Finite Group Databasebeta

← C2×#11729461 #13320890

Subgroups — C2×#11729461 #13320890

Order 1792.

1606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#525C211yesmaximal, characteristic
2C2×C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
3C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4#11729461C218yesmaximal, direct product factor
5C2×#89333C212yesmaximal, complemented
6C2×#89332C212yesmaximal, complemented
7C22×C7×#41C411yescharacteristic, complemented
8C24×C4×C7C2211yescharacteristic
9C7×#525—22no—
10C7×#525—22no—
11C7×#525C414yescomplemented
12C7×#525C2214yes—
13C2×C7×#123—21no—
14C2×C7×C22⋊Q8C411yescharacteristic, complemented
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×C7×#121C411yescharacteristic, complemented
17C2×C7×#121C2211yescharacteristic
18C2×C7×#120—21no—
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4—21no—
21C2×C7×#117C411yescharacteristic, complemented
22C22×C22⋊Dic7—21no—
23C22×C4×Dic7—21no—
24C23×C7⋊C8—21no—
25C4×C22⋊Dic7—22no—
26C4×C22⋊Dic7C2214yescomplemented
27C2×C7⋊#95C2212yes—
28C2×C22⋊(C7⋊C8)—22no—
29C2×C22⋊(C7⋊C8)—21no—
30C2×C22⋊(C7⋊C8)C2212yes—
31C2×C22⋊(C7⋊C8)C2214yescomplemented
32#89333C2218yescomplemented
33#89332C2218yescomplemented
34C2×#1501—71nomaximal, Sylow
35C25×C7D411yescharacteristic
36C2×C7×#41—22no—
37C2×C7×#41—44no—
38C2×C7×#41C2×C414yescomplemented
39C22×C7×Q8C2×C411yescharacteristic
40C22×C7×D4—21no—
41C22×C7×D4—21no—
42C22×C7×D4—21no—
43C22×C7×D4C2×C411yescharacteristic
44C23×C4×C7—21no—
45C23×C4×C7—44no—
46C23×C4×C7C2×C411yescharacteristic
47C23×C4×C7C2×C411yescharacteristic
48C23×C4×C7C2×C412yes—
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D412yes—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D412yes—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C23×Dic7—22no—
57C7×#123—24no—
58C7×#123—24no—
59C7×C22⋊Q8—22no—
60C7×C22⋊Q8—21no—
61C7×C22⋊Q8—21no—
62C7×C22⋊Q8C2×C412yes—
63C7×C22⋊Q8C2×C414yescomplemented
64C7×C22⋊Q8C2312yes—
65C7×#121—22no—
66C7×#121—21no—
67C7×#121—21no—
68C7×#121C2×C412yes—
69C7×#121C2×C414yescomplemented
70C7×#121C2312yes—
71C7×#120—24no—
72C7×#120—24no—
73C4×C7×D4—24no—
74C4×C7×D4—24no—
75C4×C7×D4—24no—
76C4×C7×D4—24no—
77C7×#117—22no—
78C7×#117C2×C414yescomplemented
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4C2×C411yescharacteristic
82C2×C7×C4⋊C4C2×C411yescharacteristic
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C42×C7—21no—
89C2×C22⋊Dic7—28no—
90C2×C22⋊Dic7—24no—
91C2×C4×Dic7—24no—
92C2×C4×Dic7—44no—
93C22×C7⋊C8—22no—
94C22×C7⋊C8—24no—
95C22×C7⋊C8—44no—
96C7⋊#95—22no—
97C7⋊#95C2314yes—
98C22⋊(C7⋊C8)—22no—
99C22⋊(C7⋊C8)—24no—
100C22⋊(C7⋊C8)C2314yes—
101C2×#525D711yescharacteristic
102C22×C22⋊C8—71no—
103C2×C4×C22⋊C4—71no—
104#1501—78no—
105C2×#328—72no—
106C2×#327—72no—
107C24×C7—24no—
108C24×C7—24no—
109C24×C7—21no—
110C24×C7—42no—
111C24×C7C22⋊C411yescharacteristic
112C24×C7C22⋊C412yes—
113C24×C7C22⋊C411yescharacteristic
114C24×C7C2×D412yes—
115C24×C7C2×D411yescharacteristic
116C7×#41—48no—
117C7×#41—48no—
118C2×C7×Q8—21no—
119C2×C7×Q8—42no—
120C2×C7×Q8C22×C412yes—
121C2×C7×D4—22no—
122C2×C7×D4—24no—
123C2×C7×D4—24no—
124C2×C7×D4—22no—
125C2×C7×D4—21no—
126C2×C7×D4—42no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4C22×C412yes—
131C22×C4×C7—24no—
132C22×C4×C7—24no—
133C22×C4×C7—24no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—24no—
138C22×C4×C7—44no—
139C22×C4×C7—44no—
140C22×C4×C7—44no—
141C22×C4×C7—44no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—44no—
146C22×C4×C7C22⋊C411yescharacteristic
147C22×C4×C7C22⋊C412yes—
148C22×C4×C7C22⋊C412yes—
149C22×C4×C7C22⋊C412yes—
150C22×C4×C7C22⋊C412yes—
151C22×C4×C7C22⋊C412yes—
152C22×C4×C7C22⋊C411yescharacteristic
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C22×C412yes—
155C22×C4×C7C22×C411yescharacteristic
156C22×C4×C7C22×C412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—28no—
165C22×Dic7—24no—
166C22×Dic7—44no—
167C22×Dic7—44no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4C22×C412yes—
174C7×C4⋊C4C22×C412yes—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—22no—
182C42×C7—24no—
183C22⋊Dic7—216no—
184C4×Dic7—48no—
185C2×C7⋊C8—24no—
186C2×C7⋊C8—44no—
187C2×C7⋊C8—48no—
188C22×#41Dic711yescharacteristic, complemented
189C24×C4C2×D711yescharacteristic
190C23×C8—141no—
191#525—22no—
192#525—22no—
193#525Dic714yescomplemented
194#525C2×D714yes—
195C2×#123—21no—
196C2×C22⋊Q8Dic711yescharacteristic, complemented
197C2×C22⋊Q8C2×D711yescharacteristic
198C2×#121Dic711yescharacteristic, complemented
199C2×#121C2×D711yescharacteristic
200C2×#120—21no—
201C2×C4×D4—21no—
202C2×C4×D4—21no—
203C2×#117Dic711yescharacteristic, complemented
204C22×C22⋊C4—141no—
205C22×C42—141no—
206C2×C22⋊C8—72no—
207C2×C22⋊C8—74no—
208C2×C22⋊C8—141no—
209C2×C22⋊C8—142no—
210C4×C22⋊C4—74no—
211C4×C22⋊C4—142no—
212C2×#95—72no—
213#328—78no—
214#327—78no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—24no—
220C23×C7—22no—
221C23×C7—24no—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—416no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7#10414yes—
234C23×C7C2×C22⋊C412yes—
235C23×C7C2×C22⋊C411yescharacteristic
236C23×C7C2×C22⋊C412yes—
237C23×C7C2×C22⋊C411yescharacteristic
238C23×C7#13514yes—
239C23×C7C22×D411yescharacteristic
240C7×Q8—42no—
241C7×Q8—42no—
242C7×D4—42no—
243C7×D4—42no—
244C7×D4—48no—
245C7×D4—44no—
246C7×D4—44no—
247C7×D4—48no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—48no—
272C2×C4×C7#9914yes—
273C2×C4×C7C2×C22⋊C411yescharacteristic
274C2×C4×C7C2×C22⋊C412yes—
275C2×C4×C7C2×C22⋊C411yescharacteristic
276C2×C4×C7C2×C22⋊C412yes—
277C2×C4×C7C23×C411yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×Dic7—28no—
280C2×Dic7—44no—
281C2×Dic7—416no—
282C2×Dic7—44no—
283C7⋊C8—44no—
284C25C22⋊D711yescharacteristic
285C2×#41—22no—
286C2×#41—44no—
287C2×#41C2×Dic714yescomplemented
288C22×Q8C2×Dic711yescharacteristic
289C22×D4—21no—
290C22×D4—21no—
291C22×D4—21no—
292C22×D4C2×Dic711yescharacteristic
293C23×C4—21no—
294C23×C4—44no—
295C23×C4—142no—
296C23×C4C2×Dic711yescharacteristic
297C23×C4C2×Dic711yescharacteristic
298C23×C4C2×Dic712yes—
299C23×C4C22⋊D711yescharacteristic
300C23×C4C22⋊D711yescharacteristic
301C23×C4C22⋊D712yes—
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22⋊D712yes—
304C23×C4C22×D711yescharacteristic
305C23×C4C22×D712yes—
306C22×C8—142no—
307C22×C8—144no—
308C22×C8—284no—
309#123—24no—
310#123—24no—
311C22⋊Q8—22no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8C2×Dic712yes—
315C22⋊Q8C2×Dic714yescomplemented
316C22⋊Q8C22×D712yes—
317#121—22no—
318#121—21no—
319#121—21no—
320#121C2×Dic712yes—
321#121C2×Dic714yescomplemented
322#121C22×D712yes—
323#120—24no—
324#120—24no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329#117—22no—
330#117C2×Dic714yescomplemented
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C2×Dic711yescharacteristic
334C2×C4⋊C4C2×Dic711yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—148no—
341C2×C22⋊C4—144no—
342C2×C42—21no—
343C2×C42—144no—
344C2×C42—284no—
345C22⋊C8—74no—
346C22⋊C8—142no—
347C22⋊C8—144no—
348#95—74no—
349#95—142no—
350C22×C7—21no—
351C22×C7—21no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—21no—
355C22×C7—21no—
356C22×C7—28no—
357C22×C7—28no—
358C22×C7—22no—
359C22×C7—22no—
360C22×C7—22no—
361C22×C7—48no—
362C22×C7—48no—
363C22×C7—44no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—84no—
368C22×C7C2×#9912yes—
369C22×C7#41612yes—
370C22×C7#41612yes—
371C22×C7C2×#10414yes—
372C22×C7C2×#10412yes—
373C22×C7C22×C22⋊C411yescharacteristic
374C22×C7C2×#13512yes—
375C4×C7—21no—
376C4×C7—21no—
377C4×C7—22no—
378C4×C7—21no—
379C4×C7—21no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—42no—
383C4×C7C2×#9914yes—
384Dic7—48no—
385C24—24no—
386C24—24no—
387C24—21no—
388C24—42no—
389C24C22⋊Dic711yescharacteristic
390C24C22⋊Dic712yes—
391C24C22⋊Dic711yescharacteristic
392C24C2×C22⋊D712yes—
393C24C2×C22⋊D711yescharacteristic
394#41—48no—
395#41—48no—
396C2×Q8—21no—
397C2×Q8—42no—
398C2×Q8C22×Dic712yes—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4C22×Dic712yes—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—148no—
425C22×C4—144no—
426C22×C4—284no—
427C22×C4—284no—
428C22×C4C22⋊Dic711yescharacteristic
429C22×C4C22⋊Dic712yes—
430C22×C4C22⋊Dic712yes—
431C22×C4C22⋊Dic712yes—
432C22×C4C22⋊Dic712yes—
433C22×C4C22⋊Dic712yes—
434C22×C4C22⋊Dic711yescharacteristic
435C22×C4C22×Dic711yescharacteristic
436C22×C4C22×Dic712yes—
437C22×C4C22×Dic711yescharacteristic
438C22×C4C22×Dic712yes—
439C22×C4C2×C22⋊D712yes—
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D712yes—
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D712yes—
445C22×C4C23×D711yescharacteristic
446C2×C8—144no—
447C2×C8—284no—
448C2×C8—288no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4C22×Dic712yes—
455C4⋊C4C22×Dic712yes—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—1416no—
464C42—24no—
465C42—288no—
466C2×C7—22no—
467C2×C7—22no—
468C2×C7—24no—
469C2×C7—42no—
470C2×C7—82no—
471C2×C7#150114yescomplemented
472C2×C7C22×#9911yescharacteristic
473C2×C7C2×#41611yescharacteristic
474C2×C7C22×#10411yescharacteristic
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—416no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23#641114yes—
494C23#651314yes—
495C23C2×C22⋊Dic712yes—
496C23C2×C22⋊Dic711yescharacteristic
497C23C2×C22⋊Dic712yes—
498C23C2×C22⋊Dic711yescharacteristic
499C23C22×C22⋊D711yescharacteristic
500Q8—42no—
501Q8—42no—
502D4—42no—
503D4—42no—
504D4—48no—
505D4—44no—
506D4—44no—
507D4—48no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—48no—
532C2×C4—148no—
533C2×C4—284no—
534C2×C4—2816no—
535C2×C4—284no—
536C2×C4#640814yes—
537C2×C4C2×C22⋊Dic711yescharacteristic
538C2×C4C2×C22⋊Dic712yes—
539C2×C4C2×C22⋊Dic711yescharacteristic
540C2×C4C2×C22⋊Dic712yes—
541C2×C4C23×Dic711yescharacteristic
542C2×C4C22×C22⋊D711yescharacteristic
543C8—284no—
544C7C2×#150111yescharacteristic, complemented, Sylow
545C22—21no—
546C22—21no—
547C22—22no—
548C22—28no—
549C22—28no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—48no—
557C22—48no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—84no—
563C22C2×#640812yes—
564C22#9000512yes—
565C22#9000512yes—
566C22C2×#641114yes—
567C22C2×#641112yes—
568C22C2×#651312yes—
569C22C22×C22⋊Dic711yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—288no—
579C4C2×#640814yes—
580C2—22no—
581C2—22no—
582C2—24no—
583C2—42no—
584C2—82no—
585C2#1172946114yesdirect product factor
586C2C22×#640811yescharacteristic
587C2C2×#9000511yescharacteristic
588C2C22×#641111yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#525C2
socleC23×C7C2×C22⋊C4
cosocleC22×C4×C7C24
hypercenterC2×#525D7
hypocenterC7C2×#1501
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊Dic7C2semidirect
3. C22×C22⋊(C7⋊C8)C2semidirect
4. #11729461C2direct (N × H)
5. C2×#89333C2semidirect
6. C2×#89332C2semidirect
7. C22×C7×#41C4semidirect
11. C7×#525C4semidirect
14. C2×C7×C22⋊Q8C4semidirect
16. C2×C7×#121C4semidirect
21. C2×C7×#117C4semidirect
26. C4×C22⋊Dic7C22semidirect
31. C2×C22⋊(C7⋊C8)C22semidirect
32. #89333C22semidirect
33. #89332C22semidirect
38. C2×C7×#41C2×C4semidirect
63. C7×C22⋊Q8C2×C4semidirect
69. C7×#121C2×C4semidirect
78. C7×#117C2×C4semidirect
188. C22×#41Dic7semidirect
193. #525Dic7semidirect
196. C2×C22⋊Q8Dic7semidirect
198. C2×#121Dic7semidirect
203. C2×#117Dic7semidirect
287. C2×#41C2×Dic7semidirect
315. C22⋊Q8C2×Dic7semidirect
321. #121C2×Dic7semidirect
330. #117C2×Dic7semidirect
471. C2×C7#1501semidirect
544. C7C2×#1501semidirect
585. C2#11729461direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5252
C2×C4×C22⋊Dic72
C22×C22⋊(C7⋊C8)2
#117294612
C2×#893332
C2×#893322
C2×#15017