Finite Group Databasebeta

← #11733362

Subgroups — #11733362

Order 896.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6445C211yesmaximal, characteristic
2C2×#6438C211yesmaximal, characteristic, complemented
3C7×#388C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89763C211yesmaximal, characteristic, complemented
6#89731C211yesmaximal, characteristic, complemented
7#89728C211yesmaximal, characteristic, complemented
8#89727C211yesmaximal, characteristic, complemented
9#89727C211yesmaximal, characteristic, complemented
10#89719C211yesmaximal, characteristic, complemented
11#89712C211yesmaximal, characteristic, complemented
12#89447C211yesmaximal, characteristic, complemented
13#89445C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89437C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×D28C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic, complemented
21C22×Dic14C2211yescharacteristic, complemented
22C2×C7×C4⋊C4C2211yescharacteristic, complemented
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C42×C7C2211yescharacteristic, complemented
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
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, complemented
40C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
42C2×Dic7⋊C4C2211yescharacteristic, complemented
43#6445—24no—
44#6438—24no—
45C7×#95C2211yescharacteristic, complemented
46C7×#95C2211yescharacteristic, complemented
47C7⋊#95C2211yescharacteristic, complemented
48C7⋊#95C2211yescharacteristic, complemented
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53#2684—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7—22no—
56C23×D7—22no—
57C23×D7—22no—
58C23×D7—41no—
59C23×D7—41no—
60C23×D7—41no—
61C23×D7—41no—
62C23×D7C2311yescharacteristic
63C23×D7C2311yescharacteristic
64C23×D7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7—41no—
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C2×D28—22no—
80C2×D28—22no—
81C2×D28—41no—
82C2×D28—41no—
83C2×D28—41no—
84C2×D28D414yes—
85C2×C4×D7—41no—
86C2×C4×D7—41no—
87C2×C4×D7—41no—
88C2×Dic14—22no—
89C2×Dic14—41no—
90C2×Dic14—41no—
91C7×C4⋊C4—41no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—22no—
94C7×C22⋊C4—41no—
95C42×C7—22no—
96C22⋊Dic7—41no—
97C22⋊Dic7—41no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—22no—
105(C2×D7)⋊C4—41no—
106(C2×D7)⋊C4—41no—
107(C2×D7)⋊C4—41no—
108(C2×D7)⋊C4—41no—
109(C2×D7)⋊C4—41no—
110(C2×D7)⋊C4—41no—
111Dic7⋊C4—41no—
112C2×#124—71no—
113C2×#120—71no—
114#397—71no—
115#397—71no—
116#394—71no—
117#394—71no—
118#393—71no—
119#393—71no—
120#388—71no—
121#388D711yescharacteristic, complemented
122#386—71no—
123#386—71no—
124#386—71no—
125#379—71no—
126#379—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7#4112yes—
134C23×C7C2411yescharacteristic
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—44no—
154C22×D7—44no—
155C22×D7—44no—
156C22×D7—44no—
157C22×D7—44no—
158C22×D7—44no—
159C22×D7—44no—
160C22×D7C2×D412yes—
161C22×D7C2×D412yes—
162C22×D7#4112yes—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—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—21no—
180C2×C4×C7C2×D412yes—
181C22⋊D7—44no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
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—
197D28—44no—
198D28—44no—
199C4×D7—44no—
200Dic14—44no—
201C25—71no—
202C22×Q8—71no—
203C22×D4—71no—
204C22×D4—71no—
205C22×D4—71no—
206C23×C4—71no—
207#124—144no—
208#120—144no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4C2×D711yescharacteristic, complemented
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4—71no—
215C2×C22⋊C4—71no—
216C2×C22⋊C4—71no—
217C2×C22⋊C4—71no—
218C2×C22⋊C4—71no—
219C2×C22⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4—71no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4—71no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4C2×D711yescharacteristic
227C2×C22⋊C4C2×D711yescharacteristic
228C2×C22⋊C4C2×D711yescharacteristic
229C2×C42C2×D711yescharacteristic, complemented
230#95—71no—
231#95—71no—
232#95—71no—
233#95—71no—
234#95—71no—
235#95—71no—
236#95C2×D711yescharacteristic, complemented
237#95C2×D711yescharacteristic, complemented
238C22×C7—22no—
239C22×C7—22no—
240C22×C7—22no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7C22×D411yescharacteristic
246C22×C7C2×#4111yescharacteristic
247C22×C7C2×#4111yescharacteristic
248C22×C7H(5,𝔽2)11yescharacteristic
249C22×C7H(5,𝔽2)11yescharacteristic
250C22×C7H(5,𝔽2)11yescharacteristic
251C22×C7H(5,𝔽2)11yescharacteristic
252C2×D7—22no—
253C2×D7—22no—
254C2×D7—22no—
255C2×D7—22no—
256C2×D7—22no—
257C2×D7—22no—
258C2×D7—22no—
259C2×D7—22no—
260C2×D7—22no—
261C2×D7—416no—
262C2×D7—41no—
263C2×D7—41no—
264C2×D7—41no—
265C2×D7—41no—
266C2×D7—41no—
267C2×D7—41no—
268C2×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—41no—
273C2×D7—41no—
274C4×C7—22no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—41no—
279C4×C7—41no—
280Dic7—41no—
281Dic7—41no—
282Dic7—41no—
283Dic7—41no—
284Dic7—41no—
285C24—71no—
286C24—71no—
287C24—71no—
288C24—142no—
289C24—142no—
290C24—142no—
291C24—281no—
292C24—281no—
293C24—281no—
294C24—281no—
295C24C22×D711yescharacteristic
296C2×Q8—142no—
297C2×Q8—281no—
298C2×Q8—281no—
299C2×D4—74no—
300C2×D4—142no—
301C2×D4—142no—
302C2×D4—142no—
303C2×D4—142no—
304C2×D4—281no—
305C2×D4—281no—
306C2×D4—281no—
307C2×D4—281no—
308C22×C4—71no—
309C22×C4—71no—
310C22×C4—71no—
311C22×C4—71no—
312C22×C4—71no—
313C22×C4—281no—
314C22×C4—281no—
315C22×C4—281no—
316C22×C4C22×D711yescharacteristic
317C22×C4C22×D711yescharacteristic
318C22×C4C22×D711yescharacteristic
319C22×C4C22×D711yescharacteristic
320C22×C4C22×D711yescharacteristic
321C22×C4C22×D711yescharacteristic
322C4⋊C4—41no—
323C4⋊C4—281no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—142no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C22⋊C4—142no—
331C22⋊C4—142no—
332C22⋊C4—142no—
333C22⋊C4—142no—
334C22⋊C4—281no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C22⋊C4—281no—
341C22⋊C4—281no—
342C42—22no—
343C2×C7—42no—
344C2×C7#53411yescharacteristic
345C2×C7#53811yescharacteristic
346C2×C7#53811yescharacteristic
347C2×C7#54611yescharacteristic
348C2×C7#54611yescharacteristic
349C2×C7#55111yescharacteristic
350C2×C7H(3,𝔽4)11yescharacteristic
351D7—42no—
352D7—42no—
353D7—42no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—72no—
361C23—72no—
362C23—72no—
363C23—141no—
364C23—141no—
365C23—141no—
366C23—141no—
367C23—141no—
368C23—141no—
369C23—141no—
370C23—141no—
371C23—141no—
372C23—141no—
373C23—141no—
374C23—141no—
375C23—141no—
376C23—141no—
377C23—141no—
378C23—141no—
379C23—141no—
380C23—141no—
381C23—284no—
382C23—284no—
383C23—284no—
384C23—284no—
385C23—284no—
386C23—284no—
387C23—284no—
388C23#115712yes—
389C23C23×D711yescharacteristic
390Q8—284no—
391D4—284no—
392D4—284no—
393D4—284no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—141no—
412C2×C4—141no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—284no—
427C2×C4D4×D712yes—
428C7#268411yescharacteristic, complemented, Sylow
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—142no—
437C22—142no—
438C22—142no—
439C22—142no—
440C22—142no—
441C22—142no—
442C22—142no—
443C22—142no—
444C22—142no—
445C22—2816no—
446C22—281no—
447C22—281no—
448C22—281no—
449C22—281no—
450C22—281no—
451C22—281no—
452C22—281no—
453C22—281no—
454C22—281no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22C2×#115711yescharacteristic
459C22C2×D4×D711yescharacteristic
460C22#654811yescharacteristic
461C22D7×#4111yescharacteristic
462C22#655311yescharacteristic
463C22#655311yescharacteristic
464C22#655311yescharacteristic
465C4—22no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—281no—
472C4—281no—
473C4—281no—
474C4—281no—
475C4—281no—
476C2—42no—
477C2—282no—
478C2—282no—
479C2—282no—
480C2#9021811yescharacteristic
481C2#9024111yescharacteristic
482C2#9025111yescharacteristic
483C2#9030311yescharacteristic
484C2#9034411yescharacteristic
485C2#9037411yescharacteristic
486C2#9039511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC2×#6445C2
hypercenter#388D7
hypocenterC7#2684
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#6438C2semidirect
3. C7×#388C2semidirect
4. #89765C2semidirect
5. #89763C2semidirect
6. #89731C2semidirect
7. #89728C2semidirect
8. #89727C2semidirect
9. #89727C2semidirect
10. #89719C2semidirect
11. #89712C2semidirect
12. #89447C2semidirect
13. #89445C2semidirect
14. #89437C2semidirect
15. #89437C2semidirect
20. C22×C4×D7C22semidirect
21. C22×Dic14C22semidirect
22. C2×C7×C4⋊C4C22semidirect
26. C2×C42×C7C22semidirect
39. C2×(C2×D7)⋊C4C22semidirect
40. C2×(C2×D7)⋊C4C22semidirect
41. C2×(C2×D7)⋊C4C22semidirect
42. C2×Dic7⋊C4C22semidirect
45. C7×#95C22semidirect
46. C7×#95C22semidirect
47. C7⋊#95C22semidirect
48. C7⋊#95C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
121. #388D7semidirect
210. C2×C4⋊C4C2×D7semidirect
229. C2×C42C2×D7semidirect
236. #95C2×D7semidirect
237. #95C2×D7semidirect
428. C7#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#64452
C2×#64382
C7×#3882
#897652
#897632
#897312
#897282
#897272
#897192
#897122
#894472
#894452
#894372
#26847