Finite Group Databasebeta

← C2×#11740809 #14226595

Subgroups — C2×#11740809 #14226595

Order 1792.

1930 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
1C2×C7×#533C211yesmaximal, characteristic, complemented
2C2×#90170C211yesmaximal, characteristic
3#11740809C2116yesmaximal, direct product factor
4C2×#89866C211yesmaximal, characteristic, complemented
5C2×#89862C211yesmaximal, characteristic, complemented
6C2×#89859C211yesmaximal, characteristic, complemented
7C2×#89855C211yesmaximal, characteristic, complemented
8C2×#89823C212yesmaximal, complemented
9C2×#89820C212yesmaximal
10C2×#89800C212yesmaximal, complemented
11C2×#89794C212yesmaximal, complemented
12C22×C4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
13C7×#533—24no—
14C7×#533C2218yescomplemented
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#125—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2212yes—
20C2×C4×C7×Q8—21no—
21C2×C4×C7×Q8C2212yes—
22C2×C7×#117—21no—
23C22×C42×C7C2211yescharacteristic
24C22×#1144C2212yes—
25C22×#1143C2212yes—
26C2×#6493C2211yescharacteristic
27C2×#6492—21no—
28C2×C4×C22⋊D7—21no—
29C2×#6487C2211yescharacteristic
30C23×C7⋊C8C2212yes—
31#90170—24no—
32#90170C2218yes—
33C2×#6439—21no—
34C2×#6438C2211yescharacteristic
35C2×C4×D28C2211yescharacteristic
36C2×#6435—21no—
37C2×#6433C2211yescharacteristic
38C2×C4×Dic14C2211yescharacteristic
39C2×#6409C2212yes—
40C2×#6409C2212yes—
41#89866C2218yescomplemented
42#89862C2218yescomplemented
43#89859C2218yescomplemented
44#89855C2218yescomplemented
45#89823C22116yescomplemented
46#89820C22116yes—
47#89800C22116yescomplemented
48#89794C22116yescomplemented
49C2×#6384C2212yes—
50C2×#6383C2212yes—
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53C2×C4⋊(C7⋊C8)C2212yes—
54C2×C4⋊(C7⋊C8)C2211yescharacteristic
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56C2×C4⋊(C7⋊C8)C2218yescomplemented
57C2×#3248—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×C7×Q8C2312yes—
60C23×C4×C7D411yescharacteristic
61C23×C4×C7D411yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C22×C22⋊D7—21no—
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×Dic14C2311yescharacteristic
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2314yes—
70C7×C4⋊Q8C2314yes—
71C7×#125—28no—
72C7×C22⋊Q8—28no—
73C7×C22⋊Q8—24no—
74C7×C22⋊Q8C2318yes—
75C4×C7×Q8—28no—
76C4×C7×Q8—24no—
77C4×C7×Q8C2318yes—
78C7×#117—28no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C22⋊C4—22no—
86C2×C42×C7—22no—
87C2×C42×C7D411yescharacteristic
88C2×C42×C7D411yescharacteristic
89C2×C42×C7D414yes—
90C2×C42×C7C2311yescharacteristic
91C2×C42×C7C2311yescharacteristic
92C2×C42×C7C2314yes—
93C2×C22⋊Dic7—21no—
94C2×#1144—44no—
95C2×#1144C2318yes—
96C2×#1143—44no—
97C2×#1143C2318yes—
98#6493—22no—
99#6493C2314yes—
100#6492—28no—
101C4×C22⋊D7—28no—
102C2×(C2×D7)⋊C4—21no—
103C2×(C2×D7)⋊C4—21no—
104C2×(C2×D7)⋊C4—21no—
105C2×(C2×D7)⋊C4—21no—
106C2×C4⋊Dic7C2311yescharacteristic
107C2×C4⋊Dic7C2311yescharacteristic
108#6487—22no—
109#6487C2314yes—
110C2×Dic7⋊C4—21no—
111C2×Dic7⋊C4—21no—
112C2×Dic7⋊C4—21no—
113C2×Dic7⋊C4—21no—
114C2×C4×Dic7—21no—
115C22×C7⋊C8—44no—
116C22×C7⋊C8C2312yes—
117C22×C7⋊C8C2312yes—
118C22×C7⋊C8C2318yes—
119#6439—28no—
120#6438—22no—
121#6438C2314yes—
122C4×D28—22no—
123C4×D28C2314yes—
124#6435—28no—
125#6433—22no—
126#6433C2314yes—
127C4×Dic14—22no—
128C4×Dic14C2314yes—
129#6409C2318yes—
130#6409C2318yes—
131#6384C2318yes—
132#6383C2318yes—
133#6382C2318yes—
134#6381C2318yes—
135C4⋊(C7⋊C8)C2314yes—
136C4⋊(C7⋊C8)C2314yes—
137C4⋊(C7⋊C8)C2318yes—
138C2×#533D711yescharacteristic, complemented
139C2×#529—71no—
140#3248—716no—
141C2×#479—72no—
142C2×#475—71no—
143C2×#474—71no—
144C2×#467—72no—
145C2×#465—72no—
146C2×#464—72no—
147C2×#462—71no—
148C2×#461—71no—
149C22×C4⋊C8—71no—
150C24×C7C2×D411yescharacteristic
151C23×D7—21no—
152C2×C7×Q8—22no—
153C2×C7×Q8—24no—
154C2×C7×Q8—24no—
155C2×C7×Q8—24no—
156C2×C7×Q8—44no—
157C2×C7×Q8C2414yes—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D414yes—
170C22×C4×C7C2×D414yes—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2412yes—
173C22×C4×C7C2411yescharacteristic
174C2×C22⋊D7—24no—
175C2×C22⋊D7—44no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C2×D28—22no—
180C2×D28—22no—
181C2×D28—21no—
182C2×D28C2412yes—
183C2×C4×D7—24no—
184C2×C4×D7—44no—
185C2×Dic14—22no—
186C2×Dic14—22no—
187C2×Dic14—21no—
188C2×Dic14C2412yes—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—28no—
193C7×C4⋊C4—28no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4C2414yes—
196C7×C22⋊C4—28no—
197C42×C7—28no—
198C42×C7C2×D412yes—
199C42×C7C2×D412yes—
200C42×C7C2×D414yes—
201C42×C7C2×D414yes—
202C42×C7C2412yes—
203C42×C7C2412yes—
204C22⋊Dic7—24no—
205#1144—48no—
206#1143—48no—
207(C2×D7)⋊C4—24no—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—24no—
210(C2×D7)⋊C4—24no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7C2412yes—
214C4⋊Dic7C2412yes—
215Dic7⋊C4—24no—
216Dic7⋊C4—24no—
217Dic7⋊C4—24no—
218Dic7⋊C4—24no—
219C4×Dic7—24no—
220C2×C7⋊C8—24no—
221C2×C7⋊C8—24no—
222C2×C7⋊C8—48no—
223C2×C7⋊C8C2414yes—
224C2×C7⋊C8C2414yes—
225C22×Q16—72no—
226C22×SD16—72no—
227C23×C8—72no—
228#533—24no—
229#533C2×D718yescomplemented
230C2×C4⋊Q8C2×D711yescharacteristic
231C2×C4⋊Q8C2×D711yescharacteristic
232#529—78no—
233#529—144no—
234C2×#126—141no—
235C2×#125—21no—
236C2×#125—71no—
237C2×#124—71no—
238C2×#123—141no—
239C2×C22⋊Q8—21no—
240C2×C22⋊Q8—71no—
241C2×C22⋊Q8C2×D712yes—
242C2×#121—71no—
243C2×C4×Q8—21no—
244C2×C4×Q8—71no—
245C2×C4×Q8C2×D712yes—
246C2×C4×D4—71no—
247C2×C4×D4—141no—
248C2×#117—21no—
249C2×#117—141no—
250C22×C42C2×D711yescharacteristic
251#479—716no—
252#475—78no—
253#474—78no—
254#467—716no—
255#465—716no—
256#464—716no—
257#462—78no—
258#461—78no—
259C2×#107—72no—
260C2×#106—72no—
261C2×C4⋊C8—71no—
262C2×C4⋊C8—72no—
263C2×C4⋊C8—71no—
264C2×C4⋊C8—78no—
265C2×#103—72no—
266C2×#103—72no—
267C2×#103—72no—
268C2×#102—72no—
269C23×C7—22no—
270C23×C7—22no—
271C23×C7#13714yes—
272C23×C7C22×D412yes—
273C23×C7C22×D411yescharacteristic
274C22×D7—22no—
275C22×D7—42no—
276C22×D7—42no—
277C22×D7—42no—
278C7×Q8—24no—
279C7×Q8—24no—
280C7×Q8—44no—
281C7×Q8—48no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—28no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—24no—
288C2×C4×C7—28no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—44no—
294C2×C4×C7—44no—
295C2×C4×C7#13514yes—
296C2×C4×C7C22×D412yes—
297C2×C4×C7C22×D412yes—
298C2×C4×C7C22×D412yes—
299C2×C4×C7C22×D412yes—
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D412yes—
302C2×C4×C7H(5,𝔽2)12yes—
303C2×C4×C72−1+412yes—
304C2×C4×C7C2511yescharacteristic
305C22⋊D7—48no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312D28—22no—
313D28—22no—
314D28—42no—
315C4×D7—48no—
316Dic14—22no—
317Dic14—22no—
318Dic14—42no—
319C7⋊C8—24no—
320C7⋊C8—24no—
321C22×Q8—21no—
322C22×Q8—71no—
323C22×Q8C22×D712yes—
324C22×D4—71no—
325C22×D4—141no—
326C23×C4—141no—
327C23×C4C22⋊D711yescharacteristic
328C23×C4C22⋊D711yescharacteristic
329C23×C4C22×D711yescharacteristic
330C2×Q16—78no—
331C2×Q16—284no—
332C2×SD16—78no—
333C2×SD16—284no—
334C22×C8—72no—
335C22×C8—72no—
336C22×C8—78no—
337C22×C8—284no—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340C4⋊Q8C22×D714yes—
341C4⋊Q8C22×D714yes—
342#126—148no—
343#125—28no—
344#125—74no—
345#125—142no—
346#124—74no—
347#124—142no—
348#123—148no—
349C22⋊Q8—28no—
350C22⋊Q8—24no—
351C22⋊Q8—74no—
352C22⋊Q8—142no—
353C22⋊Q8C22×D718yes—
354#121—74no—
355#121—142no—
356C4×Q8—28no—
357C4×Q8—24no—
358C4×Q8—74no—
359C4×Q8—142no—
360C4×Q8C22×D718yes—
361C4×D4—74no—
362C4×D4—148no—
363C4×D4—142no—
364#117—28no—
365#117—148no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D712yes—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—141no—
380C2×C22⋊C4—141no—
381C2×C22⋊C4—141no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C42—22no—
385C2×C42—141no—
386C2×C42C22⋊D711yescharacteristic
387C2×C42C22⋊D711yescharacteristic
388C2×C42C22⋊D714yes—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391C2×C42C22×D714yes—
392#107—78no—
393#106—78no—
394C4⋊C8—74no—
395C4⋊C8—74no—
396C4⋊C8—78no—
397#103—78no—
398#103—78no—
399#103—78no—
400#102—78no—
401C22×C7—22no—
402C22×C7—28no—
403C22×C7—22no—
404C22×C7#53712yes—
405C22×C7C2×#13512yes—
406C22×C7C2×#13714yes—
407C22×C7C2×#13712yes—
408C22×C7C23×D411yescharacteristic
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—42no—
412C2×D7—84no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—44no—
418C4×C7—44no—
419C4×C7#53712yes—
420C4×C7C2×#13514yes—
421C4×C7C2×H(5,𝔽2)11yescharacteristic
422C4×C7C2×2−1+411yescharacteristic
423Dic7—42no—
424Dic7—42no—
425Dic7—42no—
426C24—141no—
427C24C2×C22⋊D711yescharacteristic
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—44no—
433C2×Q8—72no—
434C2×Q8—141no—
435C2×Q8—142no—
436C2×Q8—142no—
437C2×Q8C23×D714yes—
438C2×D4—72no—
439C2×D4—144no—
440C2×D4—142no—
441C2×D4—142no—
442C2×D4—141no—
443C2×D4—284no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—144no—
454C22×C4—284no—
455C22×C4C2×C22⋊D712yes—
456C22×C4C2×C22⋊D714yes—
457C22×C4C2×C22⋊D714yes—
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C23×D712yes—
464C22×C4C23×D711yescharacteristic
465Q16—288no—
466SD16—288no—
467C2×C8—74no—
468C2×C8—74no—
469C2×C8—144no—
470C2×C8—144no—
471C2×C8—288no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—72no—
479C4⋊C4—72no—
480C4⋊C4—144no—
481C4⋊C4—144no—
482C4⋊C4—144no—
483C4⋊C4—144no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4C23×D714yes—
487C22⋊C4—28no—
488C22⋊C4—144no—
489C22⋊C4—144no—
490C22⋊C4—144no—
491C22⋊C4—144no—
492C22⋊C4—144no—
493C42—28no—
494C42—144no—
495C42C2×C22⋊D714yes—
496C42C2×C22⋊D712yes—
497C42C2×C22⋊D712yes—
498C42C2×C22⋊D714yes—
499C42C23×D712yes—
500C42C23×D712yes—
501C2×C7—24no—
502C2×C7#324814yescomplemented
503C2×C7C2×#53711yescharacteristic
504C2×C7C22×#13511yescharacteristic
505C2×C7C22×#13711yescharacteristic
506D7—82no—
507C23—22no—
508C23—22no—
509C23—142no—
510C23—282no—
511C23—282no—
512C23—282no—
513C23#650514yes—
514C23C22×C22⋊D712yes—
515C23C22×C22⋊D711yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—44no—
520Q8—142no—
521Q8—142no—
522Q8—282no—
523D4—142no—
524D4—142no—
525D4—288no—
526D4—282no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—288no—
547C2×C4#651314yes—
548C2×C4#655312yes—
549C2×C4#655412yes—
550C2×C4C22×C22⋊D712yes—
551C2×C4C22×C22⋊D712yes—
552C2×C4C22×C22⋊D712yes—
553C2×C4C22×C22⋊D712yes—
554C2×C4C22×C22⋊D712yes—
555C2×C4C22×C22⋊D711yescharacteristic
556C2×C4C24×D711yescharacteristic
557C8—144no—
558C8—144no—
559C7C2×#324811yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—28no—
563C22—282no—
564C22—282no—
565C22—282no—
566C22—564no—
567C22C2×#650514yes—
568C22C2×#650512yes—
569C22C2×#651312yes—
570C22#9052412yes—
571C22C23×C22⋊D711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—44no—
577C4—44no—
578C4—282no—
579C4—282no—
580C4—282no—
581C4C2×#651314yes—
582C4#9052412yes—
583C4C2×#655311yescharacteristic
584C4C2×#655411yescharacteristic
585C2—24no—
586C2—562no—
587C2#1174080914yesdirect product factor
588C2C22×#650511yescharacteristic
589C2C22×#651311yescharacteristic
590C2C2×#9052411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#533C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC2×#533D7
hypocenterC7C2×#3248
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C7×#533C2semidirect
3. #11740809C2direct (N × H)
4. C2×#89866C2semidirect
5. C2×#89862C2semidirect
6. C2×#89859C2semidirect
7. C2×#89855C2semidirect
8. C2×#89823C2semidirect
10. C2×#89800C2semidirect
11. C2×#89794C2semidirect
12. C22×C4⋊(C7⋊C8)C2semidirect
14. C7×#533C22semidirect
41. #89866C22semidirect
42. #89862C22semidirect
43. #89859C22semidirect
44. #89855C22semidirect
45. #89823C22semidirect
47. #89800C22semidirect
48. #89794C22semidirect
56. C2×C4⋊(C7⋊C8)C22semidirect
138. C2×#533D7semidirect
229. #533C2×D7semidirect
502. C2×C7#3248semidirect
559. C7C2×#3248semidirect
587. C2#11740809direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×#901702
#117408092
C2×#898662
C2×#898622
C2×#898592
C2×#898552
C2×#898232
C2×#898202
C2×#898002
C2×#897942
C22×C4⋊(C7⋊C8)2
C2×#32487