Finite Group Databasebeta

← C2×#11726122 #13321821

Subgroups — C2×#11726122 #13321821

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90382C211yesmaximal, characteristic
2C2×#89477C211yesmaximal, characteristic, complemented
3C2×#89477C211yesmaximal, characteristic, complemented
4C2×C7×#330C211yesmaximal, characteristic, complemented
5C2×#89338C211yesmaximal, characteristic, complemented
6C2×#89259C211yesmaximal, characteristic, complemented
7C2×#89259C211yesmaximal, characteristic, complemented
8#11726122C218yesmaximal, direct product factor
9C2×C7×#125C2211yescharacteristic
10#90382—22no—
11#90382—22no—
12#90382C414yes—
13#90382C2214yes—
14C2×#6461—21no—
15C2×#6460C411yescharacteristic
16C2×#6458C411yescharacteristic
17C2×#6457—21no—
18C2×#6456—21no—
19C2×#6456C411yescharacteristic
20C2×#6455—21no—
21C2×#6450C411yescharacteristic
22C2×#6438C2211yescharacteristic
23C2×C42×D7C2211yescharacteristic
24C2×C7×#97C2211yescharacteristic
25C2×C7×#97C2211yescharacteristic
26C2×(C2×D7)⋊C8—21no—
27C2×(C2×D7)⋊C8—21no—
28C2×Dic7⋊C8—21no—
29C2×Dic7⋊C8—21no—
30C2×#6377C2211yescharacteristic
31C2×#6377C2211yescharacteristic
32#89477—22no—
33#89477—22no—
34#89477C2214yescomplemented
35#89477C2214yescomplemented
36C7×#330C2214yescomplemented
37#89338C2214yescomplemented
38#89259C2214yescomplemented
39#89259C2214yescomplemented
40C2×#1535—71nomaximal, Sylow
41C22×D28—21no—
42C22×D28C2×C411yescharacteristic
43C22×C4×D7—21no—
44C22×C4×D7C2×C411yescharacteristic
45C22×C4×D7D411yescharacteristic
46C22×C4×D7D411yescharacteristic
47C22×Dic14C2×C411yescharacteristic
48C22×C7×C8—21no—
49C22×C7×C8—21no—
50C7×#125—21no—
51C7×#125—21no—
52C7×#125C2×C412yes—
53C7×#125C2312yes—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4C2×C411yescharacteristic
57C2×C7×C4⋊C4C2×C411yescharacteristic
58C2×C42×C7C2311yescharacteristic
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—21no—
61C2×(C2×D7)⋊C4—21no—
62C2×(C2×D7)⋊C4—21no—
63C2×(C2×D7)⋊C4—21no—
64C2×C4⋊Dic7—21no—
65C2×Dic7⋊C4—21no—
66C2×C4×Dic7—21no—
67C2×C4×Dic7C2×C411yescharacteristic
68C2×C4×Dic7D411yescharacteristic
69C2×C4×Dic7D411yescharacteristic
70C22×C7⋊C8—21no—
71C22×C7⋊C8—21no—
72#6461—24no—
73#6461—24no—
74#6460—22no—
75#6460C2×C414yes—
76#6458—22no—
77#6458C2×C414yes—
78#6457—24no—
79#6457—24no—
80#6456—24no—
81#6456—24no—
82#6456—22no—
83#6456C2×C414yes—
84#6455—24no—
85#6455—24no—
86#6450—22no—
87#6450C2×C414yes—
88#6438—21no—
89#6438—21no—
90#6438C2×C412yes—
91#6438C2312yes—
92C42×D7C2×C412yes—
93C42×D7D412yes—
94C42×D7D412yes—
95C42×D7C2312yes—
96C7×#97—21no—
97C7×#97—21no—
98C7×#97C2312yes—
99C7×#97C2312yes—
100(C2×D7)⋊C8—24no—
101(C2×D7)⋊C8—24no—
102Dic7⋊C8—24no—
103Dic7⋊C8—24no—
104#6377—21no—
105#6377—21no—
106#6377C2312yes—
107#6377C2312yes—
108C2×#529—71no—
109C2×#432—71no—
110C2×#432—71no—
111#1535—78no—
112C2×#330—71no—
113C2×#330D711yescharacteristic, complemented
114C2×#329—71no—
115C2×#329—71no—
116C23×D7—21no—
117C23×D7C22⋊C411yescharacteristic
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7C22×C411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×Dic7—21no—
124C22×Dic7C22⋊C411yescharacteristic
125C22×Dic7C22⋊C411yescharacteristic
126C22×Dic7C22⋊C411yescharacteristic
127C2×D28—24no—
128C2×D28—21no—
129C2×D28—42no—
130C2×D28—44no—
131C2×D28C22×C412yes—
132C2×C4×D7—24no—
133C2×C4×D7—42no—
134C2×C4×D7—42no—
135C2×C4×D7—42no—
136C2×C4×D7—44no—
137C2×C4×D7C22⋊C412yes—
138C2×C4×D7C22⋊C412yes—
139C2×C4×D7C22⋊C412yes—
140C2×C4×D7C22×C412yes—
141C2×C4×D7C2×D412yes—
142C2×C4×D7C2×D412yes—
143C2×Dic14—21no—
144C2×Dic14—42no—
145C2×Dic14C22×C412yes—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8—42no—
149C2×C7×C8—42no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4C22×C412yes—
157C7×C4⋊C4C22×C412yes—
158C42×C7C22×C411yescharacteristic
159C42×C7C2×D411yescharacteristic
160C42×C7C2×D411yescharacteristic
161C42×C7C2411yescharacteristic
162(C2×D7)⋊C4—24no—
163(C2×D7)⋊C4—22no—
164(C2×D7)⋊C4—22no—
165(C2×D7)⋊C4—22no—
166(C2×D7)⋊C4—22no—
167(C2×D7)⋊C4—24no—
168(C2×D7)⋊C4—24no—
169C4⋊Dic7—24no—
170Dic7⋊C4—24no—
171C4×Dic7—24no—
172C4×Dic7C22⋊C412yes—
173C4×Dic7C22⋊C412yes—
174C4×Dic7C22⋊C412yes—
175C4×Dic7C22×C412yes—
176C4×Dic7C2×D412yes—
177C4×Dic7C2×D412yes—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—42no—
181C2×C7⋊C8—42no—
182#529—74no—
183#529—74no—
184#529—142no—
185#529—142no—
186C2×#126—141no—
187C2×#125C2×D711yescharacteristic
188C2×#124—71no—
189C2×#123—141no—
190C2×C22⋊Q8—71no—
191C2×#121—71no—
192C2×C4×Q8—71no—
193C2×C4×D4—71no—
194C2×C4×D4—141no—
195C2×#117—141no—
196C22×C42—71no—
197#432—74no—
198#432—74no—
199#432—142no—
200#432—142no—
201C2×C4⋊C8—141no—
202C2×C4⋊C8—141no—
203C2×C22⋊C8—141no—
204C2×C22⋊C8—141no—
205C2×#97—71no—
206C2×#97—71no—
207C2×#97C2×D711yescharacteristic
208C2×#97C2×D711yescharacteristic
209#330—74no—
210#330C2×D714yescomplemented
211#329—74no—
212#329—74no—
213C23×C7C2×C22⋊C411yescharacteristic
214C22×D7—22no—
215C22×D7—22no—
216C22×D7—22no—
217C22×D7—22no—
218C22×D7—42no—
219C22×D7—42no—
220C22×D7—42no—
221C22×D7C2×C22⋊C412yes—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7C2×C22⋊C411yescharacteristic
230C2×C4×C7C2×C22⋊C411yescharacteristic
231C2×C4×C7C2×C22⋊C411yescharacteristic
232C2×C4×C7C23×C411yescharacteristic
233C2×C4×C7C22×D411yescharacteristic
234C2×C4×C7C22×D411yescharacteristic
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—22no—
238C2×Dic7—22no—
239C2×Dic7—42no—
240C2×Dic7C2×C22⋊C412yes—
241C2×Dic7C2×C22⋊C412yes—
242C2×Dic7C2×C22⋊C412yes—
243D28—42no—
244D28—42no—
245D28—48no—
246C4×D7—42no—
247C4×D7—42no—
248C4×D7—42no—
249C4×D7—42no—
250C4×D7—42no—
251C4×D7—42no—
252C4×D7—48no—
253Dic14—42no—
254Dic14—42no—
255C7×C8—42no—
256C7×C8—42no—
257C7⋊C8—42no—
258C7⋊C8—42no—
259C22×Q8—71no—
260C22×D4—71no—
261C22×D4—141no—
262C23×C4—71no—
263C23×C4—71no—
264C23×C4—71no—
265C23×C4—141no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—141no—
269C22×C8—141no—
270#126—144no—
271#126—144no—
272#125—21no—
273#125—21no—
274#125C4×D712yes—
275#125C22×D712yes—
276#124—72no—
277#124—72no—
278#124—141no—
279#124—141no—
280#123—144no—
281#123—144no—
282C22⋊Q8—74no—
283C22⋊Q8—142no—
284#121—74no—
285#121—142no—
286C4×Q8—74no—
287C4×Q8—142no—
288C4×D4—74no—
289C4×D4—142no—
290C4×D4—144no—
291C4×D4—144no—
292#117—144no—
293#117—144no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4C4×D711yescharacteristic
299C2×C4⋊C4C4×D711yescharacteristic
300C2×C22⋊C4—141no—
301C2×C22⋊C4—141no—
302C2×C22⋊C4—141no—
303C2×C22⋊C4—141no—
304C2×C22⋊C4—141no—
305C2×C42—71no—
306C2×C42—71no—
307C2×C42—71no—
308C2×C42—72no—
309C2×C42—72no—
310C2×C42—72no—
311C2×C42—72no—
312C2×C42—141no—
313C2×C42C22×D711yescharacteristic
314C4⋊C8—144no—
315C4⋊C8—144no—
316C22⋊C8—144no—
317C22⋊C8—144no—
318#97—21no—
319#97—21no—
320#97—72no—
321#97—72no—
322#97—141no—
323#97—141no—
324#97C22×D712yes—
325#97C22×D712yes—
326C22×C7#41312yes—
327C22×C7#42112yes—
328C22×C7#42112yes—
329C22×C7C22×C22⋊C411yescharacteristic
330C2×D7—22no—
331C2×D7—22no—
332C2×D7—22no—
333C2×D7—44no—
334C2×D7—42no—
335C2×D7—42no—
336C2×D7—42no—
337C2×D7—84no—
338C4×C7—21no—
339C4×C7—21no—
340C4×C7—21no—
341C4×C7—21no—
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—42no—
345C4×C7—42no—
346Dic7—22no—
347Dic7—22no—
348Dic7—22no—
349Dic7—42no—
350C24—71no—
351C24—141no—
352C2×Q8—72no—
353C2×Q8—141no—
354C2×Q8—282no—
355C2×D4—72no—
356C2×D4—144no—
357C2×D4—141no—
358C2×D4—282no—
359C2×D4—284no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—71no—
363C22×C4—71no—
364C22×C4—71no—
365C22×C4—72no—
366C22×C4—72no—
367C22×C4—72no—
368C22×C4—72no—
369C22×C4—72no—
370C22×C4—72no—
371C22×C4—144no—
372C22×C4—141no—
373C22×C4—282no—
374C22×C4—282no—
375C22×C4—282no—
376C22×C4—284no—
377C22×C4C2×C4×D711yescharacteristic
378C22×C4D4×D711yescharacteristic
379C22×C4D4×D711yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—142no—
385C2×C8—142no—
386C2×C8—282no—
387C2×C8—282no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—144no—
395C4⋊C4—144no—
396C4⋊C4C2×C4×D712yes—
397C4⋊C4C2×C4×D712yes—
398C22⋊C4—144no—
399C22⋊C4—144no—
400C22⋊C4—144no—
401C22⋊C4—142no—
402C22⋊C4—142no—
403C22⋊C4—142no—
404C22⋊C4—142no—
405C42—72no—
406C42—72no—
407C42—72no—
408C42—72no—
409C42—72no—
410C42—72no—
411C42—144no—
412C42C2×C4×D711yescharacteristic
413C42D4×D711yescharacteristic
414C42D4×D711yescharacteristic
415C42C23×D711yescharacteristic
416C2×C7#153514yescomplemented
417C2×C7C2×#41311yescharacteristic
418C2×C7C2×#42111yescharacteristic
419C2×C7C2×#42111yescharacteristic
420D7—42no—
421D7—82no—
422C23—72no—
423C23—142no—
424C23—142no—
425C23—142no—
426C23—142no—
427C23—282no—
428C23—282no—
429C23—282no—
430C23D7×C22⋊C411yescharacteristic
431Q8—282no—
432Q8—282no—
433D4—282no—
434D4—282no—
435D4—288no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—72no—
444C2×C4—72no—
445C2×C4—72no—
446C2×C4—142no—
447C2×C4—142no—
448C2×C4—142no—
449C2×C4—142no—
450C2×C4—282no—
451C2×C4—282no—
452C2×C4—282no—
453C2×C4—282no—
454C2×C4—282no—
455C2×C4—282no—
456C2×C4—288no—
457C2×C4—282no—
458C2×C4D7×C22⋊C411yescharacteristic
459C2×C4D7×C22⋊C411yescharacteristic
460C2×C4D7×C22⋊C411yescharacteristic
461C2×C4C22×C4×D711yescharacteristic
462C2×C4C2×D4×D711yescharacteristic
463C2×C4C2×D4×D711yescharacteristic
464C8—42no—
465C8—42no—
466C8—282no—
467C8—282no—
468C7C2×#153511yescharacteristic, complemented, Sylow
469C22—142no—
470C22—142no—
471C22—142no—
472C22—284no—
473C22—282no—
474C22—282no—
475C22—282no—
476C22—564no—
477C22#8952312yes—
478C22#8959312yes—
479C22#8959312yes—
480C22C2×D7×C22⋊C411yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—42no—
488C4—42no—
489C4—142no—
490C4—142no—
491C4—142no—
492C4—282no—
493C2—282no—
494C2—562no—
495C2#1172612214yesdirect product factor
496C2C2×#8952311yescharacteristic
497C2C2×#8959311yescharacteristic
498C2C2×#8959311yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC2×C7×#330C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6458C4
cosocleC42×C7C24
hypercenterC2×#330D7
hypocenterC7C2×#1535
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
2. C2×#89477C2semidirect
3. C2×#89477C2semidirect
4. C2×C7×#330C2semidirect
5. C2×#89338C2semidirect
6. C2×#89259C2semidirect
7. C2×#89259C2semidirect
8. #11726122C2direct (N × H)
34. #89477C22semidirect
35. #89477C22semidirect
36. C7×#330C22semidirect
37. #89338C22semidirect
38. #89259C22semidirect
39. #89259C22semidirect
113. C2×#330D7semidirect
210. #330C2×D7semidirect
416. C2×C7#1535semidirect
468. C7C2×#1535semidirect
495. C2#11726122direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#903822
C2×#894772
C2×C7×#3302
C2×#893382
C2×#892592
#117261222
C2×#15357