Finite Group Databasebeta

← C2×#11737885 #14226648

Subgroups — C2×#11737885 #14226648

Order 1792.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic, complemented
2C2×#90248C211yesmaximal, characteristic
3C2×C7×#461C211yesmaximal, characteristic, complemented
4C2×#89946C212yesmaximal
5C2×#89938C212yesmaximal, complemented
6C2×#89862C211yesmaximal, characteristic, complemented
7C2×#89780C211yesmaximal, characteristic, complemented
8#11737885C2116yesmaximal, direct product factor
9C2×#89622C211yesmaximal, characteristic, complemented
10C2×#89609C211yesmaximal, characteristic, complemented
11C2×#89588C212yesmaximal, complemented
12C2×#89568C212yesmaximal, complemented
13C22×C7×SD16C2212yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6509C2212yes—
17C2×Q8×Dic7C2212yes—
18C22×#1144C2212yes—
19C2×#6501C2211yescharacteristic
20C2×#6499C2211yescharacteristic
21C2×C4×C22⋊D7—21no—
22C2×#6489—21no—
23C22×#1131C2212yes—
24#90419—24no—
25#90419C2218yescomplemented
26#90248—24no—
27#90248C2218yes—
28C2×#6460—21no—
29C2×#6460C2211yescharacteristic
30C2×#6455—21no—
31C2×#6453—21no—
32C2×#6453C2211yescharacteristic
33C2×#6450—21no—
34C2×#6446—21no—
35C2×#6442—21no—
36C2×C42×D7C2211yescharacteristic
37C2×#6432C2211yescharacteristic
38C2×C4×Dic14C2211yescharacteristic
39C7×#461C2218yescomplemented
40C2×C7×C4⋊C8C2211yescharacteristic
41C2×C7×#103C2212yes—
42C2×#6409C2212yes—
43C2×#6406C2212yes—
44#89946C22116yes—
45#89938C22116yescomplemented
46C2×#6392C2212yes—
47C2×#6390C2212yes—
48C2×Dic7⋊C8C2212yes—
49#89862C2218yescomplemented
50#89780C2218yescomplemented
51C2×#6382C2212yes—
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89622C2218yescomplemented
54#89609C2218yescomplemented
55#89588C22116yescomplemented
56#89568C22116yescomplemented
57C2×#3252—71nomaximal, Sylow
58C22×C7×Q8C2312yes—
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C22×C22⋊D7—21no—
62C22×C4×D7D411yescharacteristic
63C22×C4×D7D411yescharacteristic
64C22×C4×D7C2311yescharacteristic
65C22×Dic14—21no—
66C22×Dic14C2311yescharacteristic
67C2×C7×SD16—44no—
68C2×C7×SD16C2318yes—
69C22×C7×C8C2312yes—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8C2314yes—
72C4×C7×D4—22no—
73C4×C7×D4C2314yes—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82#6509—24no—
83#6509C2318yes—
84Q8×Dic7—24no—
85Q8×Dic7C2318yes—
86C2×#1144—44no—
87C2×#1144C2318yes—
88#6501—22no—
89#6501C2314yes—
90#6499—22no—
91#6499C2314yes—
92C4×C22⋊D7—28no—
93C2×(C2×D7)⋊C4—21no—
94C2×(C2×D7)⋊C4—22no—
95#6489—28no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—22no—
100C2×C4⋊Dic7C2312yes—
101C2×C4⋊Dic7C2311yescharacteristic
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4—21no—
104C2×Dic7⋊C4—22no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—22no—
107C2×C4×Dic7D411yescharacteristic
108C2×C4×Dic7D411yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C22×C7⋊C8C2312yes—
111C2×#1131—44no—
112C2×#1131C2318yes—
113#6460—28no—
114#6460—22no—
115#6460C2314yes—
116#6455—28no—
117#6453—28no—
118#6453—22no—
119#6453C2314yes—
120#6450—28no—
121#6446—28no—
122#6442—28no—
123C42×D7D414yes—
124C42×D7C2314yes—
125#6432—22no—
126#6432C2314yes—
127C4×Dic14—22no—
128C4×Dic14C2314yes—
129C7×C4⋊C8C2314yes—
130C7×#103C2318yes—
131#6409C2318yes—
132#6406C2318yes—
133#6392C2318yes—
134#6390C2318yes—
135Dic7⋊C8C2318yes—
136#6382C2318yes—
137C4⋊(C7⋊C8)C2314yes—
138C2×#533—71no—
139C2×#529—71no—
140#3252—716no—
141C2×#479—72no—
142C2×#477—71no—
143C2×#476—71no—
144C2×#470—72no—
145C2×#468—72no—
146C2×#464—72no—
147C2×#462—71no—
148C2×#461D711yescharacteristic, complemented
149C2×#423—71no—
150C24×C7—21no—
151C23×D7C2×D411yescharacteristic
152C2×C7×Q8—22no—
153C2×C7×Q8—24no—
154C2×C7×Q8—24no—
155C2×C7×Q8C2414yes—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4C2412yes—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—24no—
163C22×C4×C7—44no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2411yescharacteristic
167C2×C22⋊D7—24no—
168C2×C22⋊D7—44no—
169C22×Dic7—21no—
170C22×Dic7—22no—
171C22×Dic7—21no—
172C22×Dic7C2×D411yescharacteristic
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C2×C4×D7—24no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—42no—
179C2×C4×D7C2×D412yes—
180C2×C4×D7C2×D414yes—
181C2×C4×D7C2×D414yes—
182C2×C4×D7C2412yes—
183C2×Dic14—22no—
184C2×Dic14—22no—
185C2×Dic14—21no—
186C2×Dic14—24no—
187C2×Dic14—44no—
188C2×Dic14C2412yes—
189C7×SD16—48no—
190C2×C7×C8—24no—
191C2×C7×C8C2414yes—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4C2412yes—
196C7×C22⋊C4—24no—
197C42×C7C2×D412yes—
198C42×C7C2412yes—
199C22⋊Dic7—24no—
200C22⋊Dic7—24no—
201C22⋊Dic7—24no—
202#1144—48no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—28no—
205C4⋊Dic7—21no—
206C4⋊Dic7—22no—
207C4⋊Dic7—24no—
208C4⋊Dic7—24no—
209C4⋊Dic7—24no—
210C4⋊Dic7—28no—
211C4⋊Dic7C2414yes—
212C4⋊Dic7C2412yes—
213Dic7⋊C4—24no—
214Dic7⋊C4—24no—
215Dic7⋊C4—28no—
216C4×Dic7—24no—
217C4×Dic7—28no—
218C4×Dic7C2×D412yes—
219C4×Dic7C2×D414yes—
220C4×Dic7C2×D414yes—
221C4×Dic7C2412yes—
222C2×C7⋊C8—24no—
223C2×C7⋊C8C2414yes—
224#1131—48no—
225C22×Q16—72no—
226C22×SD16C2×D712yes—
227C22×M16—72no—
228#533—78no—
229#533—144no—
230C2×C4⋊Q8—71no—
231C2×C4⋊Q8C2×D711yescharacteristic
232#529—78no—
233#529—144no—
234C2×#126—141no—
235C2×#125—71no—
236C2×#125—141no—
237C2×#124—71no—
238C2×#123—141no—
239C2×C22⋊Q8—72no—
240C2×C22⋊Q8—71no—
241C2×C22⋊Q8—141no—
242C2×#121—71no—
243C2×C4×Q8—72no—
244C2×C4×Q8—71no—
245C2×C4×Q8—141no—
246C2×C4×D4—141no—
247C2×C4×D4C2×D711yescharacteristic
248C2×#117—141no—
249C2×#117—141no—
250C22×C42—71no—
251#479—716no—
252#477—78no—
253#476—78no—
254#470—716no—
255#468—716no—
256#464—716no—
257#462—78no—
258#461C2×D718yescomplemented
259C2×#107—72no—
260C2×#106—72no—
261#423—78no—
262C2×C4⋊C8—72no—
263C2×C4⋊C8—71no—
264C2×C4⋊C8C2×D711yescharacteristic
265C2×#103—72no—
266C2×#103—72no—
267C2×#103C2×D712yes—
268C2×#102—72no—
269C23×C7—22no—
270C23×C7—42no—
271C23×C7—42no—
272C23×C7—42no—
273C23×C7C22×D411yescharacteristic
274C22×D7—22no—
275C22×D7—22no—
276C22×D7—22no—
277C22×D7C22×D412yes—
278C7×Q8—24no—
279C7×Q8—24no—
280C7×Q8—44no—
281C7×D4—22no—
282C7×D4—22no—
283C7×D4—42no—
284C2×C4×C7—22no—
285C2×C4×C7—24no—
286C2×C4×C7—22no—
287C2×C4×C7—44no—
288C2×C4×C7—42no—
289C2×C4×C7—48no—
290C2×C4×C7#13714yes—
291C2×C4×C7C22×D412yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D412yes—
294C2×C4×C7H(5,𝔽2)12yes—
295C2×C4×C72−1+412yes—
296C2×C4×C7C2511yescharacteristic
297C22⋊D7—48no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—24no—
303C2×Dic7—22no—
304C2×Dic7—42no—
305C2×Dic7—44no—
306C2×Dic7—42no—
307C2×Dic7C22×D412yes—
308C2×Dic7C22×D412yes—
309C2×Dic7C22×D412yes—
310C4×D7—24no—
311C4×D7—28no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—44no—
315Dic14—22no—
316Dic14—22no—
317Dic14—48no—
318Dic14—42no—
319C7×C8—24no—
320C7⋊C8—24no—
321C22×Q8—71no—
322C22×Q8—141no—
323C22×Q8C22×D712yes—
324C22×D4—141no—
325C22×D4C22×D711yescharacteristic
326C23×C4—21no—
327C23×C4—71no—
328C23×C4—71no—
329C23×C4—71no—
330C2×Q16—78no—
331C2×Q16—284no—
332C2×SD16—44no—
333C2×SD16C22×D718yes—
334C2×M16—78no—
335C2×M16—284no—
336C22×C8—72no—
337C22×C8C22×D712yes—
338C4⋊Q8—22no—
339C4⋊Q8—74no—
340C4⋊Q8—142no—
341C4⋊Q8C22×D714yes—
342#126—148no—
343#125—74no—
344#125—142no—
345#125—148no—
346#124—74no—
347#124—142no—
348#123—148no—
349C22⋊Q8—78no—
350C22⋊Q8—74no—
351C22⋊Q8—142no—
352C22⋊Q8—148no—
353C22⋊Q8—144no—
354#121—74no—
355#121—142no—
356C4×Q8—78no—
357C4×Q8—74no—
358C4×Q8—148no—
359C4×Q8—144no—
360C4×Q8—142no—
361C4×D4—22no—
362C4×D4—148no—
363C4×D4C22×D714yes—
364#117—148no—
365#117—148no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—72no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—142no—
380C2×C22⋊C4—141no—
381C2×C22⋊C4—141no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C42—71no—
385C2×C42—71no—
386C2×C42—71no—
387C2×C42—74no—
388C2×C42—74no—
389C2×C42—142no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392#107—78no—
393#106—78no—
394C4⋊C8—78no—
395C4⋊C8—74no—
396C4⋊C8C22×D714yes—
397#103—78no—
398#103—78no—
399#103C22×D718yes—
400#102—78no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—84no—
405C22×C7#53712yes—
406C22×C7C2×#13712yes—
407C22×C7#57512yes—
408C22×C7C23×D411yescharacteristic
409C2×D7—22no—
410C2×D7—22no—
411C2×D7—22no—
412C2×D7—44no—
413C4×C7—22no—
414C4×C7—44no—
415C4×C7—42no—
416C4×C7#53712yes—
417C4×C7C2×#13714yes—
418C4×C7C2×H(5,𝔽2)11yescharacteristic
419C4×C7C2×2−1+411yescharacteristic
420Dic7—22no—
421Dic7—22no—
422Dic7—22no—
423Dic7—42no—
424Dic7—44no—
425Dic7—42no—
426C24—21no—
427C24—71no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—72no—
432C2×Q8—141no—
433C2×Q8—144no—
434C2×Q8—142no—
435C2×Q8—142no—
436C2×Q8—284no—
437C2×Q8C23×D714yes—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—144no—
442C2×D4—284no—
443C2×D4C23×D712yes—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—44no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—74no—
452C22×C4—72no—
453C22×C4—72no—
454C22×C4—74no—
455C22×C4—144no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4—282no—
462C22×C4D4×D711yescharacteristic
463C22×C4D4×D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465Q16—288no—
466SD16—48no—
467M16—288no—
468C2×C8—24no—
469C2×C8—74no—
470C2×C8—144no—
471C2×C8C23×D714yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—74no—
476C4⋊C4—72no—
477C4⋊C4—144no—
478C4⋊C4—148no—
479C4⋊C4—144no—
480C4⋊C4—144no—
481C4⋊C4—148no—
482C4⋊C4—144no—
483C4⋊C4—141no—
484C4⋊C4—142no—
485C4⋊C4—144no—
486C4⋊C4C23×D712yes—
487C22⋊C4—24no—
488C22⋊C4—148no—
489C22⋊C4—144no—
490C22⋊C4—144no—
491C22⋊C4—144no—
492C22⋊C4—144no—
493C42—74no—
494C42—72no—
495C42—74no—
496C42—72no—
497C42—148no—
498C42—144no—
499C42D4×D712yes—
500C42C23×D712yes—
501C2×C7—82no—
502C2×C7#325214yescomplemented
503C2×C7C2×#53711yescharacteristic
504C2×C7C22×#13711yescharacteristic
505C2×C7C2×#57511yescharacteristic
506D7—42no—
507C23—22no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—72no—
512C23—142no—
513C23—142no—
514C23—142no—
515C23C2×D4×D711yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—44no—
519Q8—142no—
520Q8—142no—
521Q8—288no—
522Q8—282no—
523D4—22no—
524D4—22no—
525D4—42no—
526D4—288no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—72no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—148no—
540C2×C4—144no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—144no—
546C2×C4—284no—
547C2×C4—282no—
548C2×C4—284no—
549C2×C4—282no—
550C2×C4#647814yes—
551C2×C4C2×D4×D712yes—
552C2×C4C2×D4×D712yes—
553C2×C4C2×D4×D711yescharacteristic
554C2×C4#654812yes—
555C2×C4#655412yes—
556C2×C4C24×D711yescharacteristic
557C8—24no—
558C8—144no—
559C7C2×#325211yescharacteristic, complemented, Sylow
560C22—42no—
561C22—42no—
562C22—42no—
563C22—84no—
564C22—142no—
565C22—142no—
566C22—142no—
567C22—284no—
568C22#9029912yes—
569C22C2×#647812yes—
570C22#9046812yes—
571C22C22×D4×D711yescharacteristic
572C4—22no—
573C4—44no—
574C4—42no—
575C4—142no—
576C4—142no—
577C4—142no—
578C4—282no—
579C4—284no—
580C4—282no—
581C4#9029912yes—
582C4C2×#647814yes—
583C4C2×#654811yescharacteristic
584C4C2×#655411yescharacteristic
585C2—82no—
586C2—282no—
587C2#1173788514yesdirect product factor
588C2C2×#9029911yescharacteristic
589C2C22×#647811yescharacteristic
590C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#461C2
socleC23×C7C22×D4
prime radicalC2×#6501C22
hypercenterC2×#461D7
hypocenterC7C2×#3252
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#90419C2semidirect
3. C2×C7×#461C2semidirect
5. C2×#89938C2semidirect
6. C2×#89862C2semidirect
7. C2×#89780C2semidirect
8. #11737885C2direct (N × H)
9. C2×#89622C2semidirect
10. C2×#89609C2semidirect
11. C2×#89588C2semidirect
12. C2×#89568C2semidirect
25. #90419C22semidirect
39. C7×#461C22semidirect
45. #89938C22semidirect
49. #89862C22semidirect
50. #89780C22semidirect
53. #89622C22semidirect
54. #89609C22semidirect
55. #89588C22semidirect
56. #89568C22semidirect
148. C2×#461D7semidirect
258. #461C2×D7semidirect
502. C2×C7#3252semidirect
559. C7C2×#3252semidirect
587. C2#11737885direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904192
C2×#902482
C2×C7×#4612
C2×#899462
C2×#899382
C2×#898622
C2×#897802
#117378852
C2×#896222
C2×#896092
C2×#895882
C2×#895682
C2×#32527