Finite Group Databasebeta

← #11740041

Subgroups — #11740041

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×#120C211yesmaximal, characteristic, complemented
2C2×#6516C211yesmaximal, characteristic, complemented
3C2×#6502C211yesmaximal, characteristic, complemented
4C2×#6498C211yesmaximal, characteristic, complemented
5#90019C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89995C211yesmaximal, characteristic, complemented
8#89993C211yesmaximal, characteristic, complemented
9#89992C211yesmaximal, characteristic, complemented
10#89733C211yesmaximal, characteristic, complemented
11#89731C211yesmaximal, characteristic, complemented
12#89729C211yesmaximal, characteristic, complemented
13#89724C211yesmaximal, characteristic, complemented
14#89720C211yesmaximal, characteristic, complemented
15#89717C211yesmaximal, characteristic, complemented
16C25×C7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C7×D4C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic
23C22×C22⋊D7C2211yescharacteristic, complemented
24C23×Dic7C2211yescharacteristic, complemented
25C7×#120—24no—
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic
29#6516—24no—
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C22⋊Dic7C2211yescharacteristic
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37C2×C22⋊Dic7C2211yescharacteristic
38C2×C22⋊Dic7C2211yescharacteristic
39C2×C22⋊Dic7C2211yescharacteristic, complemented
40C2×C22⋊Dic7C2211yescharacteristic, complemented
41#6502—24no—
42#6498—24no—
43C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×C4⋊Dic7C2211yescharacteristic, complemented
47C2×Dic7⋊C4C2211yescharacteristic, 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×C7—22no—
57C24×C7—22no—
58C24×C7—22no—
59C24×C7—41no—
60C24×C7—41no—
61C24×C7—41no—
62C24×C7—41no—
63C24×C7C2311yescharacteristic
64C24×C7C2311yescharacteristic
65C24×C7C2311yescharacteristic
66C24×C7C2311yescharacteristic
67C23×D7C2311yescharacteristic
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4D414yescomplemented
77C22×C4×C7C2311yescharacteristic
78C22×C4×C7C2311yescharacteristic
79C22×C4×C7C2311yescharacteristic
80C2×C22⋊D7—22no—
81C2×C22⋊D7—22no—
82C2×C22⋊D7—22no—
83C2×C22⋊D7—22no—
84C2×C22⋊D7—22no—
85C2×C22⋊D7—22no—
86C2×C22⋊D7—41no—
87C2×C22⋊D7—41no—
88C2×C22⋊D7—41no—
89C2×C22⋊D7—41no—
90C2×C22⋊D7—41no—
91C2×C22⋊D7—41no—
92C22×Dic7—22no—
93C22×Dic7—22no—
94C22×Dic7—41no—
95C22×Dic7C2311yescharacteristic
96C22×Dic7C2311yescharacteristic
97C22×Dic7C2311yescharacteristic
98C22×Dic7C2311yescharacteristic
99C22×Dic7C2311yescharacteristic
100C22×Dic7C2311yescharacteristic
101C22×Dic7C2311yescharacteristic
102C7×C22⋊C4—22no—
103C7×C22⋊C4—22no—
104C7×C22⋊C4—22no—
105C22⋊Dic7—22no—
106C22⋊Dic7—22no—
107C22⋊Dic7—22no—
108C22⋊Dic7—22no—
109C22⋊Dic7—22no—
110C22⋊Dic7—41no—
111C22⋊Dic7—41no—
112C22⋊Dic7—41no—
113C22⋊Dic7—41no—
114C22⋊Dic7—41no—
115C22⋊Dic7D414yescomplemented
116(C2×D7)⋊C4—22no—
117(C2×D7)⋊C4—41no—
118(C2×D7)⋊C4—41no—
119C4⋊Dic7—41no—
120Dic7⋊C4—22no—
121Dic7⋊C4—22no—
122Dic7⋊C4—22no—
123C4×Dic7—41no—
124C2×#123—71no—
125C2×#121—71no—
126C2×#120—71no—
127C2×#120D711yescharacteristic, complemented
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×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—21no—
163C23×C7—44no—
164C23×C7—44no—
165C23×C7—44no—
166C23×C7—44no—
167C23×C7—44no—
168C23×C7—44no—
169C23×C7—44no—
170C23×C7C2×D412yes—
171C23×C7C2×D412yes—
172C23×C7C2×D412yes—
173C23×C7#4112yes—
174C23×C7C2411yescharacteristic
175C22×D7—21no—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—21no—
179C22×D7—21no—
180C22×D7—21no—
181C22×D7—21no—
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×C7C2×D412yes—
194C22⋊D7—44no—
195C22⋊D7—44no—
196C22⋊D7—44no—
197C22⋊D7—44no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
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—44no—
218C2×Dic7C2×D412yes—
219C2×Dic7C2×D412yes—
220C25C2×D711yescharacteristic
221C22×D4—71no—
222C22×D4—71no—
223C22×D4—71no—
224C22×D4—71no—
225C22×D4C2×D711yescharacteristic
226C22×D4C2×D711yescharacteristic
227C22×D4C2×D711yescharacteristic
228C23×C4—71no—
229#123—144no—
230#121—144no—
231#120—24no—
232#120—144no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C4⋊C4—71no—
236C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
252C2×C22⋊C4C2×D711yescharacteristic
253C2×C22⋊C4C2×D711yescharacteristic
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—22no—
266C22×C7—22no—
267C22×C7—22no—
268C22×C7—22no—
269C22×C7—22no—
270C22×C7—22no—
271C22×C7—416no—
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×C7—41no—
281C22×C7—41no—
282C22×C7—41no—
283C22×C7—41no—
284C22×C7—41no—
285C22×C7—41no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7C22×D411yescharacteristic
289C22×C7C22×D411yescharacteristic
290C22×C7C2×#4111yescharacteristic
291C22×C7H(5,𝔽2)11yescharacteristic
292C22×C7H(5,𝔽2)11yescharacteristic
293C22×C7H(5,𝔽2)11yescharacteristic
294C22×C7H(5,𝔽2)11yescharacteristic
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—41no—
304C4×C7—41no—
305Dic7—22no—
306Dic7—22no—
307Dic7—41no—
308Dic7—41no—
309Dic7—41no—
310Dic7—41no—
311Dic7—41no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—71no—
320C24C22×D711yescharacteristic
321C24C22×D711yescharacteristic
322C24C22×D711yescharacteristic
323C24C22×D711yescharacteristic
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
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—281no—
339C2×D4—281no—
340C2×D4—281no—
341C2×D4—281no—
342C2×D4—281no—
343C2×D4—281no—
344C2×D4C22⋊D714yescomplemented
345C22×C4—71no—
346C22×C4—71no—
347C22×C4—71no—
348C22×C4—71no—
349C22×C4—71no—
350C22×C4—71no—
351C22×C4—71no—
352C22×C4—142no—
353C22×C4—142no—
354C22×C4—281no—
355C22×C4C22×D711yescharacteristic
356C22×C4C22×D711yescharacteristic
357C22×C4C22×D711yescharacteristic
358C4⋊C4—142no—
359C4⋊C4—142no—
360C4⋊C4—142no—
361C4⋊C4—281no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—74no—
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—42no—
383C2×C7—42no—
384C2×C7#53411yescharacteristic
385C2×C7#53511yescharacteristic
386C2×C7#53811yescharacteristic
387C2×C7D4211yescharacteristic
388C2×C7#54611yescharacteristic
389C2×C7#54611yescharacteristic
390C2×C7#56011yescharacteristic
391D7—81no—
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—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—141no—
424C23—141no—
425C23—141no—
426C23—141no—
427C23—141no—
428C23—141no—
429C23—141no—
430C23D4×D712yes—
431C23#115912yes—
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—72no—
451C2×C4—72no—
452C2×C4—141no—
453C2×C4—141no—
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×C4—284no—
472C2×C4C2×C22⋊D712yes—
473C7#267811yescharacteristic, complemented, Sylow
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—416no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—281no—
504C22—281no—
505C22—281no—
506C22—281no—
507C22—281no—
508C22—281no—
509C22—281no—
510C22C2×D4×D711yescharacteristic
511C22C2×#115911yescharacteristic
512C22#654811yescharacteristic
513C22#654811yescharacteristic
514C22#654811yescharacteristic
515C22#654811yescharacteristic
516C22C22×C22⋊D711yescharacteristic
517C4—22no—
518C4—41no—
519C4—41no—
520C4—142no—
521C4—142no—
522C4—281no—
523C4—281no—
524C4—281no—
525C4—281no—
526C4—281no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—42no—
531C2—561no—
532C2#9028211yescharacteristic
533C2#9028311yescharacteristic
534C2#9028411yescharacteristic
535C2#9028611yescharacteristic
536C2D4×C22⋊D711yescharacteristic
537C2#9049511yescharacteristic
538C2#9049711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#120C2
hypercenterC2×#120D7
hypocenterC7#2678
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×C7×#120C2semidirect
2. C2×#6516C2semidirect
3. C2×#6502C2semidirect
4. C2×#6498C2semidirect
5. #90019C2semidirect
6. #89995C2semidirect
7. #89995C2semidirect
8. #89993C2semidirect
9. #89992C2semidirect
10. #89733C2semidirect
11. #89731C2semidirect
12. #89729C2semidirect
13. #89724C2semidirect
14. #89720C2semidirect
15. #89717C2semidirect
23. C22×C22⋊D7C22semidirect
24. C23×Dic7C22semidirect
39. C2×C22⋊Dic7C22semidirect
40. C2×C22⋊Dic7C22semidirect
43. C2×(C2×D7)⋊C4C22semidirect
44. C2×(C2×D7)⋊C4C22semidirect
45. C2×(C2×D7)⋊C4C22semidirect
46. C2×C4⋊Dic7C22semidirect
47. C2×Dic7⋊C4C22semidirect
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
115. C22⋊Dic7D4semidirect
127. C2×#120D7semidirect
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×#1202
C2×#65162
C2×#65022
C2×#64982
#900192
#899952
#899932
#899922
#897332
#897312
#897292
#897242
#897202
#897172
#26787