Finite Group Databasebeta

← C2×#11739396 #14232545

Subgroups — C2×#11739396 #14232545

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×#90421C211yesmaximal, characteristic, complemented
2C2×#90425C211yesmaximal, characteristic
3C2×C7×#500C211yesmaximal, characteristic, complemented
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89856C211yesmaximal, characteristic, complemented
6#11739396C2116yesmaximal, direct product factor
7C2×#89660C212yesmaximal, complemented
8C2×#89654C212yesmaximal, complemented
9C2×#89658C212yesmaximal, complemented
10C2×#89653C212yesmaximal, complemented
11C2×C4×#1132C211yesmaximal, characteristic, complemented
12C2×#89460C211yesmaximal, characteristic, complemented
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×#6510—22no—
16C2×#6509—22no—
17C22×#1132C2211yescharacteristic
18C22×#1131C2212yes—
19#90425—24no—
20#90425C2218yes—
21#90421—24no—
22#90421C2218yescomplemented
23C2×#6460C2212yes—
24C2×#6458—21no—
25C2×#6458C2212yes—
26C2×#6456—21no—
27C2×#6455—21no—
28C2×#6455C2212yes—
29C2×D7×C4⋊C4C2212yes—
30C2×#6453—21no—
31C2×C4×D28C2211yescharacteristic
32C2×C4×D28C2211yescharacteristic
33C2×C4×Dic14C2211yescharacteristic
34C7×#500C2218yescomplemented
35C2×C7×#107C2212yes—
36C2×C7×#107C2212yes—
37C2×C4×C7×C8C2211yescharacteristic
38C2×#6395C2211yescharacteristic
39C2×(C2×D7)⋊C8C2212yes—
40C2×#6391C2211yescharacteristic
41C2×#6390C2211yescharacteristic
42#89863C2218yescomplemented
43#89856C2218yescomplemented
44C2×#6384C2212yes—
45C2×#6383C2212yes—
46C2×#6382C2212yes—
47C2×#6381C2212yes—
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49#89660C22116yescomplemented
50#89658C22116yescomplemented
51#89654C22116yescomplemented
52#89653C22116yescomplemented
53C4×#1132C2218yescomplemented
54#89460C2218yescomplemented
55C2×#3357—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8—21no—
58C22×D28—22no—
59C22×D28D411yescharacteristic
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C4×D7—22no—
63C22×C4×D7C2312yes—
64C22×Dic14C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2314yes—
70C7×C4⋊Q8C2314yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C4⋊C4C2312yes—
75C2×C42×C7C2311yescharacteristic
76#6510—216no—
77#6509—216no—
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—22no—
82C2×(C2×D7)⋊C4D412yes—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7D411yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—22no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—22no—
91C22×C7⋊C8C2312yes—
92C2×#1132—24no—
93C2×#1132C2314yes—
94C2×#1131—28no—
95C2×#1131C2318yes—
96#6460—24no—
97#6460C2318yes—
98#6458—28no—
99#6458—24no—
100#6458C2318yes—
101#6456—28no—
102#6455—24no—
103#6455—28no—
104#6455C2318yes—
105D7×C4⋊C4—24no—
106D7×C4⋊C4C2318yes—
107#6453—28no—
108C4×D28—22no—
109C4×D28D414yes—
110C4×D28C2314yes—
111C4×D28C2314yes—
112C4×Dic14—22no—
113C4×Dic14C2314yes—
114C7×#107C2318yes—
115C7×#107C2318yes—
116C4×C7×C8C2314yes—
117#6395C2314yes—
118(C2×D7)⋊C8C2318yes—
119#6391C2314yes—
120#6390C2314yes—
121#6384C2318yes—
122#6383C2318yes—
123#6382C2318yes—
124#6381C2318yes—
125C4⋊(C7⋊C8)C2314yes—
126C2×#555—71no—
127C2×#554—71no—
128#3357—716no—
129C2×#500D711yescharacteristic, complemented
130C2×#485—72no—
131C2×#480—72no—
132C2×#477—71no—
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—24no—
141C2×C7×Q8—24no—
142C2×C7×Q8—82no—
143C2×C7×Q8—82no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7C2×D412yes—
153C2×D28—24no—
154C2×D28—22no—
155C2×D28—22no—
156C2×D28—21no—
157C2×D28—28no—
158C2×D28—84no—
159C2×D28C2×D414yes—
160C2×D28C2412yes—
161C2×C4×D7—24no—
162C2×C4×D7—24no—
163C2×C4×D7—24no—
164C2×C4×D7—28no—
165C2×C4×D7—44no—
166C2×C4×D7—84no—
167C2×C4×D7C2×D414yes—
168C2×C4×D7C2414yes—
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×Dic14—21no—
172C2×Dic14C2412yes—
173C2×C7×C8—22no—
174C2×C7×C8#4114yes—
175C2×C7×C8C2412yes—
176C2×C7×C8C2412yes—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4C2414yes—
182C7×C4⋊C4C2414yes—
183C42×C7C2×D412yes—
184C42×C7C2412yes—
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—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—28no—
193C4⋊Dic7C2×D414yes—
194C4⋊Dic7C2412yes—
195C4⋊Dic7C2412yes—
196Dic7⋊C4—24no—
197Dic7⋊C4—28no—
198C4×Dic7—24no—
199C4×Dic7—28no—
200C2×C7⋊C8—24no—
201C2×C7⋊C8C2414yes—
202#1132—28no—
203#1131—216no—
204C22×SD16—71no—
205C22×M16—72no—
206#555—78no—
207#555—144no—
208#554—78no—
209#554—144no—
210C2×C4⋊Q8C2×D711yescharacteristic
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×Q8—71no—
219C2×C4×D4—71no—
220C2×C4×D4—71no—
221C2×C4×D4—141no—
222C2×#117—72no—
223C2×#117—141no—
224C22×C4⋊C4—72no—
225#500C2×D718yescomplemented
226#485—716no—
227#480—716no—
228#477—78no—
229#476—78no—
230#468—716no—
231C4×SD16—78no—
232#436—78no—
233#428—716no—
234C2×#107—72no—
235C2×#107C2×D712yes—
236C2×#107C2×D712yes—
237C2×#106—71no—
238C2×#106—72no—
239C2×C4⋊C8—71no—
240C2×#103—71no—
241C2×#103—72no—
242C2×#102—71no—
243C2×#102—72no—
244C2×C22⋊C8—72no—
245C2×C4×C8C2×D711yescharacteristic
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—84no—
256C7×Q8—84no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—44no—
261C2×C4×C7—44no—
262C2×C4×C7#13714yes—
263C2×C4×C7C22×D412yes—
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D412yes—
266C2×C4×C7C2×#4112yes—
267C2×C4×C72−1+412yes—
268C2×C4×C7C2511yescharacteristic
269C2×Dic7—24no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7C22×D414yes—
277D28—28no—
278D28—22no—
279D28—22no—
280D28—42no—
281D28—88no—
282C4×D7—28no—
283C4×D7—24no—
284C4×D7—24no—
285C4×D7—48no—
286C4×D7—88no—
287Dic14—22no—
288Dic14—22no—
289Dic14—42no—
290C7×C8—22no—
291C7×C8C2×#4114yes—
292C7⋊C8—24no—
293C22×Q8—21no—
294C22×Q8—21no—
295C22×Q8—71no—
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—72no—
307C22×C8C22×D711yescharacteristic
308C22×C8C22×D711yescharacteristic
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8C22×D714yes—
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—74no—
322C4×Q8—142no—
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—71no—
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×D712yes—
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—142no—
349C2×C42—141no—
350C2×C42C22×D711yescharacteristic
351#107—78no—
352#107C22×D718yes—
353#107C22×D718yes—
354#106—74no—
355#106—78no—
356C4⋊C8—74no—
357#103—74no—
358#103—78no—
359#102—74no—
360#102—78no—
361C22⋊C8—78no—
362C4×C8C22×D714yes—
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—44no—
377C4×C7—44no—
378C4×C7#54712yes—
379C4×C7C2×#13714yes—
380C4×C7C22×#4111yescharacteristic
381C4×C7C2×2−1+411yescharacteristic
382Dic7—24no—
383Dic7—42no—
384Dic7—42no—
385Dic7—42no—
386C24—72no—
387C24—141no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—72no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8—141no—
394C2×Q8—142no—
395C2×Q8—142no—
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—22no—
405C22×C4—22no—
406C22×C4—72no—
407C22×C4—74no—
408C22×C4—74no—
409C22×C4—148no—
410C22×C4—144no—
411C22×C4—144no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—144no—
416C22×C4—284no—
417C22×C4—564no—
418C22×C4D4×D711yescharacteristic
419C22×C4D4×D711yescharacteristic
420C22×C4C23×D711yescharacteristic
421SD16—148no—
422M16—1416no—
423C2×C8—22no—
424C2×C8—74no—
425C2×C8—144no—
426C2×C8#116114yes—
427C2×C8C23×D712yes—
428C2×C8C23×D712yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—74no—
434C4⋊C4—72no—
435C4⋊C4—72no—
436C4⋊C4—148no—
437C4⋊C4—148no—
438C4⋊C4—144no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4C23×D714yes—
442C4⋊C4C23×D714yes—
443C22⋊C4—78no—
444C22⋊C4—148no—
445C22⋊C4—148no—
446C22⋊C4—148no—
447C22⋊C4—144no—
448C42—148no—
449C42—144no—
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—84no—
468Q8—84no—
469Q8—142no—
470Q8—142no—
471Q8—282no—
472D4—148no—
473D4—142no—
474D4—142no—
475D4—282no—
476D4—568no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—74no—
483C2×C4—144no—
484C2×C4—148no—
485C2×C4—144no—
486C2×C4—144no—
487C2×C4—142no—
488C2×C4—142no—
489C2×C4—142no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4—288no—
494C2×C4—568no—
495C2×C4#648114yes—
496C2×C4C2×D4×D712yes—
497C2×C4C2×D4×D711yescharacteristic
498C2×C4C2×D4×D712yes—
499C2×C4C2×#116112yes—
500C2×C4#655112yes—
501C2×C4C24×D711yescharacteristic
502C8—22no—
503C8—144no—
504C8C2×#116114yes—
505C7C2×#335711yescharacteristic, complemented, Sylow
506C22—144no—
507C22—144no—
508C22—144no—
509C22—288no—
510C22—282no—
511C22—282no—
512C22—282no—
513C22—564no—
514C22#9041712yes—
515C22#9044812yes—
516C22C2×#648112yes—
517C22C22×D4×D711yescharacteristic
518C4—22no—
519C4—44no—
520C4—44no—
521C4—144no—
522C4—282no—
523C4—282no—
524C4—282no—
525C4#9041712yes—
526C4C2×#648114yes—
527C4C22×#116111yescharacteristic
528C4C2×#655111yescharacteristic
529C2—284no—
530C2—562no—
531C2#1173939614yesdirect product factor
532C2C2×#9041711yescharacteristic
533C2C2×#9044811yescharacteristic
534C2C22×#648111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#500C2
socleC23×C7C22×D4
prime radicalC2×#90425C2
hypercenterC2×#500D7
hypocenterC7C2×#3357
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
1. C2×#90421C2semidirect
3. C2×C7×#500C2semidirect
4. C2×#89863C2semidirect
5. C2×#89856C2semidirect
6. #11739396C2direct (N × H)
7. C2×#89660C2semidirect
8. C2×#89654C2semidirect
9. C2×#89658C2semidirect
10. C2×#89653C2semidirect
11. C2×C4×#1132C2semidirect
12. C2×#89460C2semidirect
22. #90421C22semidirect
34. C7×#500C22semidirect
42. #89863C22semidirect
43. #89856C22semidirect
49. #89660C22semidirect
50. #89658C22semidirect
51. #89654C22semidirect
52. #89653C22semidirect
53. C4×#1132C22semidirect
54. #89460C22semidirect
129. C2×#500D7semidirect
225. #500C2×D7semidirect
452. C2×C7#3357semidirect
505. C7C2×#3357semidirect
531. C2#11739396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904212
C2×#904252
C2×C7×#5002
C2×#898632
C2×#898562
#117393962
C2×#896602
C2×#896542
C2×#896582
C2×#896532
C2×C4×#11322
C2×#894602
C2×#33577