Finite Group Databasebeta

← C2×#11731852 #13522497

Subgroups — C2×#11731852 #13522497

Order 1792.

3570 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic7C211yesmaximal, characteristic, complemented
2C22×#6449C211yesmaximal, characteristic
3C22×#6449C211yesmaximal, characteristic
4C22×#6440C211yesmaximal, characteristic, complemented
5C2×C7×#385C211yesmaximal, characteristic, complemented
6C2×#89985C211yesmaximal, characteristic
7C2×#89766C212yesmaximal, complemented
8C2×#89757C212yesmaximal, complemented
9C2×#89715C211yesmaximal, characteristic, complemented
10#11731852C2116yesmaximal, direct product factor
11C2×#89433C212yesmaximal, complemented
12C2×#89442C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C23×C22⋊D7C2211yescharacteristic
15C24×Dic7C2211yescharacteristic
16C24×Dic7C2211yescharacteristic
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C22×(C2×D7)⋊C4C2211yescharacteristic
22C22×(C2×D7)⋊C4C2212yes—
23C22×(C2×D7)⋊C4C2211yescharacteristic
24C22×(C2×D7)⋊C4C2212yes—
25C22×C4⋊Dic7—28no—
26C22×C4⋊Dic7C2212yes—
27C22×C4⋊Dic7C2212yes—
28C22×C4⋊Dic7C2211yescharacteristic
29C22×C4⋊Dic7C2212yes—
30C22×C4⋊Dic7C2211yescharacteristic
31C22×C4⋊Dic7C2218yescomplemented
32C22×Dic7⋊C4C2212yes—
33C22×C4×Dic7C2212yes—
34C2×#6449—28no—
35C2×#6449—28no—
36C2×#6449C2218yes—
37C2×#6449C2218yes—
38C2×#6440—28no—
39C2×#6440C2218yescomplemented
40C7×#385C2218yescomplemented
41C2×C7×#95C2212yes—
42#89985C2218yes—
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45C2×C7⋊#95C2212yes—
46#89766C22116yescomplemented
47#89757C22116yescomplemented
48#89715C2218yescomplemented
49#89442C22116yescomplemented
50#89433C22116yescomplemented
51C2×#2413—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C24×D7C2311yescharacteristic
54C23×C4×C7—24no—
55C23×C4×C7—28no—
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2312yes—
60C23×C4×C7C2312yes—
61C22×C22⋊D7—24no—
62C22×C22⋊D7—24no—
63C22×C22⋊D7—24no—
64C22×C22⋊D7C2314yes—
65C23×Dic7—24no—
66C23×Dic7—24no—
67C23×Dic7—24no—
68C23×Dic7—24no—
69C23×Dic7—24no—
70C23×Dic7D418yes—
71C23×Dic7C2314yes—
72C23×Dic7C2312yes—
73C23×Dic7C2311yescharacteristic
74C23×Dic7C2312yes—
75C23×Dic7C2314yes—
76C23×Dic7C2311yescharacteristic
77C23×Dic7C2311yescharacteristic
78C2×C7×C4⋊C4—28no—
79C2×C7×C4⋊C4C2318yes—
80C2×C7×C22⋊C4—24no—
81C2×C7×C22⋊C4D418yes—
82C2×C7×C22⋊C4C2314yes—
83C2×C7×C22⋊C4C2314yes—
84C2×C22⋊Dic7—24no—
85C2×C22⋊Dic7C2314yes—
86C2×(C2×D7)⋊C4—28no—
87C2×(C2×D7)⋊C4—28no—
88C2×(C2×D7)⋊C4—24no—
89C2×(C2×D7)⋊C4—24no—
90C2×(C2×D7)⋊C4C2314yes—
91C2×(C2×D7)⋊C4C2318yes—
92C2×(C2×D7)⋊C4C2314yes—
93C2×(C2×D7)⋊C4C2318yes—
94C2×C4⋊Dic7—28no—
95C2×C4⋊Dic7—24no—
96C2×C4⋊Dic7—24no—
97C2×C4⋊Dic7—28no—
98C2×C4⋊Dic7—28no—
99C2×C4⋊Dic7—232no—
100C2×C4⋊Dic7C2314yes—
101C2×C4⋊Dic7C2318yes—
102C2×C4⋊Dic7C2314yes—
103C2×C4⋊Dic7C2318yes—
104C2×C4⋊Dic7C2318yes—
105C2×Dic7⋊C4—28no—
106C2×Dic7⋊C4C2318yes—
107C2×C4×Dic7—28no—
108C2×C4×Dic7C2318yes—
109#6449—232no—
110#6449—232no—
111#6440—232no—
112C7×#95C2318yes—
113C7⋊#95C2318yes—
114C7⋊#95C2318yes—
115C7⋊#95C2318yes—
116C22×#123—71no—
117C22×#123—71no—
118C22×#117—71no—
119C23×C4⋊C4—71no—
120#2413—716no—
121C2×#398—72no—
122C2×#396—72no—
123C2×#389—72no—
124C2×#388—72no—
125C2×#386—71no—
126C2×#385D711yescharacteristic, complemented
127C2×#381—71no—
128C24×C7—24no—
129C24×C7—24no—
130C24×C7C2×D414yes—
131C24×C7C2×D414yes—
132C24×C7#4114yes—
133C24×C7C2412yes—
134C24×C7C2411yescharacteristic
135C23×D7—22no—
136C23×D7—22no—
137C23×D7—24no—
138C23×D7—22no—
139C23×D7—24no—
140C23×D7C2412yes—
141C22×C4×C7—232no—
142C22×C4×C7—28no—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—216no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7C2×D418yes—
153C22×C4×C7#4118yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2412yes—
156C22×C4×C7C2414yes—
157C22×C4×C7C2414yes—
158C2×C22⋊D7—28no—
159C2×C22⋊D7—28no—
160C2×C22⋊D7—28no—
161C2×C22⋊D7—416no—
162C22×Dic7—232no—
163C22×Dic7—28no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—28no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—28no—
170C22×Dic7—24no—
171C22×Dic7—28no—
172C22×Dic7—24no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—24no—
176C22×Dic7—24no—
177C22×Dic7—28no—
178C22×Dic7—24no—
179C22×Dic7—22no—
180C22×Dic7—24no—
181C22×Dic7—416no—
182C22×Dic7C2×D4116yes—
183C22×Dic7C2×D414yes—
184C22×Dic7C2×D414yes—
185C22×Dic7C2414yes—
186C22×Dic7C2412yes—
187C22×Dic7C2414yes—
188C22×Dic7C2412yes—
189C22×Dic7C2412yes—
190C7×C4⋊C4—216no—
191C7×C22⋊C4—28no—
192C7×C22⋊C4C2×D4116yes—
193C22⋊Dic7—28no—
194(C2×D7)⋊C4—216no—
195(C2×D7)⋊C4—216no—
196(C2×D7)⋊C4—28no—
197(C2×D7)⋊C4—28no—
198C4⋊Dic7—216no—
199C4⋊Dic7—28no—
200C4⋊Dic7—28no—
201C4⋊Dic7—216no—
202C4⋊Dic7—216no—
203Dic7⋊C4—216no—
204C4×Dic7—216no—
205C23×D4—71no—
206C24×C4—71no—
207C24×C4—71no—
208C24×C4C2×D711yescharacteristic
209C2×#123—78no—
210C2×#123—78no—
211C2×#123—148no—
212C2×#123—148no—
213C2×#117—78no—
214C2×#117—148no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4—72no—
217C22×C4⋊C4—71no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—78no—
221C22×C4⋊C4—71no—
222C22×C4⋊C4—148no—
223C22×C4⋊C4C2×D712yes—
224C22×C22⋊C4—71no—
225C22×C22⋊C4—71no—
226C22×C22⋊C4—72no—
227C22×C22⋊C4—72no—
228C22×C22⋊C4—71no—
229C22×C22⋊C4C2×D711yescharacteristic
230C22×C22⋊C4C2×D711yescharacteristic
231C22×C42—72no—
232#398—716no—
233#396—716no—
234#389—716no—
235#388—716no—
236#386—78no—
237#385C2×D718yescomplemented
238#381—78no—
239C2×#95—72no—
240C2×#95—72no—
241C2×#95—72no—
242C2×#95C2×D712yes—
243C23×C7—28no—
244C23×C7—28no—
245C23×C7—216no—
246C23×C7—216no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7#121116yes—
250C23×C7C22×D414yes—
251C23×C7C22×D412yes—
252C23×C7C22×D414yes—
253C23×C7C22×D412yes—
254C23×C7C2×#4114yes—
255C23×C7C2×#4114yes—
256C23×C7C2×#4112yes—
257C23×C72−1+414yes—
258C23×C7C2511yescharacteristic
259C22×D7—22no—
260C22×D7—22no—
261C22×D7—24no—
262C22×D7—22no—
263C22×D7—24no—
264C22×D7—44no—
265C22×D7—44no—
266C22×D7—44no—
267C22×D7—48no—
268C22×D7—48no—
269C2×C4×C7—232no—
270C2×C4×C7—216no—
271C2×C4×C7—216no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7C22×D418yes—
283C2×C4×C7C2×#4118yes—
284C22⋊D7—416no—
285C2×Dic7—232no—
286C2×Dic7—28no—
287C2×Dic7—28no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—22no—
300C2×Dic7—24no—
301C2×Dic7—416no—
302C2×Dic7—48no—
303C2×Dic7—44no—
304C2×Dic7—48no—
305C2×Dic7C22×D414yes—
306C2×Dic7C22×D414yes—
307C25—71no—
308C25C22×D711yescharacteristic
309C22×D4—74no—
310C22×D4—144no—
311C22×D4—144no—
312C22×D4—144no—
313C23×C4—24no—
314C23×C4—28no—
315C23×C4—78no—
316C23×C4—74no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—74no—
320C23×C4—72no—
321C23×C4—72no—
322C23×C4—71no—
323C23×C4—144no—
324C23×C4—144no—
325C23×C4—144no—
326C23×C4—144no—
327C23×C4—144no—
328C23×C4C22×D714yes—
329C23×C4C22×D711yescharacteristic
330C23×C4C22×D711yescharacteristic
331C23×C4C22×D712yes—
332C23×C4C22×D712yes—
333#123—1432no—
334#123—1432no—
335#117—1432no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—78no—
338C2×C4⋊C4—78no—
339C2×C4⋊C4—78no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—148no—
344C2×C4⋊C4—148no—
345C2×C4⋊C4—1432no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—148no—
349C2×C4⋊C4—148no—
350C2×C4⋊C4C22×D718yes—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—74no—
353C2×C22⋊C4—74no—
354C2×C22⋊C4—78no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—78no—
357C2×C22⋊C4—144no—
358C2×C22⋊C4—144no—
359C2×C22⋊C4—144no—
360C2×C22⋊C4—148no—
361C2×C22⋊C4—148no—
362C2×C22⋊C4D2818yes—
363C2×C22⋊C4C22×D714yes—
364C2×C22⋊C4C22×D714yes—
365C2×C42—78no—
366C2×C42—148no—
367#95—78no—
368#95—78no—
369#95—78no—
370#95C22×D718yes—
371C22×C7—24no—
372C22×C7—232no—
373C22×C7—24no—
374C22×C7—28no—
375C22×C7—28no—
376C22×C7C2×#12118yes—
377C22×C7C2×#12114yes—
378C22×C7#52514yes—
379C22×C7#53614yes—
380C22×C7#54718yes—
381C22×C7#55218yes—
382C22×C7C23×D411yescharacteristic
383C22×C7C23×D411yescharacteristic
384C22×C7C22×#4112yes—
385C22×C7C22×#4111yescharacteristic
386C22×C7C2×2−1+412yes—
387C2×D7—42no—
388C2×D7—42no—
389C2×D7—42no—
390C2×D7—44no—
391C2×D7—44no—
392C2×D7—88no—
393C4×C7—28no—
394C4×C7—28no—
395C4×C7—42no—
396C4×C7—42no—
397Dic7—24no—
398Dic7—24no—
399Dic7—44no—
400Dic7—42no—
401Dic7—44no—
402C24—24no—
403C24—24no—
404C24—72no—
405C24—142no—
406C24—142no—
407C24—144no—
408C24—142no—
409C24—144no—
410C24C2×D2814yes—
411C24D4×D714yes—
412C24#116114yes—
413C24C23×D712yes—
414C24C23×D711yescharacteristic
415C2×D4—148no—
416C2×D4—148no—
417C2×D4—148no—
418C2×D4—2816no—
419C22×C4—232no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—74no—
431C22×C4—74no—
432C22×C4—716no—
433C22×C4—72no—
434C22×C4—72no—
435C22×C4—74no—
436C22×C4—74no—
437C22×C4—72no—
438C22×C4—1432no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—148no—
442C22×C4—148no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—142no—
448C22×C4—148no—
449C22×C4—148no—
450C22×C4—144no—
451C22×C4—142no—
452C22×C4—144no—
453C22×C4—148no—
454C22×C4—144no—
455C22×C4—142no—
456C22×C4—144no—
457C22×C4—2816no—
458C22×C4C2×D2818yes—
459C22×C4#115918yes—
460C22×C4C23×D712yes—
461C22×C4C23×D712yes—
462C22×C4C23×D714yes—
463C22×C4C23×D714yes—
464C4⋊C4—216no—
465C4⋊C4—1416no—
466C4⋊C4—1416no—
467C4⋊C4—148no—
468C4⋊C4—148no—
469C4⋊C4—1416no—
470C4⋊C4—1416no—
471C22⋊C4—28no—
472C22⋊C4—148no—
473C22⋊C4—148no—
474C22⋊C4—148no—
475C22⋊C4—1416no—
476C22⋊C4—1416no—
477C22⋊C4C2×D28116yes—
478C42—1416no—
479C2×C7—28no—
480C2×C7#241318yescomplemented
481C2×C7C22×#12111yescharacteristic
482C2×C7C2×#52511yescharacteristic
483C2×C7C2×#53611yescharacteristic
484C2×C7C2×#54712yes—
485C2×C7C2×#55212yes—
486D7—82no—
487C23—216no—
488C23—28no—
489C23—216no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—142no—
494C23—142no—
495C23—144no—
496C23—142no—
497C23—144no—
498C23—284no—
499C23—288no—
500C23—284no—
501C23—288no—
502C23—284no—
503C23#6458116yes—
504C23C22×D2814yes—
505C23C22×D2812yes—
506C23C2×D4×D714yes—
507C23C2×D4×D712yes—
508C23C2×#115914yes—
509C23C2×#116114yes—
510C23C2×#116112yes—
511C23#655414yes—
512C23C24×D711yescharacteristic
513D4—2816no—
514C2×C4—232no—
515C2×C4—216no—
516C2×C4—216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—74no—
528C2×C4—74no—
529C2×C4—1432no—
530C2×C4—148no—
531C2×C4—148no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—142no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—142no—
539C2×C4—144no—
540C2×C4—142no—
541C2×C4—144no—
542C2×C4—144no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—2816no—
546C2×C4—288no—
547C2×C4—284no—
548C2×C4—288no—
549C2×C4C22×D2818yes—
550C2×C4C2×#115918yes—
551C7C2×#241311yescharacteristic, complemented, Sylow
552C22—24no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—232no—
557C22—282no—
558C22—282no—
559C22—284no—
560C22—282no—
561C22—284no—
562C22—568no—
563C22C2×#645818yes—
564C22C2×#645814yes—
565C22#9024618yes—
566C22#9027714yes—
567C22#9033318yes—
568C22#9033914yes—
569C22C23×D2811yescharacteristic
570C22C22×D4×D711yescharacteristic
571C22C22×#115912yes—
572C22C22×#116111yescharacteristic
573C22C2×#655412yes—
574C4—28no—
575C4—28no—
576C4—42no—
577C4—42no—
578C4—144no—
579C4—144no—
580C4—284no—
581C4—282no—
582C4—284no—
583C2—28no—
584C2—562no—
585C2#1173185218yesdirect product factor
586C2C22×#645811yescharacteristic
587C2C2×#9024612yes—
588C2C2×#9027711yescharacteristic
589C2C2×#9033312yes—
590C2C2×#9033911yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#385C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC2×#385D7
hypocenterC7C2×#2413
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊Dic7C2semidirect
4. C22×#6440C2semidirect
5. C2×C7×#385C2semidirect
7. C2×#89766C2semidirect
8. C2×#89757C2semidirect
9. C2×#89715C2semidirect
10. #11731852C2direct (N × H)
11. C2×#89433C2semidirect
12. C2×#89442C2semidirect
31. C22×C4⋊Dic7C22semidirect
39. C2×#6440C22semidirect
40. C7×#385C22semidirect
46. #89766C22semidirect
47. #89757C22semidirect
48. #89715C22semidirect
49. #89442C22semidirect
50. #89433C22semidirect
126. C2×#385D7semidirect
237. #385C2×D7semidirect
480. C2×C7#2413semidirect
551. C7C2×#2413semidirect
585. C2#11731852direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic72
C22×#64492
C22×#64402
C2×C7×#3852
C2×#899852
C2×#897662
C2×#897572
C2×#897152
#117318522
C2×#894332
C2×#894422
C2×#24137