Finite Group Databasebeta

← C2×#11738510 #14225873

Subgroups — C2×#11738510 #14225873

Order 1792.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 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×#476C211yesmaximal, characteristic, complemented
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89779C211yesmaximal, characteristic, complemented
6#11738510C2116yesmaximal, direct product factor
7C2×#89639C212yesmaximal, complemented
8C2×#89630C212yesmaximal, complemented
9C2×#89623C211yesmaximal, characteristic, complemented
10C2×#89605C211yesmaximal, characteristic, complemented
11C2×#89540C212yesmaximal, complemented
12C2×#89551C212yesmaximal
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×#6509—21no—
16C2×Q8×Dic7—21no—
17C2×#6501C2211yescharacteristic
18C2×#6499C2211yescharacteristic
19C2×C4×C22⋊D7—21no—
20C2×#6489—21no—
21#90419—24no—
22#90419C2218yescomplemented
23#90248—24no—
24#90248C2218yes—
25C2×#6460—21no—
26C2×#6460C2211yescharacteristic
27C2×#6455C2212yes—
28C2×#6453C2212yes—
29C2×#6453C2211yescharacteristic
30C2×#6450—21no—
31C2×#6446—21no—
32C2×#6442—21no—
33C2×C42×D7C2211yescharacteristic
34C2×#6432C2211yescharacteristic
35C2×C4×Dic14C2211yescharacteristic
36C7×#476C2218yescomplemented
37C2×C7×#106C2212yes—
38C2×C7×C4⋊C8C2211yescharacteristic
39C2×C7×#102C2212yes—
40C2×#6406C2212yes—
41C2×(C2×D7)⋊C8C2212yes—
42C2×#6391C2212yes—
43C2×#6390C2212yes—
44C2×C8×Dic7C2212yes—
45#89863C2218yescomplemented
46#89779C2218yescomplemented
47C2×#6384C2212yes—
48C2×#6381C2212yes—
49C2×#6381C2212yes—
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89639C22116yescomplemented
52#89630C22116yescomplemented
53#89623C2218yescomplemented
54#89605C2218yescomplemented
55#89551C22116yes—
56#89540C22116yescomplemented
57C2×#3233—71nomaximal, Sylow
58C22×C7×Q8—21no—
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
67C22×C7×C8C2312yes—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2314yes—
70C4×C7×D4—22no—
71C4×C7×D4C2314yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80#6509—28no—
81Q8×Dic7—28no—
82#6501—22no—
83#6501C2314yes—
84#6499—22no—
85#6499C2314yes—
86C4×C22⋊D7—28no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—22no—
89#6489—28no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7C2312yes—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—22no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—22no—
101C2×C4×Dic7D411yescharacteristic
102C2×C4×Dic7D411yescharacteristic
103C2×C4×Dic7C2311yescharacteristic
104C22×C7⋊C8C2312yes—
105#6460—28no—
106#6460—22no—
107#6460C2314yes—
108#6455—24no—
109#6455C2318yes—
110#6453—24no—
111#6453—22no—
112#6453C2314yes—
113#6453C2318yes—
114#6450—28no—
115#6446—28no—
116#6442—28no—
117C42×D7D414yes—
118C42×D7C2314yes—
119#6432—22no—
120#6432C2314yes—
121C4×Dic14—22no—
122C4×Dic14C2314yes—
123C7×#106C2318yes—
124C7×C4⋊C8C2314yes—
125C7×#102C2318yes—
126#6406C2318yes—
127(C2×D7)⋊C8C2318yes—
128#6391C2318yes—
129#6390C2318yes—
130C8×Dic7C2318yes—
131#6384C2318yes—
132#6381C2318yes—
133#6381C2318yes—
134C4⋊(C7⋊C8)C2314yes—
135C2×#533—71no—
136C2×#529—71no—
137#3233—716no—
138C2×#499—72no—
139C2×#488—72no—
140C2×#485—72no—
141C2×#482—72no—
142C2×#477—71no—
143C2×#475—71no—
144C2×#476D711yescharacteristic, complemented
145C2×#474—71no—
146C2×#431—71no—
147C24×C7—21no—
148C23×D7C2×D411yescharacteristic
149C2×C7×Q8—24no—
150C2×C7×Q8—44no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4C2412yes—
155C22×C4×C7—22no—
156C22×C4×C7—21no—
157C22×C4×C7—24no—
158C22×C4×C7—44no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C2×C22⋊D7—24no—
163C2×C22⋊D7—44no—
164C22×Dic7—21no—
165C22×Dic7—22no—
166C22×Dic7—21no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C2×C4×D7—24no—
171C2×C4×D7—22no—
172C2×C4×D7—22no—
173C2×C4×D7—42no—
174C2×C4×D7C2×D412yes—
175C2×C4×D7C2×D414yes—
176C2×C4×D7C2×D414yes—
177C2×C4×D7C2412yes—
178C2×Dic14—22no—
179C2×Dic14—22no—
180C2×Dic14—21no—
181C2×Dic14—24no—
182C2×Dic14—44no—
183C2×Dic14C2412yes—
184C2×C7×C8—24no—
185C2×C7×C8C2414yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4C2414yes—
190C7×C4⋊C4C2412yes—
191C7×C22⋊C4—24no—
192C42×C7C2×D412yes—
193C42×C7C2412yes—
194C22⋊Dic7—24no—
195C22⋊Dic7—24no—
196C22⋊Dic7—24no—
197(C2×D7)⋊C4—24no—
198(C2×D7)⋊C4—28no—
199C4⋊Dic7—21no—
200C4⋊Dic7—22no—
201C4⋊Dic7—24no—
202C4⋊Dic7—24no—
203C4⋊Dic7—24no—
204C4⋊Dic7—28no—
205C4⋊Dic7C2414yes—
206C4⋊Dic7C2412yes—
207Dic7⋊C4—24no—
208Dic7⋊C4—24no—
209Dic7⋊C4—28no—
210C4×Dic7—24no—
211C4×Dic7—28no—
212C4×Dic7C2×D412yes—
213C4×Dic7C2×D414yes—
214C4×Dic7C2×D414yes—
215C4×Dic7C2412yes—
216C2×C7⋊C8—24no—
217C2×C7⋊C8C2414yes—
218#533—78no—
219#533—144no—
220C2×C4⋊Q8—71no—
221C2×C4⋊Q8C2×D711yescharacteristic
222#529—78no—
223#529—144no—
224C2×#126—141no—
225C2×#125—72no—
226C2×#125—71no—
227C2×#124—71no—
228C2×#123—141no—
229C2×C22⋊Q8—71no—
230C2×C22⋊Q8—141no—
231C2×C22⋊Q8—141no—
232C2×#121—71no—
233C2×C4×Q8—71no—
234C2×C4×Q8—141no—
235C2×C4×Q8—141no—
236C2×C4×D4—141no—
237C2×C4×D4C2×D711yescharacteristic
238C2×#117—72no—
239C2×#117—141no—
240C22×C42—71no—
241#499—716no—
242#488—716no—
243#485—716no—
244#482—716no—
245#477—78no—
246#476C2×D718yescomplemented
247#475—78no—
248#474—78no—
249#431—78no—
250C2×#107—72no—
251C2×#107—72no—
252C2×#106—72no—
253C2×#106C2×D712yes—
254C2×C4⋊C8—71no—
255C2×C4⋊C8C2×D711yescharacteristic
256C2×#103—72no—
257C2×#103—72no—
258C2×#102—72no—
259C2×#102C2×D712yes—
260C2×C22⋊C8—72no—
261C2×C4×C8—72no—
262C23×C7—22no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—22no—
270C22×D7C22×D412yes—
271C7×Q8—48no—
272C7×D4—22no—
273C7×D4—22no—
274C7×D4—42no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—42no—
279C2×C4×C7—44no—
280C2×C4×C7—48no—
281C2×C4×C7#13514yes—
282C2×C4×C7C22×D412yes—
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D412yes—
285C2×C4×C72−1+412yes—
286C2×C4×C72−1+412yes—
287C2×C4×C7C2511yescharacteristic
288C22⋊D7—48no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—24no—
294C2×Dic7—22no—
295C2×Dic7—42no—
296C2×Dic7—44no—
297C2×Dic7—42no—
298C2×Dic7C22×D412yes—
299C2×Dic7C22×D412yes—
300C2×Dic7C22×D412yes—
301C4×D7—24no—
302C4×D7—28no—
303C4×D7—22no—
304C4×D7—22no—
305C4×D7—44no—
306Dic14—22no—
307Dic14—22no—
308Dic14—42no—
309Dic14—48no—
310C7×C8—24no—
311C7⋊C8—24no—
312C22×Q8—21no—
313C22×Q8—71no—
314C22×Q8—141no—
315C22×D4—141no—
316C22×D4C22×D711yescharacteristic
317C23×C4—21no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4—71no—
321C22×C8—72no—
322C22×C8C22×D712yes—
323C4⋊Q8—22no—
324C4⋊Q8—74no—
325C4⋊Q8—142no—
326C4⋊Q8C22×D714yes—
327#126—148no—
328#125—74no—
329#125—78no—
330#125—142no—
331#125—144no—
332#124—74no—
333#124—142no—
334#123—148no—
335C22⋊Q8—74no—
336C22⋊Q8—142no—
337C22⋊Q8—148no—
338C22⋊Q8—148no—
339#121—74no—
340#121—142no—
341C4×Q8—74no—
342C4×Q8—148no—
343C4×Q8—148no—
344C4×Q8—142no—
345C4×D4—22no—
346C4×D4—148no—
347C4×D4C22×D714yes—
348#117—78no—
349#117—148no—
350#117—144no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—72no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22×D712yes—
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—142no—
365C2×C22⋊C4—141no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C42—71no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—74no—
373C2×C42—74no—
374C2×C42—142no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377#107—78no—
378#107—78no—
379#106—78no—
380#106C22×D718yes—
381C4⋊C8—74no—
382C4⋊C8C22×D714yes—
383#103—78no—
384#103—78no—
385#102—78no—
386#102C22×D718yes—
387C22⋊C8—78no—
388C4×C8—78no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—84no—
393C22×C7#53612yes—
394C22×C7C2×#13512yes—
395C22×C7#57512yes—
396C22×C7C23×D411yescharacteristic
397C2×D7—22no—
398C2×D7—22no—
399C2×D7—22no—
400C2×D7—44no—
401C4×C7—22no—
402C4×C7—42no—
403C4×C7—44no—
404C4×C7#53612yes—
405C4×C7C2×#13514yes—
406C4×C7C2×2−1+411yescharacteristic
407C4×C7C2×2−1+411yescharacteristic
408Dic7—22no—
409Dic7—22no—
410Dic7—22no—
411Dic7—42no—
412Dic7—44no—
413Dic7—42no—
414C24—21no—
415C24—71no—
416C2×Q8—24no—
417C2×Q8—44no—
418C2×Q8—72no—
419C2×Q8—141no—
420C2×Q8—144no—
421C2×Q8—142no—
422C2×Q8—142no—
423C2×Q8—284no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—144no—
428C2×D4—284no—
429C2×D4C23×D712yes—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—72no—
438C22×C4—74no—
439C22×C4—72no—
440C22×C4—74no—
441C22×C4—144no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—142no—
447C22×C4—282no—
448C22×C4D4×D711yescharacteristic
449C22×C4D4×D711yescharacteristic
450C22×C4C23×D711yescharacteristic
451C2×C8—24no—
452C2×C8—74no—
453C2×C8—144no—
454C2×C8C23×D714yes—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—74no—
459C4⋊C4—72no—
460C4⋊C4—144no—
461C4⋊C4—148no—
462C4⋊C4—144no—
463C4⋊C4—144no—
464C4⋊C4—148no—
465C4⋊C4—144no—
466C4⋊C4—142no—
467C4⋊C4—141no—
468C4⋊C4—144no—
469C4⋊C4C23×D714yes—
470C4⋊C4C23×D712yes—
471C22⋊C4—24no—
472C22⋊C4—148no—
473C22⋊C4—144no—
474C22⋊C4—144no—
475C22⋊C4—144no—
476C22⋊C4—144no—
477C42—74no—
478C42—74no—
479C42—72no—
480C42—72no—
481C42—148no—
482C42—144no—
483C42D4×D712yes—
484C42C23×D712yes—
485C2×C7—82no—
486C2×C7#323314yescomplemented
487C2×C7C2×#53611yescharacteristic
488C2×C7C22×#13511yescharacteristic
489C2×C7C2×#57511yescharacteristic
490D7—42no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—72no—
496C23—142no—
497C23—142no—
498C23—142no—
499C23C2×D4×D711yescharacteristic
500Q8—48no—
501Q8—142no—
502Q8—142no—
503Q8—288no—
504Q8—282no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—288no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—72no—
516C2×C4—72no—
517C2×C4—72no—
518C2×C4—142no—
519C2×C4—142no—
520C2×C4—142no—
521C2×C4—148no—
522C2×C4—144no—
523C2×C4—142no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—144no—
528C2×C4—284no—
529C2×C4—284no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4#647914yes—
533C2×C4C2×D4×D712yes—
534C2×C4C2×D4×D712yes—
535C2×C4C2×D4×D711yescharacteristic
536C2×C4#655112yes—
537C2×C4#655412yes—
538C2×C4C24×D711yescharacteristic
539C8—24no—
540C8—144no—
541C7C2×#323311yescharacteristic, complemented, Sylow
542C22—42no—
543C22—42no—
544C22—42no—
545C22—84no—
546C22—142no—
547C22—142no—
548C22—142no—
549C22—284no—
550C22#9031912yes—
551C22C2×#647912yes—
552C22#9046412yes—
553C22C22×D4×D711yescharacteristic
554C4—22no—
555C4—44no—
556C4—42no—
557C4—142no—
558C4—142no—
559C4—142no—
560C4—282no—
561C4—282no—
562C4—284no—
563C4#9031912yes—
564C4C2×#647914yes—
565C4C2×#655111yescharacteristic
566C4C2×#655411yescharacteristic
567C2—82no—
568C2—282no—
569C2#1173851014yesdirect product factor
570C2C2×#9031911yescharacteristic
571C2C22×#647911yescharacteristic
572C2C2×#9046411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#476C2
socleC23×C7C22×D4
prime radicalC2×#6501C22
hypercenterC2×#476D7
hypocenterC7C2×#3233
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#90419C2semidirect
3. C2×C7×#476C2semidirect
4. C2×#89863C2semidirect
5. C2×#89779C2semidirect
6. #11738510C2direct (N × H)
7. C2×#89639C2semidirect
8. C2×#89630C2semidirect
9. C2×#89623C2semidirect
10. C2×#89605C2semidirect
11. C2×#89540C2semidirect
22. #90419C22semidirect
36. C7×#476C22semidirect
45. #89863C22semidirect
46. #89779C22semidirect
51. #89639C22semidirect
52. #89630C22semidirect
53. #89623C22semidirect
54. #89605C22semidirect
56. #89540C22semidirect
144. C2×#476D7semidirect
246. #476C2×D7semidirect
486. C2×C7#3233semidirect
541. C7C2×#3233semidirect
569. C2#11738510direct (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×#4762
C2×#898632
C2×#897792
#117385102
C2×#896392
C2×#896302
C2×#896232
C2×#896052
C2×#895402
C2×#895512
C2×#32337