Finite Group Databasebeta

← C2×#11738764 #14225900

Subgroups — C2×#11738764 #14225900

Order 1792.

2050 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90287C211yesmaximal, characteristic
2C22×#6453C211yesmaximal, characteristic, complemented
3C2×C7×#481C211yesmaximal, characteristic, complemented
4C2×#89805C211yesmaximal, characteristic, complemented
5C2×#89740C211yesmaximal, characteristic, complemented
6#11738764C2116yesmaximal, direct product factor
7C2×#89643C212yesmaximal, complemented
8C2×#89628C212yesmaximal, complemented
9C2×#89536C212yesmaximal, complemented
10C2×#89538C212yesmaximal, complemented
11C2×#89508C211yesmaximal, characteristic, complemented
12C2×#89491C211yesmaximal, characteristic, complemented
13C22×#1159—21no—
14C2×C7×#121C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6503C2211yescharacteristic
17C2×#6502—21no—
18C2×D4×Dic7C2211yescharacteristic
19C2×#6493C2211yescharacteristic
20C22×C4⋊Dic7—21no—
21C22×C4⋊Dic7C2211yescharacteristic
22C22×Dic7⋊C4—21no—
23C22×C4×Dic7C2211yescharacteristic
24#90287—24no—
25#90287C2218yes—
26C2×#6456C2211yescharacteristic
27C2×#6453—24no—
28C2×#6453—21no—
29C2×#6453—21no—
30C2×#6453C2212yes—
31C2×#6453C2212yes—
32C2×#6453C2218yescomplemented
33C2×#6451C2211yescharacteristic
34C2×#6448—21no—
35C2×(C22⋊D7)⋊C4—21no—
36C7×#481C2218yescomplemented
37C2×C7×#106C2212yes—
38C2×C7×#102C2212yes—
39C2×C7×C22⋊C8C2211yescharacteristic
40C2×#6406C2212yes—
41C2×C22⋊(C7⋊C8)C2211yescharacteristic
42C2×#6395C2212yes—
43C2×#6392C2212yes—
44C2×#6389C2212yes—
45C2×Dic7⋊C8C2212yes—
46#89805C2218yescomplemented
47#89740C2218yescomplemented
48C2×#6383C2212yes—
49C2×#6382C2212yes—
50C2×#6381C2212yes—
51#89643C22116yescomplemented
52#89628C22116yescomplemented
53#89538C22116yescomplemented
54#89536C22116yescomplemented
55#89508C2218yescomplemented
56#89491C2218yescomplemented
57C2×#3234—71nomaximal, Sylow
58C22×C7×D4—21no—
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C22×C22⋊D7—21no—
63C22×C22⋊D7—21no—
64C22×C22⋊D7—21no—
65C23×Dic7—21no—
66C23×Dic7—21no—
67C23×Dic7D411yescharacteristic
68C23×Dic7D411yescharacteristic
69C2×#1159—28no—
70C2×#1159—48no—
71C22×D28C2311yescharacteristic
72C22×C4×D7—21no—
73C22×Dic14—21no—
74C22×C7×C8C2312yes—
75C7×#121—22no—
76C7×#121C2314yes—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4C2314yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C22⋊C4—21no—
83C2×C22⋊Dic7—21no—
84#6503—22no—
85#6503C2314yes—
86#6502—28no—
87D4×Dic7—22no—
88D4×Dic7C2314yes—
89#6493—22no—
90#6493C2314yes—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—22no—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7—22no—
97C2×C4⋊Dic7—28no—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2312yes—
100C2×C4⋊Dic7C2314yes—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—22no—
103C2×Dic7⋊C4—22no—
104C2×Dic7⋊C4—28no—
105C2×C4×Dic7D411yescharacteristic
106C2×C4×Dic7D411yescharacteristic
107C2×C4×Dic7D414yes—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2314yes—
111C22×C7⋊C8C2312yes—
112#6456—22no—
113#6456C2314yes—
114#6453—24no—
115#6453—24no—
116#6453—28no—
117#6453—28no—
118#6453C2318yes—
119#6453C2318yes—
120#6451—22no—
121#6451C2314yes—
122#6448—28no—
123(C22⋊D7)⋊C4—28no—
124C7×#106C2318yes—
125C7×#102C2318yes—
126C7×C22⋊C8C2314yes—
127#6406C2318yes—
128C22⋊(C7⋊C8)C2314yes—
129#6395C2318yes—
130#6392C2318yes—
131#6389C2318yes—
132Dic7⋊C8C2318yes—
133#6383C2318yes—
134#6382C2318yes—
135#6381C2318yes—
136C2×#532—71no—
137C22×#125—71no—
138#3234—716no—
139C2×#501—72no—
140C2×#490—72no—
141C2×#481—71no—
142C2×#481D711yescharacteristic, complemented
143C2×#480—71no—
144C2×#480—71no—
145C2×#478—72no—
146C2×#478—72no—
147C2×#432—71no—
148C24×C7—21no—
149C24×C7C2×D411yescharacteristic
150C23×D7—21no—
151C2×C7×D4—24no—
152C2×C7×D4—21no—
153C2×C7×D4—24no—
154C2×C7×D4—44no—
155C2×C7×D4C2412yes—
156C22×C4×C7—24no—
157C22×C4×C7—24no—
158C22×C4×C7—22no—
159C22×C4×C7—21no—
160C22×C4×C7—44no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2412yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—24no—
166C2×C22⋊D7—24no—
167C2×C22⋊D7—24no—
168C2×C22⋊D7—44no—
169C2×C22⋊D7—44no—
170C2×C22⋊D7—44no—
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7—22no—
174C22×Dic7—21no—
175C22×Dic7—24no—
176C22×Dic7—42no—
177C22×Dic7—44no—
178C22×Dic7—44no—
179C22×Dic7C2×D411yescharacteristic
180C22×Dic7C2×D411yescharacteristic
181C22×Dic7C2×D411yescharacteristic
182C22×Dic7C2×D411yescharacteristic
183C22×Dic7C2×D414yes—
184C22×Dic7C2×D414yes—
185#1159—432no—
186C2×D28—24no—
187C2×D28—21no—
188C2×D28C2412yes—
189C2×C4×D7—24no—
190C2×C4×D7—44no—
191C2×Dic14—24no—
192C2×Dic14—44no—
193C2×C7×C8—24no—
194C2×C7×C8C2414yes—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—24no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4C2414yes—
199C7×C4⋊C4C2412yes—
200C7×C22⋊C4—24no—
201C22⋊Dic7—24no—
202(C2×D7)⋊C4—24no—
203(C2×D7)⋊C4—24no—
204C4⋊Dic7—21no—
205C4⋊Dic7—24no—
206C4⋊Dic7—22no—
207C4⋊Dic7—28no—
208C4⋊Dic7—28no—
209C4⋊Dic7C2414yes—
210C4⋊Dic7C2412yes—
211Dic7⋊C4—24no—
212Dic7⋊C4—28no—
213Dic7⋊C4—28no—
214C4×Dic7C2×D414yes—
215C4×Dic7C2×D414yes—
216C4×Dic7C2×D412yes—
217C4×Dic7C2×D412yes—
218C4×Dic7C2412yes—
219C4×Dic7C2412yes—
220C2×C7⋊C8—24no—
221C2×C7⋊C8C2414yes—
222C22×#41—141no—
223#532—78no—
224#532—144no—
225C2×C4⋊Q8—71no—
226C2×#127—71no—
227C2×#125—72no—
228C2×#125—72no—
229C2×#125—78no—
230C2×#125—141no—
231C2×#125—141no—
232C2×#125—144no—
233C2×#124—141no—
234C2×#121—71no—
235C2×#121—141no—
236C2×#121C2×D711yescharacteristic
237C2×C4×D4—71no—
238C2×C4×D4—71no—
239C2×C4×D4—141no—
240C22×C4⋊C4—71no—
241C22×C4⋊C4—141no—
242C22×C4⋊C4—141no—
243C22×C4⋊C4C2×D711yescharacteristic
244C22×C42—71no—
245#501—716no—
246#490—716no—
247#481—78no—
248#481C2×D718yescomplemented
249#480—78no—
250#480—78no—
251#478—716no—
252#478—716no—
253#432—78no—
254C2×#107—72no—
255C2×#107—72no—
256C2×#106—72no—
257C2×#106C2×D712yes—
258C2×C4⋊C8—72no—
259C2×#102—72no—
260C2×#102—72no—
261C2×#102—72no—
262C2×#102C2×D712yes—
263C2×C22⋊C8—71no—
264C2×C22⋊C8C2×D711yescharacteristic
265C2×#97—72no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—22no—
269C23×C7—42no—
270C23×C7—42no—
271C23×C7—42no—
272C23×C7AGL(1,ℤ/8)14yes—
273C23×C7C22×D412yes—
274C23×C7C22×D411yescharacteristic
275C22×D7—22no—
276C22×D7—42no—
277C22×D7—42no—
278C22×D7—42no—
279C7×D4—24no—
280C7×D4—48no—
281C7×D4—42no—
282C2×C4×C7—22no—
283C2×C4×C7—24no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—22no—
287C2×C4×C7—48no—
288C2×C4×C7—42no—
289C2×C4×C7—44no—
290C2×C4×C7C22×D412yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C72−1+414yes—
293C2×C4×C7C2511yescharacteristic
294C22⋊D7—48no—
295C22⋊D7—48no—
296C22⋊D7—48no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—28no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—24no—
303C2×Dic7—22no—
304C2×Dic7—44no—
305C2×Dic7—48no—
306C2×Dic7—42no—
307C2×Dic7—48no—
308C2×Dic7—44no—
309C2×Dic7C22×D412yes—
310C2×Dic7C22×D412yes—
311C2×Dic7C22×D412yes—
312C2×Dic7C22×D412yes—
313D28—24no—
314D28—42no—
315C4×D7—48no—
316Dic14—48no—
317C7×C8—24no—
318C7⋊C8—24no—
319C2×#41—148no—
320C2×#41—288no—
321C22×Q8—141no—
322C22×D4—21no—
323C22×D4—71no—
324C22×D4—141no—
325C22×D4—141no—
326C22×D4—141no—
327C22×D4C22×D711yescharacteristic
328C23×C4—21no—
329C23×C4—71no—
330C23×C4—71no—
331C23×C4—141no—
332C23×C4—141no—
333C23×C4—141no—
334C23×C4C22×D711yescharacteristic
335C22×C8—72no—
336C22×C8C22×D712yes—
337C4⋊Q8—74no—
338C4⋊Q8—142no—
339#127—74no—
340#127—142no—
341#125—78no—
342#125—78no—
343#125—144no—
344#125—148no—
345#125—148no—
346#125—144no—
347#124—148no—
348#121—22no—
349#121—74no—
350#121—148no—
351#121—142no—
352#121C22×D714yes—
353C4×D4—74no—
354C4×D4—74no—
355C4×D4—148no—
356C4×D4—142no—
357C4×D4—142no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—72no—
361C2×C4⋊C4—74no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—148no—
364C2×C4⋊C4—142no—
365C2×C4⋊C4—142no—
366C2×C4⋊C4—148no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4—142no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4C22×D712yes—
373C2×C4⋊C4C22×D714yes—
374C2×C4⋊C4C22×D711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C42—71no—
380C2×C42—71no—
381C2×C42—74no—
382C2×C42—71no—
383C2×C42—71no—
384C2×C42—74no—
385#107—78no—
386#107—78no—
387#106—78no—
388#106C22×D718yes—
389C4⋊C8—78no—
390#102—78no—
391#102—78no—
392#102—78no—
393#102C22×D718yes—
394C22⋊C8—74no—
395C22⋊C8C22×D714yes—
396#97—78no—
397C22×C7—22no—
398C22×C7—28no—
399C22×C7—22no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—84no—
404C22×C7#53612yes—
405C22×C7C2×AGL(1,ℤ/8)14yes—
406C22×C7C2×AGL(1,ℤ/8)12yes—
407C22×C7#57512yes—
408C22×C7C23×D411yescharacteristic
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—42no—
412C2×D7—84no—
413C4×C7—22no—
414C4×C7—42no—
415C4×C7—44no—
416C4×C7#53612yes—
417C4×C7C2×2−1+412yes—
418Dic7—22no—
419Dic7—22no—
420Dic7—22no—
421Dic7—22no—
422Dic7—42no—
423Dic7—44no—
424C24—21no—
425C24—141no—
426C24D4×D711yescharacteristic
427#41—2832no—
428C2×Q8—144no—
429C2×Q8—284no—
430C2×D4—24no—
431C2×D4—21no—
432C2×D4—24no—
433C2×D4—44no—
434C2×D4—72no—
435C2×D4—144no—
436C2×D4—144no—
437C2×D4—144no—
438C2×D4—141no—
439C2×D4—144no—
440C2×D4—284no—
441C2×D4—284no—
442C2×D4—284no—
443C2×D4C23×D712yes—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—74no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—74no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—144no—
458C22×C4—144no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4—284no—
462C22×C4—282no—
463C22×C4—284no—
464C22×C4—284no—
465C22×C4D4×D712yes—
466C22×C4D4×D711yescharacteristic
467C22×C4C23×D712yes—
468C22×C4C23×D711yescharacteristic
469C2×C8—24no—
470C2×C8—74no—
471C2×C8—144no—
472C2×C8C23×D714yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—74no—
477C4⋊C4—72no—
478C4⋊C4—148no—
479C4⋊C4—148no—
480C4⋊C4—144no—
481C4⋊C4—148no—
482C4⋊C4—148no—
483C4⋊C4—144no—
484C4⋊C4—142no—
485C4⋊C4—141no—
486C4⋊C4C23×D714yes—
487C4⋊C4C23×D712yes—
488C22⋊C4—24no—
489C22⋊C4—144no—
490C22⋊C4—144no—
491C22⋊C4—144no—
492C42—72no—
493C42—74no—
494C42—74no—
495C42—72no—
496C42—72no—
497C42—72no—
498C2×C7—24no—
499C2×C7—82no—
500C2×C7#323414yescomplemented
501C2×C7C2×#53611yescharacteristic
502C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
503C2×C7C2×#57511yescharacteristic
504D7—82no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—142no—
512C23—282no—
513C23—282no—
514C23—282no—
515C23#647714yes—
516C23C2×D4×D712yes—
517C23C2×D4×D711yescharacteristic
518Q8—288no—
519D4—24no—
520D4—48no—
521D4—42no—
522D4—144no—
523D4—288no—
524D4—288no—
525D4—288no—
526D4—282no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—72no—
536C2×C4—72no—
537C2×C4—72no—
538C2×C4—72no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—148no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—144no—
546C2×C4—288no—
547C2×C4—284no—
548C2×C4—288no—
549C2×C4—288no—
550C2×C4—284no—
551C2×C4—282no—
552C2×C4C2×D4×D712yes—
553C2×C4C2×D4×D711yescharacteristic
554C2×C4#655414yes—
555C2×C4C24×D711yescharacteristic
556C8—24no—
557C8—144no—
558C7C2×#323411yescharacteristic, complemented, Sylow
559C22—22no—
560C22—22no—
561C22—28no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—282no—
567C22—282no—
568C22—282no—
569C22—564no—
570C22#9033912yes—
571C22C2×#647714yes—
572C22C2×#647712yes—
573C22#9046412yes—
574C22C22×D4×D711yescharacteristic
575C4—22no—
576C4—44no—
577C4—42no—
578C4—142no—
579C4—142no—
580C4—142no—
581C4—142no—
582C4—282no—
583C4—284no—
584C4#9033912yes—
585C4C2×#655412yes—
586C2—24no—
587C2—82no—
588C2—562no—
589C2#1173876414yesdirect product factor
590C2C2×#9033911yescharacteristic
591C2C22×#647711yescharacteristic
592C2C2×#9046411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#481C2
socleC23×C7C22×D4
prime radicalC2×#90287C2
hypercenterC2×#481D7
hypocenterC7C2×#3234
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#6453C2semidirect
3. C2×C7×#481C2semidirect
4. C2×#89805C2semidirect
5. C2×#89740C2semidirect
6. #11738764C2direct (N × H)
7. C2×#89643C2semidirect
8. C2×#89628C2semidirect
9. C2×#89536C2semidirect
10. C2×#89538C2semidirect
11. C2×#89508C2semidirect
12. C2×#89491C2semidirect
32. C2×#6453C22semidirect
36. C7×#481C22semidirect
46. #89805C22semidirect
47. #89740C22semidirect
51. #89643C22semidirect
52. #89628C22semidirect
53. #89538C22semidirect
54. #89536C22semidirect
55. #89508C22semidirect
56. #89491C22semidirect
142. C2×#481D7semidirect
248. #481C2×D7semidirect
500. C2×C7#3234semidirect
558. C7C2×#3234semidirect
589. C2#11738764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902872
C22×#64532
C2×C7×#4812
C2×#898052
C2×#897402
#117387642
C2×#896432
C2×#896282
C2×#895362
C2×#895382
C2×#895082
C2×#894912
C2×#32347