Finite Group Databasebeta

← #11740448

Subgroups — #11740448

Order 896.

796 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×#124C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic, complemented
3C2×#6500C211yesmaximal, characteristic, complemented
4C2×#6445C211yesmaximal, characteristic, complemented
5#90003C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89733C211yesmaximal, characteristic, complemented
8#89732C211yesmaximal, characteristic, complemented
9#89732C211yesmaximal, characteristic, complemented
10#89731C211yesmaximal, characteristic, complemented
11#89729C211yesmaximal, characteristic, complemented
12#89727C211yesmaximal, characteristic, complemented
13#89723C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15#89711C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic, complemented
18C22×C7×D4C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic, complemented
23C22×D28C2211yescharacteristic
24C22×D28C2211yescharacteristic
25C7×#124—24no—
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic, complemented
29C2×C7×C22⋊C4C2211yescharacteristic, complemented
30C2×C42×C7C2211yescharacteristic, complemented
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic, complemented
34#6510—24no—
35#6500—24no—
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×(C2×D7)⋊C4C2211yescharacteristic
43C2×(C2×D7)⋊C4C2211yescharacteristic
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
47C2×C4⋊Dic7C2211yescharacteristic, complemented
48C2×Dic7⋊C4C2211yescharacteristic, complemented
49C2×C4×Dic7C2211yescharacteristic, complemented
50#6445—24no—
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2670—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—41no—
62C23×D7—41no—
63C23×D7—41no—
64C23×D7—41no—
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C23×D7C2311yescharacteristic
68C2×C7×Q8—22no—
69C2×C7×Q8—22no—
70C2×C7×Q8—41no—
71C2×C7×D4—22no—
72C2×C7×D4—22no—
73C2×C7×D4—41no—
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C22×C4×C7C2311yescharacteristic
78C22×C4×C7C2311yescharacteristic
79C2×C22⋊D7—22no—
80C2×C22⋊D7—22no—
81C2×C22⋊D7—22no—
82C2×C22⋊D7—22no—
83C2×C22⋊D7—41no—
84C2×C22⋊D7—41no—
85C2×C22⋊D7—41no—
86C2×C22⋊D7—41no—
87C2×C22⋊D7—41no—
88C2×C22⋊D7—41no—
89C2×C22⋊D7—41no—
90C2×C22⋊D7—41no—
91C22×Dic7C2311yescharacteristic
92C22×Dic7C2311yescharacteristic
93C22×Dic7C2311yescharacteristic
94C22×Dic7C2311yescharacteristic
95C22×Dic7C2311yescharacteristic
96C2×D28—22no—
97C2×D28—22no—
98C2×D28—22no—
99C2×D28—22no—
100C2×D28—41no—
101C2×D28—41no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—22no—
104C7×C22⋊C4—22no—
105C7×C22⋊C4D414yescomplemented
106C42×C7—22no—
107C22⋊Dic7—22no—
108C22⋊Dic7—41no—
109C22⋊Dic7—41no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—22no—
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)⋊C4—41no—
120(C2×D7)⋊C4D414yescomplemented
121C4⋊Dic7—41no—
122Dic7⋊C4—41no—
123C4×Dic7—22no—
124C2×#124—71no—
125C2×#124D711yescharacteristic, complemented
126C2×#120—71no—
127C2×#120—71no—
128#394—71no—
129#394—71no—
130#393—71no—
131#392—71no—
132#392—71no—
133#392—71no—
134#390—71no—
135#390—71no—
136#388—71no—
137#388—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×C7C2×D412yes—
153C23×C7C2411yescharacteristic
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
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×Q8—44no—
183C7×D4—44no—
184C2×C4×C7—21no—
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—21no—
196C2×C4×C7C2×D412yes—
197C2×C4×C7C2×D412yes—
198C2×C4×C7C2×D412yes—
199C22⋊D7—44no—
200C22⋊D7—44no—
201C22⋊D7—44no—
202C22⋊D7—44no—
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×Dic7C2×D412yes—
218D28—44no—
219D28—44no—
220C25—71no—
221C22×Q8C2×D711yescharacteristic, complemented
222C22×D4—71no—
223C22×D4—71no—
224C22×D4—71no—
225C22×D4—71no—
226C22×D4—71no—
227C22×D4—71no—
228C22×D4C2×D711yescharacteristic
229#124—24no—
230#124—144no—
231#120—144no—
232#120—144no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C22⋊C4—71no—
236C2×C22⋊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⋊C4C2×D711yescharacteristic
250C2×C22⋊C4C2×D711yescharacteristic
251C2×C22⋊C4C2×D711yescharacteristic, complemented
252C2×C22⋊C4C2×D711yescharacteristic, complemented
253C2×C42—71no—
254C2×C42C2×D711yescharacteristic, complemented
255#95—71no—
256#95—71no—
257#95—71no—
258#95—71no—
259C22×C7—22no—
260C22×C7—22no—
261C22×C7—22no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—41no—
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×C7C22×D411yescharacteristic
274C22×C7C22×D411yescharacteristic
275C22×C7C2×#4111yescharacteristic
276C22×C7H(5,𝔽2)11yescharacteristic
277C22×C7H(5,𝔽2)11yescharacteristic
278C22×C7H(5,𝔽2)11yescharacteristic
279C22×C7H(5,𝔽2)11yescharacteristic
280C2×D7—22no—
281C2×D7—22no—
282C2×D7—22no—
283C2×D7—22no—
284C2×D7—22no—
285C2×D7—22no—
286C2×D7—416no—
287C2×D7—41no—
288C2×D7—41no—
289C2×D7—41no—
290C2×D7—41no—
291C2×D7—41no—
292C2×D7—41no—
293C2×D7—41no—
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—22no—
305C4×C7—41no—
306C4×C7—41no—
307Dic7—22no—
308Dic7—41no—
309Dic7—41no—
310Dic7—41no—
311Dic7—41no—
312C24—71no—
313C24—71no—
314C24—71no—
315C24—142no—
316C24—142no—
317C24—142no—
318C24—281no—
319C24—281no—
320C24—281no—
321C24—281no—
322C24C22×D711yescharacteristic
323C24C22×D711yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—142no—
331C2×D4—142no—
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×D4—281no—
345C2×D4—281no—
346C2×D4—281no—
347C2×D4—281no—
348C22×C4—71no—
349C22×C4—71no—
350C22×C4—71no—
351C22×C4—71no—
352C22×C4—71no—
353C22×C4C22×D711yescharacteristic
354C22×C4C22×D711yescharacteristic
355C22×C4C22×D711yescharacteristic
356C22×C4C22×D711yescharacteristic
357C22×C4C22×D711yescharacteristic
358C4⋊C4—281no—
359C4⋊C4—281no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—74no—
364C22⋊C4—142no—
365C22⋊C4—142no—
366C22⋊C4—142no—
367C22⋊C4—142no—
368C22⋊C4—142no—
369C22⋊C4—142no—
370C22⋊C4—142no—
371C22⋊C4—281no—
372C22⋊C4—281no—
373C22⋊C4—281no—
374C22⋊C4—281no—
375C22⋊C4—281no—
376C22⋊C4—281no—
377C22⋊C4C22⋊D714yescomplemented
378C42—22no—
379C42—142no—
380C2×C7—42no—
381C2×C7—81no—
382C2×C7#53511yescharacteristic
383C2×C7#53511yescharacteristic
384C2×C7#53811yescharacteristic
385C2×C7D4211yescharacteristic
386C2×C7#54611yescharacteristic
387C2×C7#54611yescharacteristic
388C2×C7H(3,𝔽4)11yescharacteristic
389D7—42no—
390D7—42no—
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—72no—
406C23—72no—
407C23—141no—
408C23—141no—
409C23—141no—
410C23—141no—
411C23—141no—
412C23—141no—
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—141no—
426C23—284no—
427C23—284no—
428C23—284no—
429C23—284no—
430C23—284no—
431C23—284no—
432C23—284no—
433C23C2×C22⋊D712yes—
434C23C23×D711yescharacteristic
435Q8—44no—
436D4—44no—
437D4—284no—
438D4—284no—
439D4—284no—
440D4—284no—
441D4—284no—
442D4—284no—
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—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—72no—
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×C4D4×D712yes—
471C2×C4C2×C22⋊D712yes—
472C2×C4C2×C22⋊D712yes—
473C7#267011yescharacteristic, complemented, Sylow
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—142no—
489C22—142no—
490C22—142no—
491C22—142no—
492C22—142no—
493C22—142no—
494C22—2816no—
495C22—281no—
496C22—281no—
497C22—281no—
498C22—281no—
499C22—281no—
500C22—281no—
501C22—281no—
502C22—281no—
503C22—281no—
504C22—281no—
505C22—281no—
506C22—281no—
507C22—281no—
508C22—281no—
509C22—281no—
510C22C2×D4×D711yescharacteristic
511C22#654811yescharacteristic
512C22D7×#4111yescharacteristic
513C22#655311yescharacteristic
514C22#655311yescharacteristic
515C22#655311yescharacteristic
516C22C22×C22⋊D711yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—142no—
523C4—281no—
524C4—281no—
525C4—281no—
526C4—281no—
527C2—42no—
528C2—81no—
529C2—282no—
530C2—282no—
531C2—561no—
532C2#9036511yescharacteristic
533C2#9036711yescharacteristic
534C2#9036911yescharacteristic
535C2#9037411yescharacteristic
536C2D4×C22⋊D711yescharacteristic
537C2#9052011yescharacteristic
538C2#9052111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
hypercenterC2×#124D7
hypocenterC7#2670
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C7×#124C2semidirect
2. C2×#6510C2semidirect
3. C2×#6500C2semidirect
4. C2×#6445C2semidirect
5. #90003C2semidirect
6. #89995C2semidirect
7. #89733C2semidirect
8. #89732C2semidirect
9. #89732C2semidirect
10. #89731C2semidirect
11. #89729C2semidirect
12. #89727C2semidirect
13. #89723C2semidirect
14. #89712C2semidirect
15. #89711C2semidirect
17. C22×C7×Q8C22semidirect
22. C22×C22⋊D7C22semidirect
28. C2×C7×C22⋊C4C22semidirect
29. C2×C7×C22⋊C4C22semidirect
30. C2×C42×C7C22semidirect
33. C2×C22⋊Dic7C22semidirect
44. C2×(C2×D7)⋊C4C22semidirect
45. C2×(C2×D7)⋊C4C22semidirect
46. C2×(C2×D7)⋊C4C22semidirect
47. C2×C4⋊Dic7C22semidirect
48. C2×Dic7⋊C4C22semidirect
49. C2×C4×Dic7C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
105. C7×C22⋊C4D4semidirect
120. (C2×D7)⋊C4D4semidirect
125. C2×#124D7semidirect
221. C22×Q8C2×D7semidirect
251. C2×C22⋊C4C2×D7semidirect
252. C2×C22⋊C4C2×D7semidirect
254. C2×C42C2×D7semidirect
377. C22⋊C4C22⋊D7semidirect
473. C7#2670semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#65102
C2×#65002
C2×#64452
#900032
#899952
#897332
#897322
#897312
#897292
#897272
#897232
#897122
#897112
#26707