Finite Group Databasebeta

← #13320387

Subgroups — #13320387

Order 1792.

920 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742891C211yesmaximal, characteristic
2C7×#2965C211yesmaximal, characteristic, complemented
3#11739965C211yesmaximal, characteristic, complemented
4#11728522C212yesmaximal, complemented
5#11728521C211yesmaximal, characteristic, complemented
6#11728514C212yesmaximal, complemented
7#11728509C211yesmaximal, characteristic, complemented
8#11727610C212yesmaximal, complemented
9#11727606C212yesmaximal, complemented
10C2×#89251C211yesmaximal, characteristic, complemented
11#11727488C211yesmaximal, characteristic, complemented
12C4×C7×#41C2211yescharacteristic
13#90524C411yescharacteristic
14#90516C411yescharacteristic
15#90274—21no—
16#90269—21no—
17#90255—22no—
18#90252—21no—
19#90248—21no—
20#90229—22no—
21#90225C412yes—
22#90217C411yescharacteristic
23#90210C411yescharacteristic
24#90209C412yes—
25#90170C2212yes—
26C4×#1157C2211yescharacteristic
27#90161C2212yes—
28C2×C4×Dic14C2211yescharacteristic
29C7×#446—21no—
30C7×#436—21no—
31C7×#435—22no—
32C7×#431C2211yescharacteristic
33C7×#431C2212yes—
34C4×C7×M16C2212yes—
35C2×C4×C7×C8C2211yescharacteristic
36Q8×C7⋊C8—21no—
37D4×C7⋊C8—22no—
38D4×C7⋊C8—21no—
39#89697C2211yescharacteristic
40#89697C2212yes—
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2212yes—
43#89254C2212yes—
44#89254C2211yescharacteristic
45#89254C2211yescharacteristic
46#89251C2212yes—
47#89251C2211yescharacteristic
48#89251C2211yescharacteristic
49#89251C2212yes—
50#89251C2212yes—
51#89251C2212yes—
52#89251C2212yes—
53#17469—71nomaximal, Sylow
54C2×C7×#41C2×C411yescharacteristic
55C2×#1157C2×C411yescharacteristic
56C22×Dic14C2×C411yescharacteristic
57C2×C7×M16—22no—
58C22×C7×C8—21no—
59C4×C7×Q8D411yescharacteristic
60C4×C7×Q8D411yescharacteristic
61C4×C7×D4D412yes—
62C4×C7×D4D411yescharacteristic
63C4×C7×D4D411yescharacteristic
64C4×C7×D4D412yes—
65C7×#117C2×C412yes—
66C7×#117C2×C411yescharacteristic
67C2×C42×C7C2311yescharacteristic
68C2×C42×C7C2312yes—
69#6509—21no—
70#6507—21no—
71#6502—22no—
72#6501—21no—
73#6499—21no—
74#6498—22no—
75#6493C2×C412yes—
76#6492—22no—
77C4×C22⋊D7—21no—
78C4×C22⋊D7—21no—
79C4×C22⋊D7—22no—
80#6489—22no—
81#6489C2×C411yescharacteristic
82C2×C4⋊Dic7C2×C411yescharacteristic
83#6487—22no—
84#6487C2×C412yes—
85#6487C2×C412yes—
86#6487C2×C412yes—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C22×C7⋊C8—21no—
90C22×C7⋊C8—22no—
91#6461—22no—
92#6460—21no—
93#6460—21no—
94#6459—22no—
95#6453—21no—
96#6452—22no—
97#6452—22no—
98#6451—21no—
99#6449—22no—
100#6448—22no—
101#6446—22no—
102#6442—22no—
103#6442—22no—
104#6441—22no—
105#6439—22no—
106#6438C2×C412yes—
107C4×D28C2×C411yescharacteristic
108C4×D28C2311yescharacteristic
109#6435—21no—
110#6435—22no—
111C42×D7—21no—
112#6433—22no—
113#6433C2×C412yes—
114#6432C2×C412yes—
115#6432C2×C412yes—
116C4×Dic14—21no—
117C4×Dic14—21no—
118C4×Dic14—22no—
119C4×Dic14C2×C412yes—
120C4×Dic14C2×C411yescharacteristic
121C4×Dic14C2312yes—
122C4×Dic14C2311yescharacteristic
123C7×C4⋊C8—22no—
124C7×C4⋊C8—21no—
125C7×C22⋊C8—21no—
126C7×C22⋊C8—22no—
127C7×#97—22no—
128C4×C7×C8—21no—
129C4×C7×C8C2311yescharacteristic
130C4×C7×C8C2311yescharacteristic
131C4×C7×C8C2312yes—
132C22⋊(C7⋊C8)—21no—
133C22⋊(C7⋊C8)—22no—
134C4⋊(C7⋊C8)—21no—
135C4⋊(C7⋊C8)—22no—
136C4⋊(C7⋊C8)C2312yes—
137C4⋊(C7⋊C8)C2311yescharacteristic
138C4⋊(C7⋊C8)C2311yescharacteristic
139C4×C7⋊C8—21no—
140C4×C7⋊C8—22no—
141#3461—71no—
142#2988—71no—
143#2965D711yescharacteristic, complemented
144#1485—72no—
145#1483—72no—
146#1482—72no—
147#1481—72no—
148#1479—71no—
149#1477—71no—
150#1476—71no—
151C2×Q8⋊C8—71no—
152C7×#41—24no—
153C2×C7×Q8C22⋊C411yescharacteristic
154C2×C7×D4C22⋊C412yes—
155C2×C7×D4C22⋊C411yescharacteristic
156C22×C4×C7C22×C412yes—
157C22×C4×C7C22×C411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C2×C22⋊D7—21no—
163C2×C22⋊D7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166#1157—24no—
167C2×D28C22×C411yescharacteristic
168C2×C4×D7—21no—
169C2×C4×D7—21no—
170C2×C4×D7—21no—
171C2×Dic14—24no—
172C2×Dic14—21no—
173C2×Dic14—22no—
174C2×Dic14C22×C411yescharacteristic
175C2×Dic14C22×C412yes—
176C7×M16—44no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—22no—
180C2×C7×C8—42no—
181C7×C4⋊C4C22⋊C411yescharacteristic
182C7×C4⋊C4C22⋊C412yes—
183C7×C4⋊C4C22⋊C412yes—
184C7×C4⋊C4C22⋊C411yescharacteristic
185C7×C22⋊C4C22⋊C412yes—
186C7×C22⋊C4C22⋊C412yes—
187C7×C22⋊C4C22⋊C412yes—
188C42×C7C22×C411yescharacteristic
189C42×C7C22×C412yes—
190C42×C7C2×D411yescharacteristic
191C42×C7C2×D412yes—
192C42×C7C2×D411yescharacteristic
193C42×C7C2×D412yes—
194C42×C7C2411yescharacteristic
195C22⋊Dic7—21no—
196C22⋊Dic7—22no—
197C22⋊Dic7—22no—
198C22⋊Dic7—22no—
199(C2×D7)⋊C4—21no—
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4—22no—
202(C2×D7)⋊C4—22no—
203(C2×D7)⋊C4—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7—22no—
208C4⋊Dic7C22×C412yes—
209C4⋊Dic7C22×C411yescharacteristic
210C4⋊Dic7C22×C411yescharacteristic
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—21no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4—21no—
220C4×Dic7—22no—
221C4×Dic7—21no—
222C4×Dic7—21no—
223C4×Dic7—21no—
224C4×Dic7—22no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—22no—
228C2×C7⋊C8—81no—
229C2×C7⋊C8—82no—
230#544—71no—
231#542—72no—
232#542—141no—
233#541—72no—
234#541—141no—
235#539—142no—
236#537—71no—
237#536—71no—
238#536—71no—
239#533—72no—
240#533—141no—
241#529—72no—
242#529—142no—
243#529—141no—
244C4×#41—71no—
245C4×#41C2×D711yescharacteristic
246C2×C4×Q8—71no—
247#446—21no—
248C8×Q8—141no—
249#436—21no—
250#435—22no—
251C8×D4—141no—
252C8×D4—142no—
253#431—71no—
254#431—72no—
255#431C2×D711yescharacteristic
256#431C2×D712yes—
257C2×C4⋊C8—71no—
258C2×C4⋊C8—72no—
259C4×M16C2×D712yes—
260C2×C4×C8C2×D711yescharacteristic
261Q8⋊C8—72no—
262Q8⋊C8—72no—
263Q8⋊C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—72no—
266Q8⋊C8—72no—
267Q8⋊C8—72no—
268D4⋊C8—71no—
269D4⋊C8—71no—
270D4⋊C8—72no—
271C23×C7C2×C22⋊C412yes—
272C23×C7C2×C22⋊C411yescharacteristic
273C22×D7—21no—
274C7×Q8—22no—
275C7×D4—24no—
276C7×D4—22no—
277C2×C4×C7—24no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7C2×C22⋊C411yescharacteristic
285C2×C4×C7C2×C22⋊C411yescharacteristic
286C2×C4×C7C2×C22⋊C412yes—
287C2×C4×C7C2×C22⋊C412yes—
288C2×C4×C7C2×C22⋊C412yes—
289C2×C4×C7C2×C22⋊C411yescharacteristic
290C2×C4×C7#13114yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C22⋊D7—42no—
295C22⋊D7—82no—
296C2×Dic7—21no—
297C2×Dic7—22no—
298C2×Dic7—21no—
299C2×Dic7—22no—
300C2×Dic7—21no—
301C2×Dic7—42no—
302C2×Dic7—81no—
303D28—22no—
304C4×D7—41no—
305C4×D7—41no—
306C4×D7—42no—
307C4×D7—81no—
308Dic14—24no—
309Dic14—22no—
310Dic14—42no—
311Dic14—82no—
312C7×C8—22no—
313C7×C8—41no—
314C7×C8—42no—
315C7⋊C8—41no—
316C7⋊C8—42no—
317C7⋊C8—41no—
318C2×#41—71no—
319C2×#41C4×D711yescharacteristic
320C22×Q8—71no—
321C2×M16—22no—
322C22×C8—21no—
323C22×C8—141no—
324C22×C8—142no—
325C4⋊Q8—72no—
326C4⋊Q8—72no—
327C4⋊Q8—141no—
328C4⋊Q8—141no—
329#126—142no—
330#126—142no—
331#126—142no—
332#126—142no—
333#125—72no—
334#125—142no—
335#125—141no—
336#125—142no—
337#125—142no—
338#124—72no—
339#124—141no—
340#124—142no—
341#123—142no—
342#123—142no—
343#123—142no—
344#123—142no—
345C22⋊Q8—72no—
346C22⋊Q8—72no—
347C22⋊Q8—72no—
348C22⋊Q8—141no—
349C22⋊Q8—142no—
350C22⋊Q8—141no—
351C22⋊Q8—141no—
352C22⋊Q8—142no—
353C22⋊Q8—142no—
354#121—72no—
355#121—142no—
356#121—141no—
357C4×Q8—72no—
358C4×Q8—71no—
359C4×Q8—72no—
360C4×Q8—71no—
361C4×Q8—141no—
362C4×Q8—142no—
363C4×Q8—141no—
364C4×Q8D2811yescharacteristic
365C4×Q8C22⋊D711yescharacteristic
366C4×D4—71no—
367C4×D4—71no—
368C4×D4—141no—
369C4×D4—142no—
370C4×D4—141no—
371C4×D4D2811yescharacteristic
372C4×D4D2812yes—
373C4×D4C22⋊D712yes—
374C4×D4C22⋊D711yescharacteristic
375#117—71no—
376#117—142no—
377#117—142no—
378#117—141no—
379#117C4×D711yescharacteristic
380#117C4×D712yes—
381C2×C4⋊C4—71no—
382C2×C4⋊C4—141no—
383C2×C42—141no—
384C2×C42—141no—
385C2×C42C22×D711yescharacteristic
386C2×C42C22×D712yes—
387C4⋊C8—22no—
388C4⋊C8—21no—
389C4⋊C8—71no—
390C4⋊C8—72no—
391C4⋊C8—71no—
392C4⋊C8—141no—
393C4⋊C8—142no—
394C22⋊C8—21no—
395C22⋊C8—22no—
396C22⋊C8—141no—
397C22⋊C8—142no—
398#97—22no—
399C4×C8—21no—
400C4×C8—141no—
401C4×C8—142no—
402C4×C8C22×D711yescharacteristic
403C4×C8C22×D711yescharacteristic
404C4×C8C22×D712yes—
405C22×C7—21no—
406C22×C7—21no—
407C22×C7—22no—
408C22×C7—22no—
409C22×C7—22no—
410C22×C7#40812yes—
411C22×C7C22×C22⋊C411yescharacteristic
412C2×D7—41no—
413C2×D7—41no—
414C2×D7—41no—
415C4×C7—22no—
416C4×C7—21no—
417C4×C7—22no—
418C4×C7—21no—
419C4×C7—22no—
420C4×C7—21no—
421C4×C7#40812yes—
422C4×C7C2×#13112yes—
423C4×C7#57711yescharacteristic
424C4×C7#57811yescharacteristic
425Dic7—42no—
426Dic7—41no—
427Dic7—41no—
428Dic7—41no—
429Dic7—42no—
430#41—24no—
431#41—144no—
432C2×Q8—72no—
433C2×Q8—71no—
434C2×Q8—141no—
435C2×Q8—142no—
436C2×Q8—144no—
437C2×Q8(C2×D7)⋊C411yescharacteristic
438C2×D4—71no—
439C2×D4—141no—
440C2×D4—142no—
441C2×D4(C2×D7)⋊C412yes—
442C2×D4(C2×D7)⋊C411yescharacteristic
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4C2×C4×D712yes—
449C22×C4C2×C4×D711yescharacteristic
450C22×C4C2×D2811yescharacteristic
451C22×C4C2×D2812yes—
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D712yes—
454M16—44no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—42no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—142no—
462C2×C8—561no—
463C2×C8—562no—
464C4⋊C4—72no—
465C4⋊C4—71no—
466C4⋊C4—71no—
467C4⋊C4—142no—
468C4⋊C4—141no—
469C4⋊C4—142no—
470C4⋊C4—142no—
471C4⋊C4—142no—
472C4⋊C4—142no—
473C4⋊C4—142no—
474C4⋊C4—141no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C4⋊C4—142no—
479C4⋊C4—141no—
480C4⋊C4(C2×D7)⋊C411yescharacteristic
481C4⋊C4(C2×D7)⋊C412yes—
482C4⋊C4(C2×D7)⋊C412yes—
483C4⋊C4(C2×D7)⋊C411yescharacteristic
484C22⋊C4—141no—
485C22⋊C4—142no—
486C22⋊C4—142no—
487C22⋊C4—142no—
488C22⋊C4—141no—
489C22⋊C4—142no—
490C22⋊C4—142no—
491C22⋊C4—142no—
492C22⋊C4—141no—
493C22⋊C4(C2×D7)⋊C412yes—
494C22⋊C4(C2×D7)⋊C412yes—
495C22⋊C4(C2×D7)⋊C412yes—
496C42—141no—
497C42—141no—
498C42—142no—
499C42—141no—
500C42—142no—
501C42C2×C4×D711yescharacteristic
502C42C2×C4×D712yes—
503C42C2×D2811yescharacteristic
504C42C2×D2812yes—
505C42C2×C22⋊D711yescharacteristic
506C42C2×C22⋊D712yes—
507C42C23×D711yescharacteristic
508C2×C7—22no—
509C2×C7—42no—
510C2×C7C2×#40811yescharacteristic
511C2×C7#289811yescharacteristic
512C2×C7#290111yescharacteristic
513D7—81no—
514C23—141no—
515C23C2×(C2×D7)⋊C412yes—
516C23C2×(C2×D7)⋊C411yescharacteristic
517Q8—22no—
518Q8—144no—
519Q8—142no—
520Q8—282no—
521Q8—562no—
522D4—24no—
523D4—22no—
524D4—142no—
525D4—282no—
526D4—562no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—141no—
539C2×C4—282no—
540C2×C4—282no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—561no—
544C2×C4—561no—
545C2×C4#647014yes—
546C2×C4C2×(C2×D7)⋊C411yescharacteristic
547C2×C4C2×(C2×D7)⋊C411yescharacteristic
548C2×C4C2×(C2×D7)⋊C412yes—
549C2×C4C2×(C2×D7)⋊C412yes—
550C2×C4C2×(C2×D7)⋊C412yes—
551C2×C4C2×(C2×D7)⋊C411yescharacteristic
552C2×C4C22×C4×D711yescharacteristic
553C2×C4C22×D2811yescharacteristic
554C2×C4C22×C22⋊D711yescharacteristic
555C8—22no—
556C8—41no—
557C8—42no—
558C8—281no—
559C8—282no—
560C8—281no—
561C7#1746911yescharacteristic, complemented, Sylow
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—281no—
568C22—281no—
569C22—281no—
570C22#8990112yes—
571C22C22×(C2×D7)⋊C411yescharacteristic
572C4—22no—
573C4—21no—
574C4—22no—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—281no—
579C4—282no—
580C4—281no—
581C4—282no—
582C4—281no—
583C4#8990112yes—
584C4C2×#647012yes—
585C4#9044211yescharacteristic
586C4#9044411yescharacteristic
587C2—22no—
588C2—42no—
589C2—561no—
590C2#1173998111yescharacteristic
591C2C2×#8990111yescharacteristic
592C2#1174125111yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#2965C2
socleC22×C7C22×C22⋊C4
prime radical#90524C4
cosocleC42×C7C24
hypercenter#2965D7
hypocenterC7#17469
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2965

Complements

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

NH ≅ G/Ntype
2. C7×#2965C2semidirect
3. #11739965C2semidirect
4. #11728522C2semidirect
5. #11728521C2semidirect
6. #11728514C2semidirect
7. #11728509C2semidirect
8. #11727610C2semidirect
9. #11727606C2semidirect
10. C2×#89251C2semidirect
11. #11727488C2semidirect
143. #2965D7semidirect
561. C7#17469semidirect

Maximal subgroups

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

subgroupindex
#117428912
C7×#29652
#117399652
#117285222
#117285212
#117285142
#117285092
#117276102
#117276062
C2×#892512
#117274882
#174697