Finite Group Databasebeta

← #11739852

Subgroups — #11739852

Order 896.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×D4C211yesmaximal, characteristic, complemented
2C2×#6502C211yesmaximal, characteristic, complemented
3C2×#6500C211yesmaximal, characteristic, complemented
4C2×#6493C211yesmaximal, characteristic, complemented
5C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
6C2×#6445C211yesmaximal, characteristic, complemented
7#89985C211yesmaximal, characteristic, complemented
8#89985C211yesmaximal, characteristic, complemented
9#89766C211yesmaximal, characteristic, complemented
10#89763C211yesmaximal, characteristic, complemented
11#89733C211yesmaximal, characteristic, complemented
12#89732C211yesmaximal, characteristic, complemented
13#89722C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15#89711C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C23×C4×C7C2211yescharacteristic, complemented
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic
23C22×C22⋊D7C2211yescharacteristic, complemented
24C23×Dic7C2211yescharacteristic
25C22×D28C2211yescharacteristic
26C22×D28C2211yescharacteristic
27C4×C7×D4—24no—
28C2×C7×C4⋊C4C2211yescharacteristic, complemented
29C2×C7×C22⋊C4C2211yescharacteristic
30C2×C7×C22⋊C4C2211yescharacteristic, complemented
31C2×C42×C7C2211yescharacteristic, complemented
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic, complemented
34#6502—24no—
35#6500—24no—
36#6493—24no—
37C2×(C2×D7)⋊C4—24no—
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
45C2×(C2×D7)⋊C4C2211yescharacteristic
46C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
48C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
49C2×C4⋊Dic7C2211yescharacteristic, complemented
50C2×C4⋊Dic7C2211yescharacteristic, complemented
51C2×Dic7⋊C4C2211yescharacteristic, complemented
52#6445—24no—
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55C7⋊#95C2211yescharacteristic, complemented
56C7⋊#95C2211yescharacteristic, complemented
57#2424—71nomaximal, Sylow
58C24×C7C2311yescharacteristic
59C24×C7C2311yescharacteristic
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—22no—
63C23×D7—22no—
64C23×D7—22no—
65C23×D7—41no—
66C23×D7—41no—
67C23×D7C2311yescharacteristic
68C23×D7C2311yescharacteristic
69C23×D7C2311yescharacteristic
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4D414yescomplemented
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—41no—
78C22×C4×C7—41no—
79C22×C4×C7C2311yescharacteristic
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—22no—
92C2×C22⋊D7—22no—
93C2×C22⋊D7—41no—
94C2×C22⋊D7—41no—
95C2×C22⋊D7—41no—
96C22×Dic7—22no—
97C22×Dic7—22no—
98C22×Dic7—41no—
99C22×Dic7C2311yescharacteristic
100C22×Dic7C2311yescharacteristic
101C22×Dic7C2311yescharacteristic
102C22×Dic7C2311yescharacteristic
103C22×Dic7C2311yescharacteristic
104C2×D28—22no—
105C2×D28—22no—
106C2×D28—22no—
107C2×D28—22no—
108C2×D28—41no—
109C2×D28—41no—
110C7×C4⋊C4—22no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4D414yescomplemented
113C42×C7—22no—
114C22⋊Dic7—22no—
115C22⋊Dic7—22no—
116(C2×D7)⋊C4—22no—
117(C2×D7)⋊C4—22no—
118(C2×D7)⋊C4—22no—
119(C2×D7)⋊C4—22no—
120(C2×D7)⋊C4—22no—
121(C2×D7)⋊C4—22no—
122(C2×D7)⋊C4—22no—
123(C2×D7)⋊C4—22no—
124(C2×D7)⋊C4—22no—
125(C2×D7)⋊C4—41no—
126(C2×D7)⋊C4D414yescomplemented
127C4⋊Dic7—22no—
128C4⋊Dic7—41no—
129Dic7⋊C4—22no—
130C2×#121—71no—
131C2×#121—71no—
132C2×#120—71no—
133C2×#120—71no—
134C2×C4×D4D711yescharacteristic, complemented
135C22×C22⋊C4—71no—
136#396—71no—
137#394—71no—
138#394—71no—
139#392—71no—
140#390—71no—
141#388—71no—
142#386—71no—
143#386—71no—
144#385—71no—
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×C7C2×D412yes—
156C23×C7C2×D412yes—
157C23×C7C2×D412yes—
158C23×C7#4112yes—
159C23×C7C2411yescharacteristic
160C22×D7—28no—
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—21no—
174C22×D7—21no—
175C22×D7—21no—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—21no—
179C22×D7—21no—
180C22×D7—44no—
181C22×D7—44no—
182C22×D7—44no—
183C22×D7—44no—
184C22×D7—44no—
185C22×D7—44no—
186C22×D7C2×D412yes—
187C22×D7#4112yes—
188C7×D4—28no—
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×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—44no—
201C2×C4×C7—44no—
202C2×C4×C7C2×D412yes—
203C2×C4×C7C2×D412yes—
204C2×C4×C7C2×D412yes—
205C2×C4×C7C2×D412yes—
206C22⋊D7—44no—
207C22⋊D7—44no—
208C22⋊D7—44no—
209C22⋊D7—44no—
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×Dic7—21no—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7—21no—
224C2×Dic7—44no—
225C2×Dic7C2×D412yes—
226D28—44no—
227D28—44no—
228C25—71no—
229C22×D4—71no—
230C22×D4—71no—
231C22×D4—71no—
232C22×D4—71no—
233C22×D4—71no—
234C22×D4—71no—
235C22×D4C2×D711yescharacteristic
236C23×C4—71no—
237C23×C4C2×D711yescharacteristic
238C23×C4C2×D711yescharacteristic, complemented
239#121—144no—
240#121—144no—
241#120—144no—
242#120—144no—
243C4×D4—24no—
244C2×C4⋊C4—71no—
245C2×C4⋊C4—71no—
246C2×C4⋊C4—71no—
247C2×C4⋊C4C2×D711yescharacteristic, complemented
248C2×C22⋊C4—71no—
249C2×C22⋊C4—71no—
250C2×C22⋊C4—71no—
251C2×C22⋊C4—71no—
252C2×C22⋊C4—71no—
253C2×C22⋊C4—71no—
254C2×C22⋊C4—71no—
255C2×C22⋊C4—71no—
256C2×C22⋊C4—71no—
257C2×C22⋊C4—71no—
258C2×C22⋊C4—71no—
259C2×C22⋊C4—71no—
260C2×C22⋊C4—71no—
261C2×C22⋊C4—144no—
262C2×C22⋊C4C2×D711yescharacteristic
263C2×C22⋊C4C2×D711yescharacteristic, complemented
264C2×C42C2×D711yescharacteristic, complemented
265#95—71no—
266#95—71no—
267#95—71no—
268#95—71no—
269C22×C7—22no—
270C22×C7—22no—
271C22×C7—22no—
272C22×C7—22no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7—22no—
276C22×C7—22no—
277C22×C7—22no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7—41no—
281C22×C7—41no—
282C22×C7#12114yescomplemented
283C22×C7C22×D411yescharacteristic
284C22×C7C22×D411yescharacteristic
285C22×C7C22×D411yescharacteristic
286C22×C7C2×#4111yescharacteristic
287C22×C7C2×#4111yescharacteristic
288C22×C7H(5,𝔽2)11yescharacteristic
289C22×C7H(5,𝔽2)11yescharacteristic
290C2×D7—22no—
291C2×D7—22no—
292C2×D7—22no—
293C2×D7—22no—
294C2×D7—22no—
295C2×D7—22no—
296C2×D7—416no—
297C2×D7—41no—
298C2×D7—41no—
299C2×D7—41no—
300C2×D7—41no—
301C2×D7—41no—
302C2×D7—41no—
303C2×D7—41no—
304C2×D7—41no—
305C2×D7—41no—
306C2×D7—41no—
307C2×D7—41no—
308C2×D7—41no—
309C2×D7—41no—
310C2×D7—41no—
311C2×D7—41no—
312C4×C7—22no—
313C4×C7—22no—
314C4×C7—22no—
315C4×C7—22no—
316C4×C7—41no—
317Dic7—22no—
318Dic7—41no—
319Dic7—41no—
320Dic7—41no—
321Dic7—41no—
322C24—71no—
323C24—71no—
324C24—71no—
325C24—142no—
326C24—142no—
327C24—142no—
328C24—142no—
329C24—142no—
330C24—281no—
331C24—281no—
332C24C22×D711yescharacteristic
333C24C22×D711yescharacteristic
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—142no—
337C2×D4—142no—
338C2×D4—142no—
339C2×D4—142no—
340C2×D4—142no—
341C2×D4—142no—
342C2×D4—142no—
343C2×D4—142no—
344C2×D4—142no—
345C2×D4—142no—
346C2×D4—142no—
347C2×D4—142no—
348C2×D4—142no—
349C2×D4—281no—
350C2×D4—281no—
351C2×D4—281no—
352C2×D4—281no—
353C2×D4—281no—
354C2×D4D2814yescomplemented
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—71no—
362C22×C4—71no—
363C22×C4—71no—
364C22×C4—71no—
365C22×C4—71no—
366C22×C4—142no—
367C22×C4—142no—
368C22×C4—281no—
369C22×C4C22×D711yescharacteristic
370C22×C4C22×D711yescharacteristic
371C22×C4C22×D711yescharacteristic
372C22×C4C22×D711yescharacteristic
373C22×C4C22×D711yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—142no—
376C4⋊C4—142no—
377C4⋊C4—281no—
378C22⋊C4—22no—
379C22⋊C4—74no—
380C22⋊C4—142no—
381C22⋊C4—142no—
382C22⋊C4—142no—
383C22⋊C4—142no—
384C22⋊C4—142no—
385C22⋊C4—142no—
386C22⋊C4—142no—
387C22⋊C4—142no—
388C22⋊C4—142no—
389C22⋊C4—142no—
390C22⋊C4—142no—
391C22⋊C4—281no—
392C22⋊C4C22⋊D714yescomplemented
393C42—22no—
394C2×C7—24no—
395C2×C7—42no—
396C2×C7C2×#12111yescharacteristic
397C2×C7#52511yescharacteristic
398C2×C7#53811yescharacteristic
399C2×C7D4211yescharacteristic
400C2×C7D4211yescharacteristic
401C2×C7#54611yescharacteristic
402C2×C7#54611yescharacteristic
403D7—42no—
404D7—42no—
405D7—81no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—72no—
417C23—72no—
418C23—148no—
419C23—141no—
420C23—141no—
421C23—141no—
422C23—141no—
423C23—141no—
424C23—141no—
425C23—141no—
426C23—141no—
427C23—141no—
428C23—141no—
429C23—141no—
430C23—141no—
431C23—141no—
432C23—141no—
433C23—141no—
434C23—141no—
435C23—141no—
436C23—141no—
437C23—141no—
438C23—284no—
439C23—284no—
440C23—284no—
441C23—284no—
442C23—284no—
443C23—284no—
444C23C2×D2812yes—
445C23C2×D2812yes—
446C23#115712yes—
447C23C2×C22⋊D712yes—
448C23C23×D711yescharacteristic
449D4—28no—
450D4—284no—
451D4—284no—
452D4—284no—
453D4—284no—
454D4—284no—
455D4—284no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—72no—
470C2×C4—141no—
471C2×C4—141no—
472C2×C4—141no—
473C2×C4—141no—
474C2×C4—141no—
475C2×C4—141no—
476C2×C4—141no—
477C2×C4—141no—
478C2×C4—141no—
479C2×C4—141no—
480C2×C4—141no—
481C2×C4—141no—
482C2×C4—141no—
483C2×C4—141no—
484C2×C4—284no—
485C2×C4C2×D2812yes—
486C2×C4D4×D712yes—
487C2×C4C2×C22⋊D712yes—
488C2×C4C2×C22⋊D712yes—
489C7#242411yescharacteristic, complemented, Sylow
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—142no—
504C22—142no—
505C22—142no—
506C22—142no—
507C22—142no—
508C22—142no—
509C22—2816no—
510C22—281no—
511C22—281no—
512C22—281no—
513C22—281no—
514C22—281no—
515C22—281no—
516C22—281no—
517C22—281no—
518C22—281no—
519C22—281no—
520C22—281no—
521C22—281no—
522C22—281no—
523C22—281no—
524C22—281no—
525C22#649314yescomplemented
526C22C22×D2811yescharacteristic
527C22C2×#115711yescharacteristic
528C22C2×D4×D711yescharacteristic
529C22#654811yescharacteristic
530C22D7×#4111yescharacteristic
531C22#655311yescharacteristic
532C22C22×C22⋊D711yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—142no—
539C4—281no—
540C4—281no—
541C4—281no—
542C4—281no—
543C2—24no—
544C2—42no—
545C2—282no—
546C2—282no—
547C2—561no—
548C2#9023811yescharacteristic
549C2D4×D2811yescharacteristic
550C2#9024511yescharacteristic
551C2#9025111yescharacteristic
552C2C2×#649311yescharacteristic
553C2D4×C22⋊D711yescharacteristic
554C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
hypercenterC2×C4×D4D7
hypocenterC7#2424
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×D4C2semidirect
2. C2×#6502C2semidirect
3. C2×#6500C2semidirect
4. C2×#6493C2semidirect
5. C22×(C2×D7)⋊C4C2semidirect
6. C2×#6445C2semidirect
7. #89985C2semidirect
8. #89985C2semidirect
9. #89766C2semidirect
10. #89763C2semidirect
11. #89733C2semidirect
12. #89732C2semidirect
13. #89722C2semidirect
14. #89712C2semidirect
15. #89711C2semidirect
19. C23×C4×C7C22semidirect
23. C22×C22⋊D7C22semidirect
28. C2×C7×C4⋊C4C22semidirect
30. C2×C7×C22⋊C4C22semidirect
31. C2×C42×C7C22semidirect
33. C2×C22⋊Dic7C22semidirect
46. C2×(C2×D7)⋊C4C22semidirect
47. C2×(C2×D7)⋊C4C22semidirect
48. C2×(C2×D7)⋊C4C22semidirect
49. C2×C4⋊Dic7C22semidirect
50. C2×C4⋊Dic7C22semidirect
51. C2×Dic7⋊C4C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
55. C7⋊#95C22semidirect
56. C7⋊#95C22semidirect
72. C2×C7×D4D4semidirect
112. C7×C22⋊C4D4semidirect
126. (C2×D7)⋊C4D4semidirect
134. C2×C4×D4D7semidirect
238. C23×C4C2×D7semidirect
247. C2×C4⋊C4C2×D7semidirect
263. C2×C22⋊C4C2×D7semidirect
264. C2×C42C2×D7semidirect
282. C22×C7#121semidirect
354. C2×D4D28semidirect
392. C22⋊C4C22⋊D7semidirect
489. C7#2424semidirect
525. C22#6493semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C2×#65022
C2×#65002
C2×#64932
C22×(C2×D7)⋊C42
C2×#64452
#899852
#897662
#897632
#897332
#897322
#897222
#897122
#897112
#24247