Finite Group Databasebeta

← C2×#11740567 #14225870

Subgroups — C2×#11740567 #14225870

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×C7×#529C211yesmaximal, characteristic
2C2×#90161C211yesmaximal, characteristic, complemented
3#11740567C2116yesmaximal, direct product factor
4C2×#89836C212yesmaximal, complemented
5C2×#89828C212yesmaximal, complemented
6C2×#89817C212yesmaximal, complemented
7C2×#89807C212yesmaximal
8C2×#89793C211yesmaximal, characteristic, complemented
9C2×#89792C211yesmaximal, characteristic, complemented
10C2×#89780C211yesmaximal, characteristic, complemented
11C2×#89779C211yesmaximal, characteristic, complemented
12C2×#89697C211yesmaximal, characteristic, complemented
13C7×#529—24no—
14C7×#529C2218yes—
15C2×C7×#126—21no—
16C2×C7×#125C2211yescharacteristic
17C2×C7×#124C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117—21no—
25C22×C42×C7C2211yescharacteristic
26C2×#6489C2212yes—
27C2×#6487—21no—
28C2×#6487—21no—
29#90161—24no—
30#90161C2218yescomplemented
31C2×#6433C2212yes—
32C2×#6432C2211yescharacteristic
33C2×#6432C2211yescharacteristic
34C2×C4×Dic14—21no—
35C2×C4×Dic14—21no—
36C2×#6409C2212yes—
37C2×#6409C2212yes—
38C2×#6406C2212yes—
39C2×#6406C2212yes—
40C2×C22⋊(C7⋊C8)C2212yes—
41#89836C22116yescomplemented
42#89828C22116yescomplemented
43#89817C22116yescomplemented
44#89807C22116yes—
45#89793C2218yescomplemented
46#89792C2218yescomplemented
47#89780C2218yescomplemented
48#89779C2218yescomplemented
49C2×#6382C2212yes—
50C2×#6382C2212yes—
51C2×#6381C2212yes—
52C2×#6381C2212yes—
53#89697C2218yescomplemented
54C2×C4⋊(C7⋊C8)C2211yescharacteristic
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56C2×C4×C7⋊C8C2212yes—
57C2×#3233—71nomaximal, Sylow
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7D411yescharacteristic
63C23×C4×C7D411yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C22×Dic14—21no—
66C22×Dic14—21no—
67C7×#126—28no—
68C7×#125—22no—
69C7×#125C2314yes—
70C7×#124—22no—
71C7×#124C2314yes—
72C7×#123—28no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8C2314yes—
75C7×#121—22no—
76C7×#121C2314yes—
77C4×C7×Q8—22no—
78C4×C7×Q8C2314yes—
79C4×C7×D4—28no—
80C4×C7×D4—22no—
81C4×C7×D4C2314yes—
82C7×#117—28no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—21no—
94C2×C42×C7—21no—
95C2×C42×C7D411yescharacteristic
96C2×C42×C7D411yescharacteristic
97C2×C42×C7D414yes—
98C2×C42×C7C2311yescharacteristic
99C2×C42×C7C2311yescharacteristic
100C2×C42×C7C2314yes—
101C2×C22⋊Dic7—22no—
102#6489—24no—
103#6489C2318yes—
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7—21no—
106C2×C4⋊Dic7C2312yes—
107C2×C4⋊Dic7C2312yes—
108#6487—28no—
109#6487—28no—
110C2×Dic7⋊C4—22no—
111C2×Dic7⋊C4—22no—
112C2×C4×Dic7—22no—
113C22×C7⋊C8C2312yes—
114C22×C7⋊C8C2312yes—
115#6433—24no—
116#6433C2318yes—
117#6432—22no—
118#6432—22no—
119#6432C2314yes—
120#6432C2314yes—
121C4×Dic14—28no—
122C4×Dic14—28no—
123#6409C2318yes—
124#6409C2318yes—
125#6406C2318yes—
126#6406C2318yes—
127C22⋊(C7⋊C8)C2318yes—
128#6382C2318yes—
129#6382C2318yes—
130#6381C2318yes—
131#6381C2318yes—
132C4⋊(C7⋊C8)C2314yes—
133C4⋊(C7⋊C8)C2314yes—
134C4×C7⋊C8C2318yes—
135C2×#533—71no—
136C2×#529D711yescharacteristic
137#3233—716no—
138C2×#499—72no—
139C2×#488—72no—
140C2×#485—72no—
141C2×#482—72no—
142C2×#477—71no—
143C2×#475—71no—
144C2×#476—71no—
145C2×#474—71no—
146C2×#431—71no—
147C24×C7—21no—
148C24×C7C2×D411yescharacteristic
149C2×C7×Q8—21no—
150C2×C7×Q8—22no—
151C2×C7×Q8—22no—
152C2×C7×Q8C2412yes—
153C2×C7×D4—24no—
154C2×C7×D4—21no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—44no—
158C2×C7×D4C2412yes—
159C22×C4×C7—24no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—24no—
166C22×C4×C7—42no—
167C22×C4×C7—44no—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D414yes—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D414yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2412yes—
177C22×C4×C7C2411yescharacteristic
178C22×Dic7—22no—
179C22×Dic7—22no—
180C2×Dic14—24no—
181C2×Dic14—24no—
182C2×Dic14—44no—
183C2×Dic14—44no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4C2412yes—
191C7×C4⋊C4C2412yes—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—24no—
197C42×C7—24no—
198C42×C7C2×D414yes—
199C42×C7C2×D414yes—
200C42×C7C2×D412yes—
201C42×C7C2×D412yes—
202C42×C7C2412yes—
203C42×C7C2412yes—
204C22⋊Dic7—28no—
205C4⋊Dic7—24no—
206C4⋊Dic7—24no—
207C4⋊Dic7—22no—
208C4⋊Dic7—22no—
209C4⋊Dic7C2414yes—
210C4⋊Dic7C2414yes—
211Dic7⋊C4—28no—
212Dic7⋊C4—28no—
213C4×Dic7—28no—
214C2×C7⋊C8—24no—
215C2×C7⋊C8—24no—
216C2×C7⋊C8C2414yes—
217C2×C7⋊C8C2414yes—
218#533—78no—
219#533—144no—
220C2×C4⋊Q8—71no—
221C2×C4⋊Q8—71no—
222#529—24no—
223#529C2×D718yes—
224C2×#126—21no—
225C2×#125—72no—
226C2×#125C2×D711yescharacteristic
227C2×#124C2×D711yescharacteristic
228C2×#123—21no—
229C2×C22⋊Q8—141no—
230C2×C22⋊Q8—141no—
231C2×C22⋊Q8C2×D711yescharacteristic
232C2×#121C2×D711yescharacteristic
233C2×C4×Q8—141no—
234C2×C4×Q8—141no—
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×D4—21no—
237C2×C4×D4C2×D711yescharacteristic
238C2×#117—21no—
239C2×#117—72no—
240C22×C42C2×D711yescharacteristic
241#499—716no—
242#488—716no—
243#485—716no—
244#482—716no—
245#477—78no—
246#476—78no—
247#475—78no—
248#474—78no—
249#431—78no—
250C2×#107—72no—
251C2×#107—72no—
252C2×#106—72no—
253C2×#106—72no—
254C2×C4⋊C8—71no—
255C2×C4⋊C8—71no—
256C2×#103—72no—
257C2×#103—72no—
258C2×#102—72no—
259C2×#102—72no—
260C2×C22⋊C8—72no—
261C2×C4×C8—72no—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—22no—
265C23×C7—22no—
266C23×C7—42no—
267C23×C7—42no—
268C23×C7—42no—
269C23×C7C22×D412yes—
270C23×C7C22×D411yescharacteristic
271C7×Q8—22no—
272C7×Q8—22no—
273C7×Q8—42no—
274C7×D4—22no—
275C7×D4—22no—
276C7×D4—48no—
277C7×D4—42no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—24no—
283C2×C4×C7—28no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—44no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—48no—
294C2×C4×C7#13514yes—
295C2×C4×C7C22×D412yes—
296C2×C4×C7C22×D412yes—
297C2×C4×C7C22×D412yes—
298C2×C4×C7C22×D412yes—
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C22×D412yes—
301C2×C4×C72−1+412yes—
302C2×C4×C72−1+412yes—
303C2×C4×C7C2511yescharacteristic
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—44no—
307C2×Dic7—44no—
308Dic14—48no—
309Dic14—48no—
310C7⋊C8—24no—
311C7⋊C8—24no—
312C22×Q8—141no—
313C22×Q8—141no—
314C22×Q8C22×D711yescharacteristic
315C22×D4—21no—
316C22×D4C22×D711yescharacteristic
317C23×C4—21no—
318C23×C4C22⋊D711yescharacteristic
319C23×C4C22⋊D711yescharacteristic
320C23×C4C22×D711yescharacteristic
321C22×C8—72no—
322C22×C8—72no—
323C4⋊Q8—74no—
324C4⋊Q8—74no—
325C4⋊Q8—142no—
326C4⋊Q8—142no—
327#126—28no—
328#125—22no—
329#125—78no—
330#125—144no—
331#125C22×D714yes—
332#124—22no—
333#124C22×D714yes—
334#123—28no—
335C22⋊Q8—22no—
336C22⋊Q8—148no—
337C22⋊Q8—148no—
338C22⋊Q8C22×D714yes—
339#121—22no—
340#121C22×D714yes—
341C4×Q8—22no—
342C4×Q8—148no—
343C4×Q8—148no—
344C4×Q8C22×D714yes—
345C4×D4—28no—
346C4×D4—22no—
347C4×D4C22×D714yes—
348#117—28no—
349#117—78no—
350#117—144no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—72no—
356C2×C4⋊C4—72no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—142no—
369C2×C42—21no—
370C2×C42—142no—
371C2×C42C22⋊D711yescharacteristic
372C2×C42C22⋊D711yescharacteristic
373C2×C42C22⋊D714yes—
374C2×C42C22×D711yescharacteristic
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D714yes—
377#107—78no—
378#107—78no—
379#106—78no—
380#106—78no—
381C4⋊C8—74no—
382C4⋊C8—74no—
383#103—78no—
384#103—78no—
385#102—78no—
386#102—78no—
387C22⋊C8—78no—
388C4×C8—78no—
389C22×C7—22no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—44no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—84no—
397C22×C7#53612yes—
398C22×C7C2×#13512yes—
399C22×C7#57512yes—
400C22×C7C23×D411yescharacteristic
401C4×C7—22no—
402C4×C7—22no—
403C4×C7—22no—
404C4×C7—22no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7#53612yes—
409C4×C7C2×#13514yes—
410C4×C7C2×2−1+411yescharacteristic
411C4×C7C2×2−1+411yescharacteristic
412Dic7—44no—
413Dic7—44no—
414C24—21no—
415C24C2×C22⋊D711yescharacteristic
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—144no—
420C2×Q8—144no—
421C2×Q8—284no—
422C2×Q8—284no—
423C2×Q8C23×D712yes—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—44no—
429C2×D4C23×D712yes—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4C2×C22⋊D712yes—
442C22×C4C2×C22⋊D714yes—
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D714yes—
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C23×D712yes—
450C22×C4C23×D711yescharacteristic
451C2×C8—74no—
452C2×C8—74no—
453C2×C8—144no—
454C2×C8—144no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—74no—
462C4⋊C4—74no—
463C4⋊C4—148no—
464C4⋊C4—148no—
465C4⋊C4—142no—
466C4⋊C4—142no—
467C4⋊C4—144no—
468C4⋊C4—144no—
469C4⋊C4C23×D712yes—
470C4⋊C4C23×D712yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—148no—
477C42—24no—
478C42—148no—
479C42C2×C22⋊D714yes—
480C42C2×C22⋊D712yes—
481C42C2×C22⋊D714yes—
482C42C2×C22⋊D712yes—
483C42C23×D712yes—
484C42C23×D712yes—
485C2×C7—42no—
486C2×C7—82no—
487C2×C7#323314yescomplemented
488C2×C7C2×#53611yescharacteristic
489C2×C7C22×#13511yescharacteristic
490C2×C7C2×#57511yescharacteristic
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C22×C22⋊D712yes—
499C23C22×C22⋊D711yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—42no—
503Q8—288no—
504Q8—288no—
505D4—22no—
506D4—22no—
507D4—48no—
508D4—42no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—144no—
526C2×C4—144no—
527C2×C4—284no—
528C2×C4—284no—
529C2×C4#651314yes—
530C2×C4#655412yes—
531C2×C4#655412yes—
532C2×C4C22×C22⋊D712yes—
533C2×C4C22×C22⋊D712yes—
534C2×C4C22×C22⋊D712yes—
535C2×C4C22×C22⋊D712yes—
536C2×C4C22×C22⋊D712yes—
537C2×C4C22×C22⋊D711yescharacteristic
538C2×C4C24×D711yescharacteristic
539C8—144no—
540C8—144no—
541C7C2×#323311yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—22no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—84no—
550C22C2×#651312yes—
551C22#9051312yes—
552C22#9051612yes—
553C22C23×C22⋊D711yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—284no—
562C4—284no—
563C4C2×#651314yes—
564C4#9051612yes—
565C4C2×#655411yescharacteristic
566C4C2×#655411yescharacteristic
567C2—42no—
568C2—82no—
569C2#1174056714yesdirect product factor
570C2C22×#651311yescharacteristic
571C2C2×#9051311yescharacteristic
572C2C2×#9051611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#529C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC2×#529D7
hypocenterC7C2×#3233
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#90161C2semidirect
3. #11740567C2direct (N × H)
4. C2×#89836C2semidirect
5. C2×#89828C2semidirect
6. C2×#89817C2semidirect
8. C2×#89793C2semidirect
9. C2×#89792C2semidirect
10. C2×#89780C2semidirect
11. C2×#89779C2semidirect
12. C2×#89697C2semidirect
30. #90161C22semidirect
41. #89836C22semidirect
42. #89828C22semidirect
43. #89817C22semidirect
45. #89793C22semidirect
46. #89792C22semidirect
47. #89780C22semidirect
48. #89779C22semidirect
53. #89697C22semidirect
487. C2×C7#3233semidirect
541. C7C2×#3233semidirect
569. C2#11740567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5292
C2×#901612
#117405672
C2×#898362
C2×#898282
C2×#898172
C2×#898072
C2×#897932
C2×#897922
C2×#897802
C2×#897792
C2×#896972
C2×#32337