Finite Group Databasebeta

← C2×#11741096 #14232540

Subgroups — C2×#11741096 #14232540

Order 1792.

1930 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C2×#90234C211yesmaximal, characteristic, complemented
3#11741096C2116yesmaximal, direct product factor
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89850C211yesmaximal, characteristic, complemented
6C2×#89814C212yesmaximal, complemented
7C2×#89805C212yesmaximal, complemented
8C2×C4×#1142C211yesmaximal, characteristic, complemented
9C2×#89784C211yesmaximal, characteristic, complemented
10C2×#89780C211yesmaximal, characteristic, complemented
11C2×#89768C212yesmaximal, complemented
12C2×#89767C212yesmaximal, 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×D4×Dic7—21no—
24C22×#1142C2211yescharacteristic
25C22×C4⋊Dic7C2212yes—
26C2×#6487C2212yes—
27C22×#1136C2212yes—
28#90234—24no—
29#90234C2218yescomplemented
30C2×#6453—21no—
31C2×#6441—22no—
32C2×#6432C2211yescharacteristic
33C2×C4×Dic14C2211yescharacteristic
34C2×#6406C2212yes—
35C2×#6406C2211yescharacteristic
36C2×C22⋊(C7⋊C8)C2212yes—
37#89863C2218yescomplemented
38#89850C2218yescomplemented
39#89814C22116yescomplemented
40#89805C22116yescomplemented
41C4×#1142C2218yescomplemented
42#89784C2218yescomplemented
43#89780C2218yescomplemented
44#89768C22116yescomplemented
45#89767C22116yescomplemented
46C2×#6384C2211yescharacteristic
47C2×#6384C2212yes—
48C2×#6382C2211yescharacteristic
49C2×#6382C2212yes—
50C2×#6381C2212yes—
51C2×#6381C2212yes—
52C2×#6381C2212yes—
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3357—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×D4—22no—
58C22×C7×D4D411yescharacteristic
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C23×C4×C7C2312yes—
62C23×Dic7—22no—
63C22×Dic14—21no—
64C22×Dic14C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8C2314yes—
67C7×#124—216no—
68C7×#121—28no—
69C7×#121—24no—
70C7×#121C2318yes—
71C4×C7×D4—22no—
72C4×C7×D4D414yes—
73C4×C7×D4C2314yes—
74C4×C7×D4C2314yes—
75C7×#117—24no—
76C7×#117—28no—
77C7×#117C2318yes—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4D412yes—
86C2×C42×C7—22no—
87C2×C42×C7C2311yescharacteristic
88C2×C22⋊Dic7—22no—
89D4×Dic7—28no—
90C2×#1142—24no—
91C2×#1142C2314yes—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—24no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2318yes—
98#6487—24no—
99#6487C2318yes—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—22no—
102C2×C4×Dic7—21no—
103C2×#1136—28no—
104C2×#1136C2318yes—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2312yes—
107C22×C7⋊C8C2311yescharacteristic
108#6453—28no—
109#6441—216no—
110#6432—22no—
111#6432C2314yes—
112C4×Dic14—22no—
113C4×Dic14C2314yes—
114#6406C2318yes—
115#6406C2314yes—
116C22⋊(C7⋊C8)C2318yes—
117#6384C2318yes—
118#6384C2314yes—
119#6382C2314yes—
120#6382C2318yes—
121#6381C2318yes—
122#6381C2318yes—
123#6381C2318yes—
124C4⋊(C7⋊C8)C2314yes—
125C4×C7⋊C8C2314yes—
126C2×#555D711yescharacteristic
127C2×#554—71no—
128#3357—716no—
129C2×#500—71no—
130C2×#485—72no—
131C2×#480—72no—
132C2×#477—71no—
133C2×#476—71no—
134C2×#468—72no—
135C2×C4×SD16—71no—
136C2×#436—71no—
137C2×#428—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—
165C22×Dic7—28no—
166C22×Dic7—84no—
167C2×Dic14—22no—
168C2×Dic14—22no—
169C2×Dic14—21no—
170C2×Dic14—24no—
171C2×Dic14—82no—
172C2×Dic14C2412yes—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4C2×D414yes—
177C7×C4⋊C4C2412yes—
178C7×C4⋊C4C2414yes—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—28no—
181C7×C22⋊C4—24no—
182C7×C22⋊C4C2×D418yes—
183C42×C7—28no—
184C42×C7C2×D412yes—
185C42×C7C2412yes—
186C22⋊Dic7—28no—
187#1142—28no—
188C4⋊Dic7—21no—
189C4⋊Dic7—22no—
190C4⋊Dic7—24no—
191C4⋊Dic7—28no—
192C4⋊Dic7C2412yes—
193C4⋊Dic7C2414yes—
194Dic7⋊C4—24no—
195Dic7⋊C4—28no—
196C4×Dic7—24no—
197#1136—216no—
198C2×C7⋊C8—24no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8#4114yes—
201C2×C7⋊C8C2412yes—
202C2×C7⋊C8C2414yes—
203C2×C7⋊C8C2412yes—
204C22×SD16—71no—
205C22×M16—72no—
206#555—24no—
207#555C2×D718yes—
208#554—78no—
209#554—144no—
210C2×C4⋊Q8—71no—
211C2×C4⋊Q8C2×D711yescharacteristic
212C2×#125—141no—
213C2×#124—22no—
214C2×C22⋊Q8—72no—
215C2×C22⋊Q8—142no—
216C2×#121—21no—
217C2×#121C2×D712yes—
218C2×C4×Q8—71no—
219C2×C4×D4—141no—
220C2×C4×D4C2×D711yescharacteristic
221C2×C4×D4C2×D711yescharacteristic
222C2×#117—21no—
223C2×#117C2×D712yes—
224C22×C4⋊C4—72no—
225#500—78no—
226#485—716no—
227#480—716no—
228#477—78no—
229#476—78no—
230#468—716no—
231C4×SD16—78no—
232#436—78no—
233#428—716no—
234C2×#107—72no—
235C2×#107—72no—
236C2×#107—72no—
237C2×#106—72no—
238C2×#106—71no—
239C2×C4⋊C8—71no—
240C2×#103—72no—
241C2×#103—71no—
242C2×#102—71no—
243C2×#102—72no—
244C2×C22⋊C8—72no—
245C2×C4×C8—71no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7—22no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7C22×D414yes—
254C23×C7C22×D411yescharacteristic
255C7×Q8—84no—
256C7×D4—28no—
257C7×D4—22no—
258C7×D4—22no—
259C7×D4—42no—
260C7×D4—88no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—28no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—42no—
269C2×C4×C7—44no—
270C2×C4×C7—48no—
271C2×C4×C7#13714yes—
272C2×C4×C7C22×D414yes—
273C2×C4×C7C22×D412yes—
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C22×D412yes—
276C2×C4×C7C2×#4112yes—
277C2×C4×C72−1+412yes—
278C2×C4×C7C2511yescharacteristic
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—24no—
282C2×Dic7—42no—
283C2×Dic7—44no—
284C2×Dic7—42no—
285C2×Dic7—88no—
286Dic14—22no—
287Dic14—22no—
288Dic14—42no—
289Dic14—84no—
290C7⋊C8—24no—
291C7⋊C8—22no—
292C7⋊C8C2×#4114yes—
293C22×Q8—21no—
294C22×Q8—71no—
295C22×Q8—141no—
296C22×D4—22no—
297C22×D4C22⋊D711yescharacteristic
298C22×D4C22×D711yescharacteristic
299C23×C4—21no—
300C23×C4—142no—
301C23×C4C22×D712yes—
302C2×SD16—74no—
303C2×SD16—144no—
304C2×M16—78no—
305C2×M16—148no—
306C22×C8—71no—
307C22×C8—72no—
308C22×C8—71no—
309C4⋊Q8—22no—
310C4⋊Q8—74no—
311C4⋊Q8—142no—
312C4⋊Q8C22×D714yes—
313#125—148no—
314#124—216no—
315C22⋊Q8—78no—
316C22⋊Q8—144no—
317C22⋊Q8—1416no—
318#121—28no—
319#121—24no—
320#121C22×D718yes—
321C4×Q8—74no—
322C4×Q8—142no—
323C4×D4—22no—
324C4×D4—148no—
325C4×D4C22⋊D714yes—
326C4×D4C22×D714yes—
327C4×D4C22×D714yes—
328#117—24no—
329#117—28no—
330#117C22×D718yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—78no—
335C2×C4⋊C4—142no—
336C2×C4⋊C4—144no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4C22⋊D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D712yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—142no—
347C2×C22⋊C4C22⋊D712yes—
348C2×C42—22no—
349C2×C42—141no—
350C2×C42C22×D711yescharacteristic
351#107—78no—
352#107—78no—
353#107—78no—
354#106—74no—
355#106—78no—
356C4⋊C8—74no—
357#103—74no—
358#103—78no—
359#102—74no—
360#102—78no—
361C22⋊C8—78no—
362C4×C8—74no—
363C22×C7—24no—
364C22×C7—24no—
365C22×C7—24no—
366C22×C7—48no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—84no—
371C22×C7#54712yes—
372C22×C7C2×#13712yes—
373C22×C7#57612yes—
374C22×C7C23×D411yescharacteristic
375C4×C7—24no—
376C4×C7—22no—
377C4×C7—42no—
378C4×C7—44no—
379C4×C7#54712yes—
380C4×C7C2×#13714yes—
381C4×C7C22×#4111yescharacteristic
382C4×C7C2×2−1+411yescharacteristic
383Dic7—42no—
384Dic7—44no—
385Dic7—42no—
386C24—21no—
387C24C2×C22⋊D712yes—
388C2×Q8—24no—
389C2×Q8—72no—
390C2×Q8—82no—
391C2×Q8—141no—
392C2×Q8—144no—
393C2×Q8—142no—
394C2×Q8—142no—
395C2×Q8—562no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—84no—
402C2×D4C2×C22⋊D714yes—
403C2×D4C23×D712yes—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—142no—
413C22×C4—148no—
414C22×C4—564no—
415C22×C4C2×C22⋊D714yes—
416C22×C4C2×C22⋊D712yes—
417C22×C4C2×C22⋊D711yescharacteristic
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C23×D714yes—
420C22×C4C23×D711yescharacteristic
421SD16—148no—
422M16—1416no—
423C2×C8—74no—
424C2×C8—72no—
425C2×C8—72no—
426C2×C8—74no—
427C2×C8—144no—
428C2×C8—142no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—72no—
433C4⋊C4—74no—
434C4⋊C4—144no—
435C4⋊C4—148no—
436C4⋊C4—141no—
437C4⋊C4—148no—
438C4⋊C4—142no—
439C4⋊C4—144no—
440C4⋊C4C2×C22⋊D714yes—
441C4⋊C4C23×D712yes—
442C4⋊C4C23×D714yes—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—24no—
446C22⋊C4—148no—
447C22⋊C4C2×C22⋊D718yes—
448C42—28no—
449C42—144no—
450C42C2×C22⋊D712yes—
451C42C23×D712yes—
452C2×C7—44no—
453C2×C7—82no—
454C2×C7#335714yescomplemented
455C2×C7C2×#54711yescharacteristic
456C2×C7C22×#13711yescharacteristic
457C2×C7C2×#57611yescharacteristic
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22×C22⋊D714yes—
466C23C22×C22⋊D711yescharacteristic
467Q8—84no—
468Q8—142no—
469Q8—142no—
470Q8—282no—
471Q8—564no—
472D4—28no—
473D4—22no—
474D4—22no—
475D4—42no—
476D4—88no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—142no—
488C2×C4—142no—
489C2×C4—144no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—284no—
493C2×C4—568no—
494C2×C4#651414yes—
495C2×C4D7×#4112yes—
496C2×C4#655412yes—
497C2×C4C22×C22⋊D714yes—
498C2×C4C22×C22⋊D712yes—
499C2×C4C22×C22⋊D712yes—
500C2×C4C22×C22⋊D711yescharacteristic
501C2×C4C24×D711yescharacteristic
502C8—74no—
503C8—144no—
504C8—142no—
505C7C2×#335711yescharacteristic, complemented, Sylow
506C22—24no—
507C22—24no—
508C22—24no—
509C22—48no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22C2×#651412yes—
515C22#9051512yes—
516C22#9052612yes—
517C22C23×C22⋊D711yescharacteristic
518C4—24no—
519C4—22no—
520C4—42no—
521C4—44no—
522C4—282no—
523C4—282no—
524C4—284no—
525C4C2×#651414yes—
526C4#9051512yes—
527C4C2×D7×#4111yescharacteristic
528C4C2×#655411yescharacteristic
529C2—44no—
530C2—82no—
531C2#1174109614yesdirect product factor
532C2C22×#651411yescharacteristic
533C2C2×#9051511yescharacteristic
534C2C2×#9052611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
Fitting, prime radicalC2×C7×#555C2
socleC23×C7C22×D4
hypercenterC2×#555D7
hypocenterC7C2×#3357
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×#90234C2semidirect
3. #11741096C2direct (N × H)
4. C2×#89863C2semidirect
5. C2×#89850C2semidirect
6. C2×#89814C2semidirect
7. C2×#89805C2semidirect
8. C2×C4×#1142C2semidirect
9. C2×#89784C2semidirect
10. C2×#89780C2semidirect
11. C2×#89768C2semidirect
12. C2×#89767C2semidirect
29. #90234C22semidirect
37. #89863C22semidirect
38. #89850C22semidirect
39. #89814C22semidirect
40. #89805C22semidirect
41. C4×#1142C22semidirect
42. #89784C22semidirect
43. #89780C22semidirect
44. #89768C22semidirect
45. #89767C22semidirect
454. C2×C7#3357semidirect
505. C7C2×#3357semidirect
531. C2#11741096direct (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×#902342
#117410962
C2×#898632
C2×#898502
C2×#898142
C2×#898052
C2×C4×#11422
C2×#897842
C2×#897802
C2×#897682
C2×#897672
C2×#33577