Finite Group Databasebeta

← C2×#11726030 #13320881

Subgroups — C2×#11726030 #13320881

Order 1792.

1182 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90291C211yesmaximal, characteristic
2C2×#89490C211yesmaximal, characteristic, complemented
3C2×#89420C211yesmaximal, characteristic, complemented
4C2×C7×#327C211yesmaximal, characteristic
5C2×#89332C211yesmaximal, characteristic
6C2×#89262C211yesmaximal, characteristic, complemented
7C2×#89247C211yesmaximal, characteristic, complemented
8#11726030C218yesmaximal, direct product factor
9C2×C7×#121C2211yescharacteristic
10C2×#6499C411yescharacteristic, complemented
11C2×#6498—21no—
12C2×D4×Dic7—21no—
13C2×D4×Dic7C411yescharacteristic, complemented
14C2×#6487C2211yescharacteristic
15C22×C4×Dic7—21no—
16C22×C4×Dic7C2211yescharacteristic
17#90291—22no—
18#90291—22no—
19#90291C414yescomplemented
20#90291C2214yes—
21C2×#6453C411yescharacteristic, complemented
22C2×#6450C411yescharacteristic, complemented
23C2×#6442—21no—
24C2×#6440—21no—
25C2×C7×C22⋊C8C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C2×C7⋊#95—21no—
28C2×C7⋊#95C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×Dic7⋊C8—21no—
31C2×C8×Dic7—21no—
32#89490—22no—
33#89490C2214yescomplemented
34#89420—22no—
35#89420C2214yescomplemented
36C7×#327C2214yes—
37#89332C2214yes—
38#89262C2214yescomplemented
39#89247C2214yescomplemented
40C2×#1500—71nomaximal, Sylow
41C22×C7×D4—21no—
42C22×C7×D4C2×C411yescharacteristic
43C23×C4×C7—21no—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7—21no—
46C23×Dic7—21no—
47C23×Dic7D411yescharacteristic
48C23×Dic7D411yescharacteristic
49C22×Dic14C2×C411yescharacteristic
50C22×C7×C8—21no—
51C7×#121—21no—
52C7×#121—21no—
53C7×#121C2×C412yes—
54C7×#121C2312yes—
55C2×C7×C4⋊C4C2×C411yescharacteristic
56C2×C7×C22⋊C4—21no—
57C2×C22⋊Dic7—21no—
58C2×C22⋊Dic7—21no—
59C2×C22⋊Dic7—21no—
60C2×C22⋊Dic7—21no—
61#6499—22no—
62#6499C2×C414yescomplemented
63#6498—24no—
64#6498—24no—
65D4×Dic7—24no—
66D4×Dic7—24no—
67D4×Dic7—22no—
68D4×Dic7C2×C414yescomplemented
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7C2×C411yescharacteristic
71#6487—21no—
72#6487—21no—
73#6487C2×C412yes—
74#6487C2312yes—
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×C4×Dic7—28no—
79C2×C4×Dic7—21no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7C2×C411yescharacteristic
82C2×C4×Dic7C2×C411yescharacteristic
83C2×C4×Dic7C2×C412yes—
84C2×C4×Dic7D412yes—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D412yes—
88C2×C4×Dic7C2312yes—
89C22×C7⋊C8—21no—
90#6453—22no—
91#6453C2×C414yescomplemented
92#6450—22no—
93#6450C2×C414yescomplemented
94#6442—24no—
95#6442—24no—
96#6440—24no—
97#6440—24no—
98C7×C22⋊C8—21no—
99C7×C22⋊C8C2312yes—
100C7×#95—21no—
101C7×#95C2312yes—
102C7⋊#95—21no—
103C7⋊#95—24no—
104C7⋊#95C2312yes—
105C22⋊(C7⋊C8)—21no—
106C22⋊(C7⋊C8)C2312yes—
107Dic7⋊C8—24no—
108C8×Dic7—24no—
109C2×#529—71no—
110C2×#431—71no—
111C2×#376—71no—
112#1500—78no—
113C2×#328—71no—
114C2×#328—71no—
115C2×#327—71no—
116C2×#327D711yescharacteristic
117C24×C7—21no—
118C24×C7C2×D411yescharacteristic
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4—21no—
122C2×C7×D4—42no—
123C2×C7×D4—42no—
124C2×C7×D4—42no—
125C2×C7×D4C22×C412yes—
126C22×C4×C7—24no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—42no—
130C22×C4×C7—42no—
131C22×C4×C7—42no—
132C22×C4×C7C22×C411yescharacteristic
133C22×C4×C7C22×C411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—24no—
141C22×Dic7—24no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—22no—
145C22×Dic7—24no—
146C22×Dic7—42no—
147C22×Dic7—42no—
148C22×Dic7—44no—
149C22×Dic7C22⋊C411yescharacteristic
150C22×Dic7C22⋊C411yescharacteristic
151C22×Dic7C22⋊C411yescharacteristic
152C22×Dic7C22⋊C412yes—
153C22×Dic7C22⋊C412yes—
154C22×Dic7C22⋊C411yescharacteristic
155C22×Dic7C2×D412yes—
156C22×Dic7C2×D412yes—
157C2×Dic14—21no—
158C2×Dic14—42no—
159C2×Dic14C22×C412yes—
160C2×C7×C8—22no—
161C2×C7×C8—42no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4C22×C412yes—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C22⋊Dic7—22no—
167C22⋊Dic7—22no—
168C22⋊Dic7—24no—
169C22⋊Dic7—24no—
170C22⋊Dic7—24no—
171C4⋊Dic7—21no—
172C4⋊Dic7—24no—
173C4⋊Dic7C22×C412yes—
174Dic7⋊C4—22no—
175Dic7⋊C4—22no—
176Dic7⋊C4—24no—
177Dic7⋊C4—24no—
178C4×Dic7—216no—
179C4×Dic7—24no—
180C4×Dic7C22⋊C412yes—
181C4×Dic7C22⋊C412yes—
182C4×Dic7C22⋊C412yes—
183C4×Dic7C22⋊C412yes—
184C4×Dic7C22×C412yes—
185C4×Dic7C22×C412yes—
186C4×Dic7C2×D412yes—
187C4×Dic7C2×D412yes—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—42no—
190#529—74no—
191#529—74no—
192#529—142no—
193#529—142no—
194C2×#126—141no—
195C2×#125—71no—
196C2×#124—71no—
197C2×#123—141no—
198C2×C22⋊Q8—71no—
199C2×#121C2×D711yescharacteristic
200C2×C4×Q8—71no—
201C2×C4×D4—71no—
202C2×C4×D4—141no—
203C2×#117—141no—
204C22×C42—71no—
205C22×C42—141no—
206#431—74no—
207#431—142no—
208C2×C4⋊C8—141no—
209C2×C22⋊C8—71no—
210C2×C22⋊C8C2×D711yescharacteristic
211C2×C4×C8—141no—
212#376—74no—
213#376—142no—
214C2×#95—71no—
215C2×#95—141no—
216C2×#95C2×D711yescharacteristic
217#328—74no—
218#328—74no—
219#327—74no—
220#327C2×D714yes—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7C2×C22⋊C411yescharacteristic
228C23×C7C2×C22⋊C411yescharacteristic
229C23×C7#13512yes—
230C23×C7#13512yes—
231C23×C7C22×D411yescharacteristic
232C7×D4—42no—
233C7×D4—42no—
234C7×D4—42no—
235C7×D4—42no—
236C7×D4—44no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—48no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7C2×C22⋊C411yescharacteristic
250C2×C4×C7C2×C22⋊C411yescharacteristic
251C2×C4×C7C23×C411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—24no—
256C2×Dic7—22no—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—42no—
265C2×Dic7—48no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—48no—
270C2×Dic7#10414yes—
271C2×Dic7C2×C22⋊C412yes—
272C2×Dic7C2×C22⋊C412yes—
273C2×Dic7C2×C22⋊C412yes—
274C2×Dic7C2×C22⋊C412yes—
275Dic14—42no—
276Dic14—42no—
277C7×C8—42no—
278C7⋊C8—42no—
279C22×Q8—71no—
280C22×D4—21no—
281C22×D4C4×D711yescharacteristic
282C23×C4—21no—
283C23×C4—71no—
284C23×C4—71no—
285C23×C4—141no—
286C23×C4—141no—
287C23×C4C22×D711yescharacteristic
288C22×C8—21no—
289C22×C8—141no—
290#126—144no—
291#126—144no—
292#125—74no—
293#125—142no—
294#124—74no—
295#124—142no—
296#123—144no—
297#123—144no—
298C22⋊Q8—72no—
299C22⋊Q8—72no—
300C22⋊Q8—141no—
301C22⋊Q8—141no—
302#121—21no—
303#121—21no—
304#121C4×D712yes—
305#121C22×D712yes—
306C4×Q8—74no—
307C4×Q8—142no—
308C4×D4—74no—
309C4×D4—142no—
310C4×D4—144no—
311C4×D4—144no—
312#117—144no—
313#117—144no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4C4×D711yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—141no—
322C2×C22⋊C4—141no—
323C2×C22⋊C4—141no—
324C2×C22⋊C4—141no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42—72no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—72no—
331C2×C42—72no—
332C2×C42—72no—
333C2×C42—141no—
334C2×C42—144no—
335C2×C42—148no—
336C4⋊C8—144no—
337C22⋊C8—21no—
338C22⋊C8—72no—
339C22⋊C8—141no—
340C22⋊C8C22×D712yes—
341C4×C8—144no—
342#95—21no—
343#95—72no—
344#95—144no—
345#95—141no—
346#95C22×D712yes—
347C22×C7—21no—
348C22×C7—21no—
349C22×C7—21no—
350C22×C7—21no—
351C22×C7—24no—
352C22×C7—24no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—84no—
357C22×C7#41012yes—
358C22×C7#41612yes—
359C22×C7#41612yes—
360C22×C7C2×#10412yes—
361C22×C7C22×C22⋊C411yescharacteristic
362C22×C7C2×#13511yescharacteristic
363C22×C7C2×#13511yescharacteristic
364C4×C7—21no—
365C4×C7—21no—
366C4×C7—21no—
367C4×C7—21no—
368C4×C7—44no—
369C4×C7—42no—
370Dic7—22no—
371Dic7—22no—
372Dic7—22no—
373Dic7—44no—
374Dic7—42no—
375Dic7—42no—
376Dic7C2×#10414yes—
377C24—21no—
378C24D4×D711yescharacteristic
379C2×Q8—72no—
380C2×Q8—141no—
381C2×Q8—282no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4C2×C4×D712yes—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—71no—
396C22×C4—72no—
397C22×C4—72no—
398C22×C4—71no—
399C22×C4—71no—
400C22×C4—71no—
401C22×C4—72no—
402C22×C4—72no—
403C22×C4—144no—
404C22×C4—144no—
405C22×C4—142no—
406C22×C4—142no—
407C22×C4—144no—
408C22×C4—141no—
409C22×C4—141no—
410C22×C4—142no—
411C22×C4—284no—
412C22×C4—282no—
413C22×C4—282no—
414C22×C4C2×C4×D711yescharacteristic
415C22×C4C2×C4×D711yescharacteristic
416C22×C4D4×D711yescharacteristic
417C22×C4D4×D711yescharacteristic
418C22×C4D4×D711yescharacteristic
419C22×C4C23×D711yescharacteristic
420C2×C8—22no—
421C2×C8—42no—
422C2×C8—142no—
423C2×C8—282no—
424C4⋊C4—21no—
425C4⋊C4—72no—
426C4⋊C4—144no—
427C4⋊C4—144no—
428C4⋊C4—144no—
429C4⋊C4—142no—
430C4⋊C4—141no—
431C4⋊C4—142no—
432C4⋊C4C2×C4×D712yes—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—144no—
436C22⋊C4—144no—
437C22⋊C4—144no—
438C22⋊C4—142no—
439C22⋊C4—142no—
440C42—72no—
441C42—72no—
442C42—72no—
443C42—72no—
444C42—72no—
445C42—72no—
446C42—72no—
447C42—72no—
448C42—144no—
449C42—1416no—
450C2×C7—21no—
451C2×C7—21no—
452C2×C7—22no—
453C2×C7—82no—
454C2×C7#150014yescomplemented
455C2×C7C2×#41011yescharacteristic
456C2×C7C2×#41611yescharacteristic
457C2×C7C22×#10411yescharacteristic
458C23—22no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23D7×C22⋊C411yescharacteristic
465C23D7×C22⋊C411yescharacteristic
466C23#647512yes—
467C23#647912yes—
468C23C2×D4×D711yescharacteristic
469Q8—282no—
470Q8—282no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—44no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—72no—
489C2×C4—74no—
490C2×C4—72no—
491C2×C4—72no—
492C2×C4—72no—
493C2×C4—144no—
494C2×C4—144no—
495C2×C4—142no—
496C2×C4—142no—
497C2×C4—144no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—142no—
501C2×C4—144no—
502C2×C4—142no—
503C2×C4—142no—
504C2×C4—288no—
505C2×C4—282no—
506C2×C4—288no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4D7×C22⋊C411yescharacteristic
511C2×C4D7×C22⋊C411yescharacteristic
512C2×C4C22×C4×D711yescharacteristic
513C2×C4C2×D4×D711yescharacteristic
514C8—42no—
515C8—282no—
516C7C2×#150011yescharacteristic, complemented, Sylow
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—24no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—84no—
527C22#8951212yes—
528C22#8954412yes—
529C22#8954412yes—
530C22D7×#10412yes—
531C22C2×D7×C22⋊C411yescharacteristic
532C22C2×#647511yescharacteristic
533C22C2×#647911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—44no—
539C4—42no—
540C4—74no—
541C4—142no—
542C4—142no—
543C4—142no—
544C4—284no—
545C4—282no—
546C4—282no—
547C2—21no—
548C2—21no—
549C2—22no—
550C2—82no—
551C2#1172603014yesdirect product factor
552C2C2×#8951211yescharacteristic
553C2C2×#8954411yescharacteristic
554C2C2×D7×#10411yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#327C2
socleC23×C7C2×C22⋊C4
prime radicalC2×C7×#121C22
cosocleC22×C4×C7C24
hypercenterC2×#327D7
hypocenterC7C2×#1500
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#89490C2semidirect
3. C2×#89420C2semidirect
6. C2×#89262C2semidirect
7. C2×#89247C2semidirect
8. #11726030C2direct (N × H)
10. C2×#6499C4semidirect
13. C2×D4×Dic7C4semidirect
19. #90291C4semidirect
21. C2×#6453C4semidirect
22. C2×#6450C4semidirect
33. #89490C22semidirect
35. #89420C22semidirect
38. #89262C22semidirect
39. #89247C22semidirect
62. #6499C2×C4semidirect
68. D4×Dic7C2×C4semidirect
91. #6453C2×C4semidirect
93. #6450C2×C4semidirect
454. C2×C7#1500semidirect
516. C7C2×#1500semidirect
551. C2#11726030direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902912
C2×#894902
C2×#894202
C2×C7×#3272
C2×#893322
C2×#892622
C2×#892472
#117260302
C2×#15007