Finite Group Databasebeta

← #11734585

Subgroups — #11734585

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×#6500C211yesmaximal, characteristic
2C2×#6448C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89992C211yesmaximal, characteristic, complemented
5#89765C211yesmaximal, characteristic, complemented
6#89762C211yesmaximal, characteristic, complemented
7#89733C211yesmaximal, characteristic, complemented
8#89730C211yesmaximal, characteristic, complemented
9#89730C211yesmaximal, characteristic, complemented
10#89727C211yesmaximal, characteristic, complemented
11#89727C211yesmaximal, characteristic, complemented
12#89723C211yesmaximal, characteristic, complemented
13#89719C211yesmaximal, characteristic, complemented
14#89446C211yesmaximal, characteristic, complemented
15#89445C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic, complemented
21C22×Dic14C2211yescharacteristic, complemented
22C2×C7×C4⋊C4C2211yescharacteristic, complemented
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic, complemented
26C2×C7×C22⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic, complemented
31#6500—24no—
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
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×C4⋊Dic7C2211yescharacteristic, complemented
43C2×C4×Dic7C2211yescharacteristic, complemented
44#6448—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
55C24×C7C2311yescharacteristic
56C23×D7—22no—
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—41no—
60C23×D7—41no—
61C23×D7—41no—
62C23×D7—41no—
63C23×D7C2311yescharacteristic
64C23×D7C2311yescharacteristic
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—41no—
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—41no—
76C2×C22⋊D7—41no—
77C2×C22⋊D7—41no—
78C2×C22⋊D7D414yes—
79C22×Dic7—41no—
80C22×Dic7—41no—
81C22×Dic7—41no—
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
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—
95C7×C22⋊C4—41no—
96C22⋊Dic7—22no—
97C22⋊Dic7—22no—
98C22⋊Dic7—41no—
99C22⋊Dic7—41no—
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—
110C4⋊Dic7—41no—
111C4×Dic7—22no—
112C2×#124—71no—
113C2×#120—71no—
114#397—71no—
115#397—71no—
116#394—71no—
117#394D711yescharacteristic, complemented
118#393—71no—
119#393—71no—
120#388—71no—
121#388—71no—
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—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7C2×D412yes—
140C23×C7#4112yes—
141C23×C7C2411yescharacteristic
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—21no—
154C22×D7—44no—
155C22×D7—44no—
156C22×D7—44no—
157C22×D7—44no—
158C22×D7—44no—
159C22×D7—44no—
160C22×D7—44no—
161C22×D7C2×D412yes—
162C22×D7#4112yes—
163C7×D4—44no—
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—
179C22⋊D7—44no—
180C22⋊D7—44no—
181C2×Dic7—21no—
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—
197C2×Dic7—21no—
198C2×Dic7—44no—
199C2×Dic7C2×D412yes—
200Dic14—44no—
201C25—71no—
202C22×Q8—71no—
203C22×D4—71no—
204C22×D4—71no—
205C22×D4C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
226C2×C22⋊C4C2×D711yescharacteristic
227C2×C22⋊C4C2×D711yescharacteristic, complemented
228C2×C22⋊C4C2×D711yescharacteristic, complemented
229C2×C42—71no—
230#95—71no—
231#95—71no—
232#95—71no—
233#95—71no—
234#95—71no—
235#95—71no—
236#95—71no—
237#95C2×D711yescharacteristic, complemented
238C22×C7—22no—
239C22×C7—22no—
240C22×C7—22no—
241C22×C7—22no—
242C22×C7—22no—
243C22×C7—22no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7C22×D411yescharacteristic
253C22×C7C2×#4111yescharacteristic
254C22×C7C2×#4111yescharacteristic
255C22×C7H(5,𝔽2)11yescharacteristic
256C22×C7H(5,𝔽2)11yescharacteristic
257C22×C7H(5,𝔽2)11yescharacteristic
258C22×C7H(5,𝔽2)11yescharacteristic
259C2×D7—22no—
260C2×D7—22no—
261C2×D7—22no—
262C2×D7—22no—
263C2×D7—22no—
264C2×D7—22no—
265C2×D7—416no—
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—41no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—41no—
279Dic7—22no—
280Dic7—41no—
281Dic7—41no—
282Dic7—41no—
283Dic7—41no—
284Dic7—41no—
285C24—71no—
286C24—71no—
287C24—142no—
288C24—142no—
289C24—142no—
290C24—281no—
291C24—281no—
292C24—281no—
293C24—281no—
294C24C22×D711yescharacteristic
295C24C22×D711yescharacteristic
296C2×Q8—142no—
297C2×Q8—281no—
298C2×Q8—281no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—74no—
303C2×D4—142no—
304C2×D4—142no—
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—71no—
314C22×C4—281no—
315C22×C4—281no—
316C22×C4—281no—
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—41no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C22⋊C4—142no—
331C22⋊C4—142no—
332C22⋊C4—142no—
333C22⋊C4—142no—
334C22⋊C4—142no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C22⋊C4—281no—
341C22⋊C4—281no—
342C42—142no—
343C2×C7—42no—
344C2×C7—42no—
345C2×C7#53411yescharacteristic
346C2×C7#53811yescharacteristic
347C2×C7#53811yescharacteristic
348C2×C7#54611yescharacteristic
349C2×C7#54611yescharacteristic
350C2×C7#55111yescharacteristic
351C2×C7H(3,𝔽4)11yescharacteristic
352D7—42no—
353D7—42no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—72no—
367C23—72no—
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—284no—
381C23—284no—
382C23—284no—
383C23—284no—
384C23—284no—
385C23—284no—
386C23—284no—
387C23D4×D712yes—
388C23#115912yes—
389C23C23×D711yescharacteristic
390Q8—284no—
391D4—44no—
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—72no—
410C2×C4—141no—
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—141no—
427C2×C4—284no—
428C7#268411yescharacteristic, complemented, Sylow
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—142no—
444C22—142no—
445C22—142no—
446C22—142no—
447C22—142no—
448C22—142no—
449C22—2816no—
450C22—281no—
451C22—281no—
452C22—281no—
453C22—281no—
454C22—281no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22C2×D4×D711yescharacteristic
459C22C2×#115911yescharacteristic
460C22#654811yescharacteristic
461C22#654811yescharacteristic
462C22D7×#4111yescharacteristic
463C22#655311yescharacteristic
464C22#655311yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—142no—
471C4—281no—
472C4—281no—
473C4—281no—
474C4—281no—
475C4—281no—
476C2—42no—
477C2—42no—
478C2—282no—
479C2—282no—
480C2#9028211yescharacteristic
481C2#9029611yescharacteristic
482C2#9030411yescharacteristic
483C2#9030811yescharacteristic
484C2#9034511yescharacteristic
485C2#9034911yescharacteristic
486C2#9037411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6500C2
hypercenter#394D7
hypocenterC7#2684
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6448C2semidirect
3. C7×#394C2semidirect
4. #89992C2semidirect
5. #89765C2semidirect
6. #89762C2semidirect
7. #89733C2semidirect
8. #89730C2semidirect
9. #89730C2semidirect
10. #89727C2semidirect
11. #89727C2semidirect
12. #89723C2semidirect
13. #89719C2semidirect
14. #89446C2semidirect
15. #89445C2semidirect
20. C23×Dic7C22semidirect
21. C22×Dic14C22semidirect
22. C2×C7×C4⋊C4C22semidirect
25. C2×C7×C22⋊C4C22semidirect
26. C2×C7×C22⋊C4C22semidirect
30. C2×C22⋊Dic7C22semidirect
42. C2×C4⋊Dic7C22semidirect
43. C2×C4×Dic7C22semidirect
45. C7×#95C22semidirect
46. C7⋊#95C22semidirect
47. C7⋊#95C22semidirect
48. C7⋊#95C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
117. #394D7semidirect
210. C2×C4⋊C4C2×D7semidirect
227. C2×C22⋊C4C2×D7semidirect
228. C2×C22⋊C4C2×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×#65002
C2×#64482
C7×#3942
#899922
#897652
#897622
#897332
#897302
#897272
#897232
#897192
#894462
#894452
#26847