Finite Group Databasebeta

← C2×#11739615 #13522498

Subgroups — C2×#11739615 #13522498

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×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×#6492C211yesmaximal, characteristic
3C22×#6492C211yesmaximal, characteristic
4C22×#6489C211yesmaximal, characteristic, complemented
5C2×#89985C211yesmaximal, characteristic
6C2×#89982C211yesmaximal, characteristic, complemented
7C2×#89983C211yesmaximal, characteristic, complemented
8C2×#89766C212yesmaximal, complemented
9#11739615C2116yesmaximal, direct product factor
10C2×#89756C212yesmaximal, complemented
11C2×#89744C212yesmaximal, complemented
12C2×#89762C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C24×C4×C7C2211yescharacteristic
15C24×C4×C7C2211yescharacteristic
16C23×C22⋊D7C2211yescharacteristic
17C22×C7×C4⋊C4—28no—
18C22×C7×C4⋊C4C2212yes—
19C22×C7×C4⋊C4C2212yes—
20C22×C7×C4⋊C4C2218yescomplemented
21C22×C22⋊Dic7C2211yescharacteristic
22C22×C22⋊Dic7C2211yescharacteristic
23C22×C22⋊Dic7C2211yescharacteristic
24C2×#6492—28no—
25C2×#6492—28no—
26C2×#6492C2218yes—
27C2×#6492C2218yes—
28C22×(C2×D7)⋊C4C2212yes—
29C22×(C2×D7)⋊C4C2211yescharacteristic
30C22×(C2×D7)⋊C4C2212yes—
31C22×(C2×D7)⋊C4C2211yescharacteristic
32C2×#6489—28no—
33C2×#6489C2218yescomplemented
34C22×C4⋊Dic7C2212yes—
35C22×Dic7⋊C4C2212yes—
36C22×Dic7⋊C4C2211yescharacteristic
37C22×Dic7⋊C4C2212yes—
38C22×Dic7⋊C4C2211yescharacteristic
39C22×C4×Dic7C2212yes—
40#89985C2218yes—
41#89983C2218yescomplemented
42#89982C2218yescomplemented
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45C2×C7⋊#95C2212yes—
46C2×C7⋊#95C2212yes—
47#89766C22116yescomplemented
48#89762C22116yescomplemented
49#89756C22116yescomplemented
50#89744C22116yescomplemented
51C2×#2413—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C24×D7C2311yescharacteristic
54C23×C4×C7—24no—
55C23×C4×C7—24no—
56C23×C4×C7—24no—
57C23×C4×C7—28no—
58C23×C4×C7—24no—
59C23×C4×C7—24no—
60C23×C4×C7—24no—
61C23×C4×C7D418yes—
62C23×C4×C7C2312yes—
63C23×C4×C7C2314yes—
64C23×C4×C7C2314yes—
65C23×C4×C7C2311yescharacteristic
66C23×C4×C7C2311yescharacteristic
67C23×C4×C7C2314yes—
68C23×C4×C7C2312yes—
69C22×C22⋊D7—24no—
70C22×C22⋊D7—24no—
71C22×C22⋊D7—24no—
72C22×C22⋊D7C2314yes—
73C23×Dic7C2312yes—
74C23×Dic7C2311yescharacteristic
75C23×Dic7C2311yescharacteristic
76C23×Dic7C2311yescharacteristic
77C23×Dic7C2312yes—
78C2×C7×C4⋊C4—28no—
79C2×C7×C4⋊C4—28no—
80C2×C7×C4⋊C4—232no—
81C2×C7×C4⋊C4C2318yes—
82C2×C7×C4⋊C4C2318yes—
83C2×C22⋊Dic7—24no—
84C2×C22⋊Dic7—24no—
85C2×C22⋊Dic7D418yes—
86C2×C22⋊Dic7C2314yes—
87C2×C22⋊Dic7C2314yes—
88C2×C22⋊Dic7C2314yes—
89#6492—232no—
90#6492—232no—
91C2×(C2×D7)⋊C4—24no—
92C2×(C2×D7)⋊C4—24no—
93C2×(C2×D7)⋊C4—28no—
94C2×(C2×D7)⋊C4—28no—
95C2×(C2×D7)⋊C4C2314yes—
96C2×(C2×D7)⋊C4C2318yes—
97C2×(C2×D7)⋊C4C2314yes—
98C2×(C2×D7)⋊C4C2318yes—
99#6489—232no—
100C2×C4⋊Dic7—28no—
101C2×C4⋊Dic7C2318yes—
102C2×Dic7⋊C4—24no—
103C2×Dic7⋊C4—24no—
104C2×Dic7⋊C4—28no—
105C2×Dic7⋊C4—28no—
106C2×Dic7⋊C4C2314yes—
107C2×Dic7⋊C4C2318yes—
108C2×Dic7⋊C4C2314yes—
109C2×Dic7⋊C4C2318yes—
110C2×C4×Dic7—28no—
111C2×C4×Dic7C2318yes—
112C7⋊#95C2318yes—
113C7⋊#95C2318yes—
114C7⋊#95C2318yes—
115C7⋊#95C2318yes—
116C22×#123—71no—
117C22×#123—71no—
118C22×#117—71no—
119C23×C4⋊C4D711yescharacteristic, complemented
120#2413—716no—
121C2×#398—72no—
122C2×#396—72no—
123C2×#389—72no—
124C2×#388—72no—
125C2×#386—71no—
126C2×#385—71no—
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—232no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—28no—
146C22×C4×C7—28no—
147C22×C4×C7—24no—
148C22×C4×C7—28no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—216no—
153C22×C4×C7—24no—
154C22×C4×C7—28no—
155C22×C4×C7—28no—
156C22×C4×C7—24no—
157C22×C4×C7—28no—
158C22×C4×C7—24no—
159C22×C4×C7—416no—
160C22×C4×C7C2×D414yes—
161C22×C4×C7C2×D414yes—
162C22×C4×C7C2×D4116yes—
163C22×C4×C7#4118yes—
164C22×C4×C7C2414yes—
165C22×C4×C7C2412yes—
166C22×C4×C7C2412yes—
167C22×C4×C7C2414yes—
168C2×C22⋊D7—28no—
169C2×C22⋊D7—28no—
170C2×C22⋊D7—28no—
171C2×C22⋊D7—416no—
172C22×Dic7—24no—
173C22×Dic7—24no—
174C22×Dic7—24no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—24no—
180C22×Dic7—24no—
181C22×Dic7—22no—
182C22×Dic7—24no—
183C22×Dic7—24no—
184C22×Dic7C2×D418yes—
185C22×Dic7C2412yes—
186C22×Dic7C2414yes—
187C22×Dic7C2412yes—
188C22×Dic7C2412yes—
189C22×Dic7C2414yes—
190C7×C4⋊C4—216no—
191C7×C4⋊C4—216no—
192C22⋊Dic7—28no—
193C22⋊Dic7—28no—
194C22⋊Dic7C2×D4116yes—
195(C2×D7)⋊C4—28no—
196(C2×D7)⋊C4—28no—
197(C2×D7)⋊C4—216no—
198(C2×D7)⋊C4—216no—
199C4⋊Dic7—216no—
200Dic7⋊C4—28no—
201Dic7⋊C4—28no—
202Dic7⋊C4—216no—
203Dic7⋊C4—216no—
204C4×Dic7—216no—
205C23×D4—71no—
206C24×C4C2×D711yescharacteristic
207C24×C4C2×D711yescharacteristic
208C24×C4C2×D711yescharacteristic
209C2×#123—78no—
210C2×#123—78no—
211C2×#123—148no—
212C2×#123—148no—
213C2×#117—78no—
214C2×#117—148no—
215C22×C4⋊C4—28no—
216C22×C4⋊C4—72no—
217C22×C4⋊C4—72no—
218C22×C4⋊C4—71no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—71no—
221C22×C4⋊C4C2×D712yes—
222C22×C4⋊C4C2×D712yes—
223C22×C4⋊C4C2×D718yescomplemented
224C22×C22⋊C4—71no—
225C22×C22⋊C4—71no—
226C22×C22⋊C4—71no—
227C22×C22⋊C4—71no—
228C22×C22⋊C4—72no—
229C22×C22⋊C4—72no—
230C22×C22⋊C4—71no—
231C22×C42—72no—
232#398—716no—
233#396—716no—
234#389—716no—
235#388—716no—
236#386—78no—
237#385—78no—
238#381—78no—
239C2×#95—72no—
240C2×#95—72no—
241C2×#95—72no—
242C2×#95—72no—
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—232no—
271C2×C4×C7—28no—
272C2×C4×C7—28no—
273C2×C4×C7—216no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—416no—
284C2×C4×C7—48no—
285C2×C4×C7C22×D414yes—
286C2×C4×C7C22×D414yes—
287C2×C4×C7C2×#4118yes—
288C22⋊D7—416no—
289C2×Dic7—216no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—22no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—44no—
303C2×Dic7—44no—
304C2×Dic7—44no—
305C2×Dic7—48no—
306C2×Dic7C22×D418yes—
307C25—71no—
308C25C22×D711yescharacteristic
309C22×D4—74no—
310C22×D4—144no—
311C22×D4—144no—
312C22×D4—144no—
313C23×C4—24no—
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—28no—
317C23×C4—24no—
318C23×C4—24no—
319C23×C4—24no—
320C23×C4—72no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—72no—
324C23×C4—71no—
325C23×C4C22⋊D718yes—
326C23×C4C22×D714yes—
327C23×C4C22×D714yes—
328C23×C4C22×D712yes—
329C23×C4C22×D712yes—
330C23×C4C22×D714yes—
331C23×C4C22×D711yescharacteristic
332C23×C4C22×D711yescharacteristic
333#123—1432no—
334#123—1432no—
335#117—1432no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—232no—
339C2×C4⋊C4—78no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—78no—
344C2×C4⋊C4—148no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—148no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4C22×D718yes—
350C2×C4⋊C4C22×D718yes—
351C2×C22⋊C4—78no—
352C2×C22⋊C4—74no—
353C2×C22⋊C4—74no—
354C2×C22⋊C4—74no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—78no—
357C2×C22⋊C4—74no—
358C2×C22⋊C4—78no—
359C2×C22⋊C4—148no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—144no—
362C2×C22⋊C4—148no—
363C2×C22⋊C4—144no—
364C2×C22⋊C4—144no—
365C2×C42—78no—
366C2×C42—148no—
367#95—78no—
368#95—78no—
369#95—78no—
370#95—78no—
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—24no—
394C4×C7—24no—
395C4×C7—28no—
396C4×C7—44no—
397Dic7—28no—
398Dic7—42no—
399Dic7—42no—
400Dic7—42no—
401Dic7—44no—
402C24—24no—
403C24—24no—
404C24—72no—
405C24—144no—
406C24—142no—
407C24—142no—
408C24—144no—
409C24—142no—
410C24D4×D714yes—
411C24#115914yes—
412C24C2×C22⋊D714yes—
413C24C23×D712yes—
414C24C23×D711yescharacteristic
415C2×D4—148no—
416C2×D4—148no—
417C2×D4—148no—
418C2×D4—2816no—
419C22×C4—232no—
420C22×C4—232no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—416no—
438C22×C4—78no—
439C22×C4—74no—
440C22×C4—72no—
441C22×C4—74no—
442C22×C4—72no—
443C22×C4—72no—
444C22×C4—144no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—144no—
448C22×C4—144no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—144no—
452C22×C4—144no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4#115718yes—
457C22×C4C2×C22⋊D714yes—
458C22×C4C2×C22⋊D714yes—
459C22×C4C2×C22⋊D7116yes—
460C22×C4C23×D714yes—
461C22×C4C23×D714yes—
462C22×C4C23×D712yes—
463C22×C4C23×D712yes—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—1416no—
467C4⋊C4—1416no—
468C4⋊C4—1416no—
469C4⋊C4—148no—
470C4⋊C4—148no—
471C22⋊C4—716no—
472C22⋊C4—1416no—
473C22⋊C4—148no—
474C22⋊C4—148no—
475C22⋊C4—1416no—
476C22⋊C4—148no—
477C22⋊C4—148no—
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—216no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—144no—
494C23—142no—
495C23—142no—
496C23—144no—
497C23—142no—
498C23—288no—
499C23—288no—
500C23—284no—
501C23—284no—
502C23—284no—
503C23#6502116yes—
504C23C2×#115714yes—
505C23C2×D4×D714yes—
506C23C2×D4×D712yes—
507C23C2×#115914yes—
508C23C2×#115912yes—
509C23#655114yes—
510C23C22×C22⋊D714yes—
511C23C22×C22⋊D712yes—
512C23C24×D711yescharacteristic
513D4—2816no—
514C2×C4—232no—
515C2×C4—216no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—48no—
530C2×C4—78no—
531C2×C4—1416no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—288no—
545C2×C4—284no—
546C2×C4—284no—
547C2×C4—284no—
548C2×C4C2×#115718yes—
549C2×C4C22×C22⋊D714yes—
550C2×C4C22×C22⋊D714yes—
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—
563C22#9019818yes—
564C22#9020418yes—
565C22#9048214yes—
566C22C2×#650218yes—
567C22C2×#650214yes—
568C22#9050714yes—
569C22C22×#115712yes—
570C22C22×D4×D711yescharacteristic
571C22C22×#115911yescharacteristic
572C22C2×#655112yes—
573C22C23×C22⋊D711yescharacteristic
574C4—28no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—148no—
579C4—282no—
580C4—282no—
581C4—282no—
582C4—284no—
583C2—28no—
584C2—562no—
585C2#1173961518yesdirect product factor
586C2C2×#9020412yes—
587C2C2×#9019812yes—
588C2C2×#9048211yescharacteristic
589C2C22×#650211yescharacteristic
590C2C2×#9050711yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC23×C7×C4⋊C4C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC23×C4⋊C4D7
hypocenterC7C2×#2413
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C7×C4⋊C4C2semidirect
4. C22×#6489C2semidirect
6. C2×#89982C2semidirect
7. C2×#89983C2semidirect
8. C2×#89766C2semidirect
9. #11739615C2direct (N × H)
10. C2×#89756C2semidirect
11. C2×#89744C2semidirect
12. C2×#89762C2semidirect
20. C22×C7×C4⋊C4C22semidirect
33. C2×#6489C22semidirect
41. #89983C22semidirect
42. #89982C22semidirect
47. #89766C22semidirect
48. #89762C22semidirect
49. #89756C22semidirect
50. #89744C22semidirect
119. C23×C4⋊C4D7semidirect
223. C22×C4⋊C4C2×D7semidirect
480. C2×C7#2413semidirect
551. C7C2×#2413semidirect
585. C2#11739615direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C7×C4⋊C42
C22×#64922
C22×#64892
C2×#899852
C2×#899822
C2×#899832
C2×#897662
#117396152
C2×#897562
C2×#897442
C2×#897622
C2×#24137