Finite Group Databasebeta

← #11739554

Subgroups — #11739554

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
1C22×C7×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#6516C211yesmaximal, characteristic, complemented
3C2×#6516C211yesmaximal, characteristic, complemented
4C2×#6493C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D7C211yesmaximal, characteristic, complemented
6C2×#6447C211yesmaximal, characteristic, complemented
7#89985C211yesmaximal, characteristic, complemented
8#89982C211yesmaximal, characteristic, complemented
9#89732C211yesmaximal, characteristic, complemented
10#89731C211yesmaximal, characteristic, complemented
11#89731C211yesmaximal, characteristic, complemented
12#89729C211yesmaximal, characteristic, complemented
13#89728C211yesmaximal, characteristic, complemented
14#89723C211yesmaximal, characteristic, complemented
15#89720C211yesmaximal, characteristic, complemented
16C25×C7C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic
23C22×C22⋊D7C2211yescharacteristic
24C22×C22⋊D7C2211yescharacteristic
25C22×D28C2211yescharacteristic, complemented
26C22×C4×D7C2211yescharacteristic, complemented
27C2×C7×C22⋊C4—24no—
28C2×C7×C22⋊C4C2211yescharacteristic
29C2×C7×C22⋊C4C2211yescharacteristic
30C2×C7×C22⋊C4C2211yescharacteristic
31C2×C7×C22⋊C4C2211yescharacteristic
32#6516—24no—
33#6516—24no—
34C2×C22⋊Dic7C2211yescharacteristic
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37C2×C22⋊Dic7C2211yescharacteristic
38C2×C22⋊Dic7C2211yescharacteristic
39C2×C22⋊Dic7C2211yescharacteristic
40#6493—24no—
41C4×C22⋊D7—24no—
42C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
47C2×C4⋊Dic7C2211yescharacteristic, complemented
48C2×Dic7⋊C4C2211yescharacteristic, complemented
49C2×Dic7⋊C4C2211yescharacteristic, complemented
50C2×Dic7⋊C4C2211yescharacteristic, complemented
51C2×C4×Dic7C2211yescharacteristic, complemented
52#6447—24no—
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55C7⋊#95C2211yescharacteristic, complemented
56C7⋊#95C2211yescharacteristic, complemented
57#2424—71nomaximal, Sylow
58C24×C7—22no—
59C24×C7—22no—
60C24×C7—22no—
61C24×C7—22no—
62C24×C7—22no—
63C24×C7—41no—
64C24×C7—41no—
65C24×C7C2311yescharacteristic
66C24×C7C2311yescharacteristic
67C24×C7C2311yescharacteristic
68C23×D7C2311yescharacteristic
69C23×D7C2311yescharacteristic
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7C2311yescharacteristic
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—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—
96C2×C22⋊D7—41no—
97C2×C22⋊D7D414yescomplemented
98C22×Dic7C2311yescharacteristic
99C22×Dic7C2311yescharacteristic
100C22×Dic7C2311yescharacteristic
101C22×Dic7C2311yescharacteristic
102C22×Dic7C2311yescharacteristic
103C22×Dic7C2311yescharacteristic
104C2×D28—22no—
105C2×D28—22no—
106C2×D28—41no—
107C2×C4×D7—22no—
108C2×C4×D7—22no—
109C2×C4×D7—41no—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—22no—
113C7×C22⋊C4D414yescomplemented
114C22⋊Dic7—22no—
115C22⋊Dic7—22no—
116C22⋊Dic7—22no—
117C22⋊Dic7—22no—
118C22⋊Dic7—22no—
119C22⋊Dic7D414yescomplemented
120(C2×D7)⋊C4—22no—
121(C2×D7)⋊C4—22no—
122(C2×D7)⋊C4—22no—
123(C2×D7)⋊C4—22no—
124(C2×D7)⋊C4—41no—
125C4⋊Dic7—22no—
126Dic7⋊C4—22no—
127Dic7⋊C4—22no—
128Dic7⋊C4—41no—
129C4×Dic7—22no—
130C2×#121—71no—
131C2×#121—71no—
132C2×#120—71no—
133C2×#120—71no—
134C2×C4×D4—71no—
135C22×C22⋊C4D711yescharacteristic, complemented
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—28no—
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—44no—
163C23×C7—44no—
164C23×C7—44no—
165C23×C7—44no—
166C23×C7—44no—
167C23×C7—44no—
168C23×C7C2×D412yes—
169C23×C7C2×D412yes—
170C23×C7C2×D412yes—
171C23×C7#4112yes—
172C23×C7#4112yes—
173C23×C7C2411yescharacteristic
174C22×D7—21no—
175C22×D7—21no—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—21no—
179C22×D7—21no—
180C22×D7—21no—
181C22×D7—21no—
182C22×D7—21no—
183C22×D7—21no—
184C22×D7—21no—
185C22×D7—21no—
186C22×D7—21no—
187C22×D7C2×D412yes—
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—44no—
199C2×C4×C7—44no—
200C2×C4×C7C2×D412yes—
201C2×C4×C7C2×D412yes—
202C22⋊D7—28no—
203C22⋊D7—44no—
204C22⋊D7—44no—
205C22⋊D7—44no—
206C22⋊D7—44no—
207C22⋊D7—44no—
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—
218C2×Dic7—21no—
219C2×Dic7—21no—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7C2×D412yes—
224C2×Dic7C2×D412yes—
225C2×Dic7C2×D412yes—
226D28—44no—
227C4×D7—44no—
228C25C2×D711yescharacteristic
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×C4C2×D711yescharacteristic
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⋊C4—71no—
247C2×C4⋊C4—71no—
248C2×C22⋊C4—24no—
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⋊C4C2×D711yescharacteristic
261C2×C22⋊C4C2×D711yescharacteristic
262C2×C22⋊C4C2×D711yescharacteristic
263C2×C22⋊C4C2×D711yescharacteristic
264C2×C42—71no—
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—22no—
279C22×C7—22no—
280C22×C7—22no—
281C22×C7—416no—
282C22×C7—41no—
283C22×C7—41no—
284C22×C7—41no—
285C22×C7—41no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7—41no—
289C22×C7—41no—
290C22×C7#12114yescomplemented
291C22×C7C22×D411yescharacteristic
292C22×C7C22×D411yescharacteristic
293C22×C7C22×D411yescharacteristic
294C22×C7C2×#4111yescharacteristic
295C22×C7C2×#4111yescharacteristic
296C22×C7H(5,𝔽2)11yescharacteristic
297C22×C7H(5,𝔽2)11yescharacteristic
298C2×D7—22no—
299C2×D7—22no—
300C2×D7—22no—
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—41no—
315C4×C7—41no—
316Dic7—22no—
317Dic7—22no—
318Dic7—22no—
319Dic7—41no—
320Dic7—41no—
321Dic7—41no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—41no—
328C24—41no—
329C24—71no—
330C24—71no—
331C24C22×D711yescharacteristic
332C24C22×D711yescharacteristic
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—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—71no—
367C22×C4—142no—
368C22×C4—142no—
369C22×C4—281no—
370C22×C4C22×D711yescharacteristic
371C22×C4C22×D711yescharacteristic
372C22×C4C22×D711yescharacteristic
373C22×C4C22×D711yescharacteristic
374C4⋊C4—142no—
375C4⋊C4—142no—
376C4⋊C4—142no—
377C4⋊C4—281no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—74no—
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—142no—
394C2×C7—24no—
395C2×C7—42no—
396C2×C7—42no—
397C2×C7C2×#12111yescharacteristic
398C2×C7#52511yescharacteristic
399C2×C7#53811yescharacteristic
400C2×C7D4211yescharacteristic
401C2×C7D4211yescharacteristic
402C2×C7#54611yescharacteristic
403C2×C7#54611yescharacteristic
404D7—42no—
405D7—81no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—28no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—72no—
430C23—141no—
431C23—141no—
432C23—141no—
433C23—141no—
434C23—141no—
435C23—141no—
436C23—141no—
437C23—141no—
438C23—141no—
439C23—141no—
440C23—141no—
441C23—141no—
442C23—141no—
443C23#115712yes—
444C23#115712yes—
445C23D4×D712yes—
446C23D4×D712yes—
447C23C2×C22⋊D712yes—
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—44no—
467C2×C4—44no—
468C2×C4—72no—
469C2×C4—72no—
470C2×C4—72no—
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—141no—
485C2×C4—141no—
486C2×C4—284no—
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—22no—
500C22—22no—
501C22—22no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—142no—
512C22—142no—
513C22—142no—
514C22—281no—
515C22—281no—
516C22—281no—
517C22—281no—
518C22—281no—
519C22—281no—
520C22—281no—
521C22—281no—
522C22—281no—
523C22—281no—
524C22—281no—
525C22#644714yescomplemented
526C22C2×#115711yescharacteristic
527C22C2×#115711yescharacteristic
528C22C2×D4×D711yescharacteristic
529C22C2×D4×D711yescharacteristic
530C22#654811yescharacteristic
531C22#654811yescharacteristic
532C22C22×C22⋊D711yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—142no—
538C4—142no—
539C4—142no—
540C4—281no—
541C4—281no—
542C4—281no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—282no—
547C2—561no—
548C2C2×#644711yescharacteristic
549C2#9018111yescharacteristic
550C2#9018111yescharacteristic
551C2#9018511yescharacteristic
552C2#9048211yescharacteristic
553C2D4×C22⋊D711yescharacteristic
554C2D4×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC22×C7×C22⋊C4C2
hypercenterC22×C22⋊C4D7
hypocenterC7#2424
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊C4C2semidirect
2. C2×#6516C2semidirect
3. C2×#6516C2semidirect
4. C2×#6493C2semidirect
5. C2×C4×C22⋊D7C2semidirect
6. C2×#6447C2semidirect
7. #89985C2semidirect
8. #89982C2semidirect
9. #89732C2semidirect
10. #89731C2semidirect
11. #89731C2semidirect
12. #89729C2semidirect
13. #89728C2semidirect
14. #89723C2semidirect
15. #89720C2semidirect
25. C22×D28C22semidirect
26. C22×C4×D7C22semidirect
42. C2×(C2×D7)⋊C4C22semidirect
43. C2×(C2×D7)⋊C4C22semidirect
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×Dic7⋊C4C22semidirect
50. C2×Dic7⋊C4C22semidirect
51. C2×C4×Dic7C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
55. C7⋊#95C22semidirect
56. C7⋊#95C22semidirect
97. C2×C22⋊D7D4semidirect
113. C7×C22⋊C4D4semidirect
119. C22⋊Dic7D4semidirect
135. C22×C22⋊C4D7semidirect
290. C22×C7#121semidirect
392. C22⋊C4C22⋊D7semidirect
489. C7#2424semidirect
525. C22#6447semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊C42
C2×#65162
C2×#64932
C2×C4×C22⋊D72
C2×#64472
#899852
#899822
#897322
#897312
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