Finite Group Databasebeta

← C2×#11740566 #14225822

Subgroups — C2×#11740566 #14225822

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
1C2×C7×#529C211yesmaximal, characteristic
2C22×#6433C211yesmaximal, characteristic, complemented
3#11740566C2116yesmaximal, direct product factor
4C2×#89836C212yesmaximal, complemented
5C2×#89828C212yesmaximal, complemented
6C2×#89816C211yesmaximal, characteristic, complemented
7C2×#89815C211yesmaximal, characteristic, complemented
8C2×#89806C211yesmaximal, characteristic
9C2×#89805C211yesmaximal, characteristic
10C2×#89794C212yesmaximal, complemented
11C2×#89781C212yesmaximal, 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
26C22×C4⋊Dic7C2211yescharacteristic
27C22×C4⋊Dic7C2211yescharacteristic
28C22×Dic7⋊C4—21no—
29C22×Dic7⋊C4—21no—
30C2×#6433—24no—
31C2×#6433—21no—
32C2×#6433—21no—
33C2×#6433C2212yes—
34C2×#6433C2212yes—
35C2×#6433C2218yescomplemented
36C2×#6409C2212yes—
37C2×#6409C2212yes—
38C2×#6406C2212yes—
39C2×#6406C2212yes—
40C2×C22⋊(C7⋊C8)C2211yescharacteristic
41C2×C22⋊(C7⋊C8)C2211yescharacteristic
42#89836C22116yescomplemented
43#89828C22116yescomplemented
44#89816C2218yescomplemented
45#89815C2218yescomplemented
46#89806C2218yes—
47#89805C2218yes—
48#89794C22116yescomplemented
49#89781C22116yescomplemented
50C2×#6382C2212yes—
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53C2×#6381C2212yes—
54#89697C2218yescomplemented
55C2×C4⋊(C7⋊C8)C2212yes—
56C2×C4×C7⋊C8C2212yes—
57C2×#3232—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
65C23×Dic7—21no—
66C23×Dic7—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×C4⋊Dic7—21no—
102C2×C4⋊Dic7—21no—
103C2×C4⋊Dic7—22no—
104C2×C4⋊Dic7—22no—
105C2×C4⋊Dic7C2312yes—
106C2×C4⋊Dic7C2312yes—
107C2×C4⋊Dic7C2314yes—
108C2×C4⋊Dic7C2314yes—
109C2×Dic7⋊C4—22no—
110C2×Dic7⋊C4—22no—
111C2×Dic7⋊C4—22no—
112C2×Dic7⋊C4—22no—
113C2×Dic7⋊C4—28no—
114C2×Dic7⋊C4—28no—
115C22×C7⋊C8C2312yes—
116C22×C7⋊C8C2312yes—
117#6433—24no—
118#6433—24no—
119#6433—28no—
120#6433—28no—
121#6433C2318yes—
122#6433C2318yes—
123#6409C2318yes—
124#6409C2318yes—
125#6406C2318yes—
126#6406C2318yes—
127C22⋊(C7⋊C8)C2314yes—
128C22⋊(C7⋊C8)C2314yes—
129#6382C2318yes—
130#6382C2318yes—
131#6381C2318yes—
132#6381C2318yes—
133C4⋊(C7⋊C8)C2318yes—
134C4×C7⋊C8C2318yes—
135C2×#529D711yescharacteristic
136C22×#125—71no—
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×C7—21no—
148C24×C7C2×D411yescharacteristic
149C2×C7×Q8—21no—
150C2×C7×Q8—24no—
151C2×C7×Q8C2412yes—
152C2×C7×D4—24no—
153C2×C7×D4—21no—
154C2×C7×D4—24no—
155C2×C7×D4—44no—
156C2×C7×D4C2412yes—
157C22×C4×C7—24no—
158C22×C4×C7—24no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—24no—
163C22×C4×C7—42no—
164C22×C4×C7—44no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D414yes—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D414yes—
173C22×C4×C7C2412yes—
174C22×C4×C7C2411yescharacteristic
175C22×Dic7—24no—
176C22×Dic7—24no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—44no—
180C22×Dic7—44no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4C2412yes—
188C7×C4⋊C4C2412yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C42×C7—24no—
195C42×C7C2×D414yes—
196C42×C7C2×D412yes—
197C42×C7C2×D412yes—
198C42×C7C2×D414yes—
199C42×C7C2412yes—
200C42×C7C2412yes—
201C4⋊Dic7—24no—
202C4⋊Dic7—24no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4⋊Dic7C2414yes—
206C4⋊Dic7C2414yes—
207Dic7⋊C4—28no—
208Dic7⋊C4—28no—
209Dic7⋊C4—28no—
210Dic7⋊C4—28no—
211C2×C7⋊C8—24no—
212C2×C7⋊C8—24no—
213C2×C7⋊C8C2414yes—
214C2×C7⋊C8C2414yes—
215#529—24no—
216#529C2×D718yes—
217C2×#126—21no—
218C2×#125—72no—
219C2×#125—72no—
220C2×#125—78no—
221C2×#125—141no—
222C2×#125—141no—
223C2×#125—144no—
224C2×#125C2×D711yescharacteristic
225C2×#124C2×D711yescharacteristic
226C2×#123—21no—
227C2×C22⋊Q8C2×D711yescharacteristic
228C2×#121C2×D711yescharacteristic
229C2×C4×Q8C2×D711yescharacteristic
230C2×C4×D4—21no—
231C2×C4×D4C2×D711yescharacteristic
232C2×#117—21no—
233C22×C4⋊C4—71no—
234C22×C4⋊C4—71no—
235C22×C4⋊C4—141no—
236C22×C4⋊C4—141no—
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—22no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7#13514yes—
266C23×C7C22×D412yes—
267C23×C7C22×D411yescharacteristic
268C7×Q8—24no—
269C7×Q8—42no—
270C7×D4—24no—
271C7×D4—48no—
272C7×D4—42no—
273C2×C4×C7—22no—
274C2×C4×C7—28no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—44no—
285C2×C4×C7—48no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7C22×D412yes—
290C2×C4×C7C22×D412yes—
291C2×C4×C7C22×D412yes—
292C2×C4×C7C22×D412yes—
293C2×C4×C7C22×D412yes—
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C72−1+414yes—
296C2×C4×C7C2511yescharacteristic
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—48no—
300C2×Dic7—48no—
301C2×Dic7—44no—
302C2×Dic7—44no—
303C7⋊C8—24no—
304C7⋊C8—24no—
305C22×Q8C22×D711yescharacteristic
306C22×D4—21no—
307C22×D4C22×D711yescharacteristic
308C23×C4—21no—
309C23×C4—141no—
310C23×C4—141no—
311C23×C4C22⋊D711yescharacteristic
312C23×C4C22⋊D711yescharacteristic
313C23×C4C22×D711yescharacteristic
314C22×C8—72no—
315C22×C8—72no—
316#126—28no—
317#125—22no—
318#125—78no—
319#125—78no—
320#125—148no—
321#125—144no—
322#125—144no—
323#125—148no—
324#125C22×D714yes—
325#124—22no—
326#124C22×D714yes—
327#123—28no—
328C22⋊Q8—22no—
329C22⋊Q8C22×D714yes—
330#121—22no—
331#121C22×D714yes—
332C4×Q8—22no—
333C4×Q8C22×D714yes—
334C4×D4—28no—
335C4×D4—22no—
336C4×D4C22×D714yes—
337#117—28no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—72no—
343C2×C4⋊C4—72no—
344C2×C4⋊C4—74no—
345C2×C4⋊C4—74no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—148no—
351C2×C4⋊C4—148no—
352C2×C4⋊C4—142no—
353C2×C4⋊C4—142no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C42—21no—
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—42no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—84no—
389C22×C7#53612yes—
390C22×C7C2×#13514yes—
391C22×C7C2×#13512yes—
392C22×C7#57512yes—
393C22×C7C23×D411yescharacteristic
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—42no—
400C4×C7—42no—
401C4×C7—42no—
402C4×C7#53612yes—
403C4×C7C2×2−1+412yes—
404Dic7—44no—
405Dic7—44no—
406C24—21no—
407C24C2×C22⋊D711yescharacteristic
408C2×Q8—21no—
409C2×Q8—24no—
410C2×Q8C23×D712yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—44no—
415C2×D4C23×D712yes—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—144no—
425C22×C4—144no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—284no—
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—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—74no—
451C4⋊C4—74no—
452C4⋊C4—148no—
453C4⋊C4—148no—
454C4⋊C4—148no—
455C4⋊C4—148no—
456C4⋊C4—144no—
457C4⋊C4—144no—
458C4⋊C4—142no—
459C4⋊C4—142no—
460C4⋊C4C23×D712yes—
461C4⋊C4C23×D712yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C42—24no—
468C42C2×C22⋊D714yes—
469C42C2×C22⋊D712yes—
470C42C2×C22⋊D712yes—
471C42C2×C22⋊D714yes—
472C42C23×D712yes—
473C42C23×D712yes—
474C2×C7—24no—
475C2×C7—82no—
476C2×C7#323214yescomplemented
477C2×C7C2×#53611yescharacteristic
478C2×C7C22×#13511yescharacteristic
479C2×C7C2×#57511yescharacteristic
480C23—22no—
481C23—22no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23#651314yes—
487C23C22×C22⋊D712yes—
488C23C22×C22⋊D711yescharacteristic
489Q8—24no—
490Q8—42no—
491D4—24no—
492D4—48no—
493D4—42no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—144no—
511C2×C4—144no—
512C2×C4—288no—
513C2×C4—288no—
514C2×C4—284no—
515C2×C4—284no—
516C2×C4#655414yes—
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—22no—
528C22—22no—
529C22—28no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—84no—
534C22C2×#651314yes—
535C22C2×#651312yes—
536C22#9051312yes—
537C22#9051612yes—
538C22C23×C22⋊D711yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—284no—
548C4—284no—
549C4#9051612yes—
550C4C2×#655412yes—
551C2—24no—
552C2—82no—
553C2#1174056614yesdirect product factor
554C2C22×#651311yescharacteristic
555C2C2×#9051311yescharacteristic
556C2C2×#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×#3232
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. C22×#6433C2semidirect
3. #11740566C2direct (N × H)
4. C2×#89836C2semidirect
5. C2×#89828C2semidirect
6. C2×#89816C2semidirect
7. C2×#89815C2semidirect
10. C2×#89794C2semidirect
11. C2×#89781C2semidirect
12. C2×#89697C2semidirect
35. C2×#6433C22semidirect
42. #89836C22semidirect
43. #89828C22semidirect
44. #89816C22semidirect
45. #89815C22semidirect
48. #89794C22semidirect
49. #89781C22semidirect
54. #89697C22semidirect
476. C2×C7#3232semidirect
526. C7C2×#3232semidirect
553. C2#11740566direct (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
C22×#64332
#117405662
C2×#898362
C2×#898282
C2×#898162
C2×#898152
C2×#898062
C2×#898052
C2×#897942
C2×#897812
C2×#896972
C2×#32327