Finite Group Databasebeta

← #13353877

Subgroups — #13353877

Order 1792.

1382 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×C4×#1161C211yesmaximal, characteristic
2C2×#89477C211yesmaximal, characteristic, complemented
3#11731198C211yesmaximal, characteristic, complemented
4C7×#1763C211yesmaximal, characteristic, complemented
5#11729385C211yesmaximal, characteristic, complemented
6#11728021C212yesmaximal
7#11727975C212yesmaximal, complemented
8#11727566C211yesmaximal, characteristic, complemented
9#11727228C211yesmaximal, characteristic, complemented
10#11726646C212yesmaximal, complemented
11#11726567C212yesmaximal, complemented
12C22×#1161C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C4×#1161—24no—
16C4×#1161C418yescomplemented
17C2×#6456—21no—
18C2×#6456C2211yescharacteristic
19C2×#6455—21no—
20C2×#6455C2211yescharacteristic
21C2×C4×D28—21no—
22C2×C4×D28C2211yescharacteristic
23C2×#6435—21no—
24C2×C42×D7—21no—
25C2×C42×D7C2211yescharacteristic
26C2×C7×#103C2212yes—
27C2×C7×#97C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C2×#6409C2212yes—
30C2×#6395C2212yes—
31C2×(C2×D7)⋊C8C2212yes—
32C2×Dic7⋊C8C2212yes—
33#89759C2212yes—
34#89748C2212yes—
35C2×#6383C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37C4×Dic7⋊C4—21no—
38C2×#6377C2211yescharacteristic
39#89477—24no—
40#89439—21no—
41#89422—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#20743—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C2×#1161—24no—
49C2×#1161—24no—
50C2×#1161—22no—
51C2×#1161C2×C414yes—
52C22×D28—21no—
53C22×D28C2311yescharacteristic
54C22×C4×D7—21no—
55C22×C4×D7—21no—
56C22×C4×D7—21no—
57C22×C4×D7D411yescharacteristic
58C22×C4×D7D411yescharacteristic
59C22×C4×D7C2311yescharacteristic
60C22×C7×C8C2312yes—
61C4×C7×Q8—22no—
62C4×C7×Q8C2×C414yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2312yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C42×C7—21no—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69Q8×Dic7—22no—
70Q8×Dic7C2×C414yes—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7C2312yes—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×Dic7⋊C4—22no—
79C2×Dic7⋊C4—22no—
80C2×C4×Dic7—21no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7C2311yescharacteristic
86C22×C7⋊C8C2312yes—
87#6456—28no—
88#6456—22no—
89#6456C2×C414yes—
90#6455—22no—
91#6455—28no—
92#6455C2×C414yes—
93C4×D28—28no—
94C4×D28—22no—
95C4×D28C2×C414yes—
96#6435—44no—
97C42×D7—28no—
98C42×D7C2×C414yes—
99C42×D7D414yes—
100C7×#103—24no—
101C7×#97—22no—
102C7×#95—21no—
103#6409—24no—
104C7⋊#95—21no—
105C7⋊#95—22no—
106#6395—24no—
107(C2×D7)⋊C8—24no—
108Dic7⋊C8—24no—
109#6383—24no—
110#6377—22no—
111C2×C4×#41—71no—
112C2×#432—71no—
113#2285—71no—
114#1825—72no—
115#1803—72no—
116#1802—72no—
117#1788—72no—
118#1763—71no—
119#1763D711yescharacteristic, complemented
120#1762—71no—
121#1762—71no—
122C23×D7—21no—
123C23×D7C2×D411yescharacteristic
124C2×C7×Q8—21no—
125C2×C7×Q8—22no—
126C2×C7×Q8C22×C412yes—
127C2×C7×Q8#4114yes—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—22no—
136C22×Dic7C2×D411yescharacteristic
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139#1161—216no—
140#1161—28no—
141#1161—28no—
142C2×D28—28no—
143C2×D28—24no—
144C2×D28—21no—
145C2×D28—22no—
146C2×D28C22×C412yes—
147C2×D28#4114yes—
148C2×C4×D7—28no—
149C2×C4×D7—22no—
150C2×C4×D7—22no—
151C2×C4×D7—24no—
152C2×C4×D7—22no—
153C2×C4×D7—22no—
154C2×C4×D7—24no—
155C2×C4×D7—24no—
156C2×C4×D7—42no—
157C2×C4×D7—44no—
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C4×D7C22⋊C414yes—
161C2×C4×D7C22⋊C414yes—
162C2×C4×D7C22×C412yes—
163C2×C4×D7C2×D412yes—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4C22×C412yes—
172C42×C7—24no—
173C42×C7—42no—
174C42×C7C22×C412yes—
175C42×C7C2×D412yes—
176(C2×D7)⋊C4—24no—
177(C2×D7)⋊C4—24no—
178(C2×D7)⋊C4—24no—
179(C2×D7)⋊C4—44no—
180C4⋊Dic7—21no—
181C4⋊Dic7—24no—
182C4⋊Dic7—22no—
183C4⋊Dic7—22no—
184C4⋊Dic7C22×C412yes—
185Dic7⋊C4—44no—
186Dic7⋊C4—44no—
187C4×Dic7—24no—
188C4×Dic7—24no—
189C4×Dic7—42no—
190C4×Dic7C22⋊C414yes—
191C4×Dic7C22⋊C414yes—
192C4×Dic7C22×C412yes—
193C4×Dic7C2×D412yes—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C22×#41—71no—
198C4×#41—78no—
199C4×#41—144no—
200C2×C4×Q8—71no—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4—71no—
203C2×C4×D4—71no—
204C2×C4×D4—141no—
205C2×C4×D4—141no—
206C2×#117—71no—
207C2×#117—141no—
208C2×#117—141no—
209C22×C42—71no—
210C22×C42—141no—
211#432—144no—
212C2×C4⋊C8—72no—
213C2×#103—72no—
214C2×#103C2×D712yes—
215C2×#102—72no—
216C2×#102—72no—
217C2×C22⋊C8—72no—
218C2×#97—71no—
219C2×#97C2×D711yescharacteristic
220#384—72no—
221#382—141no—
222#380—72no—
223#380—141no—
224C4×C4⋊C4—71no—
225C4×C4⋊C4—141no—
226C4×C4⋊C4C2×D711yescharacteristic
227#340—72no—
228#340—72no—
229#340—72no—
230#340C2×D712yes—
231C23×C7C22×D411yescharacteristic
232C22×D7—22no—
233C22×D7—22no—
234C22×D7—24no—
235C22×D7—21no—
236C22×D7—21no—
237C22×D7—22no—
238C22×D7—41no—
239C22×D7—41no—
240C22×D7—42no—
241C22×D7C2×C22⋊C412yes—
242C7×Q8—22no—
243C7×Q8—24no—
244C7×Q8#11714yescomplemented
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7C2×C22⋊C412yes—
257C2×C4×C7C2×C22⋊C412yes—
258C2×C4×C7C23×C411yescharacteristic
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C2×#4112yes—
261C2×C4×C7C2×#4112yes—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7C2×C22⋊C412yes—
270C2×Dic7C2×C22⋊C412yes—
271C2×Dic7C2×C22⋊C412yes—
272C2×Dic7AGL(1,ℤ/8)12yes—
273C2×Dic7#13712yes—
274D28—216no—
275D28—22no—
276D28—24no—
277D28#11714yescomplemented
278C4×D7—216no—
279C4×D7—22no—
280C4×D7—22no—
281C4×D7—24no—
282C4×D7—28no—
283C4×D7—44no—
284C4×D7—48no—
285C4×D7—48no—
286C7×C8—42no—
287C7⋊C8—42no—
288C2×#41—74no—
289C2×#41—142no—
290C2×#41—144no—
291C2×#41—144no—
292C22×Q8C22×D711yescharacteristic
293C22×D4—71no—
294C22×D4—141no—
295C23×C4—71no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—141no—
299C23×C4—141no—
300C23×C4—141no—
301C22×C8—72no—
302C22×C8C22×D712yes—
303C4×Q8—22no—
304C4×Q8—74no—
305C4×Q8—142no—
306C4×Q8C4×D714yes—
307C4×D4—74no—
308C4×D4—74no—
309C4×D4—148no—
310C4×D4—142no—
311C4×D4—148no—
312C4×D4—142no—
313#117—74no—
314#117—148no—
315#117—142no—
316#117—284no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—72no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4C22×D712yes—
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C22⋊C4—141no—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4—142no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—71no—
332C2×C42—71no—
333C2×C42—71no—
334C2×C42—74no—
335C2×C42—74no—
336C2×C42—148no—
337C2×C42—141no—
338C2×C42—141no—
339C2×C42—141no—
340C2×C42C22×D711yescharacteristic
341C4⋊C8—144no—
342#103—24no—
343#103—144no—
344#102—144no—
345#102—144no—
346C22⋊C8—144no—
347#97—22no—
348#97—142no—
349#95—21no—
350#95—142no—
351#95—141no—
352C22×C7C22×C22⋊C411yescharacteristic
353C22×C7#52512yes—
354C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C7C2×#13711yescharacteristic
356C22×C7#57512yes—
357C2×D7—22no—
358C2×D7—22no—
359C2×D7—22no—
360C2×D7—24no—
361C2×D7—42no—
362C2×D7—44no—
363C2×D7—42no—
364C2×D7—44no—
365C2×D7—42no—
366C2×D7—42no—
367C4×C7—22no—
368C4×C7—22no—
369C4×C7—24no—
370C4×C7—42no—
371C4×C7—82no—
372C4×C7C2×#11711yescharacteristic
373C4×C7C2×#11711yescharacteristic
374C4×C7#52512yes—
375Dic7—22no—
376Dic7—22no—
377Dic7—42no—
378Dic7—82no—
379Dic7#41714yes—
380C24—71no—
381C24—141no—
382#41—148no—
383#41—1416no—
384#41—148no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8C2×C4×D712yes—
388C2×Q8#115714yes—
389C2×D4—72no—
390C2×D4—74no—
391C2×D4—144no—
392C2×D4—148no—
393C2×D4—142no—
394C2×D4—141no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—71no—
399C22×C4—71no—
400C22×C4—71no—
401C22×C4—74no—
402C22×C4—72no—
403C22×C4—72no—
404C22×C4—74no—
405C22×C4—144no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—144no—
409C22×C4—148no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—141no—
413C22×C4—144no—
414C22×C4—142no—
415C22×C4—282no—
416C22×C4—284no—
417C22×C4—282no—
418C22×C4—282no—
419C22×C4D4×D711yescharacteristic
420C22×C4D4×D711yescharacteristic
421C22×C4C23×D711yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—142no—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—72no—
433C4⋊C4—144no—
434C4⋊C4—141no—
435C4⋊C4—142no—
436C4⋊C4—142no—
437C4⋊C4—284no—
438C4⋊C4—284no—
439C4⋊C4C2×C4×D712yes—
440C22⋊C4—144no—
441C22⋊C4—144no—
442C22⋊C4—144no—
443C22⋊C4—284no—
444C42—24no—
445C42—42no—
446C42—74no—
447C42—74no—
448C42—72no—
449C42—72no—
450C42—144no—
451C42—144no—
452C42—282no—
453C42C2×C4×D712yes—
454C42D4×D712yes—
455C2×C7#228011yescharacteristic
456C2×C7C2×#41711yescharacteristic
457C2×C7#289611yescharacteristic
458C2×C7#311512yes—
459C2×C7#311612yes—
460D7—42no—
461D7—44no—
462C23—72no—
463C23—142no—
464C23—141no—
465C23—141no—
466C23—142no—
467C23—144no—
468C23—142no—
469C23—281no—
470C23—281no—
471C23—282no—
472C23C2×D4×D711yescharacteristic
473Q8—22no—
474Q8—24no—
475Q8#643514yescomplemented
476D4—74no—
477D4—1416no—
478D4—142no—
479D4—144no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—72no—
492C2×C4—72no—
493C2×C4—72no—
494C2×C4—72no—
495C2×C4—72no—
496C2×C4—148no—
497C2×C4—1416no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—142no—
501C2×C4—142no—
502C2×C4—144no—
503C2×C4—142no—
504C2×C4—284no—
505C2×C4—288no—
506C2×C4—288no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4D7×C22⋊C412yes—
512C2×C4D7×C22⋊C412yes—
513C2×C4C22×C4×D711yescharacteristic
514C2×C4C2×#115712yes—
515C2×C4C2×D4×D711yescharacteristic
516C2×C4D7×#4112yes—
517C8—42no—
518C8—282no—
519C7#2074311yescharacteristic, complemented, Sylow
520C22—142no—
521C22—144no—
522C22—142no—
523C22—142no—
524C22—282no—
525C22—282no—
526C22—282no—
527C22—284no—
528C22—282no—
529C22—284no—
530C22C2×D7×C22⋊C411yescharacteristic
531C22#9023812yes—
532C22D7×AGL(1,ℤ/8)11yescharacteristic
533C22D7×#13711yescharacteristic
534C22#9046812yes—
535C4—22no—
536C4—24no—
537C4—22no—
538C4—42no—
539C4—74no—
540C4—82no—
541C4—142no—
542C4—142no—
543C4—282no—
544C4—562no—
545C4C2×#643511yescharacteristic
546C4D7×#11711yescharacteristic
547C4#9023812yes—
548C2—282no—
549C2—284no—
550C2#1173100611yescharacteristic
551C2D7×#41711yescharacteristic
552C2#1173640111yescharacteristic
553C2#1173725612yes—
554C2#1173729312yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1763C2
socleC23×C7C22×D4
prime radicalC22×#1161C22
cosocleC22×C4×C7C24
hypercenter#1763D7
hypocenterC7#20743
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#89477C2semidirect
3. #11731198C2semidirect
4. C7×#1763C2semidirect
5. #11729385C2semidirect
7. #11727975C2semidirect
8. #11727566C2semidirect
9. #11727228C2semidirect
10. #11726646C2semidirect
11. #11726567C2semidirect
16. C4×#1161C4semidirect
119. #1763D7semidirect
244. C7×Q8#117semidirect
277. D28#117semidirect
475. Q8#6435semidirect
519. C7#20743semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11612
C2×#894772
#117311982
C7×#17632
#117293852
#117280212
#117279752
#117275662
#117272282
#117266462
#117265672
#207437