Finite Group Databasebeta

← C2×#11737856 #14221931

Subgroups — C2×#11737856 #14221931

Order 1792.

2150 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic, complemented
2C2×C4×#1159C211yesmaximal, characteristic
3C2×C7×#461C211yesmaximal, characteristic, complemented
4C2×#89945C212yesmaximal
5C2×#89936C212yesmaximal, complemented
6C2×#89862C211yesmaximal, characteristic, complemented
7C2×#89780C211yesmaximal, characteristic, complemented
8#11737856C2116yesmaximal, direct product factor
9C2×#89622C211yesmaximal, characteristic, complemented
10C2×#89607C211yesmaximal, characteristic, complemented
11C2×#89586C212yesmaximal, complemented
12C2×#89561C212yesmaximal, complemented
13C22×#1159C2211yescharacteristic
14C22×C7×SD16C2212yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C4×C7×D4C2211yescharacteristic
17C2×#6509C2212yes—
18C2×Q8×Dic7C2212yes—
19C22×#1144C2212yes—
20C2×D4×Dic7C2211yescharacteristic
21C2×C4×C22⋊D7—21no—
22C22×C4×Dic7—21no—
23#90419—24no—
24#90419C2218yescomplemented
25C4×#1159—24no—
26C4×#1159C2218yes—
27C2×#6460—21no—
28C2×#6455—21no—
29C2×#6455C2211yescharacteristic
30C2×#6453—21no—
31C2×#6450—21no—
32C2×#6450C2211yescharacteristic
33C2×(C22⋊D7)⋊C4—21no—
34C2×#6440—21no—
35C2×C42×D7C2211yescharacteristic
36C2×#6432C2211yescharacteristic
37C2×C4×Dic14C2211yescharacteristic
38C7×#461C2218yescomplemented
39C2×C7×C4⋊C8C2211yescharacteristic
40C2×C7×#103C2212yes—
41C2×#6409C2212yes—
42C2×#6406C2212yes—
43#89945C22116yes—
44#89936C22116yescomplemented
45C2×(C2×D7)⋊C8C2212yes—
46C2×#6392C2212yes—
47C2×#6390C2212yes—
48C2×#6389C2212yes—
49#89862C2218yescomplemented
50#89780C2218yescomplemented
51C2×#6382C2212yes—
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89622C2218yescomplemented
54#89607C2218yescomplemented
55#89586C22116yescomplemented
56#89561C22116yescomplemented
57C2×#3116—71nomaximal, Sylow
58C22×C7×Q8C2312yes—
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C22×C22⋊D7—21no—
62C23×Dic7—21no—
63C23×Dic7—21no—
64C2×#1159—24no—
65C2×#1159—22no—
66C2×#1159—24no—
67C2×#1159C2314yes—
68C22×C4×D7D411yescharacteristic
69C22×C4×D7D411yescharacteristic
70C22×C4×D7C2311yescharacteristic
71C22×Dic14—21no—
72C22×Dic14C2311yescharacteristic
73C2×C7×SD16—28no—
74C2×C7×SD16C2318yes—
75C22×C7×C8C2312yes—
76C7×C4⋊Q8—22no—
77C7×C4⋊Q8C2314yes—
78C4×C7×D4—22no—
79C4×C7×D4C2314yes—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C22⋊C4—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86#6509—24no—
87#6509C2318yes—
88Q8×Dic7—24no—
89Q8×Dic7C2318yes—
90C2×#1144—28no—
91C2×#1144C2318yes—
92D4×Dic7—22no—
93D4×Dic7C2314yes—
94C4×C22⋊D7—28no—
95C2×(C2×D7)⋊C4—21no—
96C2×(C2×D7)⋊C4—22no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—22no—
99C2×C4⋊Dic7C2312yes—
100C2×C4⋊Dic7C2311yescharacteristic
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4—22no—
104C2×C4×Dic7—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—22no—
108C2×C4×Dic7—28no—
109C2×C4×Dic7D411yescharacteristic
110C2×C4×Dic7D411yescharacteristic
111C2×C4×Dic7C2311yescharacteristic
112C22×C7⋊C8C2312yes—
113#6460—28no—
114#6455—28no—
115#6455—22no—
116#6455C2314yes—
117#6453—28no—
118#6450—28no—
119#6450—22no—
120#6450C2314yes—
121(C22⋊D7)⋊C4—28no—
122#6440—28no—
123C42×D7D414yes—
124C42×D7C2314yes—
125#6432—22no—
126#6432C2314yes—
127C4×Dic14—22no—
128C4×Dic14C2314yes—
129C7×C4⋊C8C2314yes—
130C7×#103C2318yes—
131#6409C2318yes—
132#6406C2318yes—
133(C2×D7)⋊C8C2318yes—
134#6392C2318yes—
135#6390C2318yes—
136#6389C2318yes—
137#6382C2318yes—
138C4⋊(C7⋊C8)C2314yes—
139C2×#533—71no—
140C2×C4×#41—71no—
141#3116—716no—
142C2×#485—72no—
143C2×#477—71no—
144C2×#476—71no—
145C2×#462—71no—
146C2×#461D711yescharacteristic, complemented
147C2×#452—72no—
148C2×#441—72no—
149C2×#440—72no—
150C2×#432—71no—
151C24×C7—21no—
152C23×D7C2×D411yescharacteristic
153C2×C7×Q8—22no—
154C2×C7×Q8—24no—
155C2×C7×Q8—24no—
156C2×C7×Q8C2414yes—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4#4114yes—
160C2×C7×D4C2412yes—
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—24no—
164C22×C4×C7—44no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C2×C22⋊D7—28no—
169C2×C22⋊D7—24no—
170C22×Dic7—28no—
171C22×Dic7—21no—
172C22×Dic7—24no—
173C22×Dic7—21no—
174C22×Dic7—22no—
175C22×Dic7—24no—
176C22×Dic7—44no—
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D411yescharacteristic
179C22×Dic7C2×D411yescharacteristic
180#1159—216no—
181#1159—28no—
182#1159—28no—
183C2×C4×D7—24no—
184C2×C4×D7—22no—
185C2×C4×D7—22no—
186C2×C4×D7—42no—
187C2×C4×D7C2×D412yes—
188C2×C4×D7C2×D414yes—
189C2×C4×D7C2×D414yes—
190C2×C4×D7C2412yes—
191C2×Dic14—21no—
192C2×Dic14—22no—
193C2×Dic14—24no—
194C2×Dic14—44no—
195C2×Dic14#4114yes—
196C2×Dic14C2412yes—
197C7×SD16—216no—
198C2×C7×C8—24no—
199C2×C7×C8C2414yes—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—24no—
202C7×C4⋊C4—24no—
203C7×C4⋊C4C2412yes—
204C7×C22⋊C4—24no—
205C42×C7C2×D412yes—
206C42×C7C2412yes—
207C22⋊Dic7—24no—
208#1144—216no—
209(C2×D7)⋊C4—24no—
210(C2×D7)⋊C4—28no—
211C4⋊Dic7—21no—
212C4⋊Dic7—22no—
213C4⋊Dic7—24no—
214C4⋊Dic7—28no—
215C4⋊Dic7C2414yes—
216C4⋊Dic7C2412yes—
217Dic7⋊C4—24no—
218Dic7⋊C4—24no—
219Dic7⋊C4—28no—
220C4×Dic7—24no—
221C4×Dic7—24no—
222C4×Dic7—24no—
223C4×Dic7—28no—
224C4×Dic7C2×D412yes—
225C4×Dic7C2×D414yes—
226C4×Dic7C2×D414yes—
227C4×Dic7C2412yes—
228C2×C7⋊C8—24no—
229C2×C7⋊C8C2414yes—
230C22×#41—71no—
231C22×Q16—72no—
232C22×SD16C2×D712yes—
233#533—78no—
234#533—144no—
235C2×C4⋊Q8—71no—
236C2×C4⋊Q8C2×D711yescharacteristic
237C2×#125—141no—
238C2×C22⋊Q8—72no—
239C2×C22⋊Q8—141no—
240C4×#41—78no—
241C4×#41—144no—
242C2×C4×Q8—72no—
243C2×C4×Q8—71no—
244C2×C4×Q8—71no—
245C2×C4×Q8—141no—
246C2×C4×D4—71no—
247C2×C4×D4—141no—
248C2×C4×D4—141no—
249C2×C4×D4C2×D711yescharacteristic
250C2×#117—71no—
251C2×#117—141no—
252C2×#117—141no—
253C22×C42—71no—
254C22×C42—141no—
255#485—716no—
256#477—78no—
257#476—78no—
258#462—78no—
259#461C2×D718yescomplemented
260#452—716no—
261#441—716no—
262#440—716no—
263#432—78no—
264C2×#107—72no—
265C2×#106—72no—
266C2×C4⋊C8—71no—
267C2×C4⋊C8C2×D711yescharacteristic
268C2×#103—72no—
269C2×#103—72no—
270C2×#103C2×D712yes—
271C2×#102—72no—
272C2×C22⋊C8—72no—
273C2×#97—72no—
274C23×C7—24no—
275C23×C7—22no—
276C23×C7—42no—
277C23×C7—42no—
278C23×C7C22×D411yescharacteristic
279C22×D7—22no—
280C22×D7—22no—
281C22×D7—22no—
282C22×D7C22×D412yes—
283C7×Q8—24no—
284C7×Q8—24no—
285C7×Q8—44no—
286C7×D4—24no—
287C7×D4—22no—
288C7×D4C2×#4114yes—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—24no—
292C2×C4×C7—44no—
293C2×C4×C7—42no—
294C2×C4×C7—48no—
295C2×C4×C7#13714yes—
296C2×C4×C7C22×D412yes—
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×D412yes—
299C2×C4×C7C2×#4112yes—
300C2×C4×C7C2×#4112yes—
301C2×C4×C7C2511yescharacteristic
302C22⋊D7—216no—
303C2×Dic7—216no—
304C2×Dic7—24no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—24no—
311C2×Dic7—48no—
312C2×Dic7—44no—
313C2×Dic7—42no—
314C2×Dic7C22×D412yes—
315C2×Dic7C22×D412yes—
316C2×Dic7C22×D412yes—
317C4×D7—24no—
318C4×D7—28no—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—44no—
322Dic14—24no—
323Dic14—22no—
324Dic14—48no—
325Dic14C2×#4114yes—
326C7×C8—24no—
327C7⋊C8—24no—
328C2×#41—74no—
329C2×#41—142no—
330C2×#41—144no—
331C2×#41—144no—
332C22×Q8—71no—
333C22×Q8—141no—
334C22×Q8C22×D712yes—
335C22×D4—141no—
336C22×D4C22×D711yescharacteristic
337C23×C4—21no—
338C23×C4—71no—
339C23×C4—71no—
340C23×C4—71no—
341C23×C4—141no—
342C23×C4—141no—
343C2×Q16—78no—
344C2×Q16—148no—
345C2×SD16—28no—
346C2×SD16C22×D718yes—
347C22×C8—72no—
348C22×C8C22×D712yes—
349C4⋊Q8—22no—
350C4⋊Q8—74no—
351C4⋊Q8—142no—
352C4⋊Q8C22×D714yes—
353#125—148no—
354C22⋊Q8—78no—
355C22⋊Q8—148no—
356C22⋊Q8—144no—
357C4×Q8—78no—
358C4×Q8—74no—
359C4×Q8—74no—
360C4×Q8—142no—
361C4×Q8—148no—
362C4×Q8—144no—
363C4×Q8—142no—
364C4×D4—22no—
365C4×D4—74no—
366C4×D4—142no—
367C4×D4—148no—
368C4×D4—148no—
369C4×D4C22×D714yes—
370#117—74no—
371#117—148no—
372#117—142no—
373#117—148no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—72no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—142no—
380C2×C4⋊C4—142no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—142no—
387C2×C22⋊C4—141no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—74no—
392C2×C42—74no—
393C2×C42—148no—
394C2×C42—141no—
395C2×C42—142no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42C22×D711yescharacteristic
399#107—78no—
400#106—78no—
401C4⋊C8—74no—
402C4⋊C8C22×D714yes—
403#103—78no—
404#103—78no—
405#103C22×D718yes—
406#102—78no—
407C22⋊C8—78no—
408#97—78no—
409C22×C7—24no—
410C22×C7—48no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7#52512yes—
414C22×C7C2×#13712yes—
415C22×C7#57512yes—
416C22×C7C23×D411yescharacteristic
417C2×D7—22no—
418C2×D7—22no—
419C2×D7—22no—
420C2×D7—44no—
421C4×C7—22no—
422C4×C7—44no—
423C4×C7—42no—
424C4×C7#52512yes—
425C4×C7C2×#13714yes—
426C4×C7C22×#4111yescharacteristic
427C4×C7C22×#4111yescharacteristic
428Dic7—24no—
429Dic7—22no—
430Dic7—22no—
431Dic7—22no—
432Dic7—44no—
433Dic7—42no—
434C24—21no—
435C24—71no—
436#41—148no—
437#41—1416no—
438#41—148no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—74no—
443C2×Q8—72no—
444C2×Q8—141no—
445C2×Q8—144no—
446C2×Q8—142no—
447C2×Q8—284no—
448C2×Q8C23×D714yes—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—144no—
452C2×D4—148no—
453C2×D4#115914yes—
454C2×D4C23×D712yes—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—74no—
463C22×C4—72no—
464C22×C4—72no—
465C22×C4—74no—
466C22×C4—144no—
467C22×C4—144no—
468C22×C4—148no—
469C22×C4—141no—
470C22×C4—144no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—141no—
474C22×C4—142no—
475C22×C4—282no—
476C22×C4—284no—
477C22×C4D4×D711yescharacteristic
478C22×C4D4×D711yescharacteristic
479C22×C4C23×D711yescharacteristic
480Q16—1416no—
481SD16—216no—
482C2×C8—24no—
483C2×C8—74no—
484C2×C8—144no—
485C2×C8C23×D714yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—74no—
490C4⋊C4—72no—
491C4⋊C4—144no—
492C4⋊C4—148no—
493C4⋊C4—148no—
494C4⋊C4—144no—
495C4⋊C4—141no—
496C4⋊C4—142no—
497C4⋊C4—144no—
498C4⋊C4C23×D712yes—
499C22⋊C4—24no—
500C22⋊C4—144no—
501C22⋊C4—148no—
502C22⋊C4—144no—
503C42—74no—
504C42—74no—
505C42—72no—
506C42—72no—
507C42—144no—
508C42—144no—
509C42—148no—
510C42—144no—
511C42D4×D712yes—
512C42C23×D712yes—
513C2×C7—44no—
514C2×C7#311614yescomplemented
515C2×C7C2×#52511yescharacteristic
516C2×C7C22×#13711yescharacteristic
517C2×C7C2×#57511yescharacteristic
518D7—42no—
519C23—24no—
520C23—22no—
521C23—42no—
522C23—42no—
523C23—72no—
524C23—142no—
525C23—142no—
526C23—142no—
527C23C2×D4×D711yescharacteristic
528Q8—24no—
529Q8—24no—
530Q8—44no—
531Q8—74no—
532Q8—144no—
533Q8—142no—
534Q8—288no—
535D4—24no—
536D4—22no—
537D4—1416no—
538D4C2×#115914yes—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—72no—
546C2×C4—72no—
547C2×C4—72no—
548C2×C4—1416no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—144no—
553C2×C4—142no—
554C2×C4—148no—
555C2×C4—144no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—144no—
560C2×C4—284no—
561C2×C4—288no—
562C2×C4—282no—
563C2×C4—284no—
564C2×C4#647814yes—
565C2×C4C2×D4×D712yes—
566C2×C4C2×D4×D712yes—
567C2×C4C2×D4×D711yescharacteristic
568C2×C4C2×#115912yes—
569C2×C4D7×#4112yes—
570C2×C4C24×D711yescharacteristic
571C8—24no—
572C8—144no—
573C7C2×#311611yescharacteristic, complemented, Sylow
574C22—24no—
575C22—48no—
576C22—42no—
577C22—42no—
578C22—142no—
579C22—142no—
580C22—142no—
581C22—284no—
582C22#9029712yes—
583C22C2×#647812yes—
584C22#9046812yes—
585C22C22×D4×D711yescharacteristic
586C4—22no—
587C4—44no—
588C4—42no—
589C4—142no—
590C4—142no—
591C4—144no—
592C4—142no—
593C4—282no—
594C4—284no—
595C4#9029712yes—
596C4C2×#647814yes—
597C4C22×#115911yescharacteristic
598C4C2×D7×#4111yescharacteristic
599C2—44no—
600C2—282no—
601C2#1173785614yesdirect product factor
602C2C2×#9029711yescharacteristic
603C2C22×#647811yescharacteristic
604C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#461C2
socleC23×C7C22×D4
prime radicalC22×#1159C22
hypercenterC2×#461D7
hypocenterC7C2×#3116
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#90419C2semidirect
3. C2×C7×#461C2semidirect
5. C2×#89936C2semidirect
6. C2×#89862C2semidirect
7. C2×#89780C2semidirect
8. #11737856C2direct (N × H)
9. C2×#89622C2semidirect
10. C2×#89607C2semidirect
11. C2×#89586C2semidirect
12. C2×#89561C2semidirect
24. #90419C22semidirect
38. C7×#461C22semidirect
44. #89936C22semidirect
49. #89862C22semidirect
50. #89780C22semidirect
53. #89622C22semidirect
54. #89607C22semidirect
55. #89586C22semidirect
56. #89561C22semidirect
146. C2×#461D7semidirect
259. #461C2×D7semidirect
514. C2×C7#3116semidirect
573. C7C2×#3116semidirect
601. C2#11737856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904192
C2×C4×#11592
C2×C7×#4612
C2×#899452
C2×#899362
C2×#898622
C2×#897802
#117378562
C2×#896222
C2×#896072
C2×#895862
C2×#895612
C2×#31167