Finite Group Databasebeta

← C2×#11739407 #14223709

Subgroups — C2×#11739407 #14223709

Order 1792.

2278 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90416C211yesmaximal, characteristic, complemented
2C2×#90382C211yesmaximal, characteristic, complemented
3C2×C7×#501C211yesmaximal, characteristic, complemented
4C2×#89861C211yesmaximal, characteristic, complemented
5C2×#89839C211yesmaximal, characteristic, complemented
6#11739407C2116yesmaximal, direct product factor
7C2×#89655C212yesmaximal, complemented
8C2×#89661C212yesmaximal, complemented
9C2×#89636C212yesmaximal, complemented
10C2×#89641C212yesmaximal, complemented
11C2×#89484C211yesmaximal, characteristic, complemented
12C2×D7×#97C211yesmaximal, characteristic, complemented
13C22×#1161—21no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125C2211yescharacteristic
16C2×#6510—21no—
17C22×D56C2212yes—
18C22×#1132C2212yes—
19C22×C8×D7C2212yes—
20#90416—24no—
21#90416C2218yescomplemented
22#90382—24no—
23#90382C2218yescomplemented
24C2×#6461—21no—
25C2×#6460C2211yescharacteristic
26C2×#6458—21no—
27C2×#6458C2211yescharacteristic
28C2×#6458C2212yes—
29C2×#6457—21no—
30C2×#6456—21no—
31C2×#6456—21no—
32C2×#6456C2212yes—
33C2×#6456C2211yescharacteristic
34C2×#6455—21no—
35C2×#6450C2211yescharacteristic
36C2×#6438C2211yescharacteristic
37C2×#6437C2211yescharacteristic
38C2×C42×D7C2211yescharacteristic
39C7×#501C2218yescomplemented
40C2×C7×#107C2212yes—
41C2×C7×#106C2212yes—
42C2×C7×#97C2211yescharacteristic
43C2×#6389C2212yes—
44#89861C2218yescomplemented
45#89839C2218yescomplemented
46C2×#6384C2212yes—
47C2×#6383C2212yes—
48C2×#6383C2212yes—
49C2×#6383C2212yes—
50C2×#6377C2211yescharacteristic
51#89661C22116yescomplemented
52#89655C22116yescomplemented
53#89641C22116yescomplemented
54#89636C22116yescomplemented
55#89484C2218yescomplemented
56D7×#97C2218yescomplemented
57C2×#3180—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C2×#1161—28no—
60C2×#1161—48no—
61C22×D28—21no—
62C22×D28—22no—
63C22×D28—21no—
64C22×D28—21no—
65C22×D28C2312yes—
66C22×D28C2311yescharacteristic
67C22×C4×D7—21no—
68C22×C4×D7—22no—
69C22×C4×D7D411yescharacteristic
70C22×C4×D7D411yescharacteristic
71C22×C4×D7C2311yescharacteristic
72C22×Dic14C2311yescharacteristic
73C22×C7×C8C2312yes—
74C7×C4⋊Q8—22no—
75C7×C4⋊Q8C2314yes—
76C7×#125—22no—
77C7×#125C2314yes—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85#6510—28no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—22no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—22no—
93C2×C4⋊Dic7—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7D411yescharacteristic
97C2×C4×Dic7D411yescharacteristic
98C2×C4×Dic7C2311yescharacteristic
99C22×C7⋊C8C2312yes—
100C2×D56—28no—
101C2×D56C2318yes—
102C2×#1132—28no—
103C2×#1132C2318yes—
104C2×C8×D7—28no—
105C2×C8×D7C2318yes—
106#6461—28no—
107#6460—22no—
108#6460C2314yes—
109#6458—22no—
110#6458—24no—
111#6458—28no—
112#6458C2314yes—
113#6458C2318yes—
114#6457—28no—
115#6456—28no—
116#6456—22no—
117#6456—24no—
118#6456—28no—
119#6456C2314yes—
120#6456C2318yes—
121#6455—28no—
122#6450—22no—
123#6450C2314yes—
124#6438—22no—
125#6438C2314yes—
126#6437—22no—
127#6437C2314yes—
128C42×D7D414yes—
129C42×D7C2314yes—
130C7×#107C2318yes—
131C7×#106C2318yes—
132C7×#97C2314yes—
133#6389C2318yes—
134#6384C2318yes—
135#6383C2318yes—
136#6383C2318yes—
137#6383C2318yes—
138#6377C2314yes—
139C2×#532—71no—
140C2×#529—71no—
141#3180—716no—
142C2×#501D711yescharacteristic, complemented
143C2×#496—71no—
144C2×#490—71no—
145C2×#489—71no—
146C2×#466—72no—
147C2×#465—72no—
148C2×#442—72no—
149C2×#440—72no—
150C22×#97—71no—
151C23×D7—21no—
152C23×D7—22no—
153C23×D7C2×D411yescharacteristic
154C2×C7×Q8—24no—
155C2×C7×Q8—44no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166#1161—432no—
167C2×D28—24no—
168C2×D28—24no—
169C2×D28—24no—
170C2×D28—24no—
171C2×D28—28no—
172C2×D28—22no—
173C2×D28—24no—
174C2×D28—24no—
175C2×D28—21no—
176C2×D28—44no—
177C2×D28—44no—
178C2×D28—44no—
179C2×D28—48no—
180C2×D28C2414yes—
181C2×D28C2412yes—
182C2×C4×D7—24no—
183C2×C4×D7—24no—
184C2×C4×D7—24no—
185C2×C4×D7—28no—
186C2×C4×D7—42no—
187C2×C4×D7—44no—
188C2×C4×D7—48no—
189C2×C4×D7C2×D412yes—
190C2×C4×D7C2×D414yes—
191C2×C4×D7C2×D414yes—
192C2×C4×D7C2412yes—
193C2×Dic14—24no—
194C2×Dic14—21no—
195C2×Dic14C2412yes—
196C2×C7×C8#4118yescomplemented
197C2×C7×C8C2414yes—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—24no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—24no—
204C7×C4⋊C4C2412yes—
205C7×C4⋊C4C2414yes—
206C7×C4⋊C4C2412yes—
207C42×C7C2×D412yes—
208C42×C7C2412yes—
209(C2×D7)⋊C4—24no—
210(C2×D7)⋊C4—24no—
211(C2×D7)⋊C4—24no—
212(C2×D7)⋊C4—24no—
213(C2×D7)⋊C4—28no—
214(C2×D7)⋊C4—24no—
215(C2×D7)⋊C4—28no—
216C4⋊Dic7—24no—
217Dic7⋊C4—24no—
218C4×Dic7—24no—
219C4×Dic7C2×D412yes—
220C4×Dic7C2×D414yes—
221C4×Dic7C2×D414yes—
222C4×Dic7C2412yes—
223C2×C7⋊C8—24no—
224C2×C7⋊C8C2414yes—
225D56—216no—
226#1132—216no—
227C8×D7—216no—
228C22×#41—141no—
229C22×SD16—72no—
230C22×D8—72no—
231C23×C8—72no—
232#532—78no—
233#532—144no—
234C2×C4⋊Q8C2×D711yescharacteristic
235C2×#127—71no—
236#529—78no—
237#529—144no—
238C2×#126—141no—
239C2×#125C2×D711yescharacteristic
240C2×#124—71no—
241C2×#124—141no—
242C2×#123—141no—
243C2×C22⋊Q8—71no—
244C2×#121—71no—
245C2×#121—72no—
246C2×#121—141no—
247C2×C4×Q8—71no—
248C2×C4×D4—71no—
249C2×C4×D4—72no—
250C2×C4×D4—141no—
251C2×C4×D4—141no—
252C2×#117—141no—
253C22×C42—71no—
254#501C2×D718yescomplemented
255#496—78no—
256#490—78no—
257#489—78no—
258#466—716no—
259#465—716no—
260#442—716no—
261#440—716no—
262C2×#107C2×D712yes—
263C2×#106C2×D712yes—
264C2×#103—72no—
265C2×#102—72no—
266C2×#102—72no—
267C2×#102—72no—
268C2×#97—72no—
269C2×#97—71no—
270C2×#97—78no—
271C2×#97C2×D711yescharacteristic
272C23×C7C22×D411yescharacteristic
273C22×D7—22no—
274C22×D7—22no—
275C22×D7—22no—
276C22×D7—24no—
277C22×D7—44no—
278C22×D7—42no—
279C22×D7—44no—
280C22×D7—42no—
281C22×D7—42no—
282C22×D7—44no—
283C22×D7AGL(1,ℤ/8)14yes—
284C22×D7C22×D412yes—
285C7×Q8—48no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—24no—
291C2×C4×C7—44no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7C22×D412yes—
295C2×C4×C7C22×D412yes—
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4114yes—
298C2×C4×C7C2511yescharacteristic
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—42no—
304C2×Dic7C22×D412yes—
305C2×Dic7C22×D412yes—
306C2×Dic7C22×D412yes—
307D28—24no—
308D28—24no—
309D28—24no—
310D28—48no—
311D28—48no—
312D28—48no—
313D28—416no—
314D28—44no—
315D28—42no—
316C4×D7—28no—
317C4×D7—24no—
318C4×D7—24no—
319C4×D7—44no—
320C4×D7—48no—
321C4×D7—416no—
322Dic14—24no—
323Dic14—42no—
324C7×C8C2×#4118yescomplemented
325C7⋊C8—24no—
326C2×#41—148no—
327C2×#41—288no—
328C22×Q8—21no—
329C22×Q8—71no—
330C22×D4—71no—
331C22×D4—72no—
332C22×D4—141no—
333C22×D4—142no—
334C22×D4—141no—
335C22×D4—141no—
336C23×C4—71no—
337C23×C4—71no—
338C23×C4—71no—
339C23×C4—141no—
340C23×C4—142no—
341C2×SD16—78no—
342C2×SD16—148no—
343C2×D8—78no—
344C2×D8—148no—
345C22×C8—72no—
346C22×C8—78no—
347C22×C8—148no—
348C22×C8C22×D712yes—
349C4⋊Q8—22no—
350C4⋊Q8C22×D714yes—
351#127—74no—
352#127—142no—
353#126—148no—
354#125—22no—
355#125C22×D714yes—
356#124—74no—
357#124—148no—
358#124—142no—
359#123—148no—
360C22⋊Q8—74no—
361C22⋊Q8—142no—
362#121—74no—
363#121—78no—
364#121—142no—
365#121—144no—
366#121—148no—
367C4×Q8—74no—
368C4×Q8—142no—
369C4×D4—74no—
370C4×D4—78no—
371C4×D4—142no—
372C4×D4—144no—
373C4×D4—148no—
374C4×D4—148no—
375#117—148no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4C22×D712yes—
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C22⋊C4—141no—
385C2×C22⋊C4—142no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4—142no—
388C2×C22⋊C4—141no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—74no—
395C2×C42—74no—
396C2×C42—141no—
397C2×C42C22×D711yescharacteristic
398#107C22×D718yes—
399#106C22×D718yes—
400#103—78no—
401#102—78no—
402#102—78no—
403#102—78no—
404#97—78no—
405#97—74no—
406#97C22×D714yes—
407C22×C7#53212yes—
408C22×C7C2×AGL(1,ℤ/8)12yes—
409C22×C7#57512yes—
410C22×C7C23×D411yescharacteristic
411C2×D7—28no—
412C2×D7—22no—
413C2×D7—22no—
414C2×D7—44no—
415C2×D7—42no—
416C2×D7—44no—
417C2×D7—42no—
418C2×D7—42no—
419C2×D7—44no—
420C2×D7—88no—
421C2×D7—84no—
422C2×D7C2×AGL(1,ℤ/8)14yes—
423C4×C7—22no—
424C4×C7—22no—
425C4×C7—44no—
426C4×C7—42no—
427C4×C7—42no—
428C4×C7#53212yes—
429C4×C7C22×#4112yes—
430Dic7—22no—
431Dic7—22no—
432Dic7—22no—
433Dic7—42no—
434C24—71no—
435C24—141no—
436C24—142no—
437#41—2832no—
438C2×Q8—24no—
439C2×Q8—44no—
440C2×Q8—72no—
441C2×Q8—141no—
442C2×Q8—144no—
443C2×D4—72no—
444C2×D4—74no—
445C2×D4—144no—
446C2×D4—148no—
447C2×D4—141no—
448C2×D4—144no—
449C2×D4—142no—
450C2×D4—144no—
451C2×D4—144no—
452C2×D4—144no—
453C2×D4—144no—
454C2×D4—284no—
455C2×D4—284no—
456C2×D4—284no—
457C2×D4—288no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—72no—
465C22×C4—74no—
466C22×C4—72no—
467C22×C4—74no—
468C22×C4—144no—
469C22×C4—148no—
470C22×C4—144no—
471C22×C4—144no—
472C22×C4—141no—
473C22×C4—282no—
474C22×C4—284no—
475C22×C4—288no—
476C22×C4D4×D711yescharacteristic
477C22×C4D4×D711yescharacteristic
478C22×C4C23×D711yescharacteristic
479SD16—1416no—
480D8—1416no—
481C2×C8—74no—
482C2×C8—144no—
483C2×C8—1416no—
484C2×C8#116118yescomplemented
485C2×C8C23×D714yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—144no—
493C4⋊C4—144no—
494C4⋊C4C23×D712yes—
495C4⋊C4C23×D712yes—
496C4⋊C4C23×D714yes—
497C22⋊C4—144no—
498C22⋊C4—148no—
499C22⋊C4—144no—
500C22⋊C4—148no—
501C22⋊C4—144no—
502C22⋊C4—144no—
503C22⋊C4—144no—
504C42—74no—
505C42—72no—
506C42—74no—
507C42—72no—
508C42—144no—
509C42D4×D712yes—
510C42C23×D712yes—
511C2×C7#318014yescomplemented
512C2×C7C2×#53211yescharacteristic
513C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
514C2×C7C2×#57511yescharacteristic
515D7—24no—
516D7—84no—
517D7—82no—
518C23—74no—
519C23—72no—
520C23—142no—
521C23—142no—
522C23—142no—
523C23—144no—
524C23—282no—
525C23—284no—
526C23—284no—
527C23—282no—
528C23—282no—
529C23—284no—
530C23C2×D4×D711yescharacteristic
531Q8—48no—
532Q8—144no—
533Q8—282no—
534D4—144no—
535D4—144no—
536D4—144no—
537D4—288no—
538D4—288no—
539D4—288no—
540D4—2816no—
541D4—282no—
542D4—284no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—148no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—142no—
561C2×C4—284no—
562C2×C4—288no—
563C2×C4—2816no—
564C2×C4—282no—
565C2×C4C2×D4×D712yes—
566C2×C4C2×D4×D712yes—
567C2×C4C2×D4×D711yescharacteristic
568C2×C4C2×#116114yes—
569C2×C4C24×D711yescharacteristic
570C8—144no—
571C8C2×#116118yescomplemented
572C7C2×#318011yescharacteristic, complemented, Sylow
573C22—74no—
574C22—142no—
575C22—148no—
576C22—142no—
577C22—284no—
578C22—282no—
579C22—282no—
580C22—284no—
581C22—282no—
582C22—284no—
583C22—568no—
584C22—564no—
585C22#9041612yes—
586C22D7×AGL(1,ℤ/8)12yes—
587C22#9046812yes—
588C22C22×D4×D711yescharacteristic
589C4—22no—
590C4—22no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—142no—
595C4—142no—
596C4—142no—
597C4—282no—
598C4#9041612yes—
599C4C22×#116112yes—
600C2—144no—
601C2—564no—
602C2—562no—
603C2#1173940714yesdirect product factor
604C2C2×#9041611yescharacteristic
605C2C2×D7×AGL(1,ℤ/8)11yescharacteristic
606C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#501C2
socleC23×C7C22×D4
hypercenterC2×#501D7
hypocenterC7C2×#3180
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×#90416C2semidirect
2. C2×#90382C2semidirect
3. C2×C7×#501C2semidirect
4. C2×#89861C2semidirect
5. C2×#89839C2semidirect
6. #11739407C2direct (N × H)
7. C2×#89655C2semidirect
8. C2×#89661C2semidirect
9. C2×#89636C2semidirect
10. C2×#89641C2semidirect
11. C2×#89484C2semidirect
12. C2×D7×#97C2semidirect
21. #90416C22semidirect
23. #90382C22semidirect
39. C7×#501C22semidirect
44. #89861C22semidirect
45. #89839C22semidirect
51. #89661C22semidirect
52. #89655C22semidirect
53. #89641C22semidirect
54. #89636C22semidirect
55. #89484C22semidirect
56. D7×#97C22semidirect
142. C2×#501D7semidirect
196. C2×C7×C8#41semidirect
254. #501C2×D7semidirect
324. C7×C8C2×#41semidirect
484. C2×C8#1161semidirect
511. C2×C7#3180semidirect
571. C8C2×#1161semidirect
572. C7C2×#3180semidirect
603. C2#11739407direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904162
C2×#903822
C2×C7×#5012
C2×#898612
C2×#898392
#117394072
C2×#896552
C2×#896612
C2×#896362
C2×#896412
C2×#894842
C2×D7×#972
C2×#31807