Finite Group Databasebeta

← C2×#11738575 #14232543

Subgroups — C2×#11738575 #14232543

Order 1792.

1930 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90425C211yesmaximal, characteristic
2C2×#90270C211yesmaximal, characteristic, complemented
3C2×C7×#477C211yesmaximal, characteristic, complemented
4C2×#89850C211yesmaximal, characteristic, complemented
5C2×C4×#1143C211yesmaximal, characteristic, complemented
6#11738575C2116yesmaximal, direct product factor
7C2×#89654C212yesmaximal, complemented
8C2×#89653C212yesmaximal, complemented
9C2×#89614C211yesmaximal, characteristic, complemented
10C2×#89610C211yesmaximal, characteristic, complemented
11C2×#89587C212yesmaximal, complemented
12C2×#89584C212yesmaximal, complemented
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6510—22no—
16C2×#6509C2212yes—
17C22×#1143C2211yescharacteristic
18C22×#1131C2212yes—
19#90425—24no—
20#90425C2218yes—
21#90270—24no—
22#90270C2218yescomplemented
23C2×#6459—22no—
24C2×#6458—21no—
25C2×#6458C2212yes—
26C2×#6455—21no—
27C2×#6455C2212yes—
28C2×D7×C4⋊C4C2212yes—
29C2×C4×D28—21no—
30C2×C4×D28C2211yescharacteristic
31C2×C4×D28C2211yescharacteristic
32C2×#6433—21no—
33C2×#6432C2211yescharacteristic
34C7×#477C2218yescomplemented
35C2×C7×#107C2212yes—
36C2×C7×C4⋊C8C2211yescharacteristic
37C2×C7×#103C2212yes—
38C2×#6409C2211yescharacteristic
39C2×#6395C2212yes—
40C2×(C2×D7)⋊C8C2212yes—
41C2×#6391C2212yes—
42#89850C2218yescomplemented
43C4×#1143C2218yescomplemented
44C2×#6383C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6381C2212yes—
47C2×#6381C2212yes—
48C2×C4×C7⋊C8C2211yescharacteristic
49#89654C22116yescomplemented
50#89653C22116yescomplemented
51#89614C2218yescomplemented
52#89610C2218yescomplemented
53#89587C22116yescomplemented
54#89584C22116yescomplemented
55C2×#3357—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8C2311yescharacteristic
58C22×D28—22no—
59C22×D28D411yescharacteristic
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C4×D7—22no—
63C22×C4×D7C2312yes—
64C22×Dic14—21no—
65C22×C7×C8C2312yes—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8C2314yes—
68C4×C7×Q8—22no—
69C4×C7×Q8C2314yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2312yes—
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76#6510—216no—
77#6509—24no—
78#6509C2318yes—
79C2×#1143—24no—
80C2×#1143C2314yes—
81C2×(C2×D7)⋊C4—22no—
82C2×(C2×D7)⋊C4—22no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4D412yes—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7D411yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2312yes—
90C2×Dic7⋊C4—22no—
91C2×Dic7⋊C4—22no—
92C2×C4×Dic7—22no—
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C2×#1131—28no—
96C2×#1131C2318yes—
97#6459—216no—
98#6458—28no—
99#6458—24no—
100#6458C2318yes—
101#6455—24no—
102#6455—28no—
103#6455C2318yes—
104D7×C4⋊C4—24no—
105D7×C4⋊C4C2318yes—
106C4×D28—28no—
107C4×D28—22no—
108C4×D28D414yes—
109C4×D28C2314yes—
110C4×D28C2314yes—
111#6433—28no—
112#6432—22no—
113#6432C2314yes—
114C7×#107C2318yes—
115C7×C4⋊C8C2314yes—
116C7×#103C2318yes—
117#6409C2314yes—
118#6395C2318yes—
119(C2×D7)⋊C8C2318yes—
120#6391C2318yes—
121#6383C2314yes—
122#6382C2314yes—
123#6381C2318yes—
124#6381C2318yes—
125C4×C7⋊C8C2314yes—
126C2×#555—71no—
127C2×#554—71no—
128#3357—716no—
129C2×#500—71no—
130C2×#485—72no—
131C2×#480—72no—
132C2×#477D711yescharacteristic, complemented
133C2×#476—71no—
134C2×#468—72no—
135C2×C4×SD16—71no—
136C2×#436—71no—
137C2×#428—72no—
138C23×D7—21no—
139C23×D7C2×D412yes—
140C2×C7×Q8—21no—
141C2×C7×Q8—24no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—82no—
145C2×C7×Q8C2412yes—
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7C2×D412yes—
155C2×D28—24no—
156C2×D28—22no—
157C2×D28—22no—
158C2×D28—21no—
159C2×D28—28no—
160C2×D28—84no—
161C2×D28C2×D414yes—
162C2×D28C2412yes—
163C2×C4×D7—24no—
164C2×C4×D7—24no—
165C2×C4×D7—24no—
166C2×C4×D7—28no—
167C2×C4×D7—44no—
168C2×C4×D7—84no—
169C2×C4×D7C2×D414yes—
170C2×C4×D7C2414yes—
171C2×Dic14—24no—
172C2×Dic14—82no—
173C2×C7×C8—24no—
174C2×C7×C8C2414yes—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4C2414yes—
180C7×C4⋊C4C2412yes—
181C42×C7—24no—
182C42×C7C2×D412yes—
183C42×C7C2412yes—
184#1143—28no—
185(C2×D7)⋊C4—28no—
186(C2×D7)⋊C4—28no—
187(C2×D7)⋊C4—24no—
188(C2×D7)⋊C4—28no—
189(C2×D7)⋊C4C2×D418yes—
190C4⋊Dic7—22no—
191C4⋊Dic7—24no—
192C4⋊Dic7—21no—
193C4⋊Dic7C2×D414yes—
194C4⋊Dic7C2412yes—
195C4⋊Dic7C2414yes—
196Dic7⋊C4—28no—
197Dic7⋊C4—28no—
198C4×Dic7—28no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8#4114yes—
201C2×C7⋊C8C2412yes—
202C2×C7⋊C8C2412yes—
203#1131—216no—
204C22×SD16—71no—
205C22×M16—72no—
206#555—78no—
207#555—144no—
208#554—78no—
209#554—144no—
210C2×C4⋊Q8—71no—
211C2×C4⋊Q8C2×D711yescharacteristic
212C2×#125—141no—
213C2×#124—142no—
214C2×C22⋊Q8—72no—
215C2×C22⋊Q8—142no—
216C2×#121—72no—
217C2×#121—141no—
218C2×C4×Q8C2×D711yescharacteristic
219C2×C4×D4—71no—
220C2×C4×D4—71no—
221C2×C4×D4—141no—
222C2×#117—72no—
223C2×#117—141no—
224C22×C4⋊C4—72no—
225#500—78no—
226#485—716no—
227#480—716no—
228#477C2×D718yescomplemented
229#476—78no—
230#468—716no—
231C4×SD16—78no—
232#436—78no—
233#428—716no—
234C2×#107—72no—
235C2×#107—72no—
236C2×#107C2×D712yes—
237C2×#106—72no—
238C2×#106—71no—
239C2×C4⋊C8C2×D711yescharacteristic
240C2×#103—71no—
241C2×#103C2×D712yes—
242C2×#102—72no—
243C2×#102—71no—
244C2×C22⋊C8—72no—
245C2×C4×C8—71no—
246C23×C7C22×D411yescharacteristic
247C22×D7—24no—
248C22×D7—24no—
249C22×D7—24no—
250C22×D7—22no—
251C22×D7—42no—
252C22×D7—42no—
253C22×D7—42no—
254C22×D7C22×D414yes—
255C7×Q8—22no—
256C7×Q8—22no—
257C7×Q8—42no—
258C7×Q8—84no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—44no—
266C2×C4×C7#13714yes—
267C2×C4×C7C22×D412yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×D412yes—
270C2×C4×C7C2×#4112yes—
271C2×C4×C72−1+412yes—
272C2×C4×C7C2511yescharacteristic
273C2×Dic7—24no—
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—42no—
277C2×Dic7—44no—
278C2×Dic7C22×D414yes—
279D28—28no—
280D28—22no—
281D28—22no—
282D28—42no—
283D28—88no—
284C4×D7—28no—
285C4×D7—24no—
286C4×D7—24no—
287C4×D7—48no—
288C4×D7—88no—
289Dic14—84no—
290C7×C8—24no—
291C7⋊C8—22no—
292C7⋊C8C2×#4114yes—
293C22×Q8—21no—
294C22×Q8—141no—
295C22×Q8C22×D711yescharacteristic
296C22×D4—71no—
297C22×D4—71no—
298C22×D4—142no—
299C23×C4—72no—
300C23×C4—142no—
301C23×C4—141no—
302C2×SD16—74no—
303C2×SD16—144no—
304C2×M16—78no—
305C2×M16—148no—
306C22×C8—71no—
307C22×C8—71no—
308C22×C8C22×D712yes—
309C4⋊Q8—22no—
310C4⋊Q8—74no—
311C4⋊Q8—142no—
312C4⋊Q8C22×D714yes—
313#125—148no—
314#124—1416no—
315C22⋊Q8—78no—
316C22⋊Q8—1416no—
317C22⋊Q8—144no—
318#121—78no—
319#121—144no—
320#121—148no—
321C4×Q8—22no—
322C4×Q8C22×D714yes—
323C4×D4—74no—
324C4×D4—74no—
325C4×D4—74no—
326C4×D4—148no—
327C4×D4—142no—
328#117—78no—
329#117—148no—
330#117—144no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—78no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D712yes—
343C2×C22⋊C4—72no—
344C2×C22⋊C4—142no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—142no—
347C2×C22⋊C4—141no—
348C2×C42—21no—
349C2×C42—142no—
350C2×C42C22×D711yescharacteristic
351#107—78no—
352#107—78no—
353#107C22×D718yes—
354#106—78no—
355#106—74no—
356C4⋊C8C22×D714yes—
357#103—74no—
358#103C22×D718yes—
359#102—78no—
360#102—74no—
361C22⋊C8—78no—
362C4×C8—74no—
363C22×C7#54712yes—
364C22×C7C2×#13712yes—
365C22×C7#57612yes—
366C22×C7C23×D411yescharacteristic
367C2×D7—24no—
368C2×D7—24no—
369C2×D7—24no—
370C2×D7—48no—
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—42no—
374C2×D7—84no—
375C4×C7—22no—
376C4×C7—42no—
377C4×C7—42no—
378C4×C7—44no—
379C4×C7#54712yes—
380C4×C7C2×#13714yes—
381C4×C7C22×#4111yescharacteristic
382C4×C7C2×2−1+411yescharacteristic
383Dic7—24no—
384Dic7—42no—
385Dic7—44no—
386C24—72no—
387C24—141no—
388C2×Q8—21no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—82no—
393C2×Q8—144no—
394C2×Q8—562no—
395C2×Q8C23×D712yes—
396C2×D4—74no—
397C2×D4—72no—
398C2×D4—148no—
399C2×D4—144no—
400C2×D4—141no—
401C2×D4—142no—
402C2×D4—142no—
403C2×D4—564no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—72no—
408C22×C4—74no—
409C22×C4—74no—
410C22×C4—148no—
411C22×C4—144no—
412C22×C4—144no—
413C22×C4—141no—
414C22×C4—142no—
415C22×C4—144no—
416C22×C4—284no—
417C22×C4—564no—
418C22×C4D4×D711yescharacteristic
419C22×C4D4×D711yescharacteristic
420C22×C4C23×D711yescharacteristic
421SD16—148no—
422M16—1416no—
423C2×C8—24no—
424C2×C8—74no—
425C2×C8—72no—
426C2×C8—72no—
427C2×C8—142no—
428C2×C8C23×D714yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—74no—
434C4⋊C4—74no—
435C4⋊C4—72no—
436C4⋊C4—148no—
437C4⋊C4—148no—
438C4⋊C4—142no—
439C4⋊C4—141no—
440C4⋊C4—144no—
441C4⋊C4C23×D712yes—
442C4⋊C4C23×D714yes—
443C22⋊C4—78no—
444C22⋊C4—148no—
445C22⋊C4—148no—
446C22⋊C4—148no—
447C22⋊C4—144no—
448C42—24no—
449C42—148no—
450C42D4×D712yes—
451C42C23×D712yes—
452C2×C7#335714yescomplemented
453C2×C7C2×#54711yescharacteristic
454C2×C7C22×#13711yescharacteristic
455C2×C7C2×#57611yescharacteristic
456D7—44no—
457D7—82no—
458C23—74no—
459C23—144no—
460C23—144no—
461C23—144no—
462C23—142no—
463C23—282no—
464C23—282no—
465C23—282no—
466C23C2×D4×D711yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—42no—
470Q8—84no—
471Q8—564no—
472D4—148no—
473D4—142no—
474D4—142no—
475D4—282no—
476D4—568no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—74no—
485C2×C4—144no—
486C2×C4—148no—
487C2×C4—144no—
488C2×C4—144no—
489C2×C4—142no—
490C2×C4—144no—
491C2×C4—282no—
492C2×C4—284no—
493C2×C4—288no—
494C2×C4—568no—
495C2×C4#648114yes—
496C2×C4C2×D4×D712yes—
497C2×C4C2×D4×D712yes—
498C2×C4C2×D4×D711yescharacteristic
499C2×C4#655112yes—
500C2×C4D7×#4112yes—
501C2×C4C24×D711yescharacteristic
502C8—24no—
503C8—74no—
504C8—142no—
505C7C2×#335711yescharacteristic, complemented, Sylow
506C22—144no—
507C22—144no—
508C22—144no—
509C22—288no—
510C22—282no—
511C22—282no—
512C22—282no—
513C22—564no—
514C22#9032212yes—
515C22C2×#648112yes—
516C22#9046512yes—
517C22C22×D4×D711yescharacteristic
518C4—22no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4—144no—
523C4—284no—
524C4—282no—
525C4#9032212yes—
526C4C2×#648114yes—
527C4C2×#655111yescharacteristic
528C4C2×D7×#4111yescharacteristic
529C2—284no—
530C2—562no—
531C2#1173857514yesdirect product factor
532C2C2×#9032211yescharacteristic
533C2C22×#648111yescharacteristic
534C2C2×#9046511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#477C2
socleC23×C7C22×D4
prime radicalC2×#90425C2
hypercenterC2×#477D7
hypocenterC7C2×#3357
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×#90270C2semidirect
3. C2×C7×#477C2semidirect
4. C2×#89850C2semidirect
5. C2×C4×#1143C2semidirect
6. #11738575C2direct (N × H)
7. C2×#89654C2semidirect
8. C2×#89653C2semidirect
9. C2×#89614C2semidirect
10. C2×#89610C2semidirect
11. C2×#89587C2semidirect
12. C2×#89584C2semidirect
22. #90270C22semidirect
34. C7×#477C22semidirect
42. #89850C22semidirect
43. C4×#1143C22semidirect
49. #89654C22semidirect
50. #89653C22semidirect
51. #89614C22semidirect
52. #89610C22semidirect
53. #89587C22semidirect
54. #89584C22semidirect
132. C2×#477D7semidirect
228. #477C2×D7semidirect
452. C2×C7#3357semidirect
505. C7C2×#3357semidirect
531. C2#11738575direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904252
C2×#902702
C2×C7×#4772
C2×#898502
C2×C4×#11432
#117385752
C2×#896542
C2×#896532
C2×#896142
C2×#896102
C2×#895872
C2×#895842
C2×#33577