Finite Group Databasebeta

← C2×#11731281 #13520139

Subgroups — C2×#11731281 #13520139

Order 1792.

4142 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic7C211yesmaximal, characteristic
2C22×#6441C211yesmaximal, characteristic, complemented
3C2×C7×#379C211yesmaximal, characteristic
4C2×#90019C211yesmaximal, characteristic
5C2×#89721C212yesmaximal, complemented
6C2×#89720C212yesmaximal, complemented
7C2×#89720C212yesmaximal, complemented
8C2×#89719C212yesmaximal, complemented
9C2×#89722C211yesmaximal, characteristic, complemented
10C2×#89718C211yesmaximal, characteristic, complemented
11C2×#89714C211yesmaximal, characteristic, complemented
12#11731281C2116yesmaximal, direct product factor
13C26×C7C2211yescharacteristic
14C24×Dic7C2211yescharacteristic
15C24×Dic7C2211yescharacteristic
16C23×Dic14C2211yescharacteristic, complemented
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C22⋊Dic7—28no—
22C22×C22⋊Dic7C2211yescharacteristic
23C22×C22⋊Dic7C2218yes—
24C22×C22⋊Dic7C2212yes—
25C22×C22⋊Dic7C2212yes—
26C22×C22⋊Dic7C2211yescharacteristic
27C22×C22⋊Dic7C2212yes—
28C22×C22⋊Dic7C2212yes—
29C22×C4⋊Dic7C2211yescharacteristic, complemented
30C22×C4⋊Dic7C2212yescomplemented
31C22×Dic7⋊C4C2211yescharacteristic, complemented
32C22×Dic7⋊C4C2211yescharacteristic, complemented
33C22×Dic7⋊C4C2212yescomplemented
34C22×Dic7⋊C4C2212yescomplemented
35C2×#6441—28no—
36C2×#6441C2218yescomplemented
37C7×#379C2218yes—
38#90019C2218yes—
39C2×C7⋊#95C2212yescomplemented
40C2×C7⋊#95C2212yescomplemented
41C2×C7⋊#95C2212yescomplemented
42#89722C2218yescomplemented
43#89721C22116yescomplemented
44#89720C22116yescomplemented
45#89720C22116yescomplemented
46#89719C22116yescomplemented
47#89718C2218yescomplemented
48#89714C2218yescomplemented
49C2×#2626—71nomaximal, Sylow
50C25×C7—24no—
51C25×C7—24no—
52C25×C7—44no—
53C25×C7—44no—
54C25×C7Q818yescomplemented
55C25×C7C2312yes—
56C25×C7C2311yescharacteristic
57C25×C7C2314yes—
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—24no—
62C23×Dic7—44no—
63C23×Dic7—44no—
64C23×Dic7D418yes—
65C23×Dic7C2311yescharacteristic
66C23×Dic7C2312yes—
67C23×Dic7C2312yes—
68C23×Dic7C2311yescharacteristic
69C23×Dic7C2311yescharacteristic
70C23×Dic7C2311yescharacteristic
71C23×Dic7C2314yes—
72C23×Dic7C2314yes—
73C22×Dic14—24no—
74C22×Dic14—44no—
75C22×Dic14C2314yescomplemented
76C2×C7×C22⋊C4—24no—
77C2×C7×C22⋊C4—24no—
78C2×C7×C22⋊C4—44no—
79C2×C7×C22⋊C4—44no—
80C2×C7×C22⋊C4C2314yes—
81C2×C7×C22⋊C4C2314yes—
82C2×C7×C22⋊C4C2318yes—
83C2×C7×C22⋊C4C2318yes—
84C2×C22⋊Dic7—24no—
85C2×C22⋊Dic7—24no—
86C2×C22⋊Dic7—28no—
87C2×C22⋊Dic7—28no—
88C2×C22⋊Dic7—232no—
89C2×C22⋊Dic7—44no—
90C2×C22⋊Dic7—44no—
91C2×C22⋊Dic7C2314yes—
92C2×C22⋊Dic7C2314yes—
93C2×C22⋊Dic7C2318yes—
94C2×C22⋊Dic7C2318yes—
95C2×C22⋊Dic7C2318yes—
96C2×C22⋊Dic7C2318yes—
97C2×C4⋊Dic7—24no—
98C2×C4⋊Dic7—44no—
99C2×C4⋊Dic7C2314yescomplemented
100C2×C4⋊Dic7C2318yescomplemented
101C2×Dic7⋊C4—24no—
102C2×Dic7⋊C4—24no—
103C2×Dic7⋊C4—44no—
104C2×Dic7⋊C4—44no—
105C2×Dic7⋊C4C2314yescomplemented
106C2×Dic7⋊C4C2314yescomplemented
107C2×Dic7⋊C4C2318yescomplemented
108C2×Dic7⋊C4C2318yescomplemented
109#6441—232no—
110C7⋊#95C2318yescomplemented
111C7⋊#95C2318yescomplemented
112C7⋊#95C2318yescomplemented
113C22×C22⋊Q8—71no—
114C23×C22⋊C4—71no—
115#2626—716no—
116C2×#399—72no—
117C2×#396—72no—
118C2×#396—72no—
119C2×#393—72no—
120C2×#385—71no—
121C2×#385—71no—
122C2×#380—71no—
123C2×#379—71no—
124C2×#379D711yescharacteristic
125C24×C7—24no—
126C24×C7—24no—
127C24×C7—232no—
128C24×C7—24no—
129C24×C7—28no—
130C24×C7—24no—
131C24×C7—28no—
132C24×C7—24no—
133C24×C7—24no—
134C24×C7—24no—
135C24×C7—24no—
136C24×C7—432no—
137C24×C7—48no—
138C24×C7—48no—
139C24×C7—416no—
140C24×C7—416no—
141C24×C7—432no—
142C24×C7C2×D414yes—
143C24×C7C2×Q818yes—
144C24×C7C2×Q814yes—
145C24×C7C2×Q8116yescomplemented
146C24×C7#4114yes—
147C24×C7C2414yes—
148C24×C7C2412yes—
149C24×C7C2411yescharacteristic
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—24no—
153C22×C4×C7—24no—
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—24no—
157C22×C4×C7C2414yes—
158C22×C4×C7C2412yes—
159C22×C4×C7C2412yes—
160C22×Dic7—24no—
161C22×Dic7—24no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—28no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—24no—
168C22×Dic7—24no—
169C22×Dic7—24no—
170C22×Dic7—24no—
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7—416no—
174C22×Dic7—48no—
175C22×Dic7—48no—
176C22×Dic7—416no—
177C22×Dic7C2×D414yes—
178C22×Dic7C2×D414yes—
179C22×Dic7C2×D4116yes—
180C22×Dic7C2414yes—
181C22×Dic7C2414yes—
182C22×Dic7C2412yes—
183C22×Dic7C2412yes—
184C22×Dic7C2412yes—
185C22×Dic7C2412yes—
186C2×Dic14—28no—
187C2×Dic14—48no—
188C2×Dic14—416no—
189C7×C22⋊C4—28no—
190C7×C22⋊C4—28no—
191C7×C22⋊C4—48no—
192C7×C22⋊C4—48no—
193C22⋊Dic7—28no—
194C22⋊Dic7—28no—
195C22⋊Dic7—216no—
196C22⋊Dic7—216no—
197C22⋊Dic7—48no—
198C22⋊Dic7—48no—
199C4⋊Dic7—28no—
200C4⋊Dic7—48no—
201Dic7⋊C4—28no—
202Dic7⋊C4—28no—
203Dic7⋊C4—48no—
204Dic7⋊C4—48no—
205C26C2×D711yescharacteristic
206C23×Q8—71no—
207C24×C4—71no—
208C24×C4—71no—
209C2×C22⋊Q8—78no—
210C2×C22⋊Q8—148no—
211C22×C4⋊C4—71no—
212C22×C4⋊C4—71no—
213C22×C4⋊C4—71no—
214C22×C4⋊C4—72no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4—72no—
217C22×C22⋊C4—78no—
218C22×C22⋊C4—72no—
219C22×C22⋊C4—72no—
220C22×C22⋊C4—71no—
221C22×C22⋊C4—71no—
222C22×C22⋊C4—72no—
223C22×C22⋊C4—72no—
224C22×C22⋊C4—148no—
225C22×C22⋊C4C2×D711yescharacteristic
226C22×C22⋊C4C2×D711yescharacteristic
227C22×C22⋊C4C2×D712yes—
228C22×C22⋊C4C2×D712yes—
229#399—716no—
230#396—716no—
231#396—716no—
232#393—716no—
233#385—78no—
234#385—78no—
235#380—78no—
236#379—78no—
237#379C2×D718yes—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95—72no—
241C23×C7—28no—
242C23×C7—28no—
243C23×C7—216no—
244C23×C7—216no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—216no—
248C23×C7—232no—
249C23×C7—216no—
250C23×C7—216no—
251C23×C7—24no—
252C23×C7—24no—
253C23×C7—24no—
254C23×C7—24no—
255C23×C7—24no—
256C23×C7—24no—
257C23×C7—432no—
258C23×C7—4128no—
259C23×C7—416no—
260C23×C7—416no—
261C23×C7—432no—
262C23×C7—48no—
263C23×C7—48no—
264C23×C7—48no—
265C23×C7—48no—
266C23×C7C22⋊Q8116yes—
267C23×C7C22×D414yes—
268C23×C7C22×D412yes—
269C23×C7C22×Q818yes—
270C23×C7C22×Q814yes—
271C23×C7C22×Q812yes—
272C23×C7C2×#4114yes—
273C23×C7C2×#4112yes—
274C23×C7H(5,𝔽2)14yes—
275C23×C7H(5,𝔽2)14yes—
276C23×C7C2511yescharacteristic
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—44no—
285C2×C4×C7—44no—
286C2×C4×C7—48no—
287C2×Dic7—28no—
288C2×Dic7—28no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—24no—
301C2×Dic7—416no—
302C2×Dic7—416no—
303C2×Dic7—44no—
304C2×Dic7—44no—
305C2×Dic7—48no—
306C2×Dic7—48no—
307C2×Dic7C22×D414yes—
308C2×Dic7C22×D414yes—
309Dic14—416no—
310C25—24no—
311C25—24no—
312C25—44no—
313C25—44no—
314C25Dic1418yescomplemented
315C25C22×D712yes—
316C25C22×D711yescharacteristic
317C25C22×D714yes—
318C22×Q8—74no—
319C22×Q8—144no—
320C22×Q8—284no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—78no—
324C23×C4—74no—
325C23×C4—71no—
326C23×C4—72no—
327C23×C4—71no—
328C23×C4—72no—
329C23×C4—74no—
330C23×C4—144no—
331C23×C4—284no—
332C23×C4—284no—
333C23×C4C22×D711yescharacteristic
334C23×C4C22×D712yes—
335C23×C4C22×D711yescharacteristic
336C22⋊Q8—1432no—
337C2×C4⋊C4—74no—
338C2×C4⋊C4—74no—
339C2×C4⋊C4—74no—
340C2×C4⋊C4—78no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—78no—
343C2×C4⋊C4—144no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—144no—
346C2×C4⋊C4—284no—
347C2×C4⋊C4—284no—
348C2×C4⋊C4—284no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—78no—
354C2×C22⋊C4—78no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—74no—
357C2×C22⋊C4—78no—
358C2×C22⋊C4—78no—
359C2×C22⋊C4—148no—
360C2×C22⋊C4—148no—
361C2×C22⋊C4—1432no—
362C2×C22⋊C4—144no—
363C2×C22⋊C4—144no—
364C2×C22⋊C4—284no—
365C2×C22⋊C4—284no—
366C2×C22⋊C4C22×D714yes—
367C2×C22⋊C4C22×D714yes—
368C2×C22⋊C4C22×D718yes—
369C2×C22⋊C4C22×D718yes—
370#95—78no—
371#95—78no—
372#95—78no—
373C22×C7—232no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—28no—
377C22×C7—28no—
378C22×C7—28no—
379C22×C7—28no—
380C22×C7—28no—
381C22×C7—464no—
382C22×C7—432no—
383C22×C7—44no—
384C22×C7—44no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7C2×C22⋊Q818yes—
388C22×C7C2×C22⋊Q814yes—
389C22×C7#53414yes—
390C22×C7#53514yes—
391C22×C7#53818yes—
392C22×C7#54318yes—
393C22×C7C23×D411yescharacteristic
394C22×C7C23×Q811yescharacteristic
395C22×C7C22×#4111yescharacteristic
396C22×C7C2×H(5,𝔽2)12yes—
397C22×C7C2×H(5,𝔽2)12yes—
398C4×C7—42no—
399C4×C7—42no—
400C4×C7—44no—
401Dic7—24no—
402Dic7—24no—
403Dic7—42no—
404Dic7—42no—
405Dic7—44no—
406Dic7—44no—
407C24—24no—
408C24—24no—
409C24—232no—
410C24—24no—
411C24—28no—
412C24—24no—
413C24—28no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—432no—
419C24—48no—
420C24—48no—
421C24—416no—
422C24—416no—
423C24—432no—
424C24C2×Dic1418yes—
425C24C2×Dic1414yes—
426C24C2×Dic14116yescomplemented
427C24D4×D714yes—
428C24#115914yes—
429C24C23×D714yes—
430C24C23×D712yes—
431C24C23×D711yescharacteristic
432C2×Q8—148no—
433C2×Q8—288no—
434C2×Q8—2816no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—74no—
443C22×C4—74no—
444C22×C4—72no—
445C22×C4—72no—
446C22×C4—716no—
447C22×C4—74no—
448C22×C4—72no—
449C22×C4—74no—
450C22×C4—72no—
451C22×C4—144no—
452C22×C4—144no—
453C22×C4—148no—
454C22×C4—144no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—144no—
458C22×C4—144no—
459C22×C4—144no—
460C22×C4—144no—
461C22×C4—144no—
462C22×C4—144no—
463C22×C4—144no—
464C22×C4—2816no—
465C22×C4—288no—
466C22×C4—288no—
467C22×C4—2816no—
468C22×C4C23×D714yes—
469C22×C4C23×D712yes—
470C22×C4C23×D712yes—
471C4⋊C4—148no—
472C4⋊C4—148no—
473C4⋊C4—148no—
474C4⋊C4—288no—
475C4⋊C4—288no—
476C4⋊C4—288no—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—1416no—
482C22⋊C4—1416no—
483C22⋊C4—148no—
484C22⋊C4—148no—
485C22⋊C4—288no—
486C22⋊C4—288no—
487C2×C7—28no—
488C2×C7—48no—
489C2×C7#262618yescomplemented
490C2×C7C22×C22⋊Q811yescharacteristic
491C2×C7C2×#53411yescharacteristic
492C2×C7C2×#53511yescharacteristic
493C2×C7C2×#53812yes—
494C2×C7C2×#54312yes—
495C23—216no—
496C23—28no—
497C23—28no—
498C23—216no—
499C23—24no—
500C23—24no—
501C23—216no—
502C23—232no—
503C23—216no—
504C23—216no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—432no—
512C23—416no—
513C23—432no—
514C23—416no—
515C23—4128no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23#6441116yes—
521C23C22×Dic1418yes—
522C23C22×Dic1414yes—
523C23C22×Dic1412yes—
524C23C2×D4×D714yes—
525C23C2×D4×D712yes—
526C23C2×#115914yes—
527C23C2×#115912yes—
528C23#654814yes—
529C23#654814yes—
530C23C24×D711yescharacteristic
531Q8—2816no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—74no—
543C2×C4—74no—
544C2×C4—148no—
545C2×C4—148no—
546C2×C4—144no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—144no—
554C2×C4—144no—
555C2×C4—144no—
556C2×C4—144no—
557C2×C4—144no—
558C2×C4—2816no—
559C2×C4—2816no—
560C2×C4—284no—
561C2×C4—284no—
562C2×C4—288no—
563C2×C4—288no—
564C7C2×#262611yescharacteristic, complemented, Sylow
565C22—24no—
566C22—28no—
567C22—24no—
568C22—28no—
569C22—232no—
570C22—28no—
571C22—28no—
572C22—28no—
573C22—464no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—432no—
579C22C2×#644118yes—
580C22C2×#644114yes—
581C22#9017418yes—
582C22#9028014yes—
583C22#9028314yes—
584C22#9028418yes—
585C22C23×Dic1411yescharacteristic
586C22C22×D4×D711yescharacteristic
587C22C22×#115911yescharacteristic
588C22C2×#654812yes—
589C22C2×#654812yes—
590C4—42no—
591C4—44no—
592C4—42no—
593C4—144no—
594C4—144no—
595C4—284no—
596C4—282no—
597C4—284no—
598C4—282no—
599C2—28no—
600C2—48no—
601C2#1173128118yesdirect product factor
602C2C22×#644111yescharacteristic
603C2C2×#9017412yes—
604C2C2×#9028311yescharacteristic
605C2C2×#9028412yes—
606C2C2×#9028011yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#379C2
socleC24×C7C24
prime radicalC26×C7C22
hypercenterC2×#379D7
hypocenterC7C2×#2626
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
2. C22×#6441C2semidirect
5. C2×#89721C2semidirect
6. C2×#89720C2semidirect
7. C2×#89720C2semidirect
8. C2×#89719C2semidirect
9. C2×#89722C2semidirect
10. C2×#89718C2semidirect
11. C2×#89714C2semidirect
12. #11731281C2direct (N × H)
16. C23×Dic14C22semidirect
29. C22×C4⋊Dic7C22semidirect
30. C22×C4⋊Dic7C22semidirect
31. C22×Dic7⋊C4C22semidirect
32. C22×Dic7⋊C4C22semidirect
33. C22×Dic7⋊C4C22semidirect
34. C22×Dic7⋊C4C22semidirect
36. C2×#6441C22semidirect
39. C2×C7⋊#95C22semidirect
40. C2×C7⋊#95C22semidirect
41. C2×C7⋊#95C22semidirect
42. #89722C22semidirect
43. #89721C22semidirect
44. #89720C22semidirect
45. #89720C22semidirect
46. #89719C22semidirect
47. #89718C22semidirect
48. #89714C22semidirect
54. C25×C7Q8semidirect
75. C22×Dic14C23semidirect
99. C2×C4⋊Dic7C23semidirect
100. C2×C4⋊Dic7C23semidirect
105. C2×Dic7⋊C4C23semidirect
106. C2×Dic7⋊C4C23semidirect
107. C2×Dic7⋊C4C23semidirect
108. C2×Dic7⋊C4C23semidirect
110. C7⋊#95C23semidirect
111. C7⋊#95C23semidirect
112. C7⋊#95C23semidirect
145. C24×C7C2×Q8semidirect
314. C25Dic14semidirect
426. C24C2×Dic14semidirect
489. C2×C7#2626semidirect
564. C7C2×#2626semidirect
601. C2#11731281direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic72
C22×#64412
C2×C7×#3792
C2×#900192
C2×#897212
C2×#897202
C2×#897192
C2×#897222
C2×#897182
C2×#897142
#117312812
C2×#26267