Finite Group Databasebeta

← C2×#11741091 #14232280

Subgroups — C2×#11741091 #14232280

Order 1792.

1918 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C2×#90233C211yesmaximal, characteristic, complemented
3#11741091C2116yesmaximal, direct product factor
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89851C211yesmaximal, characteristic, complemented
6C2×#89813C212yesmaximal, complemented
7C2×#89807C212yesmaximal, complemented
8C2×C4×#1142C211yesmaximal, characteristic, complemented
9C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
10C2×#89779C211yesmaximal, characteristic, complemented
11C2×#89768C212yesmaximal, complemented
12C2×#89770C212yesmaximal, complemented
13C7×#555—24no—
14C7×#555C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#124—22no—
17C2×C7×#121—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22C2×C7×#117C2212yes—
23C2×#6499—21no—
24C22×#1142C2211yescharacteristic
25C2×#6489C2212yes—
26C2×#6487C2212yes—
27C23×C7⋊C8C2212yes—
28#90233—24no—
29#90233C2218yescomplemented
30C2×#6450—21no—
31C2×#6442—22no—
32C2×#6432C2211yescharacteristic
33C2×C4×Dic14C2211yescharacteristic
34C2×#6406C2212yes—
35C2×#6406C2211yescharacteristic
36C2×C22⋊(C7⋊C8)C2212yes—
37#89863C2218yescomplemented
38#89851C2218yescomplemented
39#89813C22116yescomplemented
40#89807C22116yescomplemented
41C4×#1142C2218yescomplemented
42D4×C7⋊C8C2218yescomplemented
43#89779C2218yescomplemented
44#89770C22116yescomplemented
45#89768C22116yescomplemented
46C2×#6384C2211yescharacteristic
47C2×#6384C2212yes—
48C2×#6382C2211yescharacteristic
49C2×#6382C2212yes—
50C2×#6382C2212yes—
51C2×#6381C2212yes—
52C2×#6381C2212yes—
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3340—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×D4—22no—
58C22×C7×D4D411yescharacteristic
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C23×C4×C7C2312yes—
62C22×Dic14—21no—
63C22×Dic14C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8C2314yes—
66C7×#124—216no—
67C7×#121—28no—
68C7×#121—24no—
69C7×#121C2318yes—
70C4×C7×D4—22no—
71C4×C7×D4D414yes—
72C4×C7×D4C2314yes—
73C4×C7×D4C2314yes—
74C7×#117—24no—
75C7×#117—28no—
76C7×#117C2318yes—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C22⋊C4—22no—
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4D412yes—
85C2×C42×C7—22no—
86C2×C42×C7C2311yescharacteristic
87C2×C22⋊Dic7—22no—
88C2×C22⋊Dic7—22no—
89#6499—28no—
90C2×#1142—24no—
91C2×#1142C2314yes—
92#6489—24no—
93#6489C2318yes—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2312yes—
97#6487—24no—
98#6487C2318yes—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—22no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—22no—
103C22×C7⋊C8—28no—
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2312yes—
106C22×C7⋊C8C2318yes—
107C22×C7⋊C8C2311yescharacteristic
108#6450—28no—
109#6442—216no—
110#6432—22no—
111#6432C2314yes—
112C4×Dic14—22no—
113C4×Dic14C2314yes—
114#6406C2318yes—
115#6406C2314yes—
116C22⋊(C7⋊C8)C2318yes—
117#6384C2318yes—
118#6384C2314yes—
119#6382C2318yes—
120#6382C2314yes—
121#6382C2318yes—
122#6381C2318yes—
123#6381C2318yes—
124C4⋊(C7⋊C8)C2314yes—
125C4×C7⋊C8C2314yes—
126C2×#556—71no—
127C2×#555D711yescharacteristic
128#3340—716no—
129C2×#498—71no—
130C2×#485—72no—
131C2×#482—72no—
132C2×#477—71no—
133C2×#474—71no—
134C2×#465—72no—
135C2×C4×SD16—71no—
136C2×C8×D4—71no—
137C2×#427—72no—
138C24×C7—21no—
139C24×C7C2×D412yes—
140C2×C7×Q8—24no—
141C2×C7×Q8—82no—
142C2×C7×D4—28no—
143C2×C7×D4—24no—
144C2×C7×D4—21no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—84no—
148C2×C7×D4C2×D414yes—
149C2×C7×D4C2412yes—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—21no—
153C22×C4×C7—22no—
154C22×C4×C7—24no—
155C22×C4×C7—44no—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D414yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2414yes—
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—22no—
165C2×Dic14—22no—
166C2×Dic14—22no—
167C2×Dic14—21no—
168C2×Dic14—24no—
169C2×Dic14—44no—
170C2×Dic14C2412yes—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4C2×D414yes—
175C7×C4⋊C4C2412yes—
176C7×C4⋊C4C2414yes—
177C7×C22⋊C4—28no—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4C2×D418yes—
181C42×C7—28no—
182C42×C7C2×D412yes—
183C42×C7C2412yes—
184C22⋊Dic7—28no—
185C22⋊Dic7—28no—
186#1142—28no—
187C4⋊Dic7—21no—
188C4⋊Dic7—22no—
189C4⋊Dic7—24no—
190C4⋊Dic7C2412yes—
191C4⋊Dic7C2414yes—
192Dic7⋊C4—24no—
193Dic7⋊C4—28no—
194C4×Dic7—24no—
195C4×Dic7—28no—
196C2×C7⋊C8—24no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—216no—
199C2×C7⋊C8#4114yes—
200C2×C7⋊C8C2412yes—
201C2×C7⋊C8C2414yes—
202C2×C7⋊C8C2412yes—
203C22×SD16—71no—
204C23×C8—72no—
205#556—78no—
206#556—144no—
207#555—24no—
208#555C2×D718yes—
209C2×C4⋊Q8—71no—
210C2×C4⋊Q8C2×D711yescharacteristic
211C2×#126—142no—
212C2×#124—22no—
213C2×#124—141no—
214C2×C22⋊Q8—72no—
215C2×#121—21no—
216C2×#121C2×D712yes—
217C2×C4×Q8—71no—
218C2×C4×Q8—141no—
219C2×C4×D4C2×D711yescharacteristic
220C2×C4×D4C2×D711yescharacteristic
221C2×#117—21no—
222C2×#117—72no—
223C2×#117C2×D712yes—
224#498—78no—
225#485—716no—
226#482—716no—
227#477—78no—
228#474—78no—
229#465—716no—
230C4×SD16—78no—
231C8×D4—78no—
232#427—716no—
233C2×#107—72no—
234C2×#107—72no—
235C2×#106—72no—
236C2×#106—72no—
237C2×#106—71no—
238C2×C4⋊C8—71no—
239C2×#103—72no—
240C2×#103—71no—
241C2×#102—71no—
242C2×#102—72no—
243C2×C22⋊C8—72no—
244C2×C4×C8—71no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—22no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7C22×D414yes—
253C23×C7C22×D411yescharacteristic
254C7×Q8—84no—
255C7×D4—28no—
256C7×D4—22no—
257C7×D4—22no—
258C7×D4—42no—
259C7×D4—88no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—28no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—42no—
268C2×C4×C7—44no—
269C2×C4×C7—48no—
270C2×C4×C7#13514yes—
271C2×C4×C7C22×D414yes—
272C2×C4×C7C22×D412yes—
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D412yes—
275C2×C4×C7C2×#4112yes—
276C2×C4×C72−1+412yes—
277C2×C4×C7C2511yescharacteristic
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—24no—
281C2×Dic7—42no—
282C2×Dic7—44no—
283C2×Dic7—42no—
284Dic14—22no—
285Dic14—22no—
286Dic14—42no—
287Dic14—48no—
288C7⋊C8—24no—
289C7⋊C8—22no—
290C7⋊C8C2×#4114yes—
291C22×Q8—21no—
292C22×Q8—71no—
293C22×Q8—141no—
294C22×D4—22no—
295C22×D4C22⋊D711yescharacteristic
296C22×D4C22×D711yescharacteristic
297C23×C4—21no—
298C23×C4C22×D712yes—
299C2×SD16—74no—
300C2×SD16—144no—
301C22×C8—71no—
302C22×C8—72no—
303C22×C8—78no—
304C22×C8—71no—
305C22×C8—148no—
306C4⋊Q8—22no—
307C4⋊Q8—74no—
308C4⋊Q8—142no—
309C4⋊Q8C22×D714yes—
310#126—1416no—
311#124—216no—
312#124—148no—
313C22⋊Q8—78no—
314C22⋊Q8—144no—
315#121—28no—
316#121—24no—
317#121C22×D718yes—
318C4×Q8—74no—
319C4×Q8—148no—
320C4×Q8—142no—
321C4×D4—22no—
322C4×D4C22⋊D714yes—
323C4×D4C22×D714yes—
324C4×D4C22×D714yes—
325#117—24no—
326#117—28no—
327#117—78no—
328#117—144no—
329#117C22×D718yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—72no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—142no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22⋊D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D712yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4C22⋊D712yes—
345C2×C42—22no—
346C2×C42—142no—
347C2×C42—141no—
348C2×C42C22×D711yescharacteristic
349#107—78no—
350#107—78no—
351#106—74no—
352#106—78no—
353#106—78no—
354C4⋊C8—74no—
355#103—74no—
356#103—78no—
357#102—74no—
358#102—78no—
359C22⋊C8—78no—
360C4×C8—74no—
361C22×C7—24no—
362C22×C7—24no—
363C22×C7—24no—
364C22×C7—48no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—84no—
369C22×C7#54712yes—
370C22×C7C2×#13512yes—
371C22×C7#57712yes—
372C22×C7C23×D411yescharacteristic
373C4×C7—24no—
374C4×C7—22no—
375C4×C7—42no—
376C4×C7—44no—
377C4×C7#54712yes—
378C4×C7C2×#13514yes—
379C4×C7C22×#4111yescharacteristic
380C4×C7C2×2−1+411yescharacteristic
381Dic7—42no—
382Dic7—44no—
383Dic7—42no—
384C24—21no—
385C24C2×C22⋊D712yes—
386C2×Q8—24no—
387C2×Q8—72no—
388C2×Q8—82no—
389C2×Q8—141no—
390C2×Q8—144no—
391C2×Q8—142no—
392C2×Q8—142no—
393C2×Q8—284no—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—84no—
400C2×D4C2×C22⋊D714yes—
401C2×D4C23×D712yes—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—141no—
409C22×C4—142no—
410C22×C4—141no—
411C22×C4C2×C22⋊D714yes—
412C22×C4C2×C22⋊D712yes—
413C22×C4C2×C22⋊D711yescharacteristic
414C22×C4C2×C22⋊D711yescharacteristic
415C22×C4C23×D714yes—
416C22×C4C23×D711yescharacteristic
417SD16—148no—
418C2×C8—74no—
419C2×C8—72no—
420C2×C8—72no—
421C2×C8—74no—
422C2×C8—1416no—
423C2×C8—144no—
424C2×C8—142no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—74no—
429C4⋊C4—72no—
430C4⋊C4—144no—
431C4⋊C4—148no—
432C4⋊C4—141no—
433C4⋊C4—142no—
434C4⋊C4—144no—
435C4⋊C4C2×C22⋊D714yes—
436C4⋊C4C23×D714yes—
437C4⋊C4C23×D712yes—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—24no—
441C22⋊C4—148no—
442C22⋊C4—148no—
443C22⋊C4C2×C22⋊D718yes—
444C42—28no—
445C42—148no—
446C42—144no—
447C42C2×C22⋊D712yes—
448C42C23×D712yes—
449C2×C7—44no—
450C2×C7—82no—
451C2×C7#334014yescomplemented
452C2×C7C2×#54711yescharacteristic
453C2×C7C22×#13511yescharacteristic
454C2×C7C2×#57711yescharacteristic
455C23—24no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23C22×C22⋊D714yes—
463C23C22×C22⋊D711yescharacteristic
464Q8—84no—
465Q8—142no—
466Q8—142no—
467Q8—288no—
468Q8—282no—
469D4—28no—
470D4—22no—
471D4—22no—
472D4—42no—
473D4—88no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—142no—
485C2×C4—144no—
486C2×C4—142no—
487C2×C4—284no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4#651314yes—
491C2×C4D7×#4112yes—
492C2×C4#655412yes—
493C2×C4C22×C22⋊D714yes—
494C2×C4C22×C22⋊D712yes—
495C2×C4C22×C22⋊D712yes—
496C2×C4C22×C22⋊D711yescharacteristic
497C2×C4C24×D711yescharacteristic
498C8—74no—
499C8—144no—
500C8—142no—
501C7C2×#334011yescharacteristic, complemented, Sylow
502C22—24no—
503C22—24no—
504C22—24no—
505C22—48no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22C2×#651312yes—
511C22#9051512yes—
512C22#9052712yes—
513C22C23×C22⋊D711yescharacteristic
514C4—24no—
515C4—22no—
516C4—42no—
517C4—44no—
518C4—284no—
519C4—282no—
520C4—282no—
521C4C2×#651314yes—
522C4#9051512yes—
523C4C2×D7×#4111yescharacteristic
524C4C2×#655411yescharacteristic
525C2—44no—
526C2—82no—
527C2#1174109114yesdirect product factor
528C2C22×#651311yescharacteristic
529C2C2×#9051511yescharacteristic
530C2C2×#9052711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
Fitting, prime radicalC2×C7×#555C2
socleC23×C7C22×D4
hypercenterC2×#555D7
hypocenterC7C2×#3340
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
2. C2×#90233C2semidirect
3. #11741091C2direct (N × H)
4. C2×#89863C2semidirect
5. C2×#89851C2semidirect
6. C2×#89813C2semidirect
7. C2×#89807C2semidirect
8. C2×C4×#1142C2semidirect
9. C2×D4×C7⋊C8C2semidirect
10. C2×#89779C2semidirect
11. C2×#89768C2semidirect
12. C2×#89770C2semidirect
29. #90233C22semidirect
37. #89863C22semidirect
38. #89851C22semidirect
39. #89813C22semidirect
40. #89807C22semidirect
41. C4×#1142C22semidirect
42. D4×C7⋊C8C22semidirect
43. #89779C22semidirect
44. #89770C22semidirect
45. #89768C22semidirect
451. C2×C7#3340semidirect
501. C7C2×#3340semidirect
527. C2#11741091direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5552
C2×#902332
#117410912
C2×#898632
C2×#898512
C2×#898132
C2×#898072
C2×C4×#11422
C2×D4×C7⋊C82
C2×#897792
C2×#897682
C2×#897702
C2×#33407