Finite Group Databasebeta

← C2×#11731520 #13522847

Subgroups — C2×#11731520 #13522847

Order 1792.

3678 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6492C211yesmaximal, characteristic
2C23×Dic7⋊C4C211yesmaximal, characteristic, complemented
3C22×#6449C211yesmaximal, characteristic
4C22×(C22⋊D7)⋊C4C211yesmaximal, characteristic
5C2×C7×#381C211yesmaximal, characteristic, complemented
6C2×#89715C211yesmaximal, characteristic, complemented
7C2×#89714C211yesmaximal, characteristic, complemented
8#11731520C2116yesmaximal, direct product factor
9C2×#89447C212yesmaximal, complemented
10C2×#89439C212yesmaximal, complemented
11C2×#89429C212yesmaximal, complemented
12C2×#89419C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C23×C22⋊D7C2211yescharacteristic
15C24×Dic7C2211yescharacteristic
16C24×Dic7C2211yescharacteristic
17C23×C4×D7C2211yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C2×#6492—28no—
22C2×#6492C2218yes—
23C22×(C2×D7)⋊C4C2211yescharacteristic
24C22×(C2×D7)⋊C4C2212yes—
25C22×(C2×D7)⋊C4C2212yes—
26C22×C4⋊Dic7C2212yes—
27C22×Dic7⋊C4—28no—
28C22×Dic7⋊C4C2212yes—
29C22×Dic7⋊C4C2212yes—
30C22×Dic7⋊C4C2211yescharacteristic
31C22×Dic7⋊C4C2212yes—
32C22×Dic7⋊C4C2212yes—
33C22×Dic7⋊C4C2218yescomplemented
34C22×C4×Dic7C2211yescharacteristic
35C2×#6449—28no—
36C2×#6449C2218yes—
37C2×(C22⋊D7)⋊C4—28no—
38C2×(C22⋊D7)⋊C4C2218yes—
39C7×#381C2218yescomplemented
40C2×C7×#95C2212yes—
41C2×C7×#95C2212yes—
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45#89715C2218yescomplemented
46#89714C2218yescomplemented
47#89447C22116yescomplemented
48#89439C22116yescomplemented
49#89429C22116yescomplemented
50#89419C22116yescomplemented
51C2×#2486—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C24×D7C2311yescharacteristic
54C23×C4×C7—24no—
55C23×C4×C7—24no—
56C23×C4×C7—24no—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2312yes—
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C23×C4×C7C2314yes—
62C22×C22⋊D7—24no—
63C22×C22⋊D7—24no—
64C22×C22⋊D7D418yes—
65C22×C22⋊D7C2314yes—
66C23×Dic7—28no—
67C23×Dic7—24no—
68C23×Dic7—24no—
69C23×Dic7—24no—
70C23×Dic7D418yes—
71C23×Dic7C2314yes—
72C23×Dic7C2312yes—
73C23×Dic7C2312yes—
74C23×Dic7C2314yes—
75C23×Dic7C2311yescharacteristic
76C23×Dic7C2311yescharacteristic
77C23×Dic7C2311yescharacteristic
78C22×C4×D7—24no—
79C22×C4×D7—28no—
80C22×C4×D7C2314yes—
81C2×C7×C22⋊C4—24no—
82C2×C7×C22⋊C4—24no—
83C2×C7×C22⋊C4C2314yes—
84C2×C7×C22⋊C4C2314yes—
85C2×C22⋊Dic7—24no—
86C2×C22⋊Dic7C2314yes—
87#6492—232no—
88C2×(C2×D7)⋊C4—28no—
89C2×(C2×D7)⋊C4—28no—
90C2×(C2×D7)⋊C4—24no—
91C2×(C2×D7)⋊C4C2314yes—
92C2×(C2×D7)⋊C4C2318yes—
93C2×(C2×D7)⋊C4C2318yes—
94C2×C4⋊Dic7—28no—
95C2×C4⋊Dic7C2318yes—
96C2×Dic7⋊C4—24no—
97C2×Dic7⋊C4—28no—
98C2×Dic7⋊C4—28no—
99C2×Dic7⋊C4—232no—
100C2×Dic7⋊C4—28no—
101C2×Dic7⋊C4—28no—
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2318yes—
104C2×Dic7⋊C4C2318yes—
105C2×Dic7⋊C4C2318yes—
106C2×Dic7⋊C4C2318yes—
107C2×C4×Dic7—24no—
108C2×C4×Dic7C2314yes—
109#6449—232no—
110(C22⋊D7)⋊C4—232no—
111C7×#95C2318yes—
112C7×#95C2318yes—
113C7⋊#95C2318yes—
114C7⋊#95C2318yes—
115C7⋊#95C2318yes—
116C22×#123—71no—
117C22×#123—71no—
118C22×C4×D4—71no—
119C23×C4⋊C4—71no—
120#2486—716no—
121C2×#398—72no—
122C2×#397—72no—
123C2×#389—72no—
124C2×#381—71no—
125C2×#381D711yescharacteristic, complemented
126C2×#380—72no—
127C2×#380—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—24no—
137C23×D7—22no—
138C23×D7—24no—
139C23×D7C2×D414yes—
140C23×D7C2412yes—
141C22×C4×C7—22no—
142C22×C4×C7—28no—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—28no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7—28no—
151C22×C4×C7—24no—
152C22×C4×C7—24no—
153C22×C4×C7—24no—
154C22×C4×C7—416no—
155C22×C4×C7C2412yes—
156C22×C4×C7C2412yes—
157C22×C4×C7C2414yes—
158C22×C4×C7C2414yes—
159C2×C22⋊D7—232no—
160C2×C22⋊D7—28no—
161C2×C22⋊D7—28no—
162C2×C22⋊D7C2×D4116yes—
163C22×Dic7—22no—
164C22×Dic7—24no—
165C22×Dic7—22no—
166C22×Dic7—232no—
167C22×Dic7—216no—
168C22×Dic7—28no—
169C22×Dic7—28no—
170C22×Dic7—24no—
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7—24no—
174C22×Dic7—28no—
175C22×Dic7—24no—
176C22×Dic7—24no—
177C22×Dic7—416no—
178C22×Dic7C2×D4116yes—
179C22×Dic7C2×D414yes—
180C22×Dic7C2×D414yes—
181C22×Dic7C2×D414yes—
182C22×Dic7#4118yes—
183C22×Dic7C2414yes—
184C22×Dic7C2414yes—
185C22×Dic7C2412yes—
186C22×Dic7C2412yes—
187C22×Dic7C2412yes—
188C2×C4×D7—28no—
189C2×C4×D7—216no—
190C2×C4×D7—416no—
191C7×C22⋊C4—28no—
192C7×C22⋊C4—28no—
193C22⋊Dic7—28no—
194(C2×D7)⋊C4—216no—
195(C2×D7)⋊C4—216no—
196(C2×D7)⋊C4—28no—
197C4⋊Dic7—216no—
198Dic7⋊C4—28no—
199Dic7⋊C4—216no—
200Dic7⋊C4—216no—
201Dic7⋊C4—216no—
202Dic7⋊C4—216no—
203C4×Dic7—28no—
204C23×D4—71no—
205C24×C4—71no—
206C24×C4—71no—
207C24×C4—71no—
208C24×C4C2×D711yescharacteristic
209C2×#123—78no—
210C2×#123—78no—
211C2×#123—148no—
212C2×#123—148no—
213C2×C4×D4—78no—
214C2×C4×D4—148no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4—72no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—78no—
221C22×C4⋊C4—72no—
222C22×C4⋊C4—148no—
223C22×C22⋊C4—71no—
224C22×C22⋊C4—71no—
225C22×C22⋊C4—72no—
226C22×C22⋊C4—72no—
227C22×C22⋊C4C2×D711yescharacteristic
228C22×C22⋊C4C2×D711yescharacteristic
229C22×C42—71no—
230#398—716no—
231#397—716no—
232#389—716no—
233#381—78no—
234#381C2×D718yescomplemented
235#380—78no—
236#380—716no—
237C2×#95—72no—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95C2×D712yes—
241C2×#95C2×D712yes—
242C23×C7—28no—
243C23×C7—28no—
244C23×C7—216no—
245C23×C7—216no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7#121116yes—
249C23×C7C22×D414yes—
250C23×C7C22×D414yes—
251C23×C7C22×D412yes—
252C23×C7C22×D412yes—
253C23×C7C2×#4114yes—
254C23×C7C2×#4112yes—
255C23×C7C2×#4114yes—
256C23×C72−1+414yes—
257C23×C7C2511yescharacteristic
258C22×D7—28no—
259C22×D7—216no—
260C22×D7—22no—
261C22×D7—24no—
262C22×D7—22no—
263C22×D7—24no—
264C22×D7—44no—
265C22×D7—44no—
266C22×D7—48no—
267C22×D7C22×D414yes—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—22no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—416no—
279C2×C4×C7—48no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7—48no—
283C22⋊D7—232no—
284C2×Dic7—28no—
285C2×Dic7—28no—
286C2×Dic7—22no—
287C2×Dic7—24no—
288C2×Dic7—22no—
289C2×Dic7—232no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—416no—
297C2×Dic7—48no—
298C2×Dic7—48no—
299C2×Dic7#123116yescomplemented
300C2×Dic7C22×D414yes—
301C2×Dic7C22×D414yes—
302C2×Dic7C22×D414yes—
303C2×Dic7C2×#4118yes—
304C4×D7—416no—
305C25—71no—
306C25C22×D711yescharacteristic
307C22×D4—78no—
308C22×D4—74no—
309C22×D4—144no—
310C22×D4—144no—
311C23×C4—24no—
312C23×C4—24no—
313C23×C4—24no—
314C23×C4—71no—
315C23×C4—78no—
316C23×C4—74no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—74no—
320C23×C4—72no—
321C23×C4—72no—
322C23×C4—74no—
323C23×C4—144no—
324C23×C4—148no—
325C23×C4—148no—
326C23×C4—144no—
327C23×C4—144no—
328C23×C4—144no—
329C23×C4C22×D714yes—
330C23×C4C22×D712yes—
331C23×C4C22×D711yescharacteristic
332C23×C4C22×D712yes—
333C23×C4C22×D711yescharacteristic
334#123—1432no—
335#123—1432no—
336C4×D4—1432no—
337C2×C4⋊C4—78no—
338C2×C4⋊C4—78no—
339C2×C4⋊C4—74no—
340C2×C4⋊C4—78no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—78no—
343C2×C4⋊C4—148no—
344C2×C4⋊C4—148no—
345C2×C4⋊C4—1432no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4—148no—
348C2×C4⋊C4—148no—
349C2×C4⋊C4—148no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—74no—
353C2×C22⋊C4—74no—
354C2×C22⋊C4—78no—
355C2×C22⋊C4—78no—
356C2×C22⋊C4—144no—
357C2×C22⋊C4—148no—
358C2×C22⋊C4—148no—
359C2×C22⋊C4—144no—
360C2×C22⋊C4C22×D714yes—
361C2×C22⋊C4C22×D714yes—
362C2×C42—74no—
363C2×C42—144no—
364#95—78no—
365#95—78no—
366#95—78no—
367#95C22×D718yes—
368#95C22×D718yes—
369C22×C7—24no—
370C22×C7—24no—
371C22×C7—28no—
372C22×C7—28no—
373C22×C7—232no—
374C22×C7C2×#12118yes—
375C22×C7C2×#12114yes—
376C22×C7C2×#12314yes—
377C22×C7#53614yes—
378C22×C7#54718yes—
379C22×C7#55218yes—
380C22×C7C23×D411yescharacteristic
381C22×C7C23×D411yescharacteristic
382C22×C7C22×#4111yescharacteristic
383C22×C7C22×#4112yes—
384C22×C7C2×2−1+412yes—
385C2×D7—24no—
386C2×D7—28no—
387C2×D7—42no—
388C2×D7—42no—
389C2×D7—44no—
390C2×D7—416no—
391C4×C7—44no—
392C4×C7—42no—
393C4×C7—42no—
394C4×C7—44no—
395Dic7—24no—
396Dic7—24no—
397Dic7—44no—
398Dic7—44no—
399Dic7C2×#12318yescomplemented
400C24—24no—
401C24—24no—
402C24—74no—
403C24—72no—
404C24—142no—
405C24—142no—
406C24—144no—
407C24—144no—
408C24#115714yes—
409C24D4×D714yes—
410C24D4×D714yes—
411C24C23×D712yes—
412C24C23×D711yescharacteristic
413C2×D4—716no—
414C2×D4—1432no—
415C2×D4—148no—
416C2×D4—148no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—416no—
431C22×C4—74no—
432C22×C4—78no—
433C22×C4—72no—
434C22×C4—74no—
435C22×C4—74no—
436C22×C4—716no—
437C22×C4—72no—
438C22×C4—72no—
439C22×C4—74no—
440C22×C4—74no—
441C22×C4—148no—
442C22×C4—1416no—
443C22×C4—1432no—
444C22×C4—1416no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—144no—
448C22×C4—148no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—144no—
452C22×C4—148no—
453C22×C4—144no—
454C22×C4—148no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—2816no—
458C22×C4—2816no—
459C22×C4C23×D712yes—
460C22×C4C23×D712yes—
461C22×C4C23×D714yes—
462C22×C4C23×D714yes—
463C4⋊C4—1416no—
464C4⋊C4—1416no—
465C4⋊C4—1416no—
466C4⋊C4—148no—
467C4⋊C4—1416no—
468C4⋊C4—1416no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—148no—
472C22⋊C4—1416no—
473C22⋊C4—1416no—
474C22⋊C4—148no—
475C42—148no—
476C2×C7—28no—
477C2×C7#248618yescomplemented
478C2×C7C22×#12111yescharacteristic
479C2×C7C22×#12311yescharacteristic
480C2×C7C2×#53611yescharacteristic
481C2×C7C2×#54712yes—
482C2×C7C2×#55212yes—
483D7—44no—
484C23—216no—
485C23—28no—
486C23—28no—
487C23—216no—
488C23—24no—
489C23—24no—
490C23—74no—
491C23—148no—
492C23—1416no—
493C23—142no—
494C23—142no—
495C23—144no—
496C23—144no—
497C23—288no—
498C23—284no—
499C23—284no—
500C23#6447116yes—
501C23C2×#115714yes—
502C23C2×#115712yes—
503C23C2×D4×D714yes—
504C23C2×D4×D714yes—
505C23C2×D4×D712yes—
506C23C2×D4×D712yes—
507C23D7×#4114yes—
508C23#655414yes—
509C23C24×D711yescharacteristic
510D4—1432no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—416no—
522C2×C4—48no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—716no—
527C2×C4—74no—
528C2×C4—78no—
529C2×C4—74no—
530C2×C4—74no—
531C2×C4—1432no—
532C2×C4—148no—
533C2×C4—148no—
534C2×C4—142no—
535C2×C4—142no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—144no—
542C2×C4—144no—
543C2×C4—2816no—
544C2×C4—2816no—
545C2×C4—288no—
546C2×C4—288no—
547C7C2×#248611yescharacteristic, complemented, Sylow
548C22—232no—
549C22—28no—
550C22—24no—
551C22—24no—
552C22—28no—
553C22—144no—
554C22—148no—
555C22—2816no—
556C22—284no—
557C22—282no—
558C22—282no—
559C22C2×#644718yes—
560C22C2×#644714yes—
561C22#9022218yes—
562C22#9033914yes—
563C22D7×#12314yes—
564C22#9034618yes—
565C22C22×#115711yescharacteristic
566C22C22×D4×D711yescharacteristic
567C22C22×D4×D711yescharacteristic
568C22C2×D7×#4112yes—
569C22C2×#655412yes—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—44no—
574C4—78no—
575C4—144no—
576C4—144no—
577C4—284no—
578C4—284no—
579C2—28no—
580C2—284no—
581C2#1173152018yesdirect product factor
582C2C22×#644711yescharacteristic
583C2C2×#9022212yes—
584C2C2×#9033911yescharacteristic
585C2C2×D7×#12311yescharacteristic
586C2C2×#9034612yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#381C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC2×#381D7
hypocenterC7C2×#2486
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. C23×Dic7⋊C4C2semidirect
5. C2×C7×#381C2semidirect
6. C2×#89715C2semidirect
7. C2×#89714C2semidirect
8. #11731520C2direct (N × H)
9. C2×#89447C2semidirect
10. C2×#89439C2semidirect
11. C2×#89429C2semidirect
12. C2×#89419C2semidirect
33. C22×Dic7⋊C4C22semidirect
39. C7×#381C22semidirect
45. #89715C22semidirect
46. #89714C22semidirect
47. #89447C22semidirect
48. #89439C22semidirect
49. #89429C22semidirect
50. #89419C22semidirect
125. C2×#381D7semidirect
234. #381C2×D7semidirect
299. C2×Dic7#123semidirect
399. Dic7C2×#123semidirect
477. C2×C7#2486semidirect
547. C7C2×#2486semidirect
581. C2#11731520direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64922
C23×Dic7⋊C42
C22×#64492
C22×(C22⋊D7)⋊C42
C2×C7×#3812
C2×#897152
C2×#897142
#117315202
C2×#894472
C2×#894392
C2×#894292
C2×#894192
C2×#24867