Finite Group Databasebeta

← C2×#11739680 #13512538

Subgroups — C2×#11739680 #13512538

Order 1792.

3198 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#117C211yesmaximal, characteristic
2C22×#6487C211yesmaximal, characteristic
3C22×C4×Dic14C211yesmaximal, characteristic, complemented
4C2×#89984C211yesmaximal, characteristic
5#11739680C2116yesmaximal, direct product factor
6C2×#89744C212yesmaximal, complemented
7C2×#89750C212yesmaximal, complemented
8C2×#89717C212yesmaximal
9C2×#89716C212yesmaximal
10C2×#89701C211yesmaximal, characteristic, complemented
11C2×#89700C211yesmaximal, characteristic, complemented
12C2×C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
13C24×C4×C7C2211yescharacteristic
14C23×Dic14C2211yescharacteristic
15C2×C7×#117—28no—
16C2×C7×#117C2218yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2211yescharacteristic
21C22×C42×C7C2211yescharacteristic
22C22×C42×C7C2211yescharacteristic
23C22×C22⋊Dic7C2212yes—
24C22×C22⋊Dic7C2212yes—
25C22×C4⋊Dic7C2211yescharacteristic
26C22×C4⋊Dic7C2211yescharacteristic
27C22×C4⋊Dic7C2211yescharacteristic
28C2×#6487C4116yescomplemented
29C2×#6487C2218yes—
30C22×Dic7⋊C4C2212yes—
31C22×Dic7⋊C4C2212yes—
32C22×Dic7⋊C4C2212yes—
33C22×C4×Dic7C2212yes—
34C22×C4×Dic7C2212yes—
35C2×C4×Dic14—28no—
36C2×C4×Dic14C2218yescomplemented
37#89984C2218yes—
38C2×C7⋊#95C2212yes—
39C2×C7⋊#95C2212yes—
40C2×C7⋊#95C2212yes—
41C2×C7⋊#95C2212yes—
42C2×C7⋊#95C2212yes—
43#89750C22116yescomplemented
44#89744C22116yescomplemented
45#89717C22116yes—
46#89716C22116yes—
47C4×C4⋊Dic7C2218yescomplemented
48#89701C2218yescomplemented
49#89700C2218yescomplemented
50C2×#2323—71nomaximal, Sylow
51C25×C7C2311yescharacteristic
52C23×C4×C7—24no—
53C23×C4×C7—24no—
54C23×C4×C7C2×C418yes—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C23×Dic7C2312yes—
63C23×Dic7C2312yes—
64C23×Dic7C2312yes—
65C23×Dic7C2312yes—
66C22×Dic14—24no—
67C22×Dic14—24no—
68C22×Dic14C2×C418yes—
69C22×Dic14C2314yes—
70C7×#117—232no—
71C2×C7×C4⋊C4—24no—
72C2×C7×C4⋊C4—24no—
73C2×C7×C4⋊C4C2314yes—
74C2×C7×C4⋊C4C2314yes—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—24no—
77C2×C7×C22⋊C4C2314yes—
78C2×C7×C22⋊C4C2314yes—
79C2×C42×C7—24no—
80C2×C42×C7D418yes—
81C2×C42×C7C2314yes—
82C2×C42×C7C2314yes—
83C2×C22⋊Dic7—44no—
84C2×C22⋊Dic7C2×C4116yes—
85C2×C22⋊Dic7C2318yes—
86C2×C22⋊Dic7C2318yes—
87C2×C4⋊Dic7—24no—
88C2×C4⋊Dic7—24no—
89C2×C4⋊Dic7C2×C418yes—
90C2×C4⋊Dic7C2314yes—
91C2×C4⋊Dic7C2314yes—
92C2×C4⋊Dic7C2314yes—
93#6487C2×C4164yescomplemented
94C2×Dic7⋊C4—28no—
95C2×Dic7⋊C4—44no—
96C2×Dic7⋊C4C2×C4116yes—
97C2×Dic7⋊C4C2318yes—
98C2×Dic7⋊C4C2318yes—
99C2×Dic7⋊C4C2318yes—
100C2×C4×Dic7—28no—
101C2×C4×Dic7—44no—
102C2×C4×Dic7C2318yes—
103C2×C4×Dic7C2318yes—
104C4×Dic14—232no—
105C7⋊#95C2318yes—
106C7⋊#95C2318yes—
107C7⋊#95C2318yes—
108C7⋊#95C2318yes—
109C7⋊#95C2318yes—
110C22×C22⋊Q8—71no—
111C22×C4×Q8—71no—
112C22×#117D711yescharacteristic
113#2323—716no—
114C2×#391—71no—
115C2×#389—71no—
116C2×#389—72no—
117C2×#388—72no—
118C2×#388—72no—
119C2×#387—72no—
120C2×#380—71no—
121C2×C4×C4⋊C4—71no—
122C24×C7—22no—
123C24×C7—22no—
124C24×C7—22no—
125C24×C7—22no—
126C24×C7—22no—
127C24×C7—22no—
128C24×C7C22×C414yes—
129C24×C7C2412yes—
130C24×C7C2411yescharacteristic
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—232no—
138C22×C4×C7—28no—
139C22×C4×C7—28no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7C22×C414yes—
147C22×C4×C7C22×C4116yes—
148C22×C4×C7C22×C414yes—
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7#4114yes—
153C22×C4×C7C2412yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2412yes—
156C22×C4×C7C2412yes—
157C22×C4×C7C2412yes—
158C22×C4×C7C2412yes—
159C22×Dic7—24no—
160C22×Dic7—24no—
161C22×Dic7—24no—
162C22×Dic7—24no—
163C22×Dic7—24no—
164C22×Dic7—24no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—24no—
168C22×Dic7—24no—
169C22×Dic7C22×C418yes—
170C22×Dic7C22×C418yes—
171C22×Dic7C2414yes—
172C22×Dic7C2414yes—
173C22×Dic7C2414yes—
174C22×Dic7C2414yes—
175C2×Dic14—232no—
176C2×Dic14—28no—
177C2×Dic14—28no—
178C2×Dic14C22×C4116yes—
179C7×C4⋊C4—28no—
180C7×C4⋊C4—28no—
181C7×C22⋊C4—28no—
182C7×C22⋊C4—28no—
183C42×C7—28no—
184C42×C7C2×D4116yes—
185C22⋊Dic7—48no—
186C22⋊Dic7C22×C4132yes—
187C4⋊Dic7—28no—
188C4⋊Dic7—28no—
189C4⋊Dic7C22×C4116yes—
190Dic7⋊C4—216no—
191Dic7⋊C4—48no—
192Dic7⋊C4C22×C4132yes—
193C4×Dic7—216no—
194C4×Dic7—48no—
195C23×Q8—71no—
196C24×C4C2×D711yescharacteristic
197C2×C22⋊Q8—716no—
198C2×C22⋊Q8—78no—
199C2×C4×Q8—78no—
200C2×C4×Q8—148no—
201C2×#117—28no—
202C2×#117C2×D718yes—
203C22×C4⋊C4—72no—
204C22×C4⋊C4—71no—
205C22×C4⋊C4—72no—
206C22×C4⋊C4—71no—
207C22×C4⋊C4—71no—
208C22×C4⋊C4—72no—
209C22×C4⋊C4C2×D711yescharacteristic
210C22×C4⋊C4C2×D711yescharacteristic
211C22×C22⋊C4—72no—
212C22×C22⋊C4—72no—
213C22×C22⋊C4C2×D711yescharacteristic
214C22×C22⋊C4C2×D711yescharacteristic
215C22×C42—72no—
216C22×C42—72no—
217C22×C42C2×D711yescharacteristic
218C22×C42C2×D711yescharacteristic
219#391—78no—
220#389—716no—
221#389—78no—
222#388—716no—
223#388—716no—
224#387—716no—
225#380—78no—
226C4×C4⋊C4—78no—
227C2×#95—72no—
228C2×#95—72no—
229C2×#95—72no—
230C2×#95—72no—
231C2×#95—72no—
232C23×C7—28no—
233C23×C7—28no—
234C23×C7—28no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—44no—
242C23×C7—44no—
243C23×C7—44no—
244C23×C7—44no—
245C23×C7C23×C414yes—
246C23×C7C23×C412yes—
247C23×C7C22×D412yes—
248C23×C7C2×#4112yes—
249C23×C7H(5,𝔽2)12yes—
250C23×C72−1+412yes—
251C23×C72−1+412yes—
252C23×C72−1+412yes—
253C23×C7C2511yescharacteristic
254C2×C4×C7—28no—
255C2×C4×C7—28no—
256C2×C4×C7—28no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—232no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—44no—
271C2×C4×C7—44no—
272C2×C4×C7C4×D4116yes—
273C2×C4×C7C23×C414yes—
274C2×C4×C7C23×C414yes—
275C2×C4×C7C22×D414yes—
276C2×C4×C7C22×D414yes—
277C2×C4×C7C22×D414yes—
278C2×C4×C7C2×#4114yes—
279C2×Dic7—216no—
280C2×Dic7—216no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—48no—
292C2×Dic7—48no—
293C2×Dic7C23×C418yes—
294C2×Dic7C23×C418yes—
295Dic14—232no—
296C25C22×D711yescharacteristic
297C22×Q8—78no—
298C22×Q8—74no—
299C22×Q8—144no—
300C22×Q8—144no—
301C23×C4—24no—
302C23×C4—24no—
303C23×C4—72no—
304C23×C4—72no—
305C23×C4—72no—
306C23×C4—72no—
307C23×C4C4×D718yes—
308C23×C4C22×D711yescharacteristic
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D714yes—
311C23×C4C22×D711yescharacteristic
312C23×C4C22×D711yescharacteristic
313C23×C4C22×D711yescharacteristic
314C23×C4C22×D711yescharacteristic
315C22⋊Q8—764no—
316C4×Q8—1432no—
317#117—232no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—716no—
321C2×C4⋊C4—78no—
322C2×C4⋊C4—78no—
323C2×C4⋊C4—74no—
324C2×C4⋊C4—78no—
325C2×C4⋊C4—74no—
326C2×C4⋊C4—74no—
327C2×C4⋊C4—78no—
328C2×C4⋊C4—144no—
329C2×C4⋊C4—148no—
330C2×C4⋊C4—144no—
331C2×C4⋊C4—284no—
332C2×C4⋊C4C22×D714yes—
333C2×C4⋊C4C22×D714yes—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—716no—
337C2×C22⋊C4—78no—
338C2×C22⋊C4—78no—
339C2×C22⋊C4—284no—
340C2×C22⋊C4C22×D714yes—
341C2×C22⋊C4C22×D714yes—
342C2×C42—24no—
343C2×C42—78no—
344C2×C42—78no—
345C2×C42—148no—
346C2×C42—284no—
347C2×C42C22⋊D718yes—
348C2×C42C22×D714yes—
349C2×C42C22×D714yes—
350#95—78no—
351#95—78no—
352#95—78no—
353#95—78no—
354#95—78no—
355C22×C7—24no—
356C22×C7—24no—
357C22×C7—24no—
358C22×C7—416no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7C2×C4×D414yes—
364C22×C7#51914yes—
365C22×C7#52014yes—
366C22×C7#53614yes—
367C22×C7#53714yes—
368C22×C7#54114yes—
369C22×C7#54214yes—
370C22×C7C24×C411yescharacteristic
371C22×C7C23×D411yescharacteristic
372C22×C7C22×#4111yescharacteristic
373C22×C7C2×H(5,𝔽2)11yescharacteristic
374C22×C7C2×2−1+411yescharacteristic
375C22×C7C2×2−1+411yescharacteristic
376C22×C7C2×2−1+411yescharacteristic
377C4×C7—24no—
378C4×C7—24no—
379C4×C7—24no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7C2×C4×D418yes—
383Dic7—28no—
384Dic7—28no—
385Dic7—44no—
386Dic7—44no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24C2×C4×D714yes—
394C24C23×D712yes—
395C24C23×D711yescharacteristic
396C2×Q8—716no—
397C2×Q8—1432no—
398C2×Q8—148no—
399C2×Q8—148no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—232no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—78no—
416C22×C4—78no—
417C22×C4—74no—
418C22×C4—74no—
419C22×C4—74no—
420C22×C4—74no—
421C22×C4—144no—
422C22×C4—144no—
423C22×C4—144no—
424C22×C4—144no—
425C22×C4—144no—
426C22×C4—144no—
427C22×C4—144no—
428C22×C4—144no—
429C22×C4—144no—
430C22×C4—144no—
431C22×C4C2×C4×D714yes—
432C22×C4C2×C4×D7116yes—
433C22×C4C2×C4×D714yes—
434C22×C4#115714yes—
435C22×C4C2×C22⋊D714yes—
436C22×C4C2×C22⋊D714yes—
437C22×C4C2×C22⋊D714yes—
438C22×C4C23×D712yes—
439C22×C4C23×D712yes—
440C22×C4C23×D712yes—
441C22×C4C23×D712yes—
442C22×C4C23×D712yes—
443C22×C4C23×D712yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—732no—
447C4⋊C4—716no—
448C4⋊C4—148no—
449C4⋊C4—1416no—
450C4⋊C4—148no—
451C4⋊C4—288no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—732no—
455C22⋊C4—288no—
456C42—28no—
457C42—1416no—
458C42—288no—
459C42C2×C22⋊D7116yes—
460C2×C7—44no—
461C2×C7#232318yescomplemented
462C2×C7C22×C4×D411yescharacteristic
463C2×C7C2×#51911yescharacteristic
464C2×C7C2×#52011yescharacteristic
465C2×C7C2×#53611yescharacteristic
466C2×C7C2×#53711yescharacteristic
467C2×C7C2×#54111yescharacteristic
468C2×C7C2×#54211yescharacteristic
469C23—28no—
470C23—28no—
471C23—28no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23C22×C4×D714yes—
483C23C22×C4×D712yes—
484C23C2×#115712yes—
485C23#655312yes—
486C23#655412yes—
487C23#655412yes—
488C23#655412yes—
489C23C22×C22⋊D712yes—
490C23C24×D711yescharacteristic
491Q8—1432no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—232no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—78no—
511C2×C4—78no—
512C2×C4—1416no—
513C2×C4—1416no—
514C2×C4—144no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—144no—
518C2×C4—144no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—144no—
524C2×C4—288no—
525C2×C4—288no—
526C2×C4C4×C22⋊D7116yes—
527C2×C4C22×C4×D714yes—
528C2×C4C22×C4×D714yes—
529C2×C4C2×#115714yes—
530C2×C4C22×C22⋊D714yes—
531C2×C4C22×C22⋊D714yes—
532C2×C4C22×C22⋊D714yes—
533C7C2×#232311yescharacteristic, complemented, Sylow
534C22—24no—
535C22—24no—
536C22—24no—
537C22—416no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22#9020714yes—
543C22#9020914yes—
544C22#9021414yes—
545C22#9022514yes—
546C22C2×C4×C22⋊D714yes—
547C22#9051614yes—
548C22#9052414yes—
549C22C23×C4×D711yescharacteristic
550C22C22×#115711yescharacteristic
551C22C2×#655311yescharacteristic
552C22C2×#655411yescharacteristic
553C22C2×#655411yescharacteristic
554C22C2×#655411yescharacteristic
555C22C23×C22⋊D711yescharacteristic
556C4—24no—
557C4—24no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—148no—
562C4—148no—
563C4—284no—
564C4—284no—
565C4C2×C4×C22⋊D718yes—
566C2—44no—
567C2#1173968018yesdirect product factor
568C2C2×#9020711yescharacteristic
569C2C2×#9020911yescharacteristic
570C2C2×#9021411yescharacteristic
571C2C2×#9022511yescharacteristic
572C2C22×C4×C22⋊D711yescharacteristic
573C2C2×#9051611yescharacteristic
574C2C2×#9052411yescharacteristic
subgroupG/H
centerC24C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC22×C7×#117C2
socleC24×C7C24
prime radicalC25×C7C23
cosocleC23×C7C25
hypercenterC22×#117D7
hypocenterC7C2×#2323
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C22×C4×Dic14C2semidirect
5. #11739680C2direct (N × H)
6. C2×#89744C2semidirect
7. C2×#89750C2semidirect
10. C2×#89701C2semidirect
11. C2×#89700C2semidirect
12. C2×C4×C4⋊Dic7C2semidirect
28. C2×#6487C4semidirect
36. C2×C4×Dic14C22semidirect
43. #89750C22semidirect
44. #89744C22semidirect
47. C4×C4⋊Dic7C22semidirect
48. #89701C22semidirect
49. #89700C22semidirect
93. #6487C2×C4semidirect
461. C2×C7#2323semidirect
533. C7C2×#2323semidirect
567. C2#11739680direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1172
C22×#64872
C22×C4×Dic142
C2×#899842
#117396802
C2×#897442
C2×#897502
C2×#897172
C2×#897162
C2×#897012
C2×#897002
C2×C4×C4⋊Dic72
C2×#23237