Finite Group Databasebeta

← #11734555

Subgroups — #11734555

Order 896.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6500C211yesmaximal, characteristic
2C2×#6446C211yesmaximal, characteristic, complemented
3C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#394C211yesmaximal, characteristic, complemented
6#89994C211yesmaximal, characteristic, complemented
7#89761C211yesmaximal, characteristic, complemented
8#89759C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89728C211yesmaximal, characteristic, complemented
12#89725C211yesmaximal, characteristic, complemented
13#89724C211yesmaximal, characteristic, complemented
14#89439C211yesmaximal, characteristic, complemented
15#89436C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C22×C4×D7C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic, complemented
28C2×C7×C22⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic, complemented
33C2×C22⋊Dic7C2211yescharacteristic, complemented
34#6500—24no—
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×C4⋊Dic7C2211yescharacteristic, complemented
42C2×Dic7⋊C4C2211yescharacteristic, complemented
43C2×Dic7⋊C4C2211yescharacteristic, complemented
44#6446—24no—
45D7×C22⋊C4—24no—
46D7×C22⋊C4—24no—
47C7×#95C2211yescharacteristic, complemented
48C7⋊#95C2211yescharacteristic, complemented
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2412—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—41no—
63C23×D7—41no—
64C23×D7D414yes—
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C2×C22⋊D7—22no—
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C2×C22⋊D7—22no—
80C2×C22⋊D7—22no—
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C2×C4×D7—22no—
88C2×C4×D7—22no—
89C2×C4×D7—22no—
90C2×C4×D7—22no—
91C2×C4×D7—22no—
92C2×C4×D7—22no—
93C2×C4×D7—22no—
94C2×C4×D7—22no—
95C2×C4×D7—41no—
96C2×C4×D7—41no—
97C2×C4×D7—41no—
98C2×C4×D7—41no—
99C7×C4⋊C4—41no—
100C7×C22⋊C4—22no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—41no—
104C22⋊Dic7—22no—
105C22⋊Dic7—22no—
106C22⋊Dic7—22no—
107C22⋊Dic7—22no—
108C22⋊Dic7—41no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—22no—
112(C2×D7)⋊C4—22no—
113(C2×D7)⋊C4—22no—
114(C2×D7)⋊C4D414yescomplemented
115C4⋊Dic7—22no—
116Dic7⋊C4—22no—
117Dic7⋊C4—41no—
118C2×#123—71no—
119C2×#120—71no—
120C22×C22⋊C4—71no—
121C22×C22⋊C4—71no—
122#397—71no—
123#397—71no—
124#394—71no—
125#394D711yescharacteristic, complemented
126#386—71no—
127#386—71no—
128#386—71no—
129#385—71no—
130#381—71no—
131#380—71no—
132#380—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
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×C7C2×D412yes—
147C23×C7C2411yescharacteristic
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—28no—
157C22×D7—28no—
158C22×D7—44no—
159C22×D7—44no—
160C22×D7—44no—
161C22×D7—44no—
162C22×D7—44no—
163C22×D7C2×D412yes—
164C22×D7C2×D412yes—
165C22×D7C2×D412yes—
166C22×D7#4112yes—
167C22×D7#4112yes—
168C22×D7#4112yes—
169C7×D4—44no—
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×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7C2×D412yes—
185C22⋊D7—44no—
186C22⋊D7—44no—
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×Dic7C2×D412yes—
205C4×D7—44no—
206C4×D7—44no—
207C4×D7—44no—
208C4×D7—44no—
209C25—71no—
210C22×D4—71no—
211C22×D4—71no—
212C22×D4C2×D711yescharacteristic
213C23×C4—71no—
214C23×C4—71no—
215C23×C4—71no—
216C23×C4—71no—
217#123—144no—
218#120—144no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4—71no—
222C2×C4⋊C4C2×D711yescharacteristic, complemented
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—144no—
235C2×C22⋊C4—144no—
236C2×C22⋊C4C2×D711yescharacteristic
237C2×C22⋊C4C2×D711yescharacteristic
238C2×C22⋊C4C2×D711yescharacteristic, complemented
239C2×C22⋊C4C2×D711yescharacteristic, complemented
240#95—71no—
241#95—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95—71no—
246#95—71no—
247#95C2×D711yescharacteristic, complemented
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—41no—
252C22×C7—41no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7C22×D411yescharacteristic
263C22×C7C22×D411yescharacteristic
264C22×C7C2×#4111yescharacteristic
265C22×C7C2×#4111yescharacteristic
266C22×C7C2×#4111yescharacteristic
267C22×C7H(5,𝔽2)11yescharacteristic
268C22×C7H(5,𝔽2)11yescharacteristic
269C2×D7—22no—
270C2×D7—22no—
271C2×D7—22no—
272C2×D7—22no—
273C2×D7—22no—
274C2×D7—22no—
275C2×D7—22no—
276C2×D7—22no—
277C2×D7—22no—
278C2×D7—22no—
279C2×D7—22no—
280C2×D7—22no—
281C2×D7—416no—
282C2×D7#12114yes—
283C2×D7#12314yescomplemented
284C4×C7—22no—
285C4×C7—41no—
286C4×C7—41no—
287C4×C7—41no—
288C4×C7—41no—
289Dic7—22no—
290Dic7—41no—
291Dic7—41no—
292Dic7—41no—
293Dic7—41no—
294Dic7—41no—
295C24—71no—
296C24—71no—
297C24—74no—
298C24—142no—
299C24—142no—
300C24—142no—
301C24—142no—
302C24—281no—
303C24—281no—
304C24C22×D711yescharacteristic
305C24C22×D711yescharacteristic
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—142no—
314C2×D4—142no—
315C22×C4—71no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—71no—
320C22×C4—71no—
321C22×C4—142no—
322C22×C4—142no—
323C22×C4—142no—
324C22×C4—142no—
325C22×C4—142no—
326C22×C4—142no—
327C22×C4—142no—
328C22×C4—142no—
329C22×C4—281no—
330C22×C4—281no—
331C22×C4—281no—
332C22×C4—281no—
333C22×C4C22×D711yescharacteristic
334C22×C4C22×D711yescharacteristic
335C22×C4C22×D711yescharacteristic
336C22×C4C22×D711yescharacteristic
337C22×C4C22×D711yescharacteristic
338C4⋊C4—41no—
339C4⋊C4—142no—
340C4⋊C4—142no—
341C4⋊C4—281no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—74no—
347C22⋊C4—142no—
348C22⋊C4—142no—
349C22⋊C4—142no—
350C22⋊C4—142no—
351C22⋊C4—142no—
352C22⋊C4—142no—
353C22⋊C4—142no—
354C22⋊C4—142no—
355C22⋊C4—142no—
356C22⋊C4—281no—
357C2×C7—42no—
358C2×C7—81no—
359C2×C7C2×#12111yescharacteristic
360C2×C7C2×#12311yescharacteristic
361C2×C7#53411yescharacteristic
362C2×C7#54611yescharacteristic
363C2×C7#54611yescharacteristic
364C2×C7#55111yescharacteristic
365C2×C7#55111yescharacteristic
366D7—24no—
367D7—24no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—72no—
382C23—72no—
383C23—72no—
384C23—72no—
385C23—72no—
386C23—72no—
387C23—148no—
388C23—148no—
389C23—141no—
390C23—141no—
391C23—141no—
392C23—141no—
393C23—141no—
394C23—141no—
395C23—141no—
396C23—141no—
397C23—284no—
398C23—284no—
399C23—284no—
400C23—284no—
401C23—284no—
402C23D4×D712yes—
403C23C23×D711yescharacteristic
404D4—44no—
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—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—72no—
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—284no—
440C2×C4—284no—
441C2×C4—284no—
442C2×C4—284no—
443C2×C4D4×D712yes—
444C7#241211yescharacteristic, complemented, Sylow
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—74no—
460C22—74no—
461C22—142no—
462C22—142no—
463C22—142no—
464C22—142no—
465C22—142no—
466C22—142no—
467C22—142no—
468C22—142no—
469C22—142no—
470C22—142no—
471C22—142no—
472C22—142no—
473C22—2816no—
474C22C2×D4×D711yescharacteristic
475C22C2×D4×D711yescharacteristic
476C22#654811yescharacteristic
477C22#654811yescharacteristic
478C22D7×#4111yescharacteristic
479C22D7×#4111yescharacteristic
480C22D7×#4111yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—142no—
487C4—281no—
488C4—281no—
489C4—281no—
490C4—281no—
491C4—281no—
492C2—42no—
493C2—81no—
494C2—144no—
495C2—144no—
496C2#9028011yescharacteristic
497C2D7×#12111yescharacteristic
498C2#9030111yescharacteristic
499C2#9030311yescharacteristic
500C2D7×#12311yescharacteristic
501C2#9035011yescharacteristic
502C2#9037011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6500C2
hypercenter#394D7
hypocenterC7#2412
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6446C2semidirect
3. C2×D7×C22⋊C4C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#394C2semidirect
6. #89994C2semidirect
7. #89761C2semidirect
8. #89759C2semidirect
9. #89728C2semidirect
10. #89728C2semidirect
11. #89728C2semidirect
12. #89725C2semidirect
13. #89724C2semidirect
14. #89439C2semidirect
15. #89436C2semidirect
24. C2×C7×C4⋊C4C22semidirect
27. C2×C7×C22⋊C4C22semidirect
28. C2×C7×C22⋊C4C22semidirect
32. C2×C22⋊Dic7C22semidirect
33. C2×C22⋊Dic7C22semidirect
41. C2×C4⋊Dic7C22semidirect
42. C2×Dic7⋊C4C22semidirect
43. C2×Dic7⋊C4C22semidirect
47. C7×#95C22semidirect
48. C7⋊#95C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
114. (C2×D7)⋊C4D4semidirect
125. #394D7semidirect
222. C2×C4⋊C4C2×D7semidirect
238. C2×C22⋊C4C2×D7semidirect
239. C2×C22⋊C4C2×D7semidirect
247. #95C2×D7semidirect
283. C2×D7#123semidirect
444. C7#2412semidirect

Maximal subgroups

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

subgroupindex
C2×#65002
C2×#64462
C2×D7×C22⋊C42
C7×#3942
#899942
#897612
#897592
#897282
#897252
#897242
#894392
#894362
#24127