×D7—41no—
215C22×D7—41no—
216C22×D7—41no—
217C22×D7—41no—
218C22×D7—41no—
219C7×D4—28no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7#12112yes—
230C2×C4×C7#12112yes—
231C2×C4×C7C2×SD1612yes—
232C2×C4×C7#13512yes—
233C2×C4×C7C22×D411yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C2×C4×C7C2×#4111yescharacteristic
236C2×C4×C7C2×#4111yescharacteristic
237C2×C4×C7H(5,𝔽2)11yescharacteristic
238C2×C4×C72−1+411yescharacteristic
239C22⋊D7—88no—
240C2×Dic7—42no—
241C2×Dic7—41no—
242C2×Dic7—42no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—42no—
251C2×Dic7—41no—
252C2×Dic7—42no—
253C2×Dic7—41no—
254C2×Dic7—42no—
255D28—22no—
256D28—22no—
257D28—41no—
258D28—41no—
259C4×D7—84no—
260Dic14—84no—
261C7×C8—22no—
262C7×C8—24no—
263C7×C8—41no—
264C22×Q8—141no—
265C22×D4—71no—
266C22×D4—142no—
267C22×D4D2811yescharacteristic
268C23×C4—141no—
269C23×C4C22×D712yes—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—42no—
273C22×C8C22×D711yescharacteristic
274C22×C8C22×D711yescharacteristic
275C22×C8C22×D712yes—
276C4⋊Q8—142no—
277C4⋊Q8—142no—
278#121—144no—
279#121—282no—
280#121—284no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—142no—
284C4×D4—142no—
285C2×C4⋊C4—71no—
286C2×C4⋊C4—72no—
287C2×C4⋊C4—72no—
288C2×C4⋊C4—71no—
289C2×C4⋊C4—71no—
290C2×C4⋊C4—141no—
291C2×C4⋊C4—141no—
292C2×C4⋊C4—141no—
293C2×C4⋊C4D2811yescharacteristic
294C2×C22⋊C4—142no—
295C2×C22⋊C4—141no—
296C2×C22⋊C4—142no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4—142no—
299C2×C22⋊C4D2812yes—
300C2×C42—141no—
301C2×C42C22×D711yescharacteristic
302#106—144no—
303#106—282no—
304C4⋊C8C22⋊D714yescomplemented
305#102—142no—
306#102—142no—
307#102—144no—
308C22⋊C8—24no—
309C4×C8—22no—
310#95—141no—
311#95—142no—
312#95—142no—
313#95—142no—
314C22×C7—24no—
315C22×C7—24no—
316C22×C7—24no—
317C22×C7—42no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7C2×#12111yescharacteristic
322C22×C7#54611yescharacteristic
323C22×C7#54711yescharacteristic
324C22×C7C22×SD1611yescharacteristic
325C22×C7C2×#13511yescharacteristic
326C22×C7#57611yescharacteristic
327C22×C7#57711yescharacteristic
328C2×D7—42no—
329C2×D7—42no—
330C2×D7—42no—
331C2×D7—81no—
332C2×D7—81no—
333C2×D7—81no—
334C2×D7—81no—
335C4×C7—22no—
336C4×C7—42no—
337C4×C7#46514yes—
338C4×C7#52911yescharacteristic
339C4×C7#54211yescharacteristic
340C4×C7#54611yescharacteristic
341C4×C7#54811yescharacteristic
342Dic7—81no—
343Dic7—82no—
344Dic7—81no—
345Dic7—81no—
346Dic7—82no—
347C24—141no—
348C24C2×D2812yes—
349C2×Q8—282no—
350C2×Q8—561no—
351C2×Q8—561no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—72no—
355C2×D4—141no—
356C2×D4—141no—
357C2×D4—141no—
358C2×D4—141no—
359C2×D4—141no—
360C2×D4—284no—
361C2×D4—284no—
362C2×D4—562no—
363C2×D4#113214yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—141no—
371C22×C4—141no—
372C22×C4—141no—
373C22×C4—142no—
374C22×C4—142no—
375C22×C4—282no—
376C22×C4—282no—
377C22×C4—282no—
378C22×C4C2×D2812yes—
379C22×C4C2×D2811yescharacteristic
380C22×C4C2×D2811yescharacteristic
381C22×C4C23×D711yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—48no—
390C2×C8#115912yes—
391C2×C8C2×C22⋊D714yes—
392C4⋊C4—22no—
393C4⋊C4—141no—
394C4⋊C4—142no—
395C4⋊C4—141no—
396C4⋊C4—141no—
397C4⋊C4—142no—
398C4⋊C4—141no—
399C4⋊C4—142no—
400C4⋊C4—141no—
401C4⋊C4—142no—
402C4⋊C4—141no—
403C4⋊C4—282no—
404C4⋊C4—282no—
405C4⋊C4—561no—
406C22⋊C4—24no—
407C22⋊C4—284no—
408C22⋊C4—284no—
409C22⋊C4—282no—
410C22⋊C4—562no—
411C22⋊C4—562no—
412C42—282no—
413C42#115712yes—
414C42C2×C22⋊D712yes—
415C2×C7—44no—
416C2×C7#242711yescharacteristic
417C2×C7C2×#46511yescharacteristic
418C2×C7#306211yescharacteristic
419C2×C7#329511yescharacteristic
420C2×C7#330311yescharacteristic
421C2×C7#333511yescharacteristic
422C2×C7#334111yescharacteristic
423D7—82no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—142no—
431C23—281no—
432C23—281no—
433C23—281no—
434C23—281no—
435C23—281no—
436C23—281no—
437C23C2×#113214yes—
438C23C22×D2811yescharacteristic
439Q8—564no—
440D4—28no—
441D4—142no—
442D4—142no—
443D4—281no—
444D4—281no—
445D4—568no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—281no—
456C2×C4—282no—
457C2×C4—281no—
458C2×C4—281no—
459C2×C4—281no—
460C2×C4—281no—
461C2×C4—282no—
462C2×C4—281no—
463C2×C4—282no—
464C2×C4—282no—
465C2×C4—281no—
466C2×C4—282no—
467C2×C4—282no—
468C2×C4—281no—
469C2×C4—281no—
470C2×C4—564no—
471C2×C4C2×#113212yes—
472C2×C4#646712yes—
473C2×C4#649312yes—
474C2×C4#649312yes—
475C2×C4C2×#115711yescharacteristic
476C2×C4C2×#115911yescharacteristic
477C2×C4#654811yescharacteristic
478C2×C4D7×#4111yescharacteristic
479C2×C4#655411yescharacteristic
480C2×C4C22×C22⋊D711yescharacteristic
481C8—22no—
482C8—24no—
483C8—41no—
484C7#2533911yescharacteristic, complemented, Sylow
485C22—24no—
486C22—24no—
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—282no—
493C22—282no—
494C22—282no—
495C22—561no—
496C22—561no—
497C22—561no—
498C22—561no—
499C22#9024511yescharacteristic
500C22#9024611yescharacteristic
501C22C22×#113211yescharacteristic
502C22C2×#646711yescharacteristic
503C22#9044211yescharacteristic
504C22#9044311yescharacteristic
505C22C2×#649311yescharacteristic
506C4—22no—
507C4—42no—
508C4—561no—
509C4—561no—
510C4—562no—
511C4—561no—
512C4—562no—
513C4#8988614yes—
514C4#9024811yescharacteristic
515C4#9025211yescharacteristic
516C4#9049711yescharacteristic
517C4#9052211yescharacteristic
518C2—44no—
519C2—562no—
520C2#1173685711yescharacteristic
521C2#1173686311yescharacteristic
522C2#1173688411yescharacteristic
523C2#1173688811yescharacteristic
524C2#1173985311yescharacteristic
525C2C2×#8988611yescharacteristic
526C2#1174124711yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×C8×D4C2
socleC23×C7C22×D4
prime radical#11739862C2
hypercenterC2×C8×D4D7
hypocenterC7#25339
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×C7×C8×D4C2semidirect
3. #11739829C2semidirect
4. C2×#89614C2semidirect
5. #11729727C2semidirect
6. #11729724C2semidirect
7. #11728208C2semidirect
8. #11727694C2semidirect
9. #11727689C2semidirect
10. #11727531C2semidirect
11. C4×#6395C2semidirect
82. C7×C4⋊C8D4semidirect
95. C2×C8×D4D7semidirect
304. C4⋊C8C22⋊D7semidirect
484. C7#25339semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C8×D42
#117398622
#117398292
C2×#896142
#117297272
#117297242
#117282082
#117276942
#117276892
#117275312
C4×#63952
#253397