Finite Group Databasebeta

← C2×#11736586 #14221650

Subgroups — C2×#11736586 #14221650

Order 1792.

2234 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90201C211yesmaximal, characteristic
2C2×#90195C211yesmaximal, characteristic, complemented
3C22×C7×#107C211yesmaximal, characteristic, complemented
4C2×#89885C211yesmaximal, characteristic, complemented
5C2×#89882C211yesmaximal, characteristic, complemented
6C2×#89738C211yesmaximal, characteristic, complemented
7C2×#89736C211yesmaximal, characteristic, complemented
8C2×#89656C212yesmaximal, complemented
9C2×#89660C212yesmaximal, complemented
10C2×#89654C212yesmaximal, complemented
11#11736586C2116yesmaximal, direct product factor
12C2×#89657C212yesmaximal, complemented
13C22×#1157C2211yescharacteristic
14C22×#1157C2211yescharacteristic
15C23×C7×C8C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C2×#6493—22no—
19C2×C4×C22⋊D7—22no—
20C2×#6489C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22#90201—24no—
23#90201C2218yes—
24C2×#6460—21no—
25C2×#6460C2212yes—
26C2×#6458—21no—
27C2×#6458C2212yes—
28#90195—24no—
29#90195C2218yescomplemented
30C2×#6456—21no—
31C2×#6455C2212yes—
32C2×D7×C4⋊C4C2212yes—
33C2×#6450—21no—
34C2×C7×#107C2212yes—
35C2×C7×#107C2212yes—
36C2×C7×#107C2218yescomplemented
37#89885C2218yescomplemented
38C2×#6395C2211yescharacteristic
39C2×#6395C2211yescharacteristic
40#89882C2218yescomplemented
41C2×(C2×D7)⋊C8C2212yes—
42C2×(C2×D7)⋊C8C2212yes—
43C2×#6390C2211yescharacteristic
44C2×#6390C2211yescharacteristic
45C2×#6384C2212yes—
46#89738C2218yescomplemented
47C2×#6383C2212yes—
48#89736C2218yescomplemented
49C2×#6382C2212yes—
50C2×#6381C2212yes—
51#89660C22116yescomplemented
52#89657C22116yescomplemented
53#89656C22116yescomplemented
54#89654C22116yescomplemented
55C2×#3098—71nomaximal, Sylow
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C22×C22⋊D7—21no—
60C22×C22⋊D7D412yes—
61C2×#1157—24no—
62C2×#1157—24no—
63C2×#1157—22no—
64C2×#1157—24no—
65C2×#1157—24no—
66C2×#1157D414yes—
67C2×#1157C2314yes—
68C2×#1157C2314yes—
69C22×D28—22no—
70C22×D28D411yescharacteristic
71C22×D28C2311yescharacteristic
72C22×C4×D7—21no—
73C22×C4×D7—22no—
74C22×C4×D7C2312yes—
75C22×Dic14D411yescharacteristic
76C22×Dic14C2311yescharacteristic
77C22×C7×C8—24no—
78C22×C7×C8C2311yescharacteristic
79C22×C7×C8C2311yescharacteristic
80C22×C7×C8C2314yes—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C4⋊C4C2314yes—
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C4⋊C4C2314yes—
89C2×C22⋊Dic7—21no—
90#6493—216no—
91C4×C22⋊D7—216no—
92C2×(C2×D7)⋊C4—22no—
93C2×(C2×D7)⋊C4—22no—
94C2×(C2×D7)⋊C4—22no—
95#6489—22no—
96#6489C2314yes—
97C2×C4⋊Dic7—22no—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×Dic7⋊C4—22no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—22no—
103C2×#1136—24no—
104C2×#1136C2314yes—
105C22×C7⋊C8C2312yes—
106#6460—24no—
107#6460—28no—
108#6460C2318yes—
109#6458—28no—
110#6458—24no—
111#6458C2318yes—
112#6456—28no—
113#6455—24no—
114#6455C2318yes—
115D7×C4⋊C4—24no—
116D7×C4⋊C4C2318yes—
117#6450—28no—
118C7×#107C2318yes—
119C7×#107C2318yes—
120#6395C2314yes—
121#6395C2314yes—
122(C2×D7)⋊C8C2318yes—
123(C2×D7)⋊C8C2318yes—
124#6390C2314yes—
125#6390C2314yes—
126#6384C2318yes—
127#6383C2318yes—
128#6382C2318yes—
129#6381C2318yes—
130C2×#537—71no—
131C2×#520—71no—
132#3098—716no—
133C2×#484—72no—
134C2×#485—72no—
135C2×#482—72no—
136C2×#480—72no—
137C2×#428—71no—
138C22×#107D711yescharacteristic, complemented
139C2×#417—71no—
140C2×#416—71no—
141C2×#408—71no—
142C24×C7C2×D411yescharacteristic
143C23×D7—21no—
144C23×D7C2×D412yes—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—82no—
150C22×C4×C7—82no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D412yes—
153C22×C4×C7#4114yes—
154C22×C4×C7#4114yes—
155C22×C4×C7C2411yescharacteristic
156C22×C4×C7C2412yes—
157C2×C22⋊D7—28no—
158C2×C22⋊D7—24no—
159C2×C22⋊D7—44no—
160C2×C22⋊D7C2×D418yes—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7C2×D412yes—
165#1157—216no—
166#1157—28no—
167#1157—28no—
168#1157—28no—
169#1157—28no—
170#1157—48no—
171C2×D28—24no—
172C2×D28—22no—
173C2×D28—22no—
174C2×D28—21no—
175C2×D28—28no—
176C2×D28—84no—
177C2×D28C2×D414yes—
178C2×D28C2412yes—
179C2×C4×D7—24no—
180C2×C4×D7—24no—
181C2×C4×D7—24no—
182C2×C4×D7—28no—
183C2×C4×D7—44no—
184C2×C4×D7—84no—
185C2×C4×D7C2×D414yes—
186C2×C4×D7C2414yes—
187C2×Dic14—24no—
188C2×Dic14—22no—
189C2×Dic14—22no—
190C2×Dic14—21no—
191C2×Dic14C2×D414yes—
192C2×Dic14C2412yes—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—28no—
196C2×C7×C8C2412yes—
197C2×C7×C8C2412yes—
198C7×C4⋊C4—24no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—24no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4C2414yes—
203C7×C4⋊C4C2414yes—
204C22⋊Dic7—24no—
205(C2×D7)⋊C4—28no—
206(C2×D7)⋊C4—28no—
207(C2×D7)⋊C4—28no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—28no—
211C4⋊Dic7C2412yes—
212C4⋊Dic7C2412yes—
213Dic7⋊C4—28no—
214C4×Dic7—24no—
215C4×Dic7—28no—
216#1136—28no—
217C2×C7⋊C8—24no—
218C2×C7⋊C8C2414yes—
219C22×#41—71no—
220C22×#41—71no—
221C22×M16—71no—
222C23×C8C2×D711yescharacteristic
223#537—78no—
224#537—144no—
225C2×C22⋊Q8—72no—
226C2×C22⋊Q8—141no—
227C2×#121—72no—
228C2×#121—141no—
229C2×#121—142no—
230#520—78no—
231#520—144no—
232C2×C4×Q8—141no—
233C2×C4×D4—141no—
234C2×C4×D4—142no—
235C2×#117—71no—
236C2×#117—72no—
237C22×C4⋊C4—72no—
238C22×C4⋊C4C2×D711yescharacteristic
239C22×C4⋊C4C2×D711yescharacteristic
240#485—716no—
241#484—716no—
242#482—716no—
243#480—716no—
244#428—78no—
245C2×#107—72no—
246C2×#107C2×D712yes—
247C2×#107C2×D712yes—
248C2×#107C2×D718yescomplemented
249C2×#106—72no—
250#417—78no—
251#416—78no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—72no—
255C2×#102—71no—
256C2×#102—71no—
257C2×#102—72no—
258#408—78no—
259C2×C22⋊C8—72no—
260C2×C22⋊C8—72no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7#12314yes—
264C23×C7C22×D412yes—
265C23×C7C22×D411yescharacteristic
266C22×D7—24no—
267C22×D7—24no—
268C22×D7—24no—
269C22×D7—22no—
270C22×D7—42no—
271C22×D7—42no—
272C22×D7—42no—
273C22×D7C22×D414yes—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—44no—
277C2×C4×C7—44no—
278C2×C4×C7—84no—
279C2×C4×C7—84no—
280C2×C4×C7#12314yes—
281C2×C4×C7#12318yes—
282C2×C4×C7C22×D412yes—
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C2×#4112yes—
285C2×C4×C7C2×#4114yes—
286C2×C4×C7C2×#4112yes—
287C2×C4×C7C2×#4114yes—
288C2×C4×C7C2511yescharacteristic
289C22⋊D7—216no—
290C22⋊D7—48no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7C22×D414yes—
299D28—28no—
300D28—22no—
301D28—22no—
302D28—42no—
303D28—88no—
304C4×D7—28no—
305C4×D7—24no—
306C4×D7—24no—
307C4×D7—48no—
308C4×D7—88no—
309Dic14—28no—
310Dic14—22no—
311Dic14—22no—
312Dic14—42no—
313C7×C8—22no—
314C7×C8—22no—
315C7⋊C8—24no—
316C2×#41—74no—
317C2×#41—74no—
318C2×#41—74no—
319C2×#41—144no—
320C2×#41—142no—
321C2×#41—144no—
322C2×#41—144no—
323C2×#41—144no—
324C22×Q8—71no—
325C22×Q8—71no—
326C22×D4—72no—
327C22×D4—71no—
328C22×D4—71no—
329C22×D4—142no—
330C22×D4—141no—
331C23×C4—21no—
332C23×C4—21no—
333C23×C4—72no—
334C23×C4—142no—
335C23×C4—141no—
336C23×C4C22×D711yescharacteristic
337C2×M16—74no—
338C2×M16—144no—
339C22×C8—24no—
340C22×C8—72no—
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D714yes—
344C22⋊Q8—78no—
345C22⋊Q8—144no—
346C22⋊Q8—148no—
347#121—78no—
348#121—144no—
349#121—148no—
350#121—1416no—
351C4×Q8—148no—
352C4×D4—148no—
353C4×D4—1416no—
354#117—74no—
355#117—78no—
356#117—144no—
357#117—142no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—78no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4C22×D712yes—
369C2×C4⋊C4C22×D714yes—
370C2×C4⋊C4C22×D712yes—
371C2×C4⋊C4C22×D714yes—
372C2×C22⋊C4—142no—
373C2×C22⋊C4—142no—
374C2×C22⋊C4—142no—
375C2×C22⋊C4—141no—
376C2×C42—142no—
377C2×C42—141no—
378#107—78no—
379#107C22×D718yes—
380#107C22×D718yes—
381#106—78no—
382#103—74no—
383#103—74no—
384#103—78no—
385#102—74no—
386#102—74no—
387#102—78no—
388C22⋊C8—78no—
389C22⋊C8—78no—
390C22×C7—28no—
391C22×C7—22no—
392C22×C7—22no—
393C22×C7C2×#12314yes—
394C22×C7C2×#12312yes—
395C22×C7#57612yes—
396C22×C7#57812yes—
397C22×C7C23×D411yescharacteristic
398C2×D7—24no—
399C2×D7—24no—
400C2×D7—24no—
401C2×D7—48no—
402C2×D7—42no—
403C2×D7—42no—
404C2×D7—42no—
405C2×D7—84no—
406C4×C7—44no—
407C4×C7—44no—
408C4×C7C2×#12314yes—
409C4×C7C2×#12312yes—
410C4×C7C22×#4111yescharacteristic
411C4×C7C22×#4111yescharacteristic
412Dic7—24no—
413Dic7—42no—
414Dic7—42no—
415Dic7—42no—
416C24—72no—
417C24—141no—
418C24D4×D711yescharacteristic
419#41—1416no—
420#41—148no—
421#41—148no—
422#41—148no—
423#41—148no—
424#41—288no—
425C2×Q8—74no—
426C2×Q8—72no—
427C2×Q8—144no—
428C2×Q8—141no—
429C2×Q8—142no—
430C2×Q8—142no—
431C2×D4—78no—
432C2×D4—74no—
433C2×D4—72no—
434C2×D4—148no—
435C2×D4—148no—
436C2×D4—144no—
437C2×D4—144no—
438C2×D4—142no—
439C2×D4—141no—
440C2×D4—142no—
441C2×D4—284no—
442C2×D4—564no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—72no—
448C22×C4—74no—
449C22×C4—74no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—148no—
453C22×C4—144no—
454C22×C4—144no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—144no—
459C22×C4—284no—
460C22×C4—564no—
461C22×C4#115714yes—
462C22×C4D4×D712yes—
463C22×C4D4×D711yescharacteristic
464C22×C4#116114yes—
465C22×C4C23×D712yes—
466C22×C4C23×D711yescharacteristic
467M16—148no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—28no—
471C2×C8—74no—
472C2×C8—144no—
473C2×C8C23×D712yes—
474C2×C8C23×D712yes—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—72no—
480C4⋊C4—72no—
481C4⋊C4—148no—
482C4⋊C4—148no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4C23×D714yes—
486C4⋊C4C23×D714yes—
487C22⋊C4—148no—
488C22⋊C4—148no—
489C22⋊C4—148no—
490C22⋊C4—144no—
491C42—148no—
492C42—144no—
493C2×C7—24no—
494C2×C7#309814yescomplemented
495C2×C7C22×#12311yescharacteristic
496C2×C7C2×#57611yescharacteristic
497C2×C7C2×#57811yescharacteristic
498D7—44no—
499D7—82no—
500C23—22no—
501C23—22no—
502C23—74no—
503C23—144no—
504C23—144no—
505C23—144no—
506C23—142no—
507C23—282no—
508C23—282no—
509C23—282no—
510C23#645714yes—
511C23C2×D4×D712yes—
512C23C2×D4×D711yescharacteristic
513Q8—148no—
514Q8—142no—
515Q8—142no—
516Q8—282no—
517D4—1416no—
518D4—148no—
519D4—142no—
520D4—142no—
521D4—288no—
522D4—282no—
523D4—568no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—74no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—144no—
532C2×C4—148no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—282no—
539C2×C4—282no—
540C2×C4—282no—
541C2×C4—288no—
542C2×C4—568no—
543C2×C4#645714yes—
544C2×C4#645718yes—
545C2×C4C2×#115712yes—
546C2×C4C2×#115714yes—
547C2×C4C2×D4×D712yes—
548C2×C4C2×D4×D711yescharacteristic
549C2×C4C2×#116112yes—
550C2×C4C2×#116114yes—
551C2×C4C24×D711yescharacteristic
552C8—22no—
553C8—22no—
554C8—144no—
555C7C2×#309811yescharacteristic, complemented, Sylow
556C22—28no—
557C22—22no—
558C22—22no—
559C22—144no—
560C22—144no—
561C22—144no—
562C22—288no—
563C22—282no—
564C22—282no—
565C22—282no—
566C22—564no—
567C22C2×#645714yes—
568C22C2×#645712yes—
569C22#9044812yes—
570C22#9045712yes—
571C22C22×D4×D711yescharacteristic
572C4—44no—
573C4—44no—
574C4—144no—
575C4—282no—
576C4—282no—
577C4—282no—
578C4C2×#645712yes—
579C4C2×#645714yes—
580C4C22×#115711yescharacteristic
581C4C22×#116111yescharacteristic
582C2—24no—
583C2—284no—
584C2—562no—
585C2#1173658614yesdirect product factor
586C2C22×#645711yescharacteristic
587C2C2×#9044811yescharacteristic
588C2C2×#9045711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC22×C7×#107C2
socleC23×C7C22×D4
prime radicalC2×#90201C2
hypercenterC22×#107D7
hypocenterC7C2×#3098
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C2×#90195C2semidirect
3. C22×C7×#107C2semidirect
4. C2×#89885C2semidirect
5. C2×#89882C2semidirect
6. C2×#89738C2semidirect
7. C2×#89736C2semidirect
8. C2×#89656C2semidirect
9. C2×#89660C2semidirect
10. C2×#89654C2semidirect
11. #11736586C2direct (N × H)
12. C2×#89657C2semidirect
29. #90195C22semidirect
36. C2×C7×#107C22semidirect
37. #89885C22semidirect
40. #89882C22semidirect
46. #89738C22semidirect
48. #89736C22semidirect
51. #89660C22semidirect
52. #89657C22semidirect
53. #89656C22semidirect
54. #89654C22semidirect
138. C22×#107D7semidirect
248. C2×#107C2×D7semidirect
494. C2×C7#3098semidirect
555. C7C2×#3098semidirect
585. C2#11736586direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902012
C2×#901952
C22×C7×#1072
C2×#898852
C2×#898822
C2×#897382
C2×#897362
C2×#896562
C2×#896602
C2×#896542
#117365862
C2×#896572
C2×#30987