Finite Group Databasebeta

← #11739556

Subgroups — #11739556

Order 896.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 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
3C2×#6487C211yesmaximal, characteristic, complemented
4C2×#6448C211yesmaximal, characteristic, complemented
5#90019C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic, complemented
7#89985C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
9#89733C211yesmaximal, characteristic, complemented
10#89731C211yesmaximal, characteristic, complemented
11#89729C211yesmaximal, characteristic, complemented
12#89724C211yesmaximal, characteristic, complemented
13#89723C211yesmaximal, characteristic, complemented
14#89719C211yesmaximal, characteristic, complemented
15#89716C211yesmaximal, characteristic, complemented
16C25×C7C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×Dic14C2211yescharacteristic, complemented
23C2×C7×C22⋊C4—24no—
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28#6516—24no—
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C22⋊Dic7C2211yescharacteristic
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37C2×C22⋊Dic7C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
42C2×C4⋊Dic7C2211yescharacteristic, complemented
43#6487—24no—
44C2×Dic7⋊C4C2211yescharacteristic, complemented
45C2×Dic7⋊C4C2211yescharacteristic, complemented
46C2×C4×Dic7C2211yescharacteristic, complemented
47C2×C4×Dic7C2211yescharacteristic, complemented
48#6448—24no—
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2472—71nomaximal, Sylow
56C24×C7—22no—
57C24×C7—22no—
58C24×C7—22no—
59C24×C7—22no—
60C24×C7—22no—
61C24×C7—41no—
62C24×C7—41no—
63C24×C7C2311yescharacteristic
64C24×C7C2311yescharacteristic
65C24×C7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C2×C22⋊D7—22no—
80C2×C22⋊D7—22no—
81C2×C22⋊D7—22no—
82C2×C22⋊D7—22no—
83C2×C22⋊D7—22no—
84C2×C22⋊D7—41no—
85C2×C22⋊D7—41no—
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C22×Dic7C2311yescharacteristic
91C22×Dic7C2311yescharacteristic
92C22×Dic7C2311yescharacteristic
93C2×Dic14—22no—
94C2×Dic14—22no—
95C2×Dic14—41no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4D414yescomplemented
100C22⋊Dic7—22no—
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103C22⋊Dic7—22no—
104C22⋊Dic7—22no—
105C22⋊Dic7—22no—
106C22⋊Dic7—41no—
107C22⋊Dic7—41no—
108C22⋊Dic7D414yes—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—41no—
112(C2×D7)⋊C4—41no—
113C4⋊Dic7—22no—
114Dic7⋊C4—22no—
115Dic7⋊C4—22no—
116C4×Dic7—22no—
117C4×Dic7—41no—
118C2×#124—71no—
119C2×C22⋊Q8—71no—
120C2×#120—71no—
121C22×C22⋊C4D711yescharacteristic, complemented
122#397—71no—
123#394—71no—
124#394—71no—
125#393—71no—
126#390—71no—
127#388—71no—
128#388—71no—
129#386—71no—
130#386—71no—
131#379—71no—
132C4×C22⋊C4—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—28no—
138C23×C7—21no—
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—44no—
151C23×C7—44no—
152C23×C7—44no—
153C23×C7—44no—
154C23×C7—44no—
155C23×C7—44no—
156C23×C7C2×D412yes—
157C23×C7C2×D412yes—
158C23×C7#4112yes—
159C23×C7#4112yes—
160C23×C7#4112yes—
161C23×C7C2411yescharacteristic
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—21no—
167C22×D7—21no—
168C22×D7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—44no—
180C2×C4×C7—44no—
181C2×C4×C7C2×D412yes—
182C2×C4×C7C2×D412yes—
183C22⋊D7—44no—
184C22⋊D7—44no—
185C22⋊D7—44no—
186C2×Dic7—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202C2×Dic7—21no—
203C2×Dic7—21no—
204C2×Dic7—21no—
205C2×Dic7C2×D412yes—
206C2×Dic7C2×D412yes—
207Dic14—44no—
208C25C2×D711yescharacteristic
209C22×Q8—71no—
210C22×D4—71no—
211C22×D4—71no—
212C22×D4—71no—
213C23×C4C2×D711yescharacteristic
214C23×C4C2×D711yescharacteristic
215#124—144no—
216C22⋊Q8—144no—
217#120—144no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4—71no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4—71no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4—71no—
227C2×C22⋊C4—71no—
228C2×C22⋊C4—71no—
229C2×C22⋊C4—71no—
230C2×C22⋊C4—71no—
231C2×C22⋊C4—71no—
232C2×C22⋊C4—71no—
233C2×C22⋊C4—71no—
234C2×C22⋊C4—71no—
235C2×C22⋊C4C2×D711yescharacteristic
236C2×C22⋊C4C2×D711yescharacteristic
237C2×C22⋊C4C2×D711yescharacteristic
238C2×C22⋊C4C2×D711yescharacteristic
239C2×C42—71no—
240C2×C42—71no—
241#95—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95—71no—
246#95—71no—
247C22×C7—22no—
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7—22no—
254C22×C7—22no—
255C22×C7—22no—
256C22×C7—22no—
257C22×C7—22no—
258C22×C7—22no—
259C22×C7—416no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7—41no—
268C22×C7#12414yescomplemented
269C22×C7C22×D411yescharacteristic
270C22×C7C22×D411yescharacteristic
271C22×C7C2×#4111yescharacteristic
272C22×C7C2×#4111yescharacteristic
273C22×C7C2×#4111yescharacteristic
274C22×C7H(5,𝔽2)11yescharacteristic
275C22×C7H(5,𝔽2)11yescharacteristic
276C2×D7—41no—
277C2×D7—41no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C2×D7—41no—
283C4×C7—22no—
284C4×C7—22no—
285C4×C7—41no—
286C4×C7—41no—
287Dic7—22no—
288Dic7—22no—
289Dic7—41no—
290Dic7—41no—
291Dic7—41no—
292Dic7—41no—
293Dic7—41no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—71no—
302C24C22×D711yescharacteristic
303C24C22×D711yescharacteristic
304C24C22×D711yescharacteristic
305C2×Q8—142no—
306C2×Q8—142no—
307C2×Q8—281no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—142no—
314C2×D4—142no—
315C2×D4—281no—
316C2×D4—281no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—71no—
324C22×C4—71no—
325C22×C4—71no—
326C22×C4—71no—
327C22×C4—71no—
328C22×C4—71no—
329C22×C4—71no—
330C22×C4C22×D711yescharacteristic
331C22×C4C22×D711yescharacteristic
332C22×C4C22×D711yescharacteristic
333C22×C4C22×D711yescharacteristic
334C4⋊C4—142no—
335C4⋊C4—142no—
336C4⋊C4—142no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—74no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4—142no—
349C22⋊C4—281no—
350C22⋊C4—281no—
351C22⋊C4—281no—
352C22⋊C4—281no—
353C22⋊C4C22⋊D714yescomplemented
354C42—142no—
355C42—281no—
356C2×C7—24no—
357C2×C7—42no—
358C2×C7—42no—
359C2×C7#52511yescharacteristic
360C2×C7C2×#12411yescharacteristic
361C2×C7#53811yescharacteristic
362C2×C7D4211yescharacteristic
363C2×C7#54611yescharacteristic
364C2×C7#54611yescharacteristic
365C2×C7#55111yescharacteristic
366D7—81no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—28no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—141no—
391C23—141no—
392C23—141no—
393C23—141no—
394C23—141no—
395C23—141no—
396C23—141no—
397C23#115712yes—
398C23#115712yes—
399C23D4×D712yes—
400C23#115912yes—
401C23C2×C22⋊D712yes—
402C23C23×D711yescharacteristic
403Q8—284no—
404D4—284no—
405D4—284no—
406D4—284no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—72no—
420C2×C4—72no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—141no—
427C2×C4—141no—
428C2×C4—141no—
429C2×C4—141no—
430C2×C4—141no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—141no—
438C2×C4—141no—
439C2×C4—141no—
440C2×C4C2×C22⋊D712yes—
441C2×C4C2×C22⋊D712yes—
442C7#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—281no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22#644814yescomplemented
472C22C2×#115711yescharacteristic
473C22C2×#115711yescharacteristic
474C22C2×D4×D711yescharacteristic
475C22C2×#115911yescharacteristic
476C22#654811yescharacteristic
477C22#654811yescharacteristic
478C22C22×C22⋊D711yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—142no—
484C4—142no—
485C4—281no—
486C4—281no—
487C4—281no—
488C4—281no—
489C4—281no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—561no—
494C2#9018111yescharacteristic
495C2C2×#644811yescharacteristic
496C2#9018511yescharacteristic
497C2#9018611yescharacteristic
498C2#9048211yescharacteristic
499C2D4×C22⋊D711yescharacteristic
500C2#9049711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC22×C7×C22⋊C4C2
prime radicalC2×#6516C2
hypercenterC22×C22⋊C4D7
hypocenterC7#2472
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
3. C2×#6487C2semidirect
4. C2×#6448C2semidirect
5. #90019C2semidirect
6. #89985C2semidirect
7. #89985C2semidirect
8. C4×C22⋊Dic7C2semidirect
9. #89733C2semidirect
10. #89731C2semidirect
11. #89729C2semidirect
12. #89724C2semidirect
13. #89723C2semidirect
14. #89719C2semidirect
15. #89716C2semidirect
22. C22×Dic14C22semidirect
38. C2×(C2×D7)⋊C4C22semidirect
39. C2×(C2×D7)⋊C4C22semidirect
40. C2×(C2×D7)⋊C4C22semidirect
41. C2×(C2×D7)⋊C4C22semidirect
42. C2×C4⋊Dic7C22semidirect
44. C2×Dic7⋊C4C22semidirect
45. C2×Dic7⋊C4C22semidirect
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
99. C7×C22⋊C4D4semidirect
121. C22×C22⋊C4D7semidirect
268. C22×C7#124semidirect
353. C22⋊C4C22⋊D7semidirect
442. C7#2472semidirect
471. C22#6448semidirect

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×#64872
C2×#64482
#900192
#899852
C4×C22⋊Dic72
#897332
#897312
#897292
#897242
#897232
#897192
#897162
#24727