Finite Group Databasebeta

← #11731463

Subgroups — #11731463

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
1C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
2C2×#6457C211yesmaximal, characteristic, complemented
3C2×#6445C211yesmaximal, characteristic
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#380C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic, complemented
7#89765C211yesmaximal, characteristic, complemented
8#89763C211yesmaximal, characteristic, complemented
9#89761C211yesmaximal, characteristic, complemented
10#89730C211yesmaximal, characteristic, complemented
11#89714C211yesmaximal, characteristic, complemented
12#89447C211yesmaximal, characteristic, complemented
13#89444C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89436C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C22×D28C2211yescharacteristic
21C22×D28C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C22×C4×D7C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25C2×C7×C4⋊C4C2211yescharacteristic, complemented
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×(C2×D7)⋊C4—24no—
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, complemented
39C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
42C2×Dic7⋊C4C2211yescharacteristic, complemented
43C2×Dic7⋊C4C2211yescharacteristic, complemented
44#6457—24no—
45#6445—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
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—41no—
62C23×D7—41no—
63C23×D7D414yes—
64C23×D7C2311yescharacteristic
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—41no—
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C22×Dic7—22no—
80C22×Dic7—22no—
81C22×Dic7—41no—
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C2×D28—22no—
88C2×D28—22no—
89C2×D28—22no—
90C2×D28—22no—
91C2×D28—41no—
92C2×D28—41no—
93C2×C4×D7—22no—
94C2×C4×D7—22no—
95C2×C4×D7—22no—
96C2×C4×D7—22no—
97C2×C4×D7—41no—
98C2×C4×D7—41no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—41no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C22⋊Dic7—22no—
104(C2×D7)⋊C4—22no—
105(C2×D7)⋊C4—22no—
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—22no—
112(C2×D7)⋊C4—22no—
113(C2×D7)⋊C4—41no—
114(C2×D7)⋊C4—41no—
115(C2×D7)⋊C4D414yescomplemented
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#394—71no—
126#386—71no—
127#386—71no—
128#386—71no—
129#385—71no—
130#381—71no—
131#380—71no—
132#380D711yescharacteristic, complemented
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7C2×D412yes—
138C23×C7#4112yes—
139C23×C7#4112yes—
140C23×C7C2411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—28no—
146C22×D7—21no—
147C22×D7—21no—
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—21no—
160C22×D7—44no—
161C22×D7—44no—
162C22×D7—44no—
163C22×D7—44no—
164C22×D7—44no—
165C22×D7C2×D412yes—
166C22×D7C2×D412yes—
167C22×D7C2×D412yes—
168C22×D7#4112yes—
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×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—44no—
187C2×C4×C7C2×D412yes—
188C22⋊D7—44no—
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—44no—
204C2×Dic7C2×D412yes—
205D28—44no—
206D28—44no—
207C4×D7—44no—
208C4×D7—44no—
209C25—71no—
210C22×D4—71no—
211C22×D4—71no—
212C22×D4—71no—
213C23×C4—71no—
214C23×C4—71no—
215C23×C4—71no—
216C23×C4C2×D711yescharacteristic
217#123—144no—
218#120—144no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4C2×D711yescharacteristic, complemented
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—71no—
235C2×C22⋊C4—71no—
236C2×C22⋊C4—144no—
237C2×C22⋊C4—144no—
238C2×C22⋊C4C2×D711yescharacteristic
239C2×C22⋊C4C2×D711yescharacteristic
240#95—71no—
241#95—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95—71no—
246#95C2×D711yescharacteristic, complemented
247#95C2×D711yescharacteristic, complemented
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7—22no—
254C22×C7#12314yescomplemented
255C22×C7C22×D411yescharacteristic
256C22×C7C22×D411yescharacteristic
257C22×C7C2×#4111yescharacteristic
258C22×C7C2×#4111yescharacteristic
259C22×C7C2×#4111yescharacteristic
260C22×C7H(5,𝔽2)11yescharacteristic
261C22×C7H(5,𝔽2)11yescharacteristic
262C2×D7—22no—
263C2×D7—22no—
264C2×D7—22no—
265C2×D7—22no—
266C2×D7—22no—
267C2×D7—22no—
268C2×D7—22no—
269C2×D7—22no—
270C2×D7—22no—
271C2×D7—416no—
272C2×D7—41no—
273C2×D7—41no—
274C2×D7—41no—
275C2×D7—41no—
276C2×D7—41no—
277C2×D7—41no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C2×D7—41no—
283C2×D7#12114yes—
284C4×C7—22no—
285C4×C7—41no—
286C4×C7—41no—
287C4×C7—41no—
288C4×C7—41no—
289C4×C7—41no—
290Dic7—22no—
291Dic7—41no—
292Dic7—41no—
293Dic7—41no—
294Dic7—41no—
295C24—71no—
296C24—71no—
297C24—74no—
298C24—71no—
299C24—142no—
300C24—142no—
301C24—142no—
302C24—142no—
303C24—281no—
304C24—281no—
305C24C22×D711yescharacteristic
306C2×D4—142no—
307C2×D4—142no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—281no—
314C2×D4—281no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—41no—
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—142no—
328C22×C4—142no—
329C22×C4—281no—
330C22×C4—281no—
331C22×C4—281no—
332C22×C4C22×D711yescharacteristic
333C22×C4C22×D711yescharacteristic
334C22×C4C22×D711yescharacteristic
335C22×C4C22×D711yescharacteristic
336C22×C4C22×D711yescharacteristic
337C22×C4C22×D711yescharacteristic
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—142no—
341C4⋊C4—281no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—74no—
345C22⋊C4—142no—
346C22⋊C4—142no—
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—281no—
356C22⋊C4—281no—
357C2×C7—24no—
358C2×C7C2×#12111yescharacteristic
359C2×C7C2×#12311yescharacteristic
360C2×C7#53411yescharacteristic
361C2×C7#54611yescharacteristic
362C2×C7#54611yescharacteristic
363C2×C7#55111yescharacteristic
364C2×C7#55111yescharacteristic
365D7—24no—
366D7—42no—
367D7—81no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—72no—
373C23—72no—
374C23—72no—
375C23—72no—
376C23—148no—
377C23—141no—
378C23—141no—
379C23—141no—
380C23—141no—
381C23—148no—
382C23—141no—
383C23—141no—
384C23—141no—
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#115712yes—
401C23D4×D712yes—
402C23#116112yes—
403C23C23×D711yescharacteristic
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—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—44no—
425C2×C4—72no—
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×C4—284no—
441C2×C4—284no—
442C2×C4—284no—
443C2×C4D4×D712yes—
444C7#241211yescharacteristic, complemented, Sylow
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—74no—
452C22—142no—
453C22—142no—
454C22—142no—
455C22—142no—
456C22—142no—
457C22—142no—
458C22—142no—
459C22—142no—
460C22—142no—
461C22—2816no—
462C22—281no—
463C22—281no—
464C22—281no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22—281no—
472C22—281no—
473C22#645714yescomplemented
474C22C2×#115711yescharacteristic
475C22C2×D4×D711yescharacteristic
476C22C2×D4×D711yescharacteristic
477C22#654811yescharacteristic
478C22C2×#116111yescharacteristic
479C22D7×#4111yescharacteristic
480C22#655311yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—142no—
488C4—281no—
489C4—281no—
490C4—281no—
491C4—281no—
492C2—24no—
493C2—144no—
494C2—282no—
495C2—561no—
496C2#9018111yescharacteristic
497C2C2×#645711yescharacteristic
498C2#9021811yescharacteristic
499C2D7×#12111yescharacteristic
500C2#9029611yescharacteristic
501C2#9032111yescharacteristic
502C2#9032811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#380C2
prime radicalC2×#6445C2
hypercenter#380D7
hypocenterC7#2412
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C22×(C2×D7)⋊C4C2semidirect
2. C2×#6457C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#380C2semidirect
6. #89985C2semidirect
7. #89765C2semidirect
8. #89763C2semidirect
9. #89761C2semidirect
10. #89730C2semidirect
11. #89714C2semidirect
12. #89447C2semidirect
13. #89444C2semidirect
14. #89437C2semidirect
15. #89436C2semidirect
24. C2×C7×C4⋊C4C22semidirect
25. C2×C7×C4⋊C4C22semidirect
38. C2×(C2×D7)⋊C4C22semidirect
39. C2×(C2×D7)⋊C4C22semidirect
40. C2×(C2×D7)⋊C4C22semidirect
41. C2×(C2×D7)⋊C4C22semidirect
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
115. (C2×D7)⋊C4D4semidirect
132. #380D7semidirect
221. C2×C4⋊C4C2×D7semidirect
222. C2×C4⋊C4C2×D7semidirect
246. #95C2×D7semidirect
247. #95C2×D7semidirect
254. C22×C7#123semidirect
444. C7#2412semidirect
473. C22#6457semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D7)⋊C42
C2×#64572
C2×#64452
C2×D7×C22⋊C42
C7×#3802
#899852
#897652
#897632
#897612
#897302
#897142
#894472
#894442
#894372
#894362
#24127