Finite Group Databasebeta

← C2×#11739408 #14223721

Subgroups — C2×#11739408 #14223721

Order 1792.

1998 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic, complemented
2C2×#90382C211yesmaximal, characteristic
3C2×C7×#501C211yesmaximal, characteristic, complemented
4C2×#89861C211yesmaximal, characteristic, complemented
5C2×#89841C211yesmaximal, characteristic, complemented
6#11739408C2116yesmaximal, direct product factor
7C2×#89659C212yesmaximal, complemented
8C2×#89661C212yesmaximal, complemented
9C2×#89636C212yesmaximal
10C2×#89625C212yesmaximal, complemented
11C2×#89487C211yesmaximal, characteristic, complemented
12C2×D7×#97C211yesmaximal, characteristic, complemented
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×#125C2211yescharacteristic
15C2×#6509—21no—
16C2×Q8×Dic7—21no—
17C22×Dic28C2212yes—
18C22×#1132C2212yes—
19C22×C8×D7C2212yes—
20#90419—24no—
21#90419C2218yescomplemented
22#90382—24no—
23#90382C2218yes—
24C2×#6461—21no—
25C2×#6460C2211yescharacteristic
26C2×#6460C2212yes—
27C2×#6458C2211yescharacteristic
28C2×#6457—21no—
29C2×#6456—21no—
30C2×#6456C2211yescharacteristic
31C2×#6455—21no—
32C2×#6455—21no—
33C2×#6453—21no—
34C2×#6450C2212yes—
35C2×#6450C2211yescharacteristic
36C2×#6438C2211yescharacteristic
37C2×C42×D7C2211yescharacteristic
38C2×#6432C2211yescharacteristic
39C7×#501C2218yescomplemented
40C2×C7×#107C2212yes—
41C2×C7×#106C2212yes—
42C2×C7×#97C2211yescharacteristic
43C2×#6389C2212yes—
44#89861C2218yescomplemented
45#89841C2218yescomplemented
46C2×#6384C2212yes—
47C2×#6384C2212yes—
48C2×#6384C2212yes—
49C2×#6383C2212yes—
50C2×#6377C2211yescharacteristic
51#89661C22116yescomplemented
52#89659C22116yescomplemented
53#89636C22116yes—
54#89625C22116yescomplemented
55#89487C2218yescomplemented
56D7×#97C2218yescomplemented
57C2×#3181—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×D28—21no—
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C4×D7D411yescharacteristic
63C22×C4×D7D411yescharacteristic
64C22×C4×D7C2311yescharacteristic
65C22×Dic14C2312yes—
66C22×Dic14C2311yescharacteristic
67C22×C7×C8C2312yes—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2314yes—
70C7×#125—22no—
71C7×#125C2314yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79#6509—28no—
80Q8×Dic7—28no—
81C2×(C2×D7)⋊C4—22no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—21no—
91C2×Dic7⋊C4—22no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—22no—
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7D411yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2312yes—
99C2×Dic28—28no—
100C2×Dic28C2318yes—
101C2×#1132—28no—
102C2×#1132C2318yes—
103C2×C8×D7—28no—
104C2×C8×D7C2318yes—
105#6461—28no—
106#6460—22no—
107#6460—24no—
108#6460C2314yes—
109#6460C2318yes—
110#6458—22no—
111#6458C2314yes—
112#6457—28no—
113#6456—28no—
114#6456—22no—
115#6456C2314yes—
116#6455—28no—
117#6455—28no—
118#6453—28no—
119#6450—22no—
120#6450—24no—
121#6450C2314yes—
122#6450C2318yes—
123#6438—22no—
124#6438C2314yes—
125C42×D7D414yes—
126C42×D7C2314yes—
127#6432—22no—
128#6432C2314yes—
129C7×#107C2318yes—
130C7×#106C2318yes—
131C7×#97C2314yes—
132#6389C2318yes—
133#6384C2318yes—
134#6384C2318yes—
135#6384C2318yes—
136#6383C2318yes—
137#6377C2314yes—
138C2×#533—71no—
139C2×#529—71no—
140#3181—716no—
141C2×#501D711yescharacteristic, complemented
142C2×#497—71no—
143C2×#491—71no—
144C2×#489—71no—
145C2×#467—72no—
146C2×#465—72no—
147C2×#441—72no—
148C2×#440—72no—
149C22×#97—71no—
150C23×D7—21no—
151C23×D7C2×D411yescharacteristic
152C2×C7×Q8—24no—
153C2×C7×Q8—44no—
154C22×C4×C7—22no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—22no—
161C22×Dic7—21no—
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C2×D28—24no—
166C2×D28—24no—
167C2×D28—21no—
168C2×D28—44no—
169C2×D28C2412yes—
170C2×C4×D7—24no—
171C2×C4×D7—24no—
172C2×C4×D7—24no—
173C2×C4×D7—42no—
174C2×C4×D7—44no—
175C2×C4×D7C2×D412yes—
176C2×C4×D7C2×D414yes—
177C2×C4×D7C2×D414yes—
178C2×C4×D7C2412yes—
179C2×Dic14—24no—
180C2×Dic14—24no—
181C2×Dic14—24no—
182C2×Dic14—22no—
183C2×Dic14—21no—
184C2×Dic14C2414yes—
185C2×Dic14C2412yes—
186C2×C7×C8#4118yes—
187C2×C7×C8C2414yes—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4C2412yes—
195C7×C4⋊C4C2414yes—
196C7×C4⋊C4C2412yes—
197C42×C7C2×D412yes—
198C42×C7C2412yes—
199(C2×D7)⋊C4—28no—
200(C2×D7)⋊C4—24no—
201(C2×D7)⋊C4—24no—
202(C2×D7)⋊C4—24no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—24no—
205C4⋊Dic7—24no—
206C4⋊Dic7—24no—
207C4⋊Dic7—28no—
208C4⋊Dic7—24no—
209Dic7⋊C4—28no—
210Dic7⋊C4—24no—
211C4×Dic7—24no—
212C4×Dic7—28no—
213C4×Dic7C2×D412yes—
214C4×Dic7C2×D414yes—
215C4×Dic7C2×D414yes—
216C4×Dic7C2412yes—
217C2×C7⋊C8—24no—
218C2×C7⋊C8C2414yes—
219Dic28—216no—
220#1132—216no—
221C8×D7—216no—
222C22×Q16—72no—
223C22×SD16—72no—
224C23×C8—72no—
225#533—78no—
226#533—144no—
227C2×C4⋊Q8—71no—
228C2×C4⋊Q8C2×D711yescharacteristic
229#529—78no—
230#529—144no—
231C2×#126—141no—
232C2×#125—141no—
233C2×#125C2×D711yescharacteristic
234C2×#124—71no—
235C2×#123—141no—
236C2×C22⋊Q8—71no—
237C2×C22⋊Q8—72no—
238C2×C22⋊Q8—141no—
239C2×#121—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8—72no—
242C2×C4×Q8—141no—
243C2×C4×D4—71no—
244C2×C4×D4—141no—
245C2×#117—141no—
246C2×#117—141no—
247C22×C42—71no—
248#501C2×D718yescomplemented
249#497—78no—
250#491—78no—
251#489—78no—
252#467—716no—
253#465—716no—
254#441—716no—
255#440—716no—
256C2×#107C2×D712yes—
257C2×#106C2×D712yes—
258C2×#103—72no—
259C2×#103—72no—
260C2×#103—72no—
261C2×#102—72no—
262C2×#97—72no—
263C2×#97—71no—
264C2×#97—78no—
265C2×#97C2×D711yescharacteristic
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—22no—
270C22×D7—42no—
271C22×D7—42no—
272C22×D7—42no—
273C22×D7#13714yes—
274C22×D7C22×D412yes—
275C7×Q8—48no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—44no—
284C2×C4×C7C22×D412yes—
285C2×C4×C7C22×D412yes—
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C2×#4114yes—
288C2×C4×C7C2511yescharacteristic
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—24no—
293C2×Dic7—22no—
294C2×Dic7—44no—
295C2×Dic7—42no—
296C2×Dic7C22×D412yes—
297C2×Dic7C22×D412yes—
298C2×Dic7C22×D412yes—
299D28—24no—
300D28—48no—
301D28—42no—
302C4×D7—28no—
303C4×D7—24no—
304C4×D7—24no—
305C4×D7—44no—
306C4×D7—48no—
307Dic14—24no—
308Dic14—24no—
309Dic14—24no—
310Dic14—44no—
311Dic14—42no—
312C7×C8C2×#4118yes—
313C7⋊C8—24no—
314C22×Q8—21no—
315C22×Q8—71no—
316C22×Q8—72no—
317C22×D4—71no—
318C22×D4—141no—
319C23×C4—71no—
320C23×C4—71no—
321C23×C4—71no—
322C23×C4—141no—
323C2×Q16—78no—
324C2×Q16—148no—
325C2×SD16—78no—
326C2×SD16—148no—
327C22×C8—72no—
328C22×C8—78no—
329C22×C8—148no—
330C22×C8C22×D712yes—
331C4⋊Q8—22no—
332C4⋊Q8—74no—
333C4⋊Q8—142no—
334C4⋊Q8C22×D714yes—
335#126—148no—
336#125—22no—
337#125—148no—
338#125C22×D714yes—
339#124—74no—
340#124—142no—
341#123—148no—
342C22⋊Q8—74no—
343C22⋊Q8—78no—
344C22⋊Q8—142no—
345C22⋊Q8—144no—
346C22⋊Q8—148no—
347#121—74no—
348#121—142no—
349C4×Q8—74no—
350C4×Q8—78no—
351C4×Q8—142no—
352C4×Q8—144no—
353C4×Q8—148no—
354C4×D4—74no—
355C4×D4—142no—
356C4×D4—148no—
357#117—148no—
358#117—148no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—142no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—142no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D712yes—
371C2×C22⋊C4—141no—
372C2×C22⋊C4—141no—
373C2×C22⋊C4—142no—
374C2×C22⋊C4—141no—
375C2×C22⋊C4—141no—
376C2×C22⋊C4—141no—
377C2×C42—71no—
378C2×C42—71no—
379C2×C42—71no—
380C2×C42—74no—
381C2×C42—74no—
382C2×C42—141no—
383C2×C42—142no—
384C2×C42C22×D711yescharacteristic
385#107C22×D718yes—
386#106C22×D718yes—
387#103—78no—
388#103—78no—
389#103—78no—
390#102—78no—
391#97—78no—
392#97—74no—
393#97C22×D714yes—
394C22×C7#53212yes—
395C22×C7C2×#13712yes—
396C22×C7#57512yes—
397C22×C7C23×D411yescharacteristic
398C2×D7—28no—
399C2×D7—22no—
400C2×D7—22no—
401C2×D7—42no—
402C2×D7—42no—
403C2×D7—42no—
404C2×D7—84no—
405C2×D7C2×#13714yes—
406C4×C7—22no—
407C4×C7—22no—
408C4×C7—42no—
409C4×C7—42no—
410C4×C7—44no—
411C4×C7#53212yes—
412C4×C7C22×#4112yes—
413Dic7—22no—
414Dic7—22no—
415Dic7—22no—
416Dic7—44no—
417Dic7—42no—
418C24—71no—
419C24—141no—
420C2×Q8—24no—
421C2×Q8—44no—
422C2×Q8—72no—
423C2×Q8—74no—
424C2×Q8—141no—
425C2×Q8—142no—
426C2×Q8—144no—
427C2×Q8—144no—
428C2×Q8—144no—
429C2×D4—72no—
430C2×D4—144no—
431C2×D4—141no—
432C2×D4—144no—
433C2×D4—284no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—74no—
441C22×C4—72no—
442C22×C4—72no—
443C22×C4—74no—
444C22×C4—144no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—141no—
448C22×C4—142no—
449C22×C4—282no—
450C22×C4—284no—
451C22×C4D4×D711yescharacteristic
452C22×C4D4×D711yescharacteristic
453C22×C4C23×D711yescharacteristic
454Q16—1416no—
455SD16—1416no—
456C2×C8—74no—
457C2×C8—144no—
458C2×C8—1416no—
459C2×C8#116118yes—
460C2×C8C23×D714yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—148no—
468C4⋊C4—144no—
469C4⋊C4—144no—
470C4⋊C4—148no—
471C4⋊C4—144no—
472C4⋊C4—144no—
473C4⋊C4C23×D714yes—
474C4⋊C4C23×D712yes—
475C4⋊C4C23×D712yes—
476C22⋊C4—144no—
477C22⋊C4—144no—
478C22⋊C4—148no—
479C22⋊C4—144no—
480C22⋊C4—144no—
481C22⋊C4—144no—
482C42—74no—
483C42—72no—
484C42—74no—
485C42—72no—
486C42—144no—
487C42—148no—
488C42D4×D712yes—
489C42C23×D712yes—
490C2×C7#318114yescomplemented
491C2×C7C2×#53211yescharacteristic
492C2×C7C22×#13711yescharacteristic
493C2×C7C2×#57511yescharacteristic
494D7—24no—
495D7—82no—
496C23—74no—
497C23—72no—
498C23—142no—
499C23—142no—
500C23—142no—
501C23—282no—
502C23—282no—
503C23—282no—
504C23C2×D4×D711yescharacteristic
505Q8—48no—
506Q8—144no—
507Q8—144no—
508Q8—144no—
509Q8—282no—
510Q8—284no—
511D4—144no—
512D4—288no—
513D4—282no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—72no—
523C2×C4—72no—
524C2×C4—72no—
525C2×C4—148no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—142no—
532C2×C4—144no—
533C2×C4—284no—
534C2×C4—288no—
535C2×C4—282no—
536C2×C4—284no—
537C2×C4C2×D4×D712yes—
538C2×C4C2×D4×D712yes—
539C2×C4C2×D4×D711yescharacteristic
540C2×C4C2×#116114yes—
541C2×C4C24×D711yescharacteristic
542C8—144no—
543C8C2×#116118yes—
544C7C2×#318111yescharacteristic, complemented, Sylow
545C22—74no—
546C22—142no—
547C22—148no—
548C22—142no—
549C22—282no—
550C22—282no—
551C22—282no—
552C22—564no—
553C22#9041612yes—
554C22#9046412yes—
555C22D7×#13712yes—
556C22C22×D4×D711yescharacteristic
557C4—22no—
558C4—22no—
559C4—42no—
560C4—44no—
561C4—42no—
562C4—142no—
563C4—142no—
564C4—142no—
565C4—282no—
566C4—284no—
567C4#9041612yes—
568C4C22×#116112yes—
569C2—144no—
570C2—562no—
571C2#1173940814yesdirect product factor
572C2C2×#9041611yescharacteristic
573C2C2×#9046411yescharacteristic
574C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#501C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
hypercenterC2×#501D7
hypocenterC7C2×#3181
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×#90419C2semidirect
3. C2×C7×#501C2semidirect
4. C2×#89861C2semidirect
5. C2×#89841C2semidirect
6. #11739408C2direct (N × H)
7. C2×#89659C2semidirect
8. C2×#89661C2semidirect
10. C2×#89625C2semidirect
11. C2×#89487C2semidirect
12. C2×D7×#97C2semidirect
21. #90419C22semidirect
39. C7×#501C22semidirect
44. #89861C22semidirect
45. #89841C22semidirect
51. #89661C22semidirect
52. #89659C22semidirect
54. #89625C22semidirect
55. #89487C22semidirect
56. D7×#97C22semidirect
141. C2×#501D7semidirect
248. #501C2×D7semidirect
490. C2×C7#3181semidirect
544. C7C2×#3181semidirect
571. C2#11739408direct (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×#903822
C2×C7×#5012
C2×#898612
C2×#898412
#117394082
C2×#896592
C2×#896612
C2×#896362
C2×#896252
C2×#894872
C2×D7×#972
C2×#31817