Finite Group Databasebeta

← C2×#11738558 #14225879

Subgroups — C2×#11738558 #14225879

Order 1792.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic, complemented
2C2×#90271C211yesmaximal, characteristic
3C2×C7×#477C211yesmaximal, characteristic, complemented
4C2×#89856C211yesmaximal, characteristic, complemented
5C2×#89792C211yesmaximal, characteristic, complemented
6#11738558C2116yesmaximal, direct product factor
7C2×#89650C212yesmaximal, complemented
8C2×#89657C212yesmaximal
9C2×#89618C211yesmaximal, characteristic, complemented
10C2×#89605C211yesmaximal, characteristic, complemented
11C2×#89572C212yesmaximal, complemented
12C2×#89586C212yesmaximal, complemented
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6510C2211yescharacteristic
16C2×#6509—21no—
17C2×#6509C2211yescharacteristic
18C2×Q8×Dic7—21no—
19#90419—24no—
20#90419C2218yescomplemented
21#90271—24no—
22#90271C2218yes—
23C2×#6461—21no—
24C2×#6460—21no—
25C2×#6458C2211yescharacteristic
26C2×#6457—21no—
27C2×#6455C2212yes—
28C2×#6453C2211yescharacteristic
29C2×#6453C2212yes—
30C2×#6450—21no—
31C2×C4×D28—21no—
32C2×C4×D28C2211yescharacteristic
33C2×#6435—21no—
34C2×C42×D7C2211yescharacteristic
35C2×#6432C2211yescharacteristic
36C7×#477C2218yescomplemented
37C2×C7×#107C2212yes—
38C2×C7×C4⋊C8C2211yescharacteristic
39C2×C7×#103C2212yes—
40C2×#6409C2212yes—
41C2×#6395C2212yes—
42C2×(C2×D7)⋊C8C2212yes—
43C2×#6392C2212yes—
44C2×C8×Dic7C2212yes—
45#89856C2218yescomplemented
46#89792C2218yescomplemented
47C2×#6383C2212yes—
48C2×#6382C2212yes—
49C2×#6382C2212yes—
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89657C22116yes—
52#89650C22116yescomplemented
53#89618C2218yescomplemented
54#89605C2218yescomplemented
55#89586C22116yescomplemented
56#89572C22116yescomplemented
57C2×#3233—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×C7×Q8C2311yescharacteristic
60C22×D28—21no—
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D411yescharacteristic
65C22×C4×D7C2311yescharacteristic
66C22×Dic14—21no—
67C22×C7×C8C2312yes—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2314yes—
70C4×C7×Q8—22no—
71C4×C7×Q8C2314yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78#6510—22no—
79#6510C2314yes—
80#6509—28no—
81#6509—22no—
82#6509C2314yes—
83Q8×Dic7—28no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—22no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2312yes—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—22no—
98C2×C4×Dic7—22no—
99C2×C4×Dic7D411yescharacteristic
100C2×C4×Dic7D411yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C22×C7⋊C8C2312yes—
103#6461—28no—
104#6460—28no—
105#6458—22no—
106#6458C2314yes—
107#6457—28no—
108#6455—24no—
109#6455C2318yes—
110#6453—24no—
111#6453—22no—
112#6453C2314yes—
113#6453C2318yes—
114#6450—28no—
115C4×D28—28no—
116C4×D28—22no—
117C4×D28C2314yes—
118#6435—28no—
119C42×D7D414yes—
120C42×D7C2314yes—
121#6432—22no—
122#6432C2314yes—
123C7×#107C2318yes—
124C7×C4⋊C8C2314yes—
125C7×#103C2318yes—
126#6409C2318yes—
127#6395C2318yes—
128(C2×D7)⋊C8C2318yes—
129#6392C2318yes—
130C8×Dic7C2318yes—
131#6383C2318yes—
132#6382C2318yes—
133#6382C2318yes—
134C4⋊(C7⋊C8)C2314yes—
135C2×#533—71no—
136C2×#529—71no—
137#3233—716no—
138C2×#499—72no—
139C2×#488—72no—
140C2×#485—72no—
141C2×#482—72no—
142C2×#477D711yescharacteristic, complemented
143C2×#475—71no—
144C2×#476—71no—
145C2×#474—71no—
146C2×#431—71no—
147C23×D7—21no—
148C23×D7C2×D411yescharacteristic
149C2×C7×Q8—24no—
150C2×C7×Q8—21no—
151C2×C7×Q8—22no—
152C2×C7×Q8—22no—
153C2×C7×Q8—44no—
154C2×C7×Q8C2412yes—
155C22×C4×C7—22no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C2×D28—22no—
167C2×D28—22no—
168C2×D28—24no—
169C2×D28—21no—
170C2×D28—44no—
171C2×D28C2412yes—
172C2×C4×D7—24no—
173C2×C4×D7—22no—
174C2×C4×D7—22no—
175C2×C4×D7—24no—
176C2×C4×D7—42no—
177C2×C4×D7—44no—
178C2×C4×D7C2×D412yes—
179C2×C4×D7C2×D414yes—
180C2×C4×D7C2×D414yes—
181C2×C4×D7C2412yes—
182C2×Dic14—24no—
183C2×Dic14—44no—
184C2×C7×C8—24no—
185C2×C7×C8C2414yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—24no—
190C7×C4⋊C4C2414yes—
191C7×C4⋊C4C2412yes—
192C42×C7—24no—
193C42×C7C2×D412yes—
194C42×C7C2412yes—
195(C2×D7)⋊C4—24no—
196(C2×D7)⋊C4—28no—
197(C2×D7)⋊C4—24no—
198(C2×D7)⋊C4—24no—
199(C2×D7)⋊C4—24no—
200(C2×D7)⋊C4—24no—
201C4⋊Dic7—22no—
202C4⋊Dic7—24no—
203C4⋊Dic7—21no—
204C4⋊Dic7—24no—
205C4⋊Dic7—28no—
206C4⋊Dic7C2414yes—
207C4⋊Dic7C2412yes—
208Dic7⋊C4—24no—
209Dic7⋊C4—24no—
210Dic7⋊C4—28no—
211C4×Dic7—28no—
212C4×Dic7C2×D412yes—
213C4×Dic7C2×D414yes—
214C4×Dic7C2×D414yes—
215C4×Dic7C2412yes—
216C2×C7⋊C8—24no—
217C2×C7⋊C8C2414yes—
218#533—78no—
219#533—144no—
220C2×C4⋊Q8—71no—
221C2×C4⋊Q8C2×D711yescharacteristic
222#529—78no—
223#529—144no—
224C2×#126—141no—
225C2×#125—72no—
226C2×#125—71no—
227C2×#124—71no—
228C2×#123—141no—
229C2×C22⋊Q8—71no—
230C2×C22⋊Q8—141no—
231C2×C22⋊Q8—141no—
232C2×#121—71no—
233C2×C4×Q8—141no—
234C2×C4×Q8—141no—
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×D4—71no—
237C2×C4×D4—141no—
238C2×#117—72no—
239C2×#117—141no—
240C22×C42—71no—
241#499—716no—
242#488—716no—
243#485—716no—
244#482—716no—
245#477C2×D718yescomplemented
246#476—78no—
247#475—78no—
248#474—78no—
249#431—78no—
250C2×#107—72no—
251C2×#107C2×D712yes—
252C2×#106—72no—
253C2×#106—72no—
254C2×C4⋊C8—71no—
255C2×C4⋊C8C2×D711yescharacteristic
256C2×#103—72no—
257C2×#103C2×D712yes—
258C2×#102—72no—
259C2×#102—72no—
260C2×C22⋊C8—72no—
261C2×C4×C8—72no—
262C23×C7C22×D411yescharacteristic
263C22×D7—22no—
264C22×D7—22no—
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—42no—
268C22×D7—42no—
269C22×D7—42no—
270C22×D7C22×D412yes—
271C7×Q8—22no—
272C7×Q8—22no—
273C7×Q8—48no—
274C7×Q8—42no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—44no—
282C2×C4×C7#13514yes—
283C2×C4×C7C22×D412yes—
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D412yes—
286C2×C4×C72−1+412yes—
287C2×C4×C72−1+412yes—
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—22no—
300D28—22no—
301D28—48no—
302D28—42no—
303C4×D7—24no—
304C4×D7—28no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—44no—
308C4×D7—48no—
309Dic14—48no—
310C7×C8—24no—
311C7⋊C8—24no—
312C22×Q8—21no—
313C22×Q8—141no—
314C22×Q8C22×D711yescharacteristic
315C22×D4—71no—
316C22×D4—141no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4—141no—
321C22×C8—72no—
322C22×C8C22×D712yes—
323C4⋊Q8—22no—
324C4⋊Q8—74no—
325C4⋊Q8—142no—
326C4⋊Q8C22×D714yes—
327#126—148no—
328#125—74no—
329#125—78no—
330#125—142no—
331#125—144no—
332#124—74no—
333#124—142no—
334#123—148no—
335C22⋊Q8—74no—
336C22⋊Q8—148no—
337C22⋊Q8—142no—
338C22⋊Q8—148no—
339#121—74no—
340#121—142no—
341C4×Q8—22no—
342C4×Q8—148no—
343C4×Q8—148no—
344C4×Q8C22×D714yes—
345C4×D4—74no—
346C4×D4—148no—
347C4×D4—142no—
348#117—78no—
349#117—148no—
350#117—144no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—72no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22×D712yes—
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4—141no—
366C2×C22⋊C4—142no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C42—21no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—71no—
373C2×C42—74no—
374C2×C42—74no—
375C2×C42—142no—
376C2×C42C22×D711yescharacteristic
377#107—78no—
378#107C22×D718yes—
379#106—78no—
380#106—78no—
381C4⋊C8—74no—
382C4⋊C8C22×D714yes—
383#103—78no—
384#103C22×D718yes—
385#102—78no—
386#102—78no—
387C22⋊C8—78no—
388C4×C8—78no—
389C22×C7#53612yes—
390C22×C7C2×#13512yes—
391C22×C7#57512yes—
392C22×C7C23×D411yescharacteristic
393C2×D7—22no—
394C2×D7—22no—
395C2×D7—22no—
396C2×D7—44no—
397C2×D7—42no—
398C2×D7—42no—
399C2×D7—42no—
400C2×D7—84no—
401C4×C7—22no—
402C4×C7—42no—
403C4×C7—42no—
404C4×C7—44no—
405C4×C7#53612yes—
406C4×C7C2×#13514yes—
407C4×C7C2×2−1+411yescharacteristic
408C4×C7C2×2−1+411yescharacteristic
409Dic7—22no—
410Dic7—22no—
411Dic7—22no—
412Dic7—44no—
413Dic7—42no—
414C24—71no—
415C24—141no—
416C2×Q8—21no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—44no—
421C2×Q8—144no—
422C2×Q8—284no—
423C2×Q8C23×D712yes—
424C2×D4—72no—
425C2×D4—144no—
426C2×D4—141no—
427C2×D4—142no—
428C2×D4—142no—
429C2×D4—284no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—72no—
437C22×C4—74no—
438C22×C4—72no—
439C22×C4—74no—
440C22×C4—144no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—141no—
445C22×C4—144no—
446C22×C4—282no—
447C22×C4—284no—
448C22×C4D4×D711yescharacteristic
449C22×C4D4×D711yescharacteristic
450C22×C4C23×D711yescharacteristic
451C2×C8—24no—
452C2×C8—74no—
453C2×C8—144no—
454C2×C8C23×D714yes—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—74no—
460C4⋊C4—72no—
461C4⋊C4—144no—
462C4⋊C4—144no—
463C4⋊C4—148no—
464C4⋊C4—144no—
465C4⋊C4—148no—
466C4⋊C4—142no—
467C4⋊C4—141no—
468C4⋊C4—144no—
469C4⋊C4C23×D714yes—
470C4⋊C4C23×D712yes—
471C22⋊C4—144no—
472C22⋊C4—144no—
473C22⋊C4—144no—
474C22⋊C4—148no—
475C22⋊C4—144no—
476C22⋊C4—144no—
477C42—24no—
478C42—74no—
479C42—74no—
480C42—72no—
481C42—72no—
482C42—148no—
483C42D4×D712yes—
484C42C23×D712yes—
485C2×C7#323314yescomplemented
486C2×C7C2×#53611yescharacteristic
487C2×C7C22×#13511yescharacteristic
488C2×C7C2×#57511yescharacteristic
489D7—42no—
490D7—82no—
491C23—72no—
492C23—142no—
493C23—142no—
494C23—142no—
495C23—142no—
496C23—282no—
497C23—282no—
498C23—282no—
499C23C2×D4×D711yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—48no—
503Q8—42no—
504Q8—288no—
505D4—142no—
506D4—142no—
507D4—288no—
508D4—282no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—72no—
517C2×C4—72no—
518C2×C4—72no—
519C2×C4—142no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—148no—
523C2×C4—144no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—144no—
527C2×C4—142no—
528C2×C4—284no—
529C2×C4—282no—
530C2×C4—284no—
531C2×C4—288no—
532C2×C4#648214yes—
533C2×C4C2×D4×D712yes—
534C2×C4C2×D4×D712yes—
535C2×C4C2×D4×D711yescharacteristic
536C2×C4#655112yes—
537C2×C4#655412yes—
538C2×C4C24×D711yescharacteristic
539C8—24no—
540C8—144no—
541C7C2×#323311yescharacteristic, complemented, Sylow
542C22—142no—
543C22—142no—
544C22—142no—
545C22—284no—
546C22—282no—
547C22—282no—
548C22—282no—
549C22—564no—
550C22#9031912yes—
551C22C2×#648212yes—
552C22#9046412yes—
553C22C22×D4×D711yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—142no—
559C4—142no—
560C4—142no—
561C4—282no—
562C4—284no—
563C4#9031912yes—
564C4C2×#648214yes—
565C4C2×#655111yescharacteristic
566C4C2×#655411yescharacteristic
567C2—282no—
568C2—562no—
569C2#1173855814yesdirect product factor
570C2C2×#9031911yescharacteristic
571C2C22×#648211yescharacteristic
572C2C2×#9046411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#477C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
hypercenterC2×#477D7
hypocenterC7C2×#3233
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×#90419C2semidirect
3. C2×C7×#477C2semidirect
4. C2×#89856C2semidirect
5. C2×#89792C2semidirect
6. #11738558C2direct (N × H)
7. C2×#89650C2semidirect
9. C2×#89618C2semidirect
10. C2×#89605C2semidirect
11. C2×#89572C2semidirect
12. C2×#89586C2semidirect
20. #90419C22semidirect
36. C7×#477C22semidirect
45. #89856C22semidirect
46. #89792C22semidirect
52. #89650C22semidirect
53. #89618C22semidirect
54. #89605C22semidirect
55. #89586C22semidirect
56. #89572C22semidirect
142. C2×#477D7semidirect
245. #477C2×D7semidirect
485. C2×C7#3233semidirect
541. C7C2×#3233semidirect
569. C2#11738558direct (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×#902712
C2×C7×#4772
C2×#898562
C2×#897922
#117385582
C2×#896502
C2×#896572
C2×#896182
C2×#896052
C2×#895722
C2×#895862
C2×#32337