Finite Group Databasebeta

← #11735546

Subgroups — #11735546

Order 896.

800 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×#6457C211yesmaximal, characteristic, complemented
2C2×#6456C211yesmaximal, characteristic, complemented
3C2×#6445C211yesmaximal, characteristic
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#399C211yesmaximal, characteristic, complemented
6#89765C211yesmaximal, characteristic, complemented
7#89763C211yesmaximal, characteristic, complemented
8#89761C211yesmaximal, characteristic, complemented
9#89759C211yesmaximal, characteristic, complemented
10#89733C211yesmaximal, characteristic, complemented
11#89728C211yesmaximal, characteristic, complemented
12#89727C211yesmaximal, characteristic, complemented
13#89716C211yesmaximal, characteristic, complemented
14#89442C211yesmaximal, characteristic, complemented
15#89437C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×D28C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic, complemented
23C2×C7×C4⋊C4C2211yescharacteristic, complemented
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25C2×C7×C4⋊C4C2211yescharacteristic, complemented
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic, complemented
43C2×Dic7⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic7C2211yescharacteristic, complemented
45#6457—24no—
46#6456—24no—
47#6445—24no—
48D7×C22⋊C4—24no—
49C7×#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55#2418—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—41no—
63C23×D7—41no—
64C23×D7C2311yescharacteristic
65C23×D7C2311yescharacteristic
66C23×D7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
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—
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C2×D28—22no—
82C2×D28—22no—
83C2×D28—22no—
84C2×D28—22no—
85C2×D28—41no—
86C2×D28D414yes—
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—41no—
94C2×C4×D7—41no—
95C2×C4×D7—41no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—41no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—22no—
101C7×C22⋊C4—41no—
102C22⋊Dic7—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—41no—
113(C2×D7)⋊C4—41no—
114(C2×D7)⋊C4D414yescomplemented
115C4⋊Dic7—41no—
116Dic7⋊C4—22no—
117C4×Dic7—22no—
118C2×#123—71no—
119C2×#120—71no—
120C2×C4×D4—71no—
121C22×C22⋊C4—71no—
122#399D711yescharacteristic, complemented
123#397—71no—
124#394—71no—
125#394—71no—
126#388—71no—
127#388—71no—
128#386—71no—
129#386—71no—
130#385—71no—
131#380—71no—
132#379—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
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—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—44no—
159C22×D7—44no—
160C22×D7—44no—
161C22×D7—44no—
162C22×D7—44no—
163C22×D7—44no—
164C22×D7C2×D412yes—
165C22×D7C2×D412yes—
166C22×D7C2×D412yes—
167C22×D7#4112yes—
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×C7C2×D412yes—
186C2×C4×C7C2×D412yes—
187C22⋊D7—44no—
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×Dic7C2×D412yes—
203D28—28no—
204D28—44no—
205C4×D7—44no—
206C4×D7—44no—
207C4×D7—44no—
208C25—71no—
209C22×D4—71no—
210C22×D4—71no—
211C22×D4—71no—
212C23×C4—71no—
213C23×C4—71no—
214C23×C4—71no—
215#123—144no—
216#120—144no—
217C4×D4—144no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4C2×D711yescharacteristic, complemented
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⋊C4C2×D711yescharacteristic
238C2×C22⋊C4C2×D711yescharacteristic
239C2×C22⋊C4C2×D711yescharacteristic
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×C7C22×D411yescharacteristic
255C22×C7C22×D411yescharacteristic
256C22×C7C2×#4111yescharacteristic
257C22×C7C2×#4111yescharacteristic
258C22×C7C2×#4111yescharacteristic
259C22×C7H(5,𝔽2)11yescharacteristic
260C22×C7H(5,𝔽2)11yescharacteristic
261C2×D7—22no—
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—22no—
272C2×D7—22no—
273C2×D7—416no—
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#12314yescomplemented
283C4×C7—22no—
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—
294C24—71no—
295C24—71no—
296C24—71no—
297C24—142no—
298C24—142no—
299C24—142no—
300C24—142no—
301C24—142no—
302C24—281no—
303C24—281no—
304C24C22×D711yescharacteristic
305C2×D4—74no—
306C2×D4—142no—
307C2×D4—142no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—281no—
313C2×D4—281no—
314C22×C4—71no—
315C22×C4—71no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—142no—
320C22×C4—142no—
321C22×C4—142no—
322C22×C4—142no—
323C22×C4—142no—
324C22×C4—142no—
325C22×C4—281no—
326C22×C4—281no—
327C22×C4—281no—
328C22×C4C22×D711yescharacteristic
329C22×C4C22×D711yescharacteristic
330C22×C4C22×D711yescharacteristic
331C22×C4C22×D711yescharacteristic
332C22×C4C22×D711yescharacteristic
333C22×C4C22×D711yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—41no—
337C4⋊C4—142no—
338C4⋊C4—281no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—41no—
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—142no—
352C22⋊C4—142no—
353C22⋊C4—281no—
354C22⋊C4—281no—
355C42—142no—
356C2×C7—42no—
357C2×C7C2×#12311yescharacteristic
358C2×C7#52511yescharacteristic
359C2×C7#53811yescharacteristic
360C2×C7D4211yescharacteristic
361C2×C7#54611yescharacteristic
362C2×C7#54611yescharacteristic
363C2×C7#55111yescharacteristic
364D7—24no—
365D7—42no—
366D7—42no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—72no—
374C23—72no—
375C23—72no—
376C23—72no—
377C23—72no—
378C23—148no—
379C23—141no—
380C23—141no—
381C23—141no—
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—284no—
401C23#116112yes—
402C23C23×D711yescharacteristic
403D4—148no—
404D4—284no—
405D4—284no—
406C2×C4—21no—
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—72no—
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—284no—
438C2×C4—284no—
439C2×C4—284no—
440C2×C4D4×D712yes—
441C2×C4D4×D712yes—
442C7#241811yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—74no—
451C22—142no—
452C22—142no—
453C22—142no—
454C22—142no—
455C22—142no—
456C22—142no—
457C22—142no—
458C22—142no—
459C22—142no—
460C22—142no—
461C22—142no—
462C22—142no—
463C22—2816no—
464C22—281no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22—281no—
472C22C2×D4×D711yescharacteristic
473C22C2×D4×D711yescharacteristic
474C22#654811yescharacteristic
475C22C2×#116111yescharacteristic
476C22D7×#4111yescharacteristic
477C22D7×#4111yescharacteristic
478C22#655311yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—142no—
486C4—281no—
487C4—281no—
488C4—281no—
489C4—281no—
490C2—42no—
491C2—144no—
492C2—282no—
493C2—282no—
494C2#9031811yescharacteristic
495C2#9032111yescharacteristic
496C2#9032811yescharacteristic
497C2D7×#12311yescharacteristic
498C2#9034511yescharacteristic
499C2#9034811yescharacteristic
500C2#9040811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC2×#6445C2
hypercenter#399D7
hypocenterC7#2418
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6457C2semidirect
2. C2×#6456C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#399C2semidirect
6. #89765C2semidirect
7. #89763C2semidirect
8. #89761C2semidirect
9. #89759C2semidirect
10. #89733C2semidirect
11. #89728C2semidirect
12. #89727C2semidirect
13. #89716C2semidirect
14. #89442C2semidirect
15. #89437C2semidirect
22. C22×C4×D7C22semidirect
23. C2×C7×C4⋊C4C22semidirect
24. C2×C7×C4⋊C4C22semidirect
25. C2×C7×C4⋊C4C22semidirect
39. C2×(C2×D7)⋊C4C22semidirect
40. C2×(C2×D7)⋊C4C22semidirect
41. C2×(C2×D7)⋊C4C22semidirect
42. C2×C4⋊Dic7C22semidirect
43. C2×Dic7⋊C4C22semidirect
44. C2×C4×Dic7C22semidirect
49. C7×#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
114. (C2×D7)⋊C4D4semidirect
122. #399D7semidirect
220. C2×C4⋊C4C2×D7semidirect
221. C2×C4⋊C4C2×D7semidirect
222. C2×C4⋊C4C2×D7semidirect
246. #95C2×D7semidirect
282. C2×D7#123semidirect
442. C7#2418semidirect

Maximal subgroups

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

subgroupindex
C2×#64572
C2×#64562
C2×#64452
C2×D7×C22⋊C42
C7×#3992
#897652
#897632
#897612
#897592
#897332
#897282
#897272
#897162
#894422
#894372
#24187