Finite Group Databasebeta

← #11732355

Subgroups — #11732355

Order 896.

816 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×#6501C211yesmaximal, characteristic, complemented
2C2×#6500C211yesmaximal, characteristic, complemented
3C2×#6493C211yesmaximal, characteristic, complemented
4C2×#6445C211yesmaximal, characteristic, complemented
5C7×#386C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic, complemented
7#89733C211yesmaximal, characteristic, complemented
8#89733C211yesmaximal, characteristic, complemented
9#89732C211yesmaximal, characteristic, complemented
10#89727C211yesmaximal, characteristic, complemented
11#89725C211yesmaximal, characteristic, complemented
12#89449C211yesmaximal, characteristic, complemented
13#89446C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89437C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic, complemented
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic, complemented
23C22×D28C2211yescharacteristic
24C22×D28C2211yescharacteristic
25C22×C4×D7C2211yescharacteristic, complemented
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic, complemented
32#6501—24no—
33#6500—24no—
34#6493—24no—
35C2×(C2×D7)⋊C4C2211yescharacteristic
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, complemented
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×C4⋊Dic7C2211yescharacteristic, complemented
46C2×C4⋊Dic7C2211yescharacteristic, complemented
47C2×C4⋊Dic7C2211yescharacteristic, complemented
48#6445—24no—
49C7×#95C2211yescharacteristic, complemented
50C7×#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2668—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×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4D414yescomplemented
71C22×C4×C7—22no—
72C22×C4×C7—41no—
73C22×C4×C7—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—22no—
84C2×C22⋊D7—22no—
85C2×C22⋊D7—22no—
86C2×C22⋊D7—22no—
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—41no—
99C2×D28—41no—
100C2×D28—41no—
101C2×D28D414yescomplemented
102C2×C4×D7—22no—
103C2×C4×D7—41no—
104C2×C4×D7—41no—
105C7×C22⋊C4—22no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C22⋊Dic7—22no—
109C22⋊Dic7—22no—
110C22⋊Dic7—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—22no—
117(C2×D7)⋊C4—41no—
118(C2×D7)⋊C4—41no—
119(C2×D7)⋊C4—41no—
120(C2×D7)⋊C4—41no—
121C4⋊Dic7—22no—
122C4⋊Dic7—22no—
123C4⋊Dic7—41no—
124C2×#121—71no—
125C2×#121—71no—
126C2×#120—71no—
127C2×#120—71no—
128#399—71no—
129#397—71no—
130#397—71no—
131#394—71no—
132#394—71no—
133#392—71no—
134#386—71no—
135#386—71no—
136#386—71no—
137#386D711yescharacteristic, complemented
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×C7C2×D412yes—
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—44no—
173C22×D7—44no—
174C22×D7—44no—
175C22×D7—44no—
176C22×D7—44no—
177C22×D7—44no—
178C22×D7—44no—
179C22×D7C2×D412yes—
180C22×D7C2×D412yes—
181C22×D7#4112yes—
182C7×D4—44no—
183C2×C4×C7—21no—
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×C7—44no—
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×Dic7—21no—
218D28—44no—
219D28—44no—
220C4×D7—44no—
221C25—71no—
222C22×D4—71no—
223C22×D4—71no—
224C22×D4—71no—
225C22×D4—71no—
226C22×D4—71no—
227C22×D4—71no—
228C22×D4C2×D711yescharacteristic
229C23×C4—71no—
230C23×C4C2×D711yescharacteristic, complemented
231#121—144no—
232#121—144no—
233#120—144no—
234#120—144no—
235C2×C4⋊C4—71no—
236C2×C4⋊C4—71no—
237C2×C4⋊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, complemented
254#95—71no—
255#95—71no—
256#95—71no—
257#95—71no—
258#95C2×D711yescharacteristic, complemented
259#95C2×D711yescharacteristic, complemented
260C22×C7—22no—
261C22×C7—22no—
262C22×C7—22no—
263C22×C7—22no—
264C22×C7—22no—
265C22×C7—22no—
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×C7C22×D411yescharacteristic
275C22×C7C22×D411yescharacteristic
276C22×C7C2×#4111yescharacteristic
277C22×C7H(5,𝔽2)11yescharacteristic
278C22×C7H(5,𝔽2)11yescharacteristic
279C22×C7H(5,𝔽2)11yescharacteristic
280C22×C7H(5,𝔽2)11yescharacteristic
281C2×D7—22no—
282C2×D7—22no—
283C2×D7—22no—
284C2×D7—22no—
285C2×D7—22no—
286C2×D7—22no—
287C2×D7—22no—
288C2×D7—22no—
289C2×D7—22no—
290C2×D7—416no—
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—
302C2×D7—41no—
303C4×C7—22no—
304C4×C7—22no—
305C4×C7—41no—
306C4×C7—41no—
307C4×C7—41no—
308Dic7—41no—
309Dic7—41no—
310Dic7—41no—
311Dic7—41no—
312Dic7—41no—
313C24—71no—
314C24—71no—
315C24—71no—
316C24—142no—
317C24—142no—
318C24—142no—
319C24—281no—
320C24—281no—
321C24—281no—
322C24—281no—
323C24C22×D711yescharacteristic
324C24C22×D711yescharacteristic
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—74no—
328C2×D4—142no—
329C2×D4—142no—
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×D4D2814yescomplemented
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—71no—
350C22×C4—71no—
351C22×C4—71no—
352C22×C4—71no—
353C22×C4—71no—
354C22×C4—142no—
355C22×C4—281no—
356C22×C4—281no—
357C22×C4C22×D711yescharacteristic
358C22×C4C22×D711yescharacteristic
359C22×C4C22×D711yescharacteristic
360C22×C4C22×D711yescharacteristic
361C22×C4C22×D711yescharacteristic
362C4⋊C4—142no—
363C4⋊C4—142no—
364C4⋊C4—281no—
365C22⋊C4—22no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—142no—
369C22⋊C4—142no—
370C22⋊C4—142no—
371C22⋊C4—142no—
372C22⋊C4—142no—
373C22⋊C4—142no—
374C22⋊C4—142no—
375C22⋊C4—142no—
376C22⋊C4—142no—
377C22⋊C4—281no—
378C22⋊C4—281no—
379C22⋊C4—281no—
380C22⋊C4—281no—
381C2×C7—42no—
382C2×C7—42no—
383C2×C7#53411yescharacteristic
384C2×C7#53411yescharacteristic
385C2×C7#53811yescharacteristic
386C2×C7D4211yescharacteristic
387C2×C7#54611yescharacteristic
388C2×C7#54611yescharacteristic
389C2×C7#56011yescharacteristic
390D7—42no—
391D7—42no—
392D7—42no—
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—72no—
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×D2812yes—
434C23C2×D2812yes—
435C23C23×D711yescharacteristic
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—21no—
456C2×C4—44no—
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—141no—
472C2×C4—284no—
473C2×C4C2×D2812yes—
474C2×C4D4×D712yes—
475C7#266811yescharacteristic, complemented, Sylow
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—142no—
491C22—142no—
492C22—142no—
493C22—142no—
494C22—142no—
495C22—142no—
496C22—142no—
497C22—142no—
498C22—142no—
499C22—2816no—
500C22—281no—
501C22—281no—
502C22—281no—
503C22—281no—
504C22—281no—
505C22—281no—
506C22—281no—
507C22—281no—
508C22—281no—
509C22—281no—
510C22—281no—
511C22—281no—
512C22C22×D2811yescharacteristic
513C22C2×D4×D711yescharacteristic
514C22#654811yescharacteristic
515C22#654811yescharacteristic
516C22#654811yescharacteristic
517C22D7×#4111yescharacteristic
518C22#655311yescharacteristic
519C4—22no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—281no—
525C4—281no—
526C4—281no—
527C4—281no—
528C4—281no—
529C2—42no—
530C2—42no—
531C2—282no—
532C2—282no—
533C2—282no—
534C2#9018411yescharacteristic
535C2D4×D2811yescharacteristic
536C2#9024511yescharacteristic
537C2#9028011yescharacteristic
538C2#9030311yescharacteristic
539C2#9031111yescharacteristic
540C2#9034811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
hypercenter#386D7
hypocenterC7#2668
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#6501C2semidirect
2. C2×#6500C2semidirect
3. C2×#6493C2semidirect
4. C2×#6445C2semidirect
5. C7×#386C2semidirect
6. #89985C2semidirect
7. #89733C2semidirect
8. #89733C2semidirect
9. #89732C2semidirect
10. #89727C2semidirect
11. #89725C2semidirect
12. #89449C2semidirect
13. #89446C2semidirect
14. #89437C2semidirect
15. #89437C2semidirect
18. C23×C4×C7C22semidirect
22. C22×C22⋊D7C22semidirect
25. C22×C4×D7C22semidirect
28. C2×C7×C22⋊C4C22semidirect
31. C2×C22⋊Dic7C22semidirect
43. C2×(C2×D7)⋊C4C22semidirect
44. C2×(C2×D7)⋊C4C22semidirect
45. C2×C4⋊Dic7C22semidirect
46. C2×C4⋊Dic7C22semidirect
47. C2×C4⋊Dic7C22semidirect
49. C7×#95C22semidirect
50. C7×#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
70. C2×C7×D4D4semidirect
101. C2×D28D4semidirect
137. #386D7semidirect
230. C23×C4C2×D7semidirect
253. C2×C22⋊C4C2×D7semidirect
258. #95C2×D7semidirect
259. #95C2×D7semidirect
345. C2×D4D28semidirect
475. C7#2668semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×#65002
C2×#64932
C2×#64452
C7×#3862
#899852
#897332
#897322
#897272
#897252
#894492
#894462
#894372
#26687