Finite Group Databasebeta

← C2×#11740583 #14223720

Subgroups — C2×#11740583 #14223720

Order 1792.

1998 conjugacy classes of subgroups.

The subgroup lattice has 576 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#11740583C2116yesmaximal, direct product factor
4C2×#89837C211yesmaximal, characteristic, complemented
5C2×#89841C211yesmaximal, characteristic, complemented
6C2×#89832C211yesmaximal, characteristic, complemented
7C2×#89827C211yesmaximal, characteristic, complemented
8C2×#89823C212yesmaximal, complemented
9C2×#89813C212yesmaximal
10C2×#89802C212yesmaximal, complemented
11C2×#89790C212yesmaximal, complemented
12C22×#6377C211yesmaximal, 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×#1144C2212yes—
27C22×#1142C2212yes—
28C2×#6489—21no—
29C2×#6487—21no—
30C2×#6487C2212yes—
31C23×C7⋊C8C2212yes—
32#90161—24no—
33#90161C2218yescomplemented
34C2×#6433—21no—
35C2×#6432C2211yescharacteristic
36C2×#6432C2211yescharacteristic
37C2×C4×Dic14—21no—
38C2×C4×Dic14C2212yes—
39C2×#6409C2212yes—
40C2×#6406C2212yes—
41#89841C2218yescomplemented
42#89837C2218yescomplemented
43#89832C2218yescomplemented
44#89827C2218yescomplemented
45#89823C22116yescomplemented
46#89813C22116yes—
47#89802C22116yescomplemented
48#89790C22116yescomplemented
49C2×#6384C2212yes—
50C2×#6384C2212yes—
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53C2×#6377C2211yescharacteristic
54C2×#6377C2211yescharacteristic
55C2×#6377C2212yes—
56C2×#6377C2218yescomplemented
57C2×#3181—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×Dic14C2312yes—
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—
102C2×#1144—28no—
103C2×#1144C2318yes—
104C2×#1142—28no—
105C2×#1142C2318yes—
106#6489—28no—
107C2×C4⋊Dic7—21no—
108C2×C4⋊Dic7—21no—
109C2×C4⋊Dic7—21no—
110C2×C4⋊Dic7C2312yes—
111#6487—24no—
112#6487—28no—
113#6487C2318yes—
114C2×Dic7⋊C4—22no—
115C2×Dic7⋊C4—22no—
116C2×C4×Dic7—22no—
117C22×C7⋊C8—28no—
118C22×C7⋊C8C2312yes—
119C22×C7⋊C8C2318yes—
120C22×C7⋊C8C2312yes—
121#6433—28no—
122#6432—22no—
123#6432—22no—
124#6432C2314yes—
125#6432C2314yes—
126C4×Dic14—28no—
127C4×Dic14—24no—
128C4×Dic14C2318yes—
129#6409C2318yes—
130#6406C2318yes—
131#6384C2318yes—
132#6384C2318yes—
133#6382C2318yes—
134#6381C2318yes—
135#6377C2318yes—
136#6377C2314yes—
137#6377C2314yes—
138C2×#533—71no—
139C2×#529D711yescharacteristic
140#3181—716no—
141C2×#501—71no—
142C2×#497—71no—
143C2×#491—71no—
144C2×#489—71no—
145C2×#467—72no—
146C2×#465—72no—
147C2×#441—72no—
148C2×#440—72no—
149C22×#97—71no—
150C24×C7—21no—
151C24×C7C2×D411yescharacteristic
152C2×C7×Q8—21no—
153C2×C7×Q8—24no—
154C2×C7×Q8C2412yes—
155C2×C7×D4—24no—
156C2×C7×D4—21no—
157C2×C7×D4—24no—
158C2×C7×D4—44no—
159C2×C7×D4C2412yes—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
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×D412yes—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D414yes—
175C22×C4×C7C2×D414yes—
176C22×C4×C7C2412yes—
177C22×C4×C7C2411yescharacteristic
178C22×Dic7—22no—
179C22×Dic7—22no—
180C2×Dic14—24no—
181C2×Dic14—24no—
182C2×Dic14—22no—
183C2×Dic14—24no—
184C2×Dic14—44no—
185C2×Dic14C2414yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4C2412yes—
193C7×C4⋊C4C2412yes—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C42×C7—24no—
200C42×C7C2×D412yes—
201C42×C7C2×D414yes—
202C42×C7C2×D412yes—
203C42×C7C2×D414yes—
204C42×C7C2412yes—
205C42×C7C2412yes—
206C22⋊Dic7—28no—
207#1144—216no—
208#1142—216no—
209C4⋊Dic7—22no—
210C4⋊Dic7—24no—
211C4⋊Dic7—24no—
212C4⋊Dic7—24no—
213C4⋊Dic7C2414yes—
214Dic7⋊C4—28no—
215Dic7⋊C4—28no—
216C4×Dic7—28no—
217C2×C7⋊C8—24no—
218C2×C7⋊C8—216no—
219C2×C7⋊C8#4118yes—
220C2×C7⋊C8C2414yes—
221C2×C7⋊C8C2414yes—
222C22×Q16—72no—
223C22×SD16—72no—
224C23×C8—72no—
225#533—78no—
226#533—144no—
227C2×C4⋊Q8—71no—
228C2×C4⋊Q8—71no—
229#529—24no—
230#529C2×D718yes—
231C2×#126—21no—
232C2×#125—141no—
233C2×#125C2×D711yescharacteristic
234C2×#124C2×D711yescharacteristic
235C2×#123—21no—
236C2×C22⋊Q8—72no—
237C2×C22⋊Q8—141no—
238C2×C22⋊Q8C2×D711yescharacteristic
239C2×#121C2×D711yescharacteristic
240C2×C4×Q8—72no—
241C2×C4×Q8—141no—
242C2×C4×Q8C2×D711yescharacteristic
243C2×C4×D4—21no—
244C2×C4×D4C2×D711yescharacteristic
245C2×#117—21no—
246C2×#117—141no—
247C22×C42C2×D711yescharacteristic
248#501—78no—
249#497—78no—
250#491—78no—
251#489—78no—
252#467—716no—
253#465—716no—
254#441—716no—
255#440—716no—
256C2×#107—72no—
257C2×#106—72no—
258C2×#103—72no—
259C2×#103—72no—
260C2×#103—72no—
261C2×#102—72no—
262C2×#97—71no—
263C2×#97—71no—
264C2×#97—72no—
265C2×#97—78no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—22no—
269C23×C7—42no—
270C23×C7—42no—
271C23×C7—42no—
272C23×C7#13714yes—
273C23×C7C22×D412yes—
274C23×C7C22×D411yescharacteristic
275C7×Q8—24no—
276C7×Q8—42no—
277C7×D4—24no—
278C7×D4—48no—
279C7×D4—42no—
280C2×C4×C7—22no—
281C2×C4×C7—28no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—24no—
287C2×C4×C7—24no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—44no—
292C2×C4×C7—48no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7C22×D412yes—
297C2×C4×C7C22×D412yes—
298C2×C4×C7C22×D412yes—
299C2×C4×C7C22×D412yes—
300C2×C4×C7C22×D412yes—
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C2×#4114yes—
303C2×C4×C7C2511yescharacteristic
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—44no—
307C2×Dic7—44no—
308Dic14—24no—
309Dic14—24no—
310Dic14—48no—
311Dic14—44no—
312C7⋊C8—24no—
313C7⋊C8C2×#4118yes—
314C22×Q8—72no—
315C22×Q8—141no—
316C22×Q8C22×D711yescharacteristic
317C22×D4—21no—
318C22×D4C22×D711yescharacteristic
319C23×C4—21no—
320C23×C4C22⋊D711yescharacteristic
321C23×C4C22⋊D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C2×Q16—78no—
324C2×Q16—148no—
325C2×SD16—78no—
326C2×SD16—148no—
327C22×C8—72no—
328C22×C8—72no—
329C22×C8—78no—
330C22×C8—148no—
331C4⋊Q8—74no—
332C4⋊Q8—74no—
333C4⋊Q8—142no—
334C4⋊Q8—142no—
335#126—28no—
336#125—22no—
337#125—148no—
338#125C22×D714yes—
339#124—22no—
340#124C22×D714yes—
341#123—28no—
342C22⋊Q8—22no—
343C22⋊Q8—78no—
344C22⋊Q8—144no—
345C22⋊Q8—148no—
346C22⋊Q8C22×D714yes—
347#121—22no—
348#121C22×D714yes—
349C4×Q8—22no—
350C4×Q8—78no—
351C4×Q8—144no—
352C4×Q8—148no—
353C4×Q8C22×D714yes—
354C4×D4—28no—
355C4×D4—22no—
356C4×D4C22×D714yes—
357#117—28no—
358#117—148no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—72no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—142no—
377C2×C42—21no—
378C2×C42—142no—
379C2×C42C22⋊D711yescharacteristic
380C2×C42C22⋊D711yescharacteristic
381C2×C42C22⋊D714yes—
382C2×C42C22×D711yescharacteristic
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D714yes—
385#107—78no—
386#106—78no—
387#103—78no—
388#103—78no—
389#103—78no—
390#102—78no—
391#97—78no—
392#97—74no—
393#97—74no—
394C22×C7—28no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—84no—
401C22×C7#53212yes—
402C22×C7C2×#13714yes—
403C22×C7C2×#13712yes—
404C22×C7#57512yes—
405C22×C7C23×D411yescharacteristic
406C4×C7—22no—
407C4×C7—22no—
408C4×C7—22no—
409C4×C7—22no—
410C4×C7—22no—
411C4×C7—42no—
412C4×C7—42no—
413C4×C7—42no—
414C4×C7#53212yes—
415C4×C7C22×#4112yes—
416Dic7—44no—
417Dic7—44no—
418C24—21no—
419C24C2×C22⋊D711yescharacteristic
420C2×Q8—21no—
421C2×Q8—24no—
422C2×Q8—74no—
423C2×Q8—144no—
424C2×Q8—142no—
425C2×Q8—144no—
426C2×Q8—144no—
427C2×Q8—284no—
428C2×Q8C23×D712yes—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—21no—
432C2×D4—44no—
433C2×D4C23×D712yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4C2×C22⋊D712yes—
445C22×C4C2×C22⋊D714yes—
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D714yes—
452C22×C4C23×D712yes—
453C22×C4C23×D711yescharacteristic
454Q16—1416no—
455SD16—1416no—
456C2×C8—78no—
457C2×C8—74no—
458C2×C8—74no—
459C2×C8—1416no—
460C2×C8—144no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—74no—
468C4⋊C4—148no—
469C4⋊C4—148no—
470C4⋊C4—142no—
471C4⋊C4—144no—
472C4⋊C4—144no—
473C4⋊C4—144no—
474C4⋊C4C23×D712yes—
475C4⋊C4C23×D712yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—148no—
482C42—24no—
483C42—148no—
484C42C2×C22⋊D714yes—
485C42C2×C22⋊D712yes—
486C42C2×C22⋊D714yes—
487C42C2×C22⋊D712yes—
488C42C23×D712yes—
489C42C23×D712yes—
490C2×C7—24no—
491C2×C7—82no—
492C2×C7#318114yescomplemented
493C2×C7C2×#53211yescharacteristic
494C2×C7C22×#13711yescharacteristic
495C2×C7C2×#57511yescharacteristic
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#651414yes—
503C23C22×C22⋊D712yes—
504C23C22×C22⋊D711yescharacteristic
505Q8—24no—
506Q8—42no—
507Q8—144no—
508Q8—144no—
509Q8—288no—
510Q8—284no—
511D4—24no—
512D4—48no—
513D4—42no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—284no—
533C2×C4—284no—
534C2×C4D7×#4114yes—
535C2×C4C22×C22⋊D712yes—
536C2×C4C22×C22⋊D712yes—
537C2×C4C22×C22⋊D712yes—
538C2×C4C22×C22⋊D712yes—
539C2×C4C22×C22⋊D712yes—
540C2×C4C22×C22⋊D711yescharacteristic
541C2×C4C24×D711yescharacteristic
542C8—78no—
543C8—144no—
544C7C2×#318111yescharacteristic, complemented, Sylow
545C22—22no—
546C22—22no—
547C22—28no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—84no—
552C22#9051312yes—
553C22C2×#651414yes—
554C22C2×#651412yes—
555C22#9052312yes—
556C22C23×C22⋊D711yescharacteristic
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—284no—
566C4—284no—
567C4#9052312yes—
568C4C2×D7×#4112yes—
569C2—24no—
570C2—82no—
571C2#1174058314yesdirect product factor
572C2C2×#9051311yescharacteristic
573C2C22×#651411yescharacteristic
574C2C2×#9052311yescharacteristic
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×#3181
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. #11740583C2direct (N × H)
4. C2×#89837C2semidirect
5. C2×#89841C2semidirect
6. C2×#89832C2semidirect
7. C2×#89827C2semidirect
8. C2×#89823C2semidirect
10. C2×#89802C2semidirect
11. C2×#89790C2semidirect
12. C22×#6377C2semidirect
33. #90161C22semidirect
41. #89841C22semidirect
42. #89837C22semidirect
43. #89832C22semidirect
44. #89827C22semidirect
45. #89823C22semidirect
47. #89802C22semidirect
48. #89790C22semidirect
56. C2×#6377C22semidirect
492. C2×C7#3181semidirect
544. C7C2×#3181semidirect
571. C2#11740583direct (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
#117405832
C2×#898372
C2×#898412
C2×#898322
C2×#898272
C2×#898232
C2×#898132
C2×#898022
C2×#897902
C22×#63772
C2×#31817