Finite Group Databasebeta

← #11740165

Subgroups — #11740165

Order 896.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#121C211yesmaximal, characteristic, complemented
2C2×#6500C211yesmaximal, characteristic, complemented
3C2×#6500C211yesmaximal, characteristic, complemented
4C2×#6492C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89994C211yesmaximal, characteristic, complemented
8#89993C211yesmaximal, characteristic, complemented
9#89985C211yesmaximal, characteristic, complemented
10#89766C211yesmaximal, characteristic, complemented
11#89762C211yesmaximal, characteristic, complemented
12#89733C211yesmaximal, characteristic, complemented
13#89729C211yesmaximal, characteristic, complemented
14#89727C211yesmaximal, characteristic, complemented
15#89724C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C7×D4C2211yescharacteristic, complemented
20C23×C4×C7C2211yescharacteristic, complemented
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic
23C22×C22⋊D7C2211yescharacteristic
24C22×C22⋊D7C2211yescharacteristic
25C7×#121—24no—
26C2×C7×C4⋊C4C2211yescharacteristic, complemented
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic, complemented
33C2×C22⋊Dic7C2211yescharacteristic, complemented
34C2×C22⋊Dic7C2211yescharacteristic, complemented
35C2×C22⋊Dic7C2211yescharacteristic, complemented
36#6500—24no—
37#6500—24no—
38#6492—24no—
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×(C2×D7)⋊C4C2211yescharacteristic
43C2×(C2×D7)⋊C4C2211yescharacteristic
44C2×(C2×D7)⋊C4C2211yescharacteristic
45C2×(C2×D7)⋊C4C2211yescharacteristic
46C2×(C2×D7)⋊C4C2211yescharacteristic
47C2×C4⋊Dic7C2211yescharacteristic, complemented
48C2×Dic7⋊C4C2211yescharacteristic, complemented
49C2×Dic7⋊C4C2211yescharacteristic, complemented
50C2×C4×Dic7C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2678—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C24×C7C2311yescharacteristic
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—41no—
63C23×D7—41no—
64C23×D7—41no—
65C23×D7—41no—
66C23×D7C2311yescharacteristic
67C23×D7C2311yescharacteristic
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—22no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4D414yescomplemented
77C22×C4×C7—22no—
78C22×C4×C7—22no—
79C22×C4×C7—41no—
80C22×C4×C7C2311yescharacteristic
81C22×C4×C7C2311yescharacteristic
82C22×C4×C7C2311yescharacteristic
83C22×C4×C7C2311yescharacteristic
84C2×C22⋊D7—22no—
85C2×C22⋊D7—22no—
86C2×C22⋊D7—22no—
87C2×C22⋊D7—22no—
88C2×C22⋊D7—22no—
89C2×C22⋊D7—22no—
90C2×C22⋊D7—22no—
91C2×C22⋊D7—41no—
92C2×C22⋊D7—41no—
93C2×C22⋊D7—41no—
94C2×C22⋊D7—41no—
95C2×C22⋊D7—41no—
96C22×Dic7C2311yescharacteristic
97C22×Dic7C2311yescharacteristic
98C22×Dic7C2311yescharacteristic
99C22×Dic7C2311yescharacteristic
100C22×Dic7C2311yescharacteristic
101C22×Dic7C2311yescharacteristic
102C7×C4⋊C4—22no—
103C7×C22⋊C4—22no—
104C7×C22⋊C4—22no—
105C22⋊Dic7—22no—
106C22⋊Dic7—22no—
107C22⋊Dic7—22no—
108C22⋊Dic7—41no—
109C22⋊Dic7—41no—
110C22⋊Dic7—41no—
111C22⋊Dic7—41no—
112(C2×D7)⋊C4—22no—
113(C2×D7)⋊C4—22no—
114(C2×D7)⋊C4—22no—
115(C2×D7)⋊C4—22no—
116(C2×D7)⋊C4—41no—
117(C2×D7)⋊C4—41no—
118(C2×D7)⋊C4—41no—
119(C2×D7)⋊C4D414yescomplemented
120C4⋊Dic7—41no—
121Dic7⋊C4—22no—
122Dic7⋊C4—22no—
123C4×Dic7—41no—
124C2×#123—71no—
125C2×#121D711yescharacteristic, complemented
126C2×#120—71no—
127C2×#120—71no—
128#397—71no—
129#396—71no—
130#394—71no—
131#394—71no—
132#392—71no—
133#392—71no—
134#390—71no—
135#390—71no—
136#388—71no—
137#386—71no—
138#379—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7C2×D412yes—
159C23×C7C2×D412yes—
160C23×C7C2411yescharacteristic
161C22×D7—21no—
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—21no—
167C22×D7—21no—
168C22×D7—21no—
169C22×D7—21no—
170C22×D7—21no—
171C22×D7—21no—
172C22×D7—21no—
173C22×D7—44no—
174C22×D7—44no—
175C22×D7—44no—
176C22×D7—44no—
177C22×D7—44no—
178C22×D7—44no—
179C22×D7—44no—
180C22×D7C2×D412yes—
181C22×D7#4112yes—
182C7×D4—44no—
183C7×D4—44no—
184C7×D4—44no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—44no—
196C2×C4×C7C2×D412yes—
197C2×C4×C7C2×D412yes—
198C22⋊D7—44no—
199C22⋊D7—44no—
200C22⋊D7—44no—
201C22⋊D7—44no—
202C2×Dic7—21no—
203C2×Dic7—21no—
204C2×Dic7—21no—
205C2×Dic7—21no—
206C2×Dic7—21no—
207C2×Dic7—21no—
208C2×Dic7—21no—
209C2×Dic7—21no—
210C2×Dic7—21no—
211C2×Dic7—21no—
212C2×Dic7—21no—
213C2×Dic7—21no—
214C2×Dic7—21no—
215C2×Dic7—21no—
216C2×Dic7—21no—
217C2×Dic7—21no—
218C2×Dic7—21no—
219C2×Dic7C2×D412yes—
220C25—71no—
221C22×D4—71no—
222C22×D4—71no—
223C22×D4—71no—
224C22×D4—71no—
225C22×D4C2×D711yescharacteristic
226C22×D4C2×D711yescharacteristic
227C22×D4C2×D711yescharacteristic, complemented
228C23×C4C2×D711yescharacteristic, complemented
229#123—144no—
230#121—24no—
231#120—144no—
232#120—144no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C4⋊C4—71no—
236C2×C4⋊C4C2×D711yescharacteristic, complemented
237C2×C22⋊C4—71no—
238C2×C22⋊C4—71no—
239C2×C22⋊C4—71no—
240C2×C22⋊C4—71no—
241C2×C22⋊C4—71no—
242C2×C22⋊C4—71no—
243C2×C22⋊C4—71no—
244C2×C22⋊C4—71no—
245C2×C22⋊C4—71no—
246C2×C22⋊C4—71no—
247C2×C22⋊C4—71no—
248C2×C22⋊C4—71no—
249C2×C22⋊C4—71no—
250C2×C22⋊C4—71no—
251C2×C22⋊C4—71no—
252C2×C22⋊C4C2×D711yescharacteristic
253C2×C22⋊C4C2×D711yescharacteristic, complemented
254C2×C42—71no—
255#95—71no—
256#95—71no—
257#95—71no—
258#95—71no—
259C22×C7—22no—
260C22×C7—22no—
261C22×C7—22no—
262C22×C7—22no—
263C22×C7—22no—
264C22×C7—22no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7—41no—
268C22×C7—41no—
269C22×C7—41no—
270C22×C7—41no—
271C22×C7—41no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7—41no—
275C22×C7—41no—
276C22×C7—41no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7C22×D411yescharacteristic
281C22×C7C22×D411yescharacteristic
282C22×C7C2×#4111yescharacteristic
283C22×C7H(5,𝔽2)11yescharacteristic
284C22×C7H(5,𝔽2)11yescharacteristic
285C22×C7H(5,𝔽2)11yescharacteristic
286C22×C7H(5,𝔽2)11yescharacteristic
287C2×D7—22no—
288C2×D7—22no—
289C2×D7—22no—
290C2×D7—22no—
291C2×D7—22no—
292C2×D7—22no—
293C2×D7—416no—
294C2×D7—41no—
295C2×D7—41no—
296C2×D7—41no—
297C2×D7—41no—
298C2×D7—41no—
299C2×D7—41no—
300C2×D7—41no—
301C2×D7—41no—
302C4×C7—22no—
303C4×C7—22no—
304C4×C7—41no—
305C4×C7—41no—
306Dic7—22no—
307Dic7—41no—
308Dic7—41no—
309Dic7—41no—
310Dic7—41no—
311Dic7—41no—
312C24—71no—
313C24—71no—
314C24—142no—
315C24—142no—
316C24—142no—
317C24—281no—
318C24—281no—
319C24—281no—
320C24—281no—
321C24C22×D711yescharacteristic
322C24C22×D711yescharacteristic
323C24C22×D711yescharacteristic
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—142no—
333C2×D4—142no—
334C2×D4—142no—
335C2×D4—142no—
336C2×D4—142no—
337C2×D4—142no—
338C2×D4—142no—
339C2×D4—281no—
340C2×D4—281no—
341C2×D4—281no—
342C2×D4—281no—
343C2×D4—281no—
344C2×D4C22⋊D714yescomplemented
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—71no—
349C22×C4—71no—
350C22×C4—71no—
351C22×C4—71no—
352C22×C4—71no—
353C22×C4—71no—
354C22×C4C22×D711yescharacteristic
355C22×C4C22×D711yescharacteristic
356C22×C4C22×D711yescharacteristic
357C22×C4C22×D711yescharacteristic
358C4⋊C4—22no—
359C4⋊C4—142no—
360C4⋊C4—142no—
361C4⋊C4—281no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—74no—
365C22⋊C4—142no—
366C22⋊C4—142no—
367C22⋊C4—142no—
368C22⋊C4—142no—
369C22⋊C4—142no—
370C22⋊C4—142no—
371C22⋊C4—142no—
372C22⋊C4—281no—
373C22⋊C4—281no—
374C22⋊C4—281no—
375C22⋊C4—281no—
376C22⋊C4—281no—
377C22⋊C4—281no—
378C22⋊C4—281no—
379C42—281no—
380C2×C7—42no—
381C2×C7—42no—
382C2×C7—81no—
383C2×C7#53411yescharacteristic
384C2×C7#53511yescharacteristic
385C2×C7#53811yescharacteristic
386C2×C7D4211yescharacteristic
387C2×C7#54611yescharacteristic
388C2×C7#54611yescharacteristic
389C2×C7#56011yescharacteristic
390D7—42no—
391D7—42no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—72no—
412C23—72no—
413C23—141no—
414C23—141no—
415C23—141no—
416C23—141no—
417C23—141no—
418C23—141no—
419C23—141no—
420C23—141no—
421C23—141no—
422C23—141no—
423C23—141no—
424C23—141no—
425C23—284no—
426C23—284no—
427C23—284no—
428C23—284no—
429C23—284no—
430C23—284no—
431C23—284no—
432C23C2×C22⋊D712yes—
433C23C2×C22⋊D712yes—
434C23C23×D711yescharacteristic
435D4—44no—
436D4—44no—
437D4—44no—
438D4—284no—
439D4—284no—
440D4—284no—
441D4—284no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—72no—
454C2×C4—141no—
455C2×C4—141no—
456C2×C4—141no—
457C2×C4—141no—
458C2×C4—141no—
459C2×C4—141no—
460C2×C4—141no—
461C2×C4—141no—
462C2×C4—141no—
463C2×C4—141no—
464C2×C4—141no—
465C2×C4—141no—
466C2×C4—141no—
467C2×C4—141no—
468C2×C4—141no—
469C2×C4—141no—
470C2×C4—141no—
471C2×C4D4×D712yes—
472C2×C4C2×C22⋊D712yes—
473C7#267811yescharacteristic, complemented, Sylow
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—142no—
496C22—142no—
497C22—142no—
498C22—142no—
499C22—142no—
500C22—142no—
501C22—2816no—
502C22—281no—
503C22—281no—
504C22—281no—
505C22—281no—
506C22—281no—
507C22—281no—
508C22—281no—
509C22—281no—
510C22C2×D4×D711yescharacteristic
511C22#654811yescharacteristic
512C22#654811yescharacteristic
513C22#654811yescharacteristic
514C22D7×#4111yescharacteristic
515C22#655311yescharacteristic
516C22C22×C22⋊D711yescharacteristic
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—142no—
522C4—281no—
523C4—281no—
524C4—281no—
525C4—281no—
526C4—281no—
527C2—42no—
528C2—42no—
529C2—81no—
530C2—282no—
531C2—282no—
532C2#9029811yescharacteristic
533C2#9030311yescharacteristic
534C2#9030411yescharacteristic
535C2#9031111yescharacteristic
536C2D4×C22⋊D711yescharacteristic
537C2#9049511yescharacteristic
538C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2678
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C7×#121C2semidirect
2. C2×#6500C2semidirect
3. C2×#6500C2semidirect
4. C2×#6492C2semidirect
5. #89995C2semidirect
6. #89995C2semidirect
7. #89994C2semidirect
8. #89993C2semidirect
9. #89985C2semidirect
10. #89766C2semidirect
11. #89762C2semidirect
12. #89733C2semidirect
13. #89729C2semidirect
14. #89727C2semidirect
15. #89724C2semidirect
19. C22×C7×D4C22semidirect
20. C23×C4×C7C22semidirect
26. C2×C7×C4⋊C4C22semidirect
28. C2×C7×C22⋊C4C22semidirect
32. C2×C22⋊Dic7C22semidirect
33. C2×C22⋊Dic7C22semidirect
34. C2×C22⋊Dic7C22semidirect
35. C2×C22⋊Dic7C22semidirect
47. C2×C4⋊Dic7C22semidirect
48. C2×Dic7⋊C4C22semidirect
49. C2×Dic7⋊C4C22semidirect
50. C2×C4×Dic7C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
76. C2×C7×D4D4semidirect
119. (C2×D7)⋊C4D4semidirect
125. C2×#121D7semidirect
227. C22×D4C2×D7semidirect
228. C23×C4C2×D7semidirect
236. C2×C4⋊C4C2×D7semidirect
253. C2×C22⋊C4C2×D7semidirect
344. C2×D4C22⋊D7semidirect
473. C7#2678semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#65002
C2×#64922
#899952
#899942
#899932
#899852
#897662
#897622
#897332
#897292
#897272
#897242
#26787