Finite Group Databasebeta

← #13370063

Subgroups — #13370063

Order 1792.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90254C211yesmaximal, characteristic
2C2×C7×C8×D4C211yesmaximal, characteristic, complemented
3#11739782C211yesmaximal, characteristic, complemented
4#11729727C212yesmaximal, complemented
5#11729723C212yesmaximal, complemented
6#11728233C211yesmaximal, characteristic, complemented
7#11728213C211yesmaximal, characteristic, complemented
8#11727687C212yesmaximal, complemented
9#11727680C212yesmaximal, complemented
10#11727531C211yesmaximal, characteristic, complemented
11C4×#6395C211yesmaximal, characteristic, complemented
12C23×C7×C8C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6501—21no—
15C2×#6493C2212yes—
16C2×#6489C2212yes—
17C2×#6489C2212yes—
18#90254—24no—
19#90254C418yescomplemented
20C2×#6455—21no—
21C2×#6448—22no—
22C2×C4×D28C2211yescharacteristic
23C2×#6432C2211yescharacteristic
24C7×C8×D4—24no—
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×C22⋊C8C2212yes—
27C2×C4×C7×C8C2211yescharacteristic
28#89993—21no—
29#89984C2212yes—
30C2×#6395C2211yescharacteristic
31C2×#6395C2211yescharacteristic
32C2×#6395C2212yes—
33C4⋊C4×Dic7—21no—
34#89723—22no—
35#89707C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89346C2211yescharacteristic
38#89346C2212yes—
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2211yescharacteristic
43#89346C2212yes—
44#22489—71nomaximal, Sylow
45C22×C7×D4D411yescharacteristic
46C23×C4×C7C2312yes—
47C22×C22⋊D7—22no—
48C22×D28C2311yescharacteristic
49C22×C4×D7—21no—
50C22×Dic14—21no—
51C22×C7×C8—24no—
52C22×C7×C8—24no—
53C22×C7×C8—42no—
54C22×C7×C8C2311yescharacteristic
55C22×C7×C8C2312yes—
56C22×C7×C8C2311yescharacteristic
57C4×C7×D4C2×C414yes—
58C4×C7×D4D414yes—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4D412yes—
61C2×C42×C7C2311yescharacteristic
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—22no—
65#6501—28no—
66#6493—24no—
67#6493C2×C418yes—
68C2×(C2×D7)⋊C4—21no—
69C2×(C2×D7)⋊C4—22no—
70#6489—24no—
71#6489—24no—
72#6489—42no—
73#6489C2×C418yes—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2312yes—
79C2×C4⋊Dic7C2312yes—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4×Dic7—22no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—22no—
85#6455—28no—
86#6448—216no—
87C4×D28—22no—
88C4×D28C2×C414yes—
89#6432—22no—
90#6432C2×C414yes—
91C7×C4⋊C8—22no—
92C7×C22⋊C8—24no—
93C4×C7×C8—22no—
94C7⋊#95—21no—
95C7⋊#95—22no—
96#6395—22no—
97#6395—22no—
98#6395—42no—
99C2×#555—71no—
100C2×C8×D4D711yescharacteristic, complemented
101#2354—71no—
102#1990—71no—
103#1987—71no—
104#1893—72no—
105#1875—72no—
106#1825—72no—
107#1802—71no—
108C4×#102—71no—
109#1735—72no—
110C24×C7C2×D412yes—
111C23×D7—21no—
112C2×C7×D4—22no—
113C2×C7×D4—22no—
114C2×C7×D4C22⋊C414yes—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7C22×C414yes—
120C22×C4×C7C2×D412yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D414yes—
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2411yescharacteristic
125C2×C22⋊D7—28no—
126C2×C22⋊D7—82no—
127C2×C22⋊D7—82no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C22×Dic7—22no—
131C22×Dic7—21no—
132C22×Dic7—22no—
133C2×D28—21no—
134C2×D28—21no—
135C2×D28—21no—
136C2×D28—21no—
137C2×D28—21no—
138C2×D28C22×C412yes—
139C2×C4×D7—24no—
140C2×C4×D7—42no—
141C2×C4×D7—42no—
142C2×Dic14—24no—
143C2×Dic14—81no—
144C2×Dic14—81no—
145C2×C7×C8—21no—
146C2×C7×C8—21no—
147C2×C7×C8—21no—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C2×C7×C8—21no—
151C2×C7×C8—22no—
152C2×C7×C8—21no—
153C2×C7×C8—48no—
154C2×C7×C8#4112yes—
155C7×C4⋊C4C22⋊C414yes—
156C7×C22⋊C4C22⋊C418yes—
157C42×C7C22×C412yes—
158C42×C7C2×D412yes—
159C22⋊Dic7—28no—
160C22⋊Dic7—44no—
161C22⋊Dic7—82no—
162(C2×D7)⋊C4—24no—
163(C2×D7)⋊C4—28no—
164C4⋊Dic7—24no—
165C4⋊Dic7—22no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—22no—
169C4⋊Dic7—21no—
170C4⋊Dic7—22no—
171C4⋊Dic7—21no—
172C4⋊Dic7—42no—
173C4⋊Dic7—81no—
174C4⋊Dic7C22×C414yes—
175C4⋊Dic7C22×C412yes—
176C4⋊Dic7#4112yes—
177C4×Dic7—28no—
178C4×Dic7—44no—
179C4×Dic7—42no—
180C23×C8C2×D712yes—
181#555—78no—
182#555—144no—
183C2×C4⋊Q8—71no—
184C2×#124—142no—
185C2×#121—72no—
186C2×#121—141no—
187C2×C4×D4—71no—
188C2×C4×D4C2×D711yescharacteristic
189C2×#117—72no—
190C2×#117—72no—
191C2×#117—141no—
192C8×D4—24no—
193C2×C4⋊C8C2×D711yescharacteristic
194C2×#102—71no—
195C2×#102—71no—
196C2×#102—72no—
197C2×C22⋊C8C2×D712yes—
198C2×C4×C8C2×D711yescharacteristic
199#390—141no—
200#388—142no—
201#384—71no—
202#380—72no—
203C4×C4⋊C4—71no—
204C4×C4⋊C4—141no—
205#340—72no—
206#340—71no—
207#340—72no—
208#340—72no—
209#340—72no—
210#340—72no—
211#340—71no—
212C23×C7—22no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7C2×C22⋊C414yes—
219C23×C7C22×D411yescharacteristic
220C22×D7—22no—
221C22×D7—41no—
222C22×D7—41no—
223C22×D7—41no—
224C22×D7—41no—
225C22×D7—41no—
226C22×D7—41no—
227C7×D4—28no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7C2×C22⋊C414yes—
237C2×C4×C7C2×C22⋊C412yes—
238C2×C4×C7C2×C22⋊C412yes—
239C2×C4×C7#13512yes—
240C2×C4×C7#13512yes—
241C2×C4×C7C23×C411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C2×#4111yescharacteristic
244C2×C4×C7C2×#4111yescharacteristic
245C2×C4×C7H(5,𝔽2)11yescharacteristic
246C2×C4×C72−1+411yescharacteristic
247C22⋊D7—88no—
248C2×Dic7—22no—
249C2×Dic7—24no—
250C2×Dic7—22no—
251C2×Dic7—41no—
252C2×Dic7—41no—
253C2×Dic7—41no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—41no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—41no—
263D28—22no—
264D28—22no—
265D28—41no—
266D28—41no—
267C4×D7—48no—
268Dic14—84no—
269C7×C8—22no—
270C7×C8—42no—
271C7×C8—41no—
272C22×Q8—141no—
273C22×D4—71no—
274C22×D4—142no—
275C22×D4D2811yescharacteristic
276C23×C4—141no—
277C23×C4C22×D712yes—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8—42no—
281C22×C8C22×D712yes—
282C22×C8C22×D711yescharacteristic
283C22×C8C22×D711yescharacteristic
284C4⋊Q8—74no—
285C4⋊Q8—142no—
286#124—1416no—
287#121—78no—
288#121—148no—
289#121—144no—
290C4×D4—74no—
291C4×D4—142no—
292C4×D4C4×D714yes—
293C4×D4C22⋊D714yes—
294#117—78no—
295#117—144no—
296#117—148no—
297#117—144no—
298#117—282no—
299C2×C4⋊C4—71no—
300C2×C4⋊C4—71no—
301C2×C4⋊C4—72no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—71no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4D2811yescharacteristic
308C2×C22⋊C4—142no—
309C2×C22⋊C4—142no—
310C2×C22⋊C4—141no—
311C2×C22⋊C4—142no—
312C2×C22⋊C4—142no—
313C2×C22⋊C4D2812yes—
314C2×C42—142no—
315C2×C42—142no—
316C2×C42—141no—
317C2×C42C22×D711yescharacteristic
318C4⋊C8—22no—
319#102—142no—
320#102—142no—
321#102—282no—
322C22⋊C8—24no—
323C4×C8—22no—
324#95—141no—
325#95—142no—
326C22×C7—24no—
327C22×C7—24no—
328C22×C7—24no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7C22×C22⋊C411yescharacteristic
334C22×C7#54611yescharacteristic
335C22×C7#54711yescharacteristic
336C22×C7C2×#13511yescharacteristic
337C22×C7C2×#13511yescharacteristic
338C22×C7#57611yescharacteristic
339C22×C7#57711yescharacteristic
340C2×D7—42no—
341C2×D7—42no—
342C2×D7—42no—
343C2×D7—81no—
344C2×D7—81no—
345C2×D7—81no—
346C2×D7—81no—
347C4×C7—24no—
348C4×C7—22no—
349C4×C7#41614yes—
350C4×C7C4×#4111yescharacteristic
351C4×C7#52011yescharacteristic
352C4×C7#54611yescharacteristic
353C4×C7#54711yescharacteristic
354Dic7—42no—
355Dic7—44no—
356Dic7—42no—
357Dic7—81no—
358Dic7—82no—
359C24—141no—
360C24C2×D2812yes—
361C2×Q8—144no—
362C2×Q8—561no—
363C2×Q8—561no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—72no—
367C2×D4—148no—
368C2×D4—141no—
369C2×D4—141no—
370C2×D4—141no—
371C2×D4—141no—
372C2×D4—141no—
373C2×D4—562no—
374C2×D4—562no—
375C2×D4(C2×D7)⋊C414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—141no—
381C22×C4—141no—
382C22×C4—142no—
383C22×C4—144no—
384C22×C4—141no—
385C22×C4—142no—
386C22×C4—282no—
387C22×C4—282no—
388C22×C4C2×C4×D714yes—
389C22×C4C2×D2812yes—
390C22×C4C2×D2811yescharacteristic
391C22×C4C2×D2811yescharacteristic
392C22×C4C2×C22⋊D714yes—
393C22×C4C23×D711yescharacteristic
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—48no—
403C2×C8#115912yes—
404C4⋊C4—72no—
405C4⋊C4—74no—
406C4⋊C4—72no—
407C4⋊C4—144no—
408C4⋊C4—141no—
409C4⋊C4—141no—
410C4⋊C4—142no—
411C4⋊C4—141no—
412C4⋊C4—142no—
413C4⋊C4—142no—
414C4⋊C4—141no—
415C4⋊C4—282no—
416C4⋊C4—561no—
417C4⋊C4(C2×D7)⋊C414yes—
418C22⋊C4—148no—
419C22⋊C4—148no—
420C22⋊C4—144no—
421C22⋊C4—284no—
422C22⋊C4—562no—
423C22⋊C4(C2×D7)⋊C418yes—
424C42—148no—
425C42—284no—
426C42—282no—
427C42C2×C4×D712yes—
428C42C2×C22⋊D712yes—
429C2×C7—44no—
430C2×C7#233911yescharacteristic
431C2×C7C2×#41611yescharacteristic
432C2×C7#289711yescharacteristic
433C2×C7#330311yescharacteristic
434C2×C7#330511yescharacteristic
435C2×C7#334011yescharacteristic
436C2×C7#334111yescharacteristic
437D7—82no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—142no—
445C23—281no—
446C23—281no—
447C23—281no—
448C23—281no—
449C23—281no—
450C23—281no—
451C23C2×(C2×D7)⋊C414yes—
452C23C22×D2811yescharacteristic
453Q8—564no—
454D4—28no—
455D4—142no—
456D4—142no—
457D4—281no—
458D4—281no—
459D4—568no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—142no—
469C2×C4—144no—
470C2×C4—142no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4—281no—
474C2×C4—281no—
475C2×C4—282no—
476C2×C4—282no—
477C2×C4—288no—
478C2×C4—281no—
479C2×C4—282no—
480C2×C4—281no—
481C2×C4—282no—
482C2×C4—282no—
483C2×C4—281no—
484C2×C4#646712yes—
485C2×C4#646712yes—
486C2×C4C2×(C2×D7)⋊C414yes—
487C2×C4C2×(C2×D7)⋊C412yes—
488C2×C4C2×(C2×D7)⋊C412yes—
489C2×C4C22×C4×D711yescharacteristic
490C2×C4C2×#115911yescharacteristic
491C2×C4#654811yescharacteristic
492C2×C4D7×#4111yescharacteristic
493C2×C4#655411yescharacteristic
494C2×C4C22×C22⋊D711yescharacteristic
495C8—22no—
496C8—42no—
497C8—41no—
498C7#2248911yescharacteristic, complemented, Sylow
499C22—24no—
500C22—24no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—282no—
507C22—282no—
508C22—282no—
509C22—561no—
510C22—561no—
511C22—561no—
512C22—561no—
513C22#9024511yescharacteristic
514C22#9024611yescharacteristic
515C22C2×#646711yescharacteristic
516C22C2×#646711yescharacteristic
517C22#9044211yescharacteristic
518C22#9044311yescharacteristic
519C22C22×(C2×D7)⋊C411yescharacteristic
520C4—22no—
521C4—24no—
522C4—282no—
523C4—282no—
524C4—284no—
525C4—561no—
526C4—562no—
527C4#8988514yes—
528C4C4×#115911yescharacteristic
529C4#9023711yescharacteristic
530C4#9049711yescharacteristic
531C4#9051511yescharacteristic
532C2—44no—
533C2—562no—
534C2#1173686211yescharacteristic
535C2#1173686311yescharacteristic
536C2#1173688811yescharacteristic
537C2#1173689711yescharacteristic
538C2#1173977511yescharacteristic
539C2C2×#8988511yescharacteristic
540C2#1174124211yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×C8×D4C2
socleC23×C7C22×D4
prime radicalC2×#90254C2
cosocleC22×C4×C7C24
hypercenterC2×C8×D4D7
hypocenterC7#22489
LCSG ▷ C2×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
2. C2×C7×C8×D4C2semidirect
3. #11739782C2semidirect
4. #11729727C2semidirect
5. #11729723C2semidirect
6. #11728233C2semidirect
7. #11728213C2semidirect
8. #11727687C2semidirect
9. #11727680C2semidirect
10. #11727531C2semidirect
11. C4×#6395C2semidirect
19. #90254C4semidirect
100. C2×C8×D4D7semidirect
498. C7#22489semidirect

Maximal subgroups

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

subgroupindex
C2×#902542
C2×C7×C8×D42
#117397822
#117297272
#117297232
#117282332
#117282132
#117276872
#117276802
#117275312
C4×#63952
#224897