Finite Group Databasebeta

← #11734370

Subgroups — #11734370

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
1C2×#6500C211yesmaximal, characteristic
2C2×#6460C211yesmaximal, characteristic, complemented
3C2×#6448C211yesmaximal, characteristic, complemented
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#394C211yesmaximal, characteristic, complemented
6#89993C211yesmaximal, characteristic, complemented
7#89762C211yesmaximal, characteristic, complemented
8#89733C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89727C211yesmaximal, characteristic, complemented
12#89723C211yesmaximal, characteristic, complemented
13#89719C211yesmaximal, characteristic, complemented
14#89446C211yesmaximal, characteristic, complemented
15#89438C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C22×Dic14C2211yescharacteristic, complemented
23C2×C7×C4⋊C4C2211yescharacteristic, complemented
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic, complemented
27C2×C7×C22⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic, complemented
32C2×C22⋊Dic7C2211yescharacteristic, complemented
33#6500—24no—
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
44C2×C4×Dic7C2211yescharacteristic, complemented
45C2×C4×Dic7C2211yescharacteristic, complemented
46#6460—24no—
47#6448—24no—
48D7×C22⋊C4—24no—
49C7×#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2472—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—22no—
63C23×D7—41no—
64C23×D7—41no—
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—22no—
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—41no—
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—41no—
92C2×C4×D7—41no—
93C2×Dic14—22no—
94C2×Dic14—22no—
95C2×Dic14—41no—
96C7×C4⋊C4—22no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—41no—
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103C22⋊Dic7—22no—
104C22⋊Dic7—41no—
105C22⋊Dic7—41no—
106(C2×D7)⋊C4—22no—
107(C2×D7)⋊C4—22no—
108(C2×D7)⋊C4—22no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—41no—
112(C2×D7)⋊C4D414yes—
113(C2×D7)⋊C4D414yescomplemented
114C4⋊Dic7—22no—
115C4⋊Dic7—22no—
116C4×Dic7—22no—
117C4×Dic7—41no—
118C2×#124—71no—
119C2×C22⋊Q8—71no—
120C2×#120—71no—
121C22×C22⋊C4—71no—
122#397—71no—
123#394—71no—
124#394D711yescharacteristic, complemented
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—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×C7#4112yes—
147C23×C7C2411yescharacteristic
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—28no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—44no—
160C22×D7—44no—
161C22×D7—44no—
162C22×D7—44no—
163C22×D7—44no—
164C22×D7—44no—
165C22×D7C2×D412yes—
166C22×D7C2×D412yes—
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×C7C2×D412yes—
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×Dic7C2×D412yes—
204C2×Dic7C2×D412yes—
205C4×D7—44no—
206C4×D7—44no—
207Dic14—44no—
208C25—71no—
209C22×Q8—71no—
210C22×D4—71no—
211C22×D4—71no—
212C22×D4C2×D711yescharacteristic
213C23×C4—71no—
214C23×C4—71no—
215#124—144no—
216C22⋊Q8—144no—
217#120—144no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4C2×D711yescharacteristic, complemented
221C2×C22⋊C4—71no—
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—144no—
235C2×C22⋊C4C2×D711yescharacteristic
236C2×C22⋊C4C2×D711yescharacteristic
237C2×C22⋊C4C2×D711yescharacteristic, complemented
238C2×C22⋊C4C2×D711yescharacteristic, complemented
239C2×C42—71no—
240C2×C42—71no—
241#95—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95—71no—
246#95C2×D711yescharacteristic, complemented
247C22×C7—22no—
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—41no—
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×C7C22×D411yescharacteristic
262C22×C7C22×D411yescharacteristic
263C22×C7C2×#4111yescharacteristic
264C22×C7C2×#4111yescharacteristic
265C22×C7C2×#4111yescharacteristic
266C22×C7H(5,𝔽2)11yescharacteristic
267C22×C7H(5,𝔽2)11yescharacteristic
268C2×D7—22no—
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—416no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C2×D7#12414yescomplemented
283C4×C7—22no—
284C4×C7—22no—
285C4×C7—41no—
286C4×C7—41no—
287C4×C7—41no—
288Dic7—22no—
289Dic7—22no—
290Dic7—41no—
291Dic7—41no—
292Dic7—41no—
293Dic7—41no—
294C24—71no—
295C24—71no—
296C24—142no—
297C24—142no—
298C24—142no—
299C24—142no—
300C24—142no—
301C24—281no—
302C24—281no—
303C24C22×D711yescharacteristic
304C24C22×D711yescharacteristic
305C2×Q8—142no—
306C2×Q8—142no—
307C2×Q8—281no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—142no—
314C2×D4—142no—
315C2×D4—142no—
316C2×D4—281no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—71no—
320C22×C4—71no—
321C22×C4—71no—
322C22×C4—71no—
323C22×C4—142no—
324C22×C4—142no—
325C22×C4—142no—
326C22×C4—142no—
327C22×C4—281no—
328C22×C4—281no—
329C22×C4C22×D711yescharacteristic
330C22×C4C22×D711yescharacteristic
331C22×C4C22×D711yescharacteristic
332C22×C4C22×D711yescharacteristic
333C22×C4C22×D711yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—142no—
336C4⋊C4—142no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—74no—
342C22⋊C4—74no—
343C22⋊C4—142no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4—142no—
349C22⋊C4—142no—
350C22⋊C4—142no—
351C22⋊C4—281no—
352C22⋊C4—281no—
353C22⋊C4—281no—
354C42—142no—
355C42—281no—
356C2×C7—42no—
357C2×C7—81no—
358C2×C7#52511yescharacteristic
359C2×C7C2×#12411yescharacteristic
360C2×C7#53811yescharacteristic
361C2×C7D4211yescharacteristic
362C2×C7#54611yescharacteristic
363C2×C7#54611yescharacteristic
364C2×C7#55111yescharacteristic
365D7—24no—
366D7—42no—
367C23—21no—
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—72no—
381C23—72no—
382C23—72no—
383C23—72no—
384C23—148no—
385C23—141no—
386C23—141no—
387C23—141no—
388C23—141no—
389C23—141no—
390C23—141no—
391C23—141no—
392C23—141no—
393C23—141no—
394C23—141no—
395C23—284no—
396C23—284no—
397C23—284no—
398C23—284no—
399C23—284no—
400C23—284no—
401C23#115912yes—
402C23C23×D711yescharacteristic
403Q8—284no—
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—72no—
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—284no—
439C2×C4—284no—
440C2×C4D4×D712yes—
441C2×C4D4×D712yes—
442C7#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—74no—
458C22—142no—
459C22—142no—
460C22—142no—
461C22—142no—
462C22—142no—
463C22—142no—
464C22—142no—
465C22—142no—
466C22—142no—
467C22—2816no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22—281no—
472C22C2×D4×D711yescharacteristic
473C22C2×D4×D711yescharacteristic
474C22C2×#115911yescharacteristic
475C22#654811yescharacteristic
476C22D7×#4111yescharacteristic
477C22D7×#4111yescharacteristic
478C22#655311yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—142no—
485C4—142no—
486C4—281no—
487C4—281no—
488C4—281no—
489C4—281no—
490C2—42no—
491C2—81no—
492C2—144no—
493C2—282no—
494C2#9028211yescharacteristic
495C2#9029711yescharacteristic
496C2#9030011yescharacteristic
497C2#9030311yescharacteristic
498C2#9034811yescharacteristic
499C2#9034911yescharacteristic
500C2D7×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6500C2
hypercenter#394D7
hypocenterC7#2472
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6460C2semidirect
3. C2×#6448C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#394C2semidirect
6. #89993C2semidirect
7. #89762C2semidirect
8. #89733C2semidirect
9. #89728C2semidirect
10. #89728C2semidirect
11. #89727C2semidirect
12. #89723C2semidirect
13. #89719C2semidirect
14. #89446C2semidirect
15. #89438C2semidirect
22. C22×Dic14C22semidirect
23. C2×C7×C4⋊C4C22semidirect
26. C2×C7×C22⋊C4C22semidirect
27. C2×C7×C22⋊C4C22semidirect
31. C2×C22⋊Dic7C22semidirect
32. C2×C22⋊Dic7C22semidirect
42. C2×C4⋊Dic7C22semidirect
43. C2×C4⋊Dic7C22semidirect
44. C2×C4×Dic7C22semidirect
45. C2×C4×Dic7C22semidirect
49. C7×#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
113. (C2×D7)⋊C4D4semidirect
124. #394D7semidirect
220. C2×C4⋊C4C2×D7semidirect
237. C2×C22⋊C4C2×D7semidirect
238. C2×C22⋊C4C2×D7semidirect
246. #95C2×D7semidirect
282. C2×D7#124semidirect
442. C7#2472semidirect

Maximal subgroups

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

subgroupindex
C2×#65002
C2×#64602
C2×#64482
C2×D7×C22⋊C42
C7×#3942
#899932
#897622
#897332
#897282
#897272
#897232
#897192
#894462
#894382
#24727