Finite Group Databasebeta

← C2×#11740463 #14225826

Subgroups — C2×#11740463 #14225826

Order 1792.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#125C211yesmaximal, characteristic, complemented
2C2×#90170C211yesmaximal, characteristic
3C2×#89835C212yesmaximal, complemented
4#11740463C2116yesmaximal, direct product factor
5C2×#89838C212yesmaximal, complemented
6C2×#89836C212yesmaximal, complemented
7C2×#89840C212yesmaximal, complemented
8C2×#89742C211yesmaximal, characteristic, complemented
9C2×#89743C211yesmaximal, characteristic, complemented
10C2×#89740C211yesmaximal, characteristic
11C2×#89739C211yesmaximal, characteristic
12C2×#89697C211yesmaximal, characteristic, complemented
13C2×C7×#125—21no—
14C2×C7×#125—21no—
15C2×C7×#125—24no—
16C2×C7×#125C2212yes—
17C2×C7×#125C2212yes—
18C2×C7×#125C2218yescomplemented
19C22×C7×C4⋊C4—21no—
20C22×C7×C4⋊C4—21no—
21C22×C7×C4⋊C4C2211yescharacteristic
22C22×C7×C4⋊C4C2211yescharacteristic
23C22×C42×C7C2211yescharacteristic
24C2×#6493C2211yescharacteristic
25C2×#6492—21no—
26C2×C4×C22⋊D7—21no—
27C2×#6487C2211yescharacteristic
28#90170—24no—
29#90170C2218yes—
30C2×#6439—21no—
31C2×#6438C2211yescharacteristic
32C2×C4×D28C2211yescharacteristic
33C2×#6435—21no—
34C2×#6433C2211yescharacteristic
35C2×C4×Dic14C2211yescharacteristic
36C2×C22⋊(C7⋊C8)C2211yescharacteristic
37C2×C22⋊(C7⋊C8)C2211yescharacteristic
38#89840C22116yescomplemented
39#89838C22116yescomplemented
40#89836C22116yescomplemented
41#89835C22116yescomplemented
42#89743C2218yescomplemented
43#89742C2218yescomplemented
44C2×#6384C2212yes—
45C2×#6384C2212yes—
46#89740C2218yes—
47#89739C2218yes—
48C2×#6383C2212yes—
49C2×#6383C2212yes—
50C2×#6382C2212yes—
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53C2×#6381C2212yes—
54#89697C2218yescomplemented
55C2×C4⋊(C7⋊C8)C2212yes—
56C2×C4×C7⋊C8C2212yes—
57C2×#3232—71nomaximal, Sylow
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7D411yescharacteristic
61C23×C4×C7D411yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C22×C22⋊D7—21no—
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×Dic14C2311yescharacteristic
67C7×#125—24no—
68C7×#125—24no—
69C7×#125—28no—
70C7×#125—28no—
71C7×#125C2318yes—
72C7×#125C2318yes—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—28no—
78C2×C7×C4⋊C4—28no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C4⋊C4C2314yes—
86C2×C7×C4⋊C4C2314yes—
87C2×C42×C7D411yescharacteristic
88C2×C42×C7D411yescharacteristic
89C2×C42×C7D414yes—
90C2×C42×C7C2311yescharacteristic
91C2×C42×C7C2311yescharacteristic
92C2×C42×C7C2314yes—
93C2×C22⋊Dic7—21no—
94#6493—22no—
95#6493C2314yes—
96#6492—28no—
97C4×C22⋊D7—28no—
98C2×(C2×D7)⋊C4—21no—
99C2×(C2×D7)⋊C4—21no—
100C2×(C2×D7)⋊C4—21no—
101C2×(C2×D7)⋊C4—21no—
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104#6487—22no—
105#6487C2314yes—
106C2×Dic7⋊C4—21no—
107C2×Dic7⋊C4—21no—
108C2×Dic7⋊C4—21no—
109C2×Dic7⋊C4—21no—
110C2×C4×Dic7—21no—
111C22×C7⋊C8C2312yes—
112C22×C7⋊C8C2312yes—
113#6439—28no—
114#6438—22no—
115#6438C2314yes—
116C4×D28—22no—
117C4×D28C2314yes—
118#6435—28no—
119#6433—22no—
120#6433C2314yes—
121C4×Dic14—22no—
122C4×Dic14C2314yes—
123C22⋊(C7⋊C8)C2314yes—
124C22⋊(C7⋊C8)C2314yes—
125#6384C2318yes—
126#6384C2318yes—
127#6383C2318yes—
128#6383C2318yes—
129#6382C2318yes—
130#6382C2318yes—
131#6381C2318yes—
132#6381C2318yes—
133C4⋊(C7⋊C8)C2318yes—
134C4×C7⋊C8C2318yes—
135C2×#529—71no—
136C22×#125D711yescharacteristic, complemented
137#3232—716no—
138C2×#499—72no—
139C2×#488—72no—
140C2×#484—71no—
141C2×#481—71no—
142C2×#483—71no—
143C2×#480—71no—
144C2×#479—72no—
145C2×#478—72no—
146C2×#431—71no—
147C24×C7C2×D411yescharacteristic
148C23×D7—21no—
149C22×C4×C7—24no—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—24no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—42no—
156C22×C4×C7—44no—
157C22×C4×C7—44no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D414yes—
165C22×C4×C7C2×D414yes—
166C22×C4×C7C2412yes—
167C22×C4×C7C2411yescharacteristic
168C2×C22⋊D7—24no—
169C2×C22⋊D7—44no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C2×D28—24no—
174C2×D28—21no—
175C2×D28C2412yes—
176C2×C4×D7—24no—
177C2×C4×D7—44no—
178C2×Dic14—24no—
179C2×Dic14—21no—
180C2×Dic14C2412yes—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—28no—
183C7×C4⋊C4—28no—
184C7×C4⋊C4—28no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4C2414yes—
190C7×C4⋊C4C2414yes—
191C42×C7C2×D412yes—
192C42×C7C2×D412yes—
193C42×C7C2×D414yes—
194C42×C7C2×D414yes—
195C42×C7C2412yes—
196C42×C7C2412yes—
197C22⋊Dic7—24no—
198(C2×D7)⋊C4—24no—
199(C2×D7)⋊C4—24no—
200(C2×D7)⋊C4—24no—
201(C2×D7)⋊C4—24no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7C2412yes—
205C4⋊Dic7C2412yes—
206Dic7⋊C4—24no—
207Dic7⋊C4—24no—
208Dic7⋊C4—24no—
209Dic7⋊C4—24no—
210C4×Dic7—24no—
211C2×C7⋊C8—24no—
212C2×C7⋊C8—24no—
213C2×C7⋊C8C2414yes—
214C2×C7⋊C8C2414yes—
215#529—78no—
216#529—144no—
217C2×#126—141no—
218C2×#125—21no—
219C2×#125—21no—
220C2×#125—24no—
221C2×#125—71no—
222C2×#125C2×D712yes—
223C2×#125C2×D712yes—
224C2×#125C2×D718yescomplemented
225C2×#124—71no—
226C2×#123—141no—
227C2×C22⋊Q8—71no—
228C2×#121—71no—
229C2×C4×Q8—71no—
230C2×C4×D4—71no—
231C2×C4×D4—141no—
232C2×#117—141no—
233C22×C4⋊C4—21no—
234C22×C4⋊C4—21no—
235C22×C4⋊C4C2×D711yescharacteristic
236C22×C4⋊C4C2×D711yescharacteristic
237C22×C42C2×D711yescharacteristic
238#499—716no—
239#488—716no—
240#484—78no—
241#483—78no—
242#481—78no—
243#480—78no—
244#479—716no—
245#478—716no—
246#431—78no—
247C2×#107—72no—
248C2×#107—72no—
249C2×#106—72no—
250C2×#106—72no—
251C2×C4⋊C8—72no—
252C2×#103—72no—
253C2×#103—72no—
254C2×#102—72no—
255C2×#102—72no—
256C2×C22⋊C8—71no—
257C2×C22⋊C8—71no—
258C2×C4×C8—72no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7#13514yes—
262C23×C7C22×D412yes—
263C23×C7C22×D411yescharacteristic
264C22×D7—22no—
265C22×D7—42no—
266C22×D7—42no—
267C22×D7—42no—
268C2×C4×C7—22no—
269C2×C4×C7—28no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—44no—
279C2×C4×C7—48no—
280C2×C4×C7—48no—
281C2×C4×C7—44no—
282C2×C4×C7—44no—
283C2×C4×C7C22×D412yes—
284C2×C4×C7C22×D412yes—
285C2×C4×C7C22×D412yes—
286C2×C4×C7C22×D412yes—
287C2×C4×C7C22×D412yes—
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C72−1+414yes—
290C2×C4×C7C2511yescharacteristic
291C22⋊D7—48no—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298D28—24no—
299D28—42no—
300C4×D7—48no—
301Dic14—24no—
302Dic14—42no—
303C7⋊C8—24no—
304C7⋊C8—24no—
305C22×Q8—71no—
306C22×D4—71no—
307C22×D4—141no—
308C23×C4—21no—
309C23×C4—21no—
310C23×C4—141no—
311C23×C4C22⋊D711yescharacteristic
312C23×C4C22⋊D711yescharacteristic
313C23×C4C22×D711yescharacteristic
314C22×C8—72no—
315C22×C8—72no—
316#126—148no—
317#125—24no—
318#125—24no—
319#125—28no—
320#125—28no—
321#125—74no—
322#125—142no—
323#125C22×D718yes—
324#125C22×D718yes—
325#124—74no—
326#124—142no—
327#123—148no—
328C22⋊Q8—74no—
329C22⋊Q8—142no—
330#121—74no—
331#121—142no—
332C4×Q8—74no—
333C4×Q8—142no—
334C4×D4—74no—
335C4×D4—148no—
336C4×D4—142no—
337#117—148no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—28no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22×D712yes—
355C2×C4⋊C4C22×D712yes—
356C2×C4⋊C4C22×D714yes—
357C2×C4⋊C4C22×D714yes—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C42—141no—
364C2×C42C22⋊D711yescharacteristic
365C2×C42C22⋊D711yescharacteristic
366C2×C42C22⋊D714yes—
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369C2×C42C22×D714yes—
370#107—78no—
371#107—78no—
372#106—78no—
373#106—78no—
374C4⋊C8—78no—
375#103—78no—
376#103—78no—
377#102—78no—
378#102—78no—
379C22⋊C8—74no—
380C22⋊C8—74no—
381C4×C8—78no—
382C22×C7—22no—
383C22×C7—28no—
384C22×C7—22no—
385C22×C7#53612yes—
386C22×C7C2×#13514yes—
387C22×C7C2×#13512yes—
388C22×C7#57512yes—
389C22×C7C23×D411yescharacteristic
390C2×D7—42no—
391C2×D7—42no—
392C2×D7—42no—
393C2×D7—84no—
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—44no—
400C4×C7—44no—
401C4×C7#53612yes—
402C4×C7C2×2−1+412yes—
403Dic7—42no—
404Dic7—42no—
405Dic7—42no—
406C24—141no—
407C24C2×C22⋊D711yescharacteristic
408C2×Q8—72no—
409C2×Q8—141no—
410C2×Q8—144no—
411C2×D4—72no—
412C2×D4—144no—
413C2×D4—144no—
414C2×D4—141no—
415C2×D4—284no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—144no—
429C22×C4—284no—
430C22×C4C2×C22⋊D712yes—
431C22×C4C2×C22⋊D714yes—
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C2×C22⋊D711yescharacteristic
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C2×C22⋊D714yes—
438C22×C4C23×D712yes—
439C22×C4C23×D711yescharacteristic
440C2×C8—74no—
441C2×C8—74no—
442C2×C8—144no—
443C2×C8—144no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—72no—
453C4⋊C4—72no—
454C4⋊C4—144no—
455C4⋊C4—144no—
456C4⋊C4—144no—
457C4⋊C4—144no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4C23×D714yes—
461C4⋊C4C23×D714yes—
462C22⋊C4—144no—
463C22⋊C4—144no—
464C22⋊C4—144no—
465C22⋊C4—144no—
466C22⋊C4—144no—
467C42—144no—
468C42C2×C22⋊D714yes—
469C42C2×C22⋊D712yes—
470C42C2×C22⋊D712yes—
471C42C2×C22⋊D714yes—
472C42C23×D712yes—
473C42C23×D712yes—
474C2×C7—24no—
475C2×C7#323214yescomplemented
476C2×C7C2×#53611yescharacteristic
477C2×C7C22×#13511yescharacteristic
478C2×C7C2×#57511yescharacteristic
479D7—82no—
480C23—22no—
481C23—22no—
482C23—142no—
483C23—282no—
484C23—282no—
485C23—282no—
486C23#651314yes—
487C23C22×C22⋊D712yes—
488C23C22×C22⋊D711yescharacteristic
489Q8—144no—
490Q8—282no—
491D4—144no—
492D4—288no—
493D4—282no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—142no—
512C2×C4—288no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4#655114yes—
517C2×C4C22×C22⋊D712yes—
518C2×C4C22×C22⋊D712yes—
519C2×C4C22×C22⋊D712yes—
520C2×C4C22×C22⋊D712yes—
521C2×C4C22×C22⋊D712yes—
522C2×C4C22×C22⋊D711yescharacteristic
523C2×C4C24×D711yescharacteristic
524C8—144no—
525C8—144no—
526C7C2×#323211yescharacteristic, complemented, Sylow
527C22—28no—
528C22—22no—
529C22—22no—
530C22—282no—
531C22—282no—
532C22—282no—
533C22—564no—
534C22#9050712yes—
535C22C2×#651314yes—
536C22C2×#651312yes—
537C22#9051312yes—
538C22C23×C22⋊D711yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—44no—
546C4—282no—
547C4—282no—
548C4—282no—
549C4#9050712yes—
550C4C2×#655112yes—
551C2—24no—
552C2—562no—
553C2#1174046314yesdirect product factor
554C2C2×#9050711yescharacteristic
555C2C22×#651311yescharacteristic
556C2C2×#9051311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC22×C7×#125C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC22×#125D7
hypocenterC7C2×#3232
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C22×C7×#125C2semidirect
3. C2×#89835C2semidirect
4. #11740463C2direct (N × H)
5. C2×#89838C2semidirect
6. C2×#89836C2semidirect
7. C2×#89840C2semidirect
8. C2×#89742C2semidirect
9. C2×#89743C2semidirect
12. C2×#89697C2semidirect
18. C2×C7×#125C22semidirect
38. #89840C22semidirect
39. #89838C22semidirect
40. #89836C22semidirect
41. #89835C22semidirect
42. #89743C22semidirect
43. #89742C22semidirect
54. #89697C22semidirect
136. C22×#125D7semidirect
224. C2×#125C2×D7semidirect
475. C2×C7#3232semidirect
526. C7C2×#3232semidirect
553. C2#11740463direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1252
C2×#901702
C2×#898352
#117404632
C2×#898382
C2×#898362
C2×#898402
C2×#897422
C2×#897432
C2×#897402
C2×#897392
C2×#896972
C2×#32327