Finite Group Databasebeta

← C2×#11741112 #14233364

Subgroups — C2×#11741112 #14233364

Order 1792.

1906 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×#556C211yesmaximal, characteristic, complemented
2C2×#90270C211yesmaximal, characteristic, complemented
3#11741112C2116yesmaximal, direct product factor
4C2×#89855C211yesmaximal, characteristic, complemented
5C2×#89853C211yesmaximal, characteristic, complemented
6C2×#89829C212yesmaximal
7C2×#89828C212yesmaximal, complemented
8C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
9C2×#89794C212yesmaximal, complemented
10C2×C4×#1143C212yesmaximal, complemented
11C2×#89780C211yesmaximal, characteristic, complemented
12C2×C4×#1141C211yesmaximal, characteristic
13C7×#556—24no—
14C7×#556C2218yescomplemented
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#126—22no—
17C2×C7×#124C2212yes—
18C2×C7×C22⋊Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C7×#117—21no—
23C2×#6509—21no—
24C22×#1143C2212yes—
25C22×#1141C2211yescharacteristic
26#90270—24no—
27#90270C2218yescomplemented
28C2×#6459—22no—
29C2×D7×C4⋊C4—21no—
30C2×C4×D28C2211yescharacteristic
31C2×C4×D28C2212yes—
32C2×#6433C2212yes—
33C2×#6432C2211yescharacteristic
34C2×#6409C2212yes—
35C2×#6409C2212yes—
36C2×#6409C2212yes—
37C2×#6406C2211yescharacteristic
38C2×#6406C2212yes—
39#89855C2218yescomplemented
40#89853C2218yescomplemented
41#89829C22116yes—
42#89828C22116yescomplemented
43C4×#1143C22116yescomplemented
44Q8×C7⋊C8C2218yescomplemented
45#89794C22116yescomplemented
46C4×#1141C2218yes—
47#89780C2218yescomplemented
48C2×#6383C2211yescharacteristic
49C2×#6382C2212yes—
50C2×#6381C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2212yes—
53C2×C4×C7⋊C8C2211yescharacteristic
54C2×C4×C7⋊C8C2212yes—
55C2×#3360—71nomaximal, Sylow
56C22×C7×Q8D411yescharacteristic
57C22×C7×Q8C2312yes—
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C4×D7—22no—
63C22×Dic14—21no—
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8C2314yes—
66C7×#126—216no—
67C7×#124—24no—
68C7×#124C2318yes—
69C7×C22⋊Q8—28no—
70C4×C7×Q8—24no—
71C4×C7×Q8D414yes—
72C4×C7×Q8C2318yes—
73C4×C7×Q8C2314yes—
74C4×C7×D4—22no—
75C4×C7×D4C2314yes—
76C7×#117—28no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4D412yes—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—22no—
86C2×C42×C7—22no—
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2312yes—
89#6509—28no—
90C2×#1143—28no—
91C2×#1143C2318yes—
92C2×#1141—24no—
93C2×#1141C2314yes—
94C2×(C2×D7)⋊C4—21no—
95C2×(C2×D7)⋊C4—22no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×Dic7⋊C4—22no—
100C2×Dic7⋊C4—22no—
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2312yes—
104#6459—216no—
105D7×C4⋊C4—28no—
106C4×D28—24no—
107C4×D28—22no—
108C4×D28C2314yes—
109C4×D28C2318yes—
110#6433—24no—
111#6433C2318yes—
112#6432—22no—
113#6432C2314yes—
114#6409C2318yes—
115#6409C2318yes—
116#6409C2318yes—
117#6406C2318yes—
118#6406C2314yes—
119#6383C2314yes—
120#6382C2318yes—
121#6381C2314yes—
122C4⋊(C7⋊C8)C2314yes—
123C4⋊(C7⋊C8)C2318yes—
124C4×C7⋊C8C2314yes—
125C4×C7⋊C8C2318yes—
126C2×#556D711yescharacteristic, complemented
127C2×#554—71no—
128#3360—716no—
129C2×#488—72no—
130C2×#487—71no—
131C2×#479—72no—
132C2×#476—71no—
133C2×#463—72no—
134C2×#461—71no—
135C2×C8×Q8—71no—
136C2×C4×SD16—72no—
137C2×C4×D8—71no—
138C24×C7—21no—
139C23×D7—21no—
140C2×C7×Q8—24no—
141C2×C7×Q8—22no—
142C2×C7×Q8—24no—
143C2×C7×Q8—24no—
144C2×C7×Q8C2×D414yes—
145C2×C7×Q8C2414yes—
146C2×C7×D4—21no—
147C2×C7×D4—22no—
148C2×C7×D4—22no—
149C2×C7×D4C2412yes—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—24no—
153C22×C4×C7—44no—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—22no—
161C2×D28—21no—
162C2×D28—22no—
163C2×D28#4114yes—
164C2×D28C2412yes—
165C2×C4×D7—24no—
166C2×C4×D7—28no—
167C2×C4×D7—44no—
168C2×C4×D7—84no—
169C2×Dic14—24no—
170C2×Dic14—82no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4C2×D414yes—
176C7×C4⋊C4C2×D418yes—
177C7×C4⋊C4C2412yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—28no—
181C42×C7—28no—
182C42×C7C2×D412yes—
183C42×C7C2×D414yes—
184C42×C7C2412yes—
185C42×C7C2414yes—
186#1143—216no—
187#1141—28no—
188(C2×D7)⋊C4—24no—
189(C2×D7)⋊C4—28no—
190C4⋊Dic7—22no—
191C4⋊Dic7—24no—
192C4⋊Dic7—21no—
193C4⋊Dic7C2414yes—
194C4⋊Dic7C2412yes—
195Dic7⋊C4—28no—
196Dic7⋊C4—28no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—24no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8C2412yes—
201C2×C7⋊C8C2414yes—
202C2×C7⋊C8C2412yes—
203C22×SD16—72no—
204C22×D8—71no—
205#556—24no—
206#556C2×D718yescomplemented
207#554—78no—
208#554—144no—
209C2×C4⋊Q8—71no—
210C2×C4⋊Q8C2×D711yescharacteristic
211C2×#126—22no—
212C2×#125—72no—
213C2×#124C2×D712yes—
214C2×C22⋊Q8—21no—
215C2×C22⋊Q8—141no—
216C2×C22⋊Q8—142no—
217C2×C4×Q8C2×D712yes—
218C2×C4×Q8C2×D711yescharacteristic
219C2×C4×D4—71no—
220C2×C4×D4—72no—
221C2×C4×D4C2×D711yescharacteristic
222C2×#117—21no—
223C22×C4⋊C4—141no—
224#488—716no—
225#487—78no—
226#479—716no—
227#476—78no—
228#463—716no—
229#461—78no—
230C8×Q8—78no—
231C4×SD16—716no—
232C4×D8—78no—
233C2×#107—71no—
234C2×#106—72no—
235C2×C4⋊C8—72no—
236C2×C4⋊C8—71no—
237C2×#103—72no—
238C2×#103—72no—
239C2×#103—72no—
240C2×#102—71no—
241C2×#102—72no—
242C2×#102—71no—
243C2×C4×C8—71no—
244C2×C4×C8—72no—
245C23×C7—22no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7C22×D411yescharacteristic
250C22×D7—24no—
251C22×D7—22no—
252C22×D7—42no—
253C22×D7—42no—
254C7×Q8—28no—
255C7×Q8—24no—
256C7×Q8—24no—
257C7×Q8—44no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—42no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—24no—
266C2×C4×C7—44no—
267C2×C4×C7—42no—
268C2×C4×C7—48no—
269C2×C4×C7#13514yes—
270C2×C4×C7C22×D414yes—
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—24no—
280C2×Dic7—42no—
281C2×Dic7—44no—
282D28—24no—
283D28—22no—
284D28C2×#4114yes—
285C4×D7—48no—
286C4×D7—88no—
287Dic14—84no—
288C7⋊C8—22no—
289C7⋊C8—24no—
290C7⋊C8—22no—
291C22×Q8—141no—
292C22×Q8C22⋊D711yescharacteristic
293C22×Q8C22×D712yes—
294C22×D4—71no—
295C22×D4C22×D711yescharacteristic
296C23×C4—21no—
297C23×C4—141no—
298C23×C4—142no—
299C2×SD16—78no—
300C2×SD16—148no—
301C2×D8—74no—
302C2×D8—144no—
303C22×C8—71no—
304C22×C8—72no—
305C22×C8—71no—
306C4⋊Q8—22no—
307C4⋊Q8—74no—
308C4⋊Q8—142no—
309C4⋊Q8C22×D714yes—
310#126—216no—
311#125—78no—
312#125—144no—
313#124—24no—
314#124C22×D718yes—
315C22⋊Q8—28no—
316C22⋊Q8—148no—
317C22⋊Q8—1416no—
318C4×Q8—24no—
319C4×Q8C22⋊D714yes—
320C4×Q8C22×D718yes—
321C4×Q8C22×D714yes—
322C4×D4—22no—
323C4×D4—74no—
324C4×D4—78no—
325C4×D4—142no—
326C4×D4—144no—
327C4×D4C22×D714yes—
328#117—28no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—142no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—148no—
338C2×C4⋊C4C22⋊D711yescharacteristic
339C2×C4⋊C4C22⋊D712yes—
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—142no—
346C2×C42—22no—
347C2×C42C22×D711yescharacteristic
348C2×C42C22×D712yes—
349#107—74no—
350#106—78no—
351C4⋊C8—74no—
352C4⋊C8—78no—
353#103—78no—
354#103—78no—
355#103—78no—
356#102—74no—
357#102—78no—
358#102—74no—
359C4×C8—74no—
360C4×C8—78no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—84no—
365C22×C7#54812yes—
366C22×C7C2×#13512yes—
367C22×C7#57712yes—
368C22×C7C23×D411yescharacteristic
369C2×D7—24no—
370C2×D7—48no—
371C2×D7—42no—
372C2×D7—42no—
373C4×C7—24no—
374C4×C7—22no—
375C4×C7—24no—
376C4×C7—44no—
377C4×C7—42no—
378C4×C7#54812yes—
379C4×C7C2×#13514yes—
380C4×C7C22×#4111yescharacteristic
381C4×C7C2×2−1+411yescharacteristic
382Dic7—42no—
383Dic7—44no—
384C24—21no—
385C24—141no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—144no—
391C2×Q8—562no—
392C2×Q8C2×C22⋊D714yes—
393C2×Q8C23×D714yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—74no—
398C2×D4—72no—
399C2×D4—142no—
400C2×D4—141no—
401C2×D4C23×D712yes—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—44no—
406C22×C4—141no—
407C22×C4—142no—
408C22×C4—144no—
409C22×C4—148no—
410C22×C4—284no—
411C22×C4—564no—
412C22×C4C2×C22⋊D712yes—
413C22×C4C2×C22⋊D712yes—
414C22×C4C2×C22⋊D711yescharacteristic
415C22×C4C2×C22⋊D711yescharacteristic
416C22×C4C23×D711yescharacteristic
417SD16—1416no—
418D8—148no—
419C2×C8—72no—
420C2×C8—74no—
421C2×C8—72no—
422C2×C8—142no—
423C2×C8—144no—
424C2×C8—142no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—74no—
430C4⋊C4—72no—
431C4⋊C4—148no—
432C4⋊C4—148no—
433C4⋊C4—141no—
434C4⋊C4—142no—
435C4⋊C4—144no—
436C4⋊C4C2×C22⋊D714yes—
437C4⋊C4C2×C22⋊D718yes—
438C4⋊C4C23×D712yes—
439C22⋊C4—28no—
440C22⋊C4—24no—
441C22⋊C4—28no—
442C22⋊C4—144no—
443C22⋊C4—148no—
444C42—28no—
445C42C2×C22⋊D712yes—
446C42C2×C22⋊D714yes—
447C42C23×D712yes—
448C42C23×D714yes—
449C2×C7—82no—
450C2×C7#336014yescomplemented
451C2×C7C2×#54811yescharacteristic
452C2×C7C22×#13511yescharacteristic
453C2×C7C2×#57711yescharacteristic
454D7—44no—
455C23—22no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—144no—
460C23—142no—
461C23—282no—
462C23—282no—
463C23C22×C22⋊D711yescharacteristic
464Q8—28no—
465Q8—24no—
466Q8—24no—
467Q8—44no—
468Q8—564no—
469D4—22no—
470D4—22no—
471D4—42no—
472D4—74no—
473D4—144no—
474D4—142no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—48no—
483C2×C4—142no—
484C2×C4—144no—
485C2×C4—282no—
486C2×C4—284no—
487C2×C4—288no—
488C2×C4—568no—
489C2×C4#651314yes—
490C2×C4D7×#4112yes—
491C2×C4#655412yes—
492C2×C4C22×C22⋊D714yes—
493C2×C4C22×C22⋊D714yes—
494C2×C4C22×C22⋊D712yes—
495C2×C4C22×C22⋊D712yes—
496C2×C4C22×C22⋊D711yescharacteristic
497C2×C4C24×D711yescharacteristic
498C8—142no—
499C8—144no—
500C8—142no—
501C7C2×#336011yescharacteristic, complemented, Sylow
502C22—42no—
503C22—42no—
504C22—42no—
505C22—84no—
506C22—144no—
507C22—282no—
508C22—282no—
509C22—288no—
510C22C2×#651312yes—
511C22#9052212yes—
512C22#9052812yes—
513C22C23×C22⋊D711yescharacteristic
514C4—24no—
515C4—24no—
516C4—22no—
517C4—42no—
518C4—44no—
519C4—282no—
520C4—284no—
521C4C2×#651314yes—
522C4#9052212yes—
523C4C2×D7×#4111yescharacteristic
524C4C2×#655411yescharacteristic
525C2—82no—
526C2—284no—
527C2#1174111214yesdirect product factor
528C2C22×#651311yescharacteristic
529C2C2×#9052211yescharacteristic
530C2C2×#9052811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenterC2×#556D7
hypocenterC7C2×#3360
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×C7×#556C2semidirect
2. C2×#90270C2semidirect
3. #11741112C2direct (N × H)
4. C2×#89855C2semidirect
5. C2×#89853C2semidirect
7. C2×#89828C2semidirect
8. C2×Q8×C7⋊C8C2semidirect
9. C2×#89794C2semidirect
10. C2×C4×#1143C2semidirect
11. C2×#89780C2semidirect
14. C7×#556C22semidirect
27. #90270C22semidirect
39. #89855C22semidirect
40. #89853C22semidirect
42. #89828C22semidirect
43. C4×#1143C22semidirect
44. Q8×C7⋊C8C22semidirect
45. #89794C22semidirect
47. #89780C22semidirect
126. C2×#556D7semidirect
206. #556C2×D7semidirect
450. C2×C7#3360semidirect
501. C7C2×#3360semidirect
527. C2#11741112direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#902702
#117411122
C2×#898552
C2×#898532
C2×#898292
C2×#898282
C2×Q8×C7⋊C82
C2×#897942
C2×C4×#11432
C2×#897802
C2×C4×#11412
C2×#33607