Finite Group Databasebeta

← C2×#11737480 #14225068

Subgroups — C2×#11737480 #14225068

Order 1792.

2146 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6507C211yesmaximal, characteristic, complemented
2C2×#90291C211yesmaximal, characteristic
3C2×C7×#448C211yesmaximal, characteristic
4C2×#90000C211yesmaximal, characteristic, complemented
5C2×#89939C212yesmaximal, complemented
6C2×#89934C212yesmaximal, complemented
7C2×#89812C211yesmaximal, characteristic
8#11737480C2116yesmaximal, direct product factor
9C2×#89565C212yesmaximal, complemented
10C2×#89569C212yesmaximal, complemented
11C2×#89511C211yesmaximal, characteristic, complemented
12C2×#89491C211yesmaximal, characteristic, complemented
13C23×C7×Q8C2211yescharacteristic
14C23×Dic14C2211yescharacteristic
15C22×C7×SD16C2212yes—
16C2×C7×#121C2211yescharacteristic
17C2×#6507—21no—
18C2×#6507—21no—
19C2×#6507—24no—
20C2×#6507C2212yes—
21C2×#6507C2212yes—
22C2×#6507C2218yescomplemented
23C22×#1144C2212yes—
24C2×#6499C2211yescharacteristic
25C2×#6498—21no—
26C2×D4×Dic7—21no—
27C2×D4×Dic7C2211yescharacteristic
28C22×#1142C2212yes—
29C2×#6487C2211yescharacteristic
30C22×Dic7⋊C4—21no—
31C22×Dic7⋊C4—21no—
32C22×C4×Dic7C2211yescharacteristic
33C22×Dic28C2212yes—
34#90291—24no—
35#90291C2218yes—
36C2×#6453C2211yescharacteristic
37C2×#6450C2211yescharacteristic
38C2×#6442—21no—
39C2×#6440—21no—
40C7×#448C2218yes—
41C2×C7×#103C2212yes—
42C2×C7×C22⋊C8C2211yescharacteristic
43#90000C2218yescomplemented
44C2×#6409C2212yes—
45C2×C22⋊(C7⋊C8)C2211yescharacteristic
46#89939C22116yescomplemented
47#89934C22116yescomplemented
48C2×#6390C2212yes—
49C2×#6389C2212yes—
50C2×Dic7⋊C8C2212yes—
51#89812C2218yes—
52C2×#6384C2212yes—
53#89569C22116yescomplemented
54#89565C22116yescomplemented
55#89511C2218yescomplemented
56#89491C2218yescomplemented
57C2×#3218—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×C7×Q8—22no—
60C22×C7×Q8—44no—
61C22×C7×Q8C2312yes—
62C22×C7×Q8C2314yes—
63C22×C7×D4—21no—
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C23×Dic7—21no—
68C23×Dic7—21no—
69C23×Dic7D411yescharacteristic
70C23×Dic7D411yescharacteristic
71C22×Dic14—21no—
72C22×Dic14—22no—
73C22×Dic14—44no—
74C22×Dic14C2312yes—
75C22×Dic14C2311yescharacteristic
76C22×Dic14C2314yes—
77C2×C7×SD16—44no—
78C2×C7×SD16C2318yes—
79C22×C7×C8C2312yes—
80C7×#121—22no—
81C7×#121C2314yes—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C22⋊C4—21no—
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88#6507—24no—
89#6507—24no—
90#6507—28no—
91#6507—28no—
92#6507C2318yes—
93#6507C2318yes—
94C2×#1144—44no—
95C2×#1144C2318yes—
96#6499—22no—
97#6499C2314yes—
98#6498—28no—
99D4×Dic7—28no—
100D4×Dic7—22no—
101D4×Dic7C2314yes—
102C2×#1142—44no—
103C2×#1142C2318yes—
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7C2311yescharacteristic
106#6487—22no—
107#6487C2314yes—
108C2×Dic7⋊C4—21no—
109C2×Dic7⋊C4—21no—
110C2×Dic7⋊C4—21no—
111C2×Dic7⋊C4—22no—
112C2×Dic7⋊C4—22no—
113C2×Dic7⋊C4—22no—
114C2×Dic7⋊C4—22no—
115C2×Dic7⋊C4—28no—
116C2×Dic7⋊C4—28no—
117C2×C4×Dic7—21no—
118C2×C4×Dic7D411yescharacteristic
119C2×C4×Dic7D411yescharacteristic
120C2×C4×Dic7D414yes—
121C2×C4×Dic7C2311yescharacteristic
122C2×C4×Dic7C2311yescharacteristic
123C2×C4×Dic7C2314yes—
124C22×C7⋊C8C2312yes—
125C2×Dic28—44no—
126C2×Dic28C2318yes—
127#6453—22no—
128#6453C2314yes—
129#6450—22no—
130#6450C2314yes—
131#6442—28no—
132#6440—28no—
133C7×#103C2318yes—
134C7×C22⋊C8C2314yes—
135#6409C2318yes—
136C22⋊(C7⋊C8)C2314yes—
137#6390C2318yes—
138#6389C2318yes—
139Dic7⋊C8C2318yes—
140#6384C2318yes—
141C22×C4⋊Q8—71no—
142C2×#529—71no—
143#3218—716no—
144C2×#497—72no—
145C2×#491—72no—
146C2×#462—72no—
147C2×#461—72no—
148C2×#451—71no—
149C2×#451—71no—
150C2×#448—71no—
151C2×#448D711yescharacteristic
152C2×#432—71no—
153C24×C7—21no—
154C24×C7C2×D411yescharacteristic
155C2×C7×Q8—24no—
156C2×C7×Q8—22no—
157C2×C7×Q8—24no—
158C2×C7×Q8—24no—
159C2×C7×Q8—48no—
160C2×C7×Q8—44no—
161C2×C7×Q8—48no—
162C2×C7×Q8C2414yes—
163C2×C7×D4—24no—
164C2×C7×D4—21no—
165C2×C7×D4—24no—
166C2×C7×D4—44no—
167C2×C7×D4C2412yes—
168C22×C4×C7—24no—
169C22×C4×C7—21no—
170C22×C4×C7—24no—
171C22×C4×C7—22no—
172C22×C4×C7—44no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2412yes—
176C22×C4×C7C2411yescharacteristic
177C22×Dic7—24no—
178C22×Dic7—24no—
179C22×Dic7—21no—
180C22×Dic7—24no—
181C22×Dic7—22no—
182C22×Dic7—21no—
183C22×Dic7—24no—
184C22×Dic7—44no—
185C22×Dic7—44no—
186C22×Dic7C2×D411yescharacteristic
187C22×Dic7C2×D411yescharacteristic
188C22×Dic7C2×D411yescharacteristic
189C22×Dic7C2×D414yes—
190C22×Dic7C2×D414yes—
191C22×Dic7C2×D411yescharacteristic
192C2×Dic14—24no—
193C2×Dic14—24no—
194C2×Dic14—24no—
195C2×Dic14—21no—
196C2×Dic14—24no—
197C2×Dic14—22no—
198C2×Dic14—48no—
199C2×Dic14—44no—
200C2×Dic14—48no—
201C2×Dic14C2412yes—
202C2×Dic14C2414yes—
203C7×SD16—48no—
204C2×C7×C8—24no—
205C2×C7×C8C2414yes—
206C7×C4⋊C4—21no—
207C7×C4⋊C4C2412yes—
208C7×C22⋊C4—24no—
209C22⋊Dic7—24no—
210C22⋊Dic7—24no—
211C22⋊Dic7—24no—
212C22⋊Dic7—24no—
213#1144—48no—
214#1142—48no—
215C4⋊Dic7—21no—
216C4⋊Dic7—24no—
217C4⋊Dic7C2412yes—
218Dic7⋊C4—24no—
219Dic7⋊C4—24no—
220Dic7⋊C4—24no—
221Dic7⋊C4—28no—
222Dic7⋊C4—28no—
223Dic7⋊C4—28no—
224Dic7⋊C4—28no—
225C4×Dic7—24no—
226C4×Dic7C2×D414yes—
227C4×Dic7C2×D414yes—
228C4×Dic7C2×D412yes—
229C4×Dic7C2×D412yes—
230C4×Dic7C2412yes—
231C4×Dic7C2412yes—
232C2×C7⋊C8—24no—
233C2×C7⋊C8C2414yes—
234Dic28—48no—
235C23×Q8—71no—
236C23×Q8C2×D711yescharacteristic
237C22×Q16—72no—
238C22×Q16—72no—
239C22×SD16—72no—
240C22×SD16C2×D712yes—
241C2×C4⋊Q8—72no—
242C2×C4⋊Q8—72no—
243C2×C4⋊Q8—78no—
244C2×C4⋊Q8—141no—
245C2×C4⋊Q8—141no—
246C2×C4⋊Q8—144no—
247#529—78no—
248#529—144no—
249C2×#126—141no—
250C2×#125—71no—
251C2×#124—71no—
252C2×#123—141no—
253C2×C22⋊Q8—71no—
254C2×#121C2×D711yescharacteristic
255C2×C4×Q8—71no—
256C2×C4×D4—71no—
257C2×C4×D4—141no—
258C2×#117—141no—
259C22×C4⋊C4—141no—
260C22×C4⋊C4—141no—
261C22×C42—71no—
262#497—716no—
263#491—716no—
264#462—716no—
265#461—716no—
266#451—78no—
267#451—78no—
268#448—78no—
269#448C2×D718yes—
270#432—78no—
271C2×C4⋊C8—72no—
272C2×#103—72no—
273C2×#103—72no—
274C2×#103—72no—
275C2×#103C2×D712yes—
276C2×C22⋊C8—71no—
277C2×C22⋊C8C2×D711yescharacteristic
278C2×#97—72no—
279C23×C7—22no—
280C23×C7—22no—
281C23×C7—22no—
282C23×C7—42no—
283C23×C7—42no—
284C23×C7—42no—
285C23×C7#13714yes—
286C23×C7C22×D412yes—
287C23×C7C22×D411yescharacteristic
288C7×Q8—24no—
289C7×Q8—24no—
290C7×Q8—48no—
291C7×Q8—44no—
292C7×D4—24no—
293C7×D4—48no—
294C7×D4—42no—
295C2×C4×C7—22no—
296C2×C4×C7—24no—
297C2×C4×C7—24no—
298C2×C4×C7—22no—
299C2×C4×C7—24no—
300C2×C4×C7—48no—
301C2×C4×C7—44no—
302C2×C4×C7—42no—
303C2×C4×C7C22×D412yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7H(5,𝔽2)14yes—
306C2×C4×C7C2511yescharacteristic
307C2×Dic7—28no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—28no—
312C2×Dic7—22no—
313C2×Dic7—24no—
314C2×Dic7—22no—
315C2×Dic7—48no—
316C2×Dic7—44no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—48no—
320C2×Dic7#13714yes—
321C2×Dic7C22×D412yes—
322C2×Dic7C22×D412yes—
323C2×Dic7C22×D412yes—
324C2×Dic7C22×D412yes—
325Dic14—24no—
326Dic14—24no—
327Dic14—24no—
328Dic14—48no—
329Dic14—42no—
330Dic14—44no—
331C7×C8—24no—
332C7⋊C8—24no—
333C22×Q8—22no—
334C22×Q8—21no—
335C22×Q8—44no—
336C22×Q8—71no—
337C22×Q8—72no—
338C22×Q8—74no—
339C22×Q8—142no—
340C22×Q8—141no—
341C22×Q8—284no—
342C22×Q8C22×D712yes—
343C22×Q8C22×D714yes—
344C22×D4—21no—
345C22×D4C22×D711yescharacteristic
346C23×C4—21no—
347C23×C4—71no—
348C23×C4—71no—
349C23×C4—141no—
350C23×C4—141no—
351C23×C4C22×D711yescharacteristic
352C2×Q16—78no—
353C2×Q16—78no—
354C2×Q16—284no—
355C2×Q16—284no—
356C2×SD16—44no—
357C2×SD16—78no—
358C2×SD16—284no—
359C2×SD16C22×D718yes—
360C22×C8—72no—
361C22×C8C22×D712yes—
362C4⋊Q8—78no—
363C4⋊Q8—78no—
364C4⋊Q8—148no—
365C4⋊Q8—144no—
366C4⋊Q8—144no—
367C4⋊Q8—148no—
368#126—148no—
369#125—74no—
370#125—142no—
371#124—74no—
372#124—142no—
373#123—148no—
374C22⋊Q8—74no—
375C22⋊Q8—142no—
376#121—22no—
377#121C22×D714yes—
378C4×Q8—74no—
379C4×Q8—142no—
380C4×D4—74no—
381C4×D4—142no—
382C4×D4—148no—
383#117—148no—
384C2×C4⋊C4—71no—
385C2×C4⋊C4—148no—
386C2×C4⋊C4—148no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4—142no—
389C2×C4⋊C4—142no—
390C2×C4⋊C4—142no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—141no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—74no—
404C2×C42—71no—
405C2×C42—71no—
406C2×C42—74no—
407C2×C42—141no—
408C4⋊C8—78no—
409#103—78no—
410#103—78no—
411#103—78no—
412#103C22×D718yes—
413C22⋊C8—74no—
414C22⋊C8C22×D714yes—
415#97—78no—
416C22×C7—22no—
417C22×C7—28no—
418C22×C7—22no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—42no—
422C22×C7—84no—
423C22×C7#53512yes—
424C22×C7C2×#13714yes—
425C22×C7C2×#13712yes—
426C22×C7C2×#13712yes—
427C22×C7C23×D411yescharacteristic
428C4×C7—22no—
429C4×C7—44no—
430C4×C7—42no—
431C4×C7#53512yes—
432C4×C7C2×H(5,𝔽2)12yes—
433Dic7—22no—
434Dic7—22no—
435Dic7—22no—
436Dic7—44no—
437Dic7—42no—
438Dic7—42no—
439Dic7C2×#13714yes—
440C24—21no—
441C24D4×D711yescharacteristic
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—48no—
447C2×Q8—44no—
448C2×Q8—48no—
449C2×Q8—74no—
450C2×Q8—72no—
451C2×Q8—141no—
452C2×Q8—144no—
453C2×Q8—142no—
454C2×Q8—144no—
455C2×Q8—144no—
456C2×Q8—144no—
457C2×Q8—288no—
458C2×Q8—284no—
459C2×Q8—288no—
460C2×Q8C23×D714yes—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—24no—
464C2×D4—44no—
465C2×D4C23×D712yes—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—71no—
472C22×C4—74no—
473C22×C4—74no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—144no—
478C22×C4—144no—
479C22×C4—144no—
480C22×C4—144no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—142no—
484C22×C4—284no—
485C22×C4—284no—
486C22×C4D4×D712yes—
487C22×C4D4×D711yescharacteristic
488C22×C4C23×D712yes—
489C22×C4C23×D711yescharacteristic
490Q16—288no—
491Q16—288no—
492SD16—48no—
493SD16—288no—
494C2×C8—24no—
495C2×C8—74no—
496C2×C8—144no—
497C2×C8C23×D714yes—
498C4⋊C4—21no—
499C4⋊C4—72no—
500C4⋊C4—148no—
501C4⋊C4—148no—
502C4⋊C4—148no—
503C4⋊C4—148no—
504C4⋊C4—144no—
505C4⋊C4—144no—
506C4⋊C4—144no—
507C4⋊C4—144no—
508C4⋊C4—141no—
509C4⋊C4C23×D712yes—
510C22⋊C4—24no—
511C22⋊C4—144no—
512C22⋊C4—144no—
513C22⋊C4—144no—
514C22⋊C4—144no—
515C42—74no—
516C42—72no—
517C42—72no—
518C42—74no—
519C42—72no—
520C42—72no—
521C42—144no—
522C2×C7—24no—
523C2×C7—82no—
524C2×C7#321814yescomplemented
525C2×C7C2×#53511yescharacteristic
526C2×C7C22×#13711yescharacteristic
527C2×C7C22×#13711yescharacteristic
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#647814yes—
535C23C2×D4×D712yes—
536C23C2×D4×D711yescharacteristic
537Q8—24no—
538Q8—24no—
539Q8—48no—
540Q8—44no—
541Q8—144no—
542Q8—144no—
543Q8—144no—
544Q8—288no—
545Q8—282no—
546Q8—284no—
547D4—24no—
548D4—48no—
549D4—42no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—74no—
559C2×C4—72no—
560C2×C4—72no—
561C2×C4—72no—
562C2×C4—72no—
563C2×C4—148no—
564C2×C4—148no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—144no—
571C2×C4—288no—
572C2×C4—288no—
573C2×C4—282no—
574C2×C4—284no—
575C2×C4—282no—
576C2×C4C2×D4×D712yes—
577C2×C4C2×D4×D711yescharacteristic
578C2×C4#654814yes—
579C2×C4C24×D711yescharacteristic
580C8—24no—
581C8—144no—
582C7C2×#321811yescharacteristic, complemented, Sylow
583C22—22no—
584C22—22no—
585C22—28no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—84no—
590C22#9028312yes—
591C22C2×#647814yes—
592C22C2×#647812yes—
593C22D7×#13712yes—
594C22C22×D4×D711yescharacteristic
595C4—22no—
596C4—44no—
597C4—42no—
598C4—74no—
599C4—142no—
600C4—142no—
601C4—142no—
602C4—284no—
603C4—282no—
604C4—282no—
605C4#9028312yes—
606C4C2×#654812yes—
607C2—24no—
608C2—82no—
609C2#1173748014yesdirect product factor
610C2C2×#9028311yescharacteristic
611C2C22×#647811yescharacteristic
612C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#448C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC2×#448D7
hypocenterC7C2×#3218
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×#6507C2semidirect
4. C2×#90000C2semidirect
5. C2×#89939C2semidirect
6. C2×#89934C2semidirect
8. #11737480C2direct (N × H)
9. C2×#89565C2semidirect
10. C2×#89569C2semidirect
11. C2×#89511C2semidirect
12. C2×#89491C2semidirect
22. C2×#6507C22semidirect
43. #90000C22semidirect
46. #89939C22semidirect
47. #89934C22semidirect
53. #89569C22semidirect
54. #89565C22semidirect
55. #89511C22semidirect
56. #89491C22semidirect
524. C2×C7#3218semidirect
582. C7C2×#3218semidirect
609. C2#11737480direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#65072
C2×#902912
C2×C7×#4482
C2×#900002
C2×#899392
C2×#899342
C2×#898122
#117374802
C2×#895652
C2×#895692
C2×#895112
C2×#894912
C2×#32187