Finite Group Databasebeta

← #11740214

Subgroups — #11740214

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×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic, complemented
3C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
4C2×#6445C211yesmaximal, characteristic
5#90003C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic, complemented
7#89765C211yesmaximal, characteristic, complemented
8#89763C211yesmaximal, characteristic, complemented
9#89763C211yesmaximal, characteristic, complemented
10#89762C211yesmaximal, characteristic, complemented
11#89762C211yesmaximal, characteristic, complemented
12#89760C211yesmaximal, characteristic, complemented
13#89730C211yesmaximal, characteristic, complemented
14#89727C211yesmaximal, characteristic, complemented
15C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic, complemented
18C23×C4×C7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×D28C2211yescharacteristic
22C22×D28C2211yescharacteristic
23C7×C22⋊Q8—24no—
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25C2×C7×C4⋊C4C2211yescharacteristic, complemented
26C2×C7×C4⋊C4C2211yescharacteristic, complemented
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30#6510—24no—
31C2×(C2×D7)⋊C4—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×(C2×D7)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×C4×Dic7C2211yescharacteristic, complemented
47C2×C4×Dic7C2211yescharacteristic, complemented
48#6445—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
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
67C2×C7×Q8—22no—
68C2×C7×Q8—22no—
69C2×C7×Q8—41no—
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—41no—
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C22×C4×C7C2311yescharacteristic
78C22×C4×C7C2311yescharacteristic
79C2×C22⋊D7—22no—
80C2×C22⋊D7—22no—
81C2×C22⋊D7—22no—
82C22×Dic7—22no—
83C22×Dic7—22no—
84C22×Dic7—41no—
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C2×D28—22no—
91C2×D28—22no—
92C2×D28—22no—
93C2×D28—22no—
94C2×D28—41no—
95C2×D28—41no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4D414yes—
101C22⋊Dic7—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—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—41no—
112(C2×D7)⋊C4—41no—
113(C2×D7)⋊C4—41no—
114(C2×D7)⋊C4—41no—
115(C2×D7)⋊C4D414yescomplemented
116C4×Dic7—22no—
117C4×Dic7—41no—
118C2×#124—71no—
119C2×C22⋊Q8D711yescharacteristic, complemented
120C2×#120—71no—
121C22×C22⋊C4—71no—
122#397—71no—
123#394—71no—
124#394—71no—
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×C7C2×D412yes—
138C23×C7#4112yes—
139C23×C7#4112yes—
140C23×C7C2411yescharacteristic
141C22×D7—28no—
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—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7—44no—
162C22×D7—44no—
163C22×D7—44no—
164C22×D7—44no—
165C22×D7—44no—
166C22×D7—44no—
167C22×D7C2×D412yes—
168C22×D7#4112yes—
169C7×Q8—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×C7—21no—
185C2×C4×C7—44no—
186C2×C4×C7C2×D412yes—
187C2×C4×C7C2×D412yes—
188C2×C4×C7C2×D412yes—
189C22⋊D7—44no—
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×Dic7—44no—
205C2×Dic7C2×D412yes—
206D28—44no—
207D28—44no—
208C25—71no—
209C22×Q8C2×D711yescharacteristic, complemented
210C22×D4—71no—
211C22×D4—71no—
212C22×D4—71no—
213C23×C4—71no—
214C23×C4C2×D711yescharacteristic
215#124—144no—
216C22⋊Q8—24no—
217#120—144no—
218C2×C4⋊C4C2×D711yescharacteristic, complemented
219C2×C4⋊C4C2×D711yescharacteristic, complemented
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—71no—
235C2×C22⋊C4—71no—
236C2×C22⋊C4—144no—
237C2×C22⋊C4C2×D711yescharacteristic
238C2×C22⋊C4C2×D711yescharacteristic
239C2×C42—71no—
240C2×C42—71no—
241#95—71no—
242#95—71no—
243#95—71no—
244#95—71no—
245#95—71no—
246#95—71no—
247C22×C7—22no—
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7#12414yescomplemented
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—416no—
268C2×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
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—
283C4×C7—22no—
284C4×C7—22no—
285C4×C7—22no—
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×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—142no—
314C2×D4—142no—
315C2×D4—281no—
316C2×D4—281no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—71no—
321C22×C4—71no—
322C22×C4—71no—
323C22×C4—71no—
324C22×C4—71no—
325C22×C4—142no—
326C22×C4—142no—
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—22no—
337C22⋊C4—22no—
338C22⋊C4—74no—
339C22⋊C4—142no—
340C22⋊C4—142no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4—142no—
349C22⋊C4—281no—
350C22⋊C4—281no—
351C22⋊C4—281no—
352C22⋊C4—281no—
353C22⋊C4C22⋊D714yes—
354C42—142no—
355C42—281no—
356C2×C7—24no—
357C2×C7#52511yescharacteristic
358C2×C7C2×#12411yescharacteristic
359C2×C7#53811yescharacteristic
360C2×C7D4211yescharacteristic
361C2×C7#54611yescharacteristic
362C2×C7#54611yescharacteristic
363C2×C7#55111yescharacteristic
364D7—42no—
365D7—42no—
366D7—81no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—72no—
372C23—72no—
373C23—148no—
374C23—141no—
375C23—141no—
376C23—141no—
377C23—141no—
378C23—141no—
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—284no—
394C23—284no—
395C23—284no—
396C23—284no—
397C23—284no—
398C23—284no—
399C23#116112yes—
400C23#116112yes—
401C23C2×C22⋊D712yes—
402C23C23×D711yescharacteristic
403Q8—44no—
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—44no—
423C2×C4—72no—
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×C4D4×D712yes—
440C2×C4C2×C22⋊D712yes—
441C2×C4C2×C22⋊D712yes—
442C7#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—142no—
450C22—142no—
451C22—142no—
452C22—142no—
453C22—142no—
454C22—142no—
455C22—2816no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22—281no—
460C22—281no—
461C22—281no—
462C22—281no—
463C22—281no—
464C22—281no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22#651014yescomplemented
472C22C2×D4×D711yescharacteristic
473C22#654811yescharacteristic
474C22C2×#116111yescharacteristic
475C22C2×#116111yescharacteristic
476C22D7×#4111yescharacteristic
477C22#655311yescharacteristic
478C22C22×C22⋊D711yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—142no—
486C4—281no—
487C4—281no—
488C4—281no—
489C4—281no—
490C2—24no—
491C2—282no—
492C2—282no—
493C2—561no—
494C2#9031811yescharacteristic
495C2#9032111yescharacteristic
496C2#9032811yescharacteristic
497C2#9033511yescharacteristic
498C2D4×C22⋊D711yescharacteristic
499C2C2×#651011yescharacteristic
500C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC2×#6445C2
hypercenterC2×C22⋊Q8D7
hypocenterC7#2472
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6510C2semidirect
3. C22×(C2×D7)⋊C4C2semidirect
5. #90003C2semidirect
6. #89985C2semidirect
7. #89765C2semidirect
8. #89763C2semidirect
9. #89763C2semidirect
10. #89762C2semidirect
11. #89762C2semidirect
12. #89760C2semidirect
13. #89730C2semidirect
14. #89727C2semidirect
15. C22⋊C4×Dic7C2semidirect
17. C22×C7×Q8C22semidirect
24. C2×C7×C4⋊C4C22semidirect
25. C2×C7×C4⋊C4C22semidirect
26. C2×C7×C4⋊C4C22semidirect
42. C2×(C2×D7)⋊C4C22semidirect
43. C2×(C2×D7)⋊C4C22semidirect
44. C2×(C2×D7)⋊C4C22semidirect
45. C2×(C2×D7)⋊C4C22semidirect
46. C2×C4×Dic7C22semidirect
47. C2×C4×Dic7C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
115. (C2×D7)⋊C4D4semidirect
119. C2×C22⋊Q8D7semidirect
209. C22×Q8C2×D7semidirect
218. C2×C4⋊C4C2×D7semidirect
219. C2×C4⋊C4C2×D7semidirect
220. C2×C4⋊C4C2×D7semidirect
253. C22×C7#124semidirect
442. C7#2472semidirect
471. C22#6510semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65102
C22×(C2×D7)⋊C42
C2×#64452
#900032
#899852
#897652
#897632
#897622
#897602
#897302
#897272
C22⋊C4×Dic72
#24727