Finite Group Databasebeta

← #11732050

Subgroups — #11732050

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×#6493C211yesmaximal, characteristic, complemented
2C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
3C2×#6458C211yesmaximal, characteristic, complemented
4C2×#6445C211yesmaximal, characteristic, complemented
5C2×#6445C211yesmaximal, characteristic, complemented
6C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
7C7×#385C211yesmaximal, characteristic, complemented
8#89766C211yesmaximal, characteristic, complemented
9#89763C211yesmaximal, characteristic, complemented
10#89760C211yesmaximal, characteristic, complemented
11#89730C211yesmaximal, characteristic, complemented
12#89444C211yesmaximal, characteristic, complemented
13#89443C211yesmaximal, characteristic, complemented
14#89442C211yesmaximal, characteristic, complemented
15#89437C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×D28C2211yescharacteristic
22C22×D28C2211yescharacteristic
23C22×D28C2211yescharacteristic
24C22×D28C2211yescharacteristic
25C22×D28C2211yescharacteristic, complemented
26C22×C4×D7C2211yescharacteristic, complemented
27C2×C7×C4⋊C4C2211yescharacteristic, complemented
28C2×C7×C4⋊C4C2211yescharacteristic, complemented
29C2×C7×C22⋊C4C2211yescharacteristic
30C2×C7×C22⋊C4C2211yescharacteristic
31#6493—24no—
32C2×(C2×D7)⋊C4—24no—
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
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, complemented
42C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×C4⋊Dic7C2211yescharacteristic, complemented
47C2×Dic7⋊C4C2211yescharacteristic, complemented
48C2×C4×Dic7C2211yescharacteristic, complemented
49#6458—24no—
50#6445—24no—
51#6445—24no—
52(C22⋊D7)⋊C4—24no—
53C7×#95C2211yescharacteristic, complemented
54C7×#95C2211yescharacteristic, complemented
55C7⋊#95C2211yescharacteristic, complemented
56C7⋊#95C2211yescharacteristic, complemented
57#2424—71nomaximal, Sylow
58C24×C7C2311yescharacteristic
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—22no—
63C23×D7—22no—
64C23×D7—41no—
65C23×D7—41no—
66C23×D7C2311yescharacteristic
67C23×D7C2311yescharacteristic
68C23×D7C2311yescharacteristic
69C23×D7C2311yescharacteristic
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—41no—
73C22×C4×C7C2311yescharacteristic
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⋊D7D414yescomplemented
85C22×Dic7—22no—
86C22×Dic7—22no—
87C22×Dic7—41no—
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C22×Dic7C2311yescharacteristic
91C22×Dic7C2311yescharacteristic
92C2×D28—22no—
93C2×D28—22no—
94C2×D28—22no—
95C2×D28—22no—
96C2×D28—22no—
97C2×D28—22no—
98C2×D28—22no—
99C2×D28—22no—
100C2×D28—22no—
101C2×D28—22no—
102C2×D28—41no—
103C2×D28—41no—
104C2×D28—41no—
105C2×D28—41no—
106C2×D28—41no—
107C2×C4×D7—22no—
108C2×C4×D7—22no—
109C2×C4×D7—41no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—41no—
112C7×C22⋊C4—22no—
113C7×C22⋊C4D414yescomplemented
114(C2×D7)⋊C4—22no—
115(C2×D7)⋊C4—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—
128Dic7⋊C4—22no—
129C4×Dic7—22no—
130C2×#121—71no—
131C2×#121—71no—
132C2×#120—71no—
133C2×#120—71no—
134C2×C4×D4—71no—
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#385D711yescharacteristic, complemented
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7C2×D412yes—
150C23×C7C2×D412yes—
151C23×C7#4112yes—
152C23×C7C2411yescharacteristic
153C22×D7—28no—
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—21no—
174C22×D7—21no—
175C22×D7—21no—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—21no—
179C22×D7—44no—
180C22×D7—44no—
181C22×D7—44no—
182C22×D7—44no—
183C22×D7—44no—
184C22×D7—44no—
185C22×D7C2×D412yes—
186C22×D7C2×D412yes—
187C22×D7#4112yes—
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—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—44no—
204C2×C4×C7C2×D412yes—
205C2×C4×C7C2×D412yes—
206C2×C4×C7C2×D412yes—
207C22⋊D7—28no—
208C22⋊D7—44no—
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×Dic7—44no—
220C2×Dic7C2×D412yes—
221C2×Dic7C2×D412yes—
222D28—44no—
223D28—44no—
224D28—44no—
225D28—44no—
226D28—44no—
227C4×D7—44no—
228C25—71no—
229C22×D4—71no—
230C22×D4—71no—
231C22×D4—71no—
232C22×D4—71no—
233C22×D4—71no—
234C22×D4—71no—
235C22×D4—71no—
236C23×C4—71no—
237C23×C4—71no—
238C23×C4C2×D711yescharacteristic
239#121—144no—
240#121—144no—
241#120—144no—
242#120—144no—
243C4×D4—144no—
244C2×C4⋊C4—71no—
245C2×C4⋊C4—71no—
246C2×C4⋊C4C2×D711yescharacteristic, complemented
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
264C2×C42—71no—
265#95—71no—
266#95—71no—
267#95C2×D711yescharacteristic, complemented
268#95C2×D711yescharacteristic, complemented
269C22×C7—22no—
270C22×C7—22no—
271C22×C7—22no—
272C22×C7—22no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7#12114yescomplemented
276C22×C7C22×D411yescharacteristic
277C22×C7C22×D411yescharacteristic
278C22×C7C22×D411yescharacteristic
279C22×C7C2×#4111yescharacteristic
280C22×C7C2×#4111yescharacteristic
281C22×C7H(5,𝔽2)11yescharacteristic
282C22×C7H(5,𝔽2)11yescharacteristic
283C2×D7—22no—
284C2×D7—22no—
285C2×D7—22no—
286C2×D7—22no—
287C2×D7—22no—
288C2×D7—22no—
289C2×D7—22no—
290C2×D7—22no—
291C2×D7—22no—
292C2×D7—416no—
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—
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—41no—
316C4×C7—41no—
317C4×C7—41no—
318Dic7—22no—
319Dic7—22no—
320Dic7—41no—
321Dic7—41no—
322C24—71no—
323C24—71no—
324C24—71no—
325C24—71no—
326C24—142no—
327C24—142no—
328C24—142no—
329C24—142no—
330C24—142no—
331C24—281no—
332C24—281no—
333C24C22×D711yescharacteristic
334C2×D4—74no—
335C2×D4—142no—
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—142no—
350C2×D4—281no—
351C2×D4—281no—
352C2×D4—281no—
353C2×D4—281no—
354C2×D4—281no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—71no—
359C22×C4—71no—
360C22×C4—71no—
361C22×C4—71no—
362C22×C4—142no—
363C22×C4—142no—
364C22×C4—142no—
365C22×C4—142no—
366C22×C4—281no—
367C22×C4—281no—
368C22×C4C22×D711yescharacteristic
369C22×C4C22×D711yescharacteristic
370C22×C4C22×D711yescharacteristic
371C22×C4C22×D711yescharacteristic
372C22×C4C22×D711yescharacteristic
373C22×C4C22×D711yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—41no—
376C4⋊C4—142no—
377C4⋊C4—142no—
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⋊C4D2814yescomplemented
393C42—142no—
394C2×C7—24no—
395C2×C7C2×#12111yescharacteristic
396C2×C7#52511yescharacteristic
397C2×C7#53811yescharacteristic
398C2×C7D4211yescharacteristic
399C2×C7D4211yescharacteristic
400C2×C7#54611yescharacteristic
401C2×C7#54611yescharacteristic
402D7—42no—
403D7—42no—
404D7—42no—
405D7—81no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—72no—
411C23—72no—
412C23—72no—
413C23—148no—
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—141no—
427C23—141no—
428C23—141no—
429C23—141no—
430C23—141no—
431C23—141no—
432C23—141no—
433C23—141no—
434C23—141no—
435C23—141no—
436C23—141no—
437C23—141no—
438C23—141no—
439C23—284no—
440C23—284no—
441C23—284no—
442C23—284no—
443C23—284no—
444C23—284no—
445C23C2×D2812yes—
446C23D4×D712yes—
447C23#116112yes—
448C23C23×D711yescharacteristic
449D4—148no—
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—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—44no—
472C2×C4—72no—
473C2×C4—72no—
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×C4—284no—
486C2×C4C2×D2812yes—
487C2×C4C2×D2812yes—
488C2×C4D4×D712yes—
489C7#242411yescharacteristic, complemented, Sylow
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—142no—
497C22—142no—
498C22—142no—
499C22—142no—
500C22—142no—
501C22—142no—
502C22—142no—
503C22—142no—
504C22—142no—
505C22—2816no—
506C22—281no—
507C22—281no—
508C22—281no—
509C22—281no—
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#645814yescomplemented
526C22C22×D2811yescharacteristic
527C22C2×D4×D711yescharacteristic
528C22C2×D4×D711yescharacteristic
529C22#654811yescharacteristic
530C22C2×#116111yescharacteristic
531C22D7×#4111yescharacteristic
532C22#655311yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—142no—
540C4—142no—
541C4—281no—
542C4—281no—
543C2—24no—
544C2—282no—
545C2—282no—
546C2—282no—
547C2—561no—
548C2C2×#645811yescharacteristic
549C2D4×D2811yescharacteristic
550C2#9024511yescharacteristic
551C2#9028111yescharacteristic
552C2#9031811yescharacteristic
553C2#9033511yescharacteristic
554C2#9034511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
hypercenter#385D7
hypocenterC7#2424
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#6493C2semidirect
2. C22×(C2×D7)⋊C4C2semidirect
3. C2×#6458C2semidirect
4. C2×#6445C2semidirect
5. C2×#6445C2semidirect
6. C2×(C22⋊D7)⋊C4C2semidirect
7. C7×#385C2semidirect
8. #89766C2semidirect
9. #89763C2semidirect
10. #89760C2semidirect
11. #89730C2semidirect
12. #89444C2semidirect
13. #89443C2semidirect
14. #89442C2semidirect
15. #89437C2semidirect
25. C22×D28C22semidirect
26. C22×C4×D7C22semidirect
27. C2×C7×C4⋊C4C22semidirect
28. C2×C7×C4⋊C4C22semidirect
41. C2×(C2×D7)⋊C4C22semidirect
42. C2×(C2×D7)⋊C4C22semidirect
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×C4×Dic7C22semidirect
53. C7×#95C22semidirect
54. C7×#95C22semidirect
55. C7⋊#95C22semidirect
56. C7⋊#95C22semidirect
84. C2×C22⋊D7D4semidirect
113. C7×C22⋊C4D4semidirect
126. (C2×D7)⋊C4D4semidirect
144. #385D7semidirect
246. C2×C4⋊C4C2×D7semidirect
247. C2×C4⋊C4C2×D7semidirect
267. #95C2×D7semidirect
268. #95C2×D7semidirect
275. C22×C7#121semidirect
392. C22⋊C4D28semidirect
489. C7#2424semidirect
525. C22#6458semidirect

Maximal subgroups

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

subgroupindex
C2×#64932
C22×(C2×D7)⋊C42
C2×#64582
C2×#64452
C2×(C22⋊D7)⋊C42
C7×#3852
#897662
#897632
#897602
#897302
#894442
#894432
#894422
#894372
#24247