Finite Group Databasebeta

← #13322194

Subgroups — #13322194

Order 1792.

1252 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90416C211yesmaximal, characteristic
2D7×#431C211yesmaximal, characteristic, complemented
3#11736483C211yesmaximal, characteristic, complemented
4C7×#1548C211yesmaximal, characteristic, complemented
5#11729528C211yesmaximal, characteristic, complemented
6#11728528C211yesmaximal, characteristic, complemented
7#11728256C211yesmaximal, characteristic, complemented
8#11726239C214yesmaximal, complemented
9#11726236C214yesmaximal, complemented
10C22×#1161C412yes—
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×#6510C412yes—
13#90416—21no—
14#90416—21no—
15#90416—21no—
16#90416—21no—
17#90416C412yes—
18#90416C412yes—
19#90416C2212yes—
20#90416C2212yes—
21C2×#6458—21no—
22C2×#6458—21no—
23C2×#6456—21no—
24C2×#6456—21no—
25C2×#6437C2211yescharacteristic
26C2×C42×D7C2211yescharacteristic
27C7×#431C2211yescharacteristic
28C7×#423C2211yescharacteristic
29#89898—21no—
30M16×Dic7—21no—
31#89697C2211yescharacteristic
32#89696C2211yescharacteristic
33#89607—21no—
34#89607C2212yescomplemented
35#89605—21no—
36#89605C2212yescomplemented
37#89605C2212yescomplemented
38D7×C4⋊C8C2212yescomplemented
39D7×C22⋊C8—21no—
40#89490—21no—
41#89457—21no—
42C4×C8×D7—21no—
43C7×#333C2214yescomplemented
44#89341C2214yescomplemented
45#89280C2214yescomplemented
46#89280C2214yescomplemented
47#17714—71nomaximal, Sylow
48C22×C7×Q8C2×C412yes—
49C2×#1161—22no—
50C2×#1161—22no—
51C2×#1161—48no—
52C2×#1161C2×C414yes—
53C2×#1161C2×C414yes—
54C22×D28—22no—
55C22×D28—21no—
56C22×D28—21no—
57C22×D28C2×C412yes—
58C22×C4×D7—22no—
59C22×C4×D7C2×C411yescharacteristic
60C22×C4×D7D411yescharacteristic
61C22×C4×D7D411yescharacteristic
62C2×C7×M16—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8C2×C412yes—
66C7×C4⋊Q8C2312yes—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C42×C7C2311yescharacteristic
70#6510—22no—
71#6510—22no—
72#6510C2×C414yes—
73#6510C2×C414yes—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—22no—
76C2×C4×Dic7C2×C411yescharacteristic
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×#1136—21no—
80C2×C8×D7—22no—
81#6458—22no—
82#6458—22no—
83#6458—22no—
84#6458—22no—
85#6458—22no—
86#6458—22no—
87#6458—22no—
88#6458—22no—
89#6456—22no—
90#6456—22no—
91#6456—22no—
92#6456—22no—
93#6456—22no—
94#6456—22no—
95#6456—22no—
96#6456—22no—
97#6437—21no—
98#6437—21no—
99#6437C2×C412yes—
100#6437C2312yes—
101C42×D7C2×C411yescharacteristic
102C42×D7C2×C411yescharacteristic
103C42×D7D411yescharacteristic
104C42×D7D411yescharacteristic
105C42×D7D411yescharacteristic
106C42×D7D411yescharacteristic
107C42×D7C2311yescharacteristic
108C42×D7C2311yescharacteristic
109C7×C4⋊C8—21no—
110C7×C4⋊C8C2312yes—
111C7×C4⋊C8C2312yes—
112C7×C22⋊C8—21no—
113C4×C7×C8—21no—
114C22⋊(C7⋊C8)—21no—
115(C2×D7)⋊C8—22no—
116(C2×D7)⋊C8—22no—
117(C2×D7)⋊C8—22no—
118#6389—22no—
119Dic7⋊C8—22no—
120C8×Dic7—22no—
121C8×Dic7—22no—
122C4⋊(C7⋊C8)—21no—
123C4⋊(C7⋊C8)C2312yes—
124C4⋊(C7⋊C8)C2312yes—
125C4×C7⋊C8—21no—
126C2×#532—71no—
127#2920—71no—
128C2×#431—71no—
129#1551—74no—
130#1548—71no—
131#1548D711yescharacteristic, complemented
132#1545—71no—
133#1545—71no—
134#1543—74no—
135C23×D7—22no—
136C23×D7C22⋊C411yescharacteristic
137C2×C7×Q8—22no—
138C2×C7×Q8—44no—
139C2×C7×Q8C22×C414yes—
140C22×C4×C7—22no—
141C22×C4×C7C22×C411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×Dic7C22⋊C411yescharacteristic
145C22×Dic7C22⋊C411yescharacteristic
146C22×Dic7C22⋊C411yescharacteristic
147#1161—48no—
148#1161—48no—
149#1161—48no—
150#1161—48no—
151C2×D28—24no—
152C2×D28—24no—
153C2×D28—22no—
154C2×D28—22no—
155C2×D28—22no—
156C2×D28—22no—
157C2×D28—22no—
158C2×D28—44no—
159C2×D28—48no—
160C2×D28—42no—
161C2×D28—42no—
162C2×D28—44no—
163C2×D28C22×C414yes—
164C2×C4×D7—22no—
165C2×C4×D7—22no—
166C2×C4×D7—24no—
167C2×C4×D7—24no—
168C2×C4×D7—24no—
169C2×C4×D7—42no—
170C2×C4×D7—48no—
171C2×C4×D7C22⋊C411yescharacteristic
172C2×C4×D7C22⋊C411yescharacteristic
173C2×C4×D7C22⋊C411yescharacteristic
174C2×C4×D7C22⋊C411yescharacteristic
175C2×C4×D7C22⋊C411yescharacteristic
176C2×C4×D7C22⋊C411yescharacteristic
177C2×C4×D7C22×C411yescharacteristic
178C2×C4×D7C22×C411yescharacteristic
179C2×C4×D7C2×D411yescharacteristic
180C2×C4×D7C2×D411yescharacteristic
181C2×C4×D7C2×D411yescharacteristic
182C2×C4×D7C2×D411yescharacteristic
183C7×M16—42no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C42×C7C22×C411yescharacteristic
191C42×C7C2×D411yescharacteristic
192C42×C7C2×D411yescharacteristic
193C42×C7C2411yescharacteristic
194(C2×D7)⋊C4—24no—
195(C2×D7)⋊C4—24no—
196(C2×D7)⋊C4—24no—
197(C2×D7)⋊C4—24no—
198C4×Dic7C22⋊C411yescharacteristic
199C4×Dic7C22⋊C411yescharacteristic
200C4×Dic7C22⋊C411yescharacteristic
201C4×Dic7C22⋊C411yescharacteristic
202C4×Dic7C22⋊C411yescharacteristic
203C4×Dic7C22⋊C411yescharacteristic
204C4×Dic7C22×C411yescharacteristic
205C4×Dic7C22×C411yescharacteristic
206C4×Dic7C2×D411yescharacteristic
207C4×Dic7C2×D411yescharacteristic
208C4×Dic7C2×D411yescharacteristic
209C4×Dic7C2×D411yescharacteristic
210#1136—42no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C8×D7—44no—
214C22×#41—72no—
215#532—72no—
216#532—72no—
217#532—72no—
218#532—72no—
219#532—141no—
220#532—141no—
221#532—141no—
222#532—141no—
223C2×C4⋊Q8C2×D711yescharacteristic
224C2×#127—71no—
225C2×#124—72no—
226C2×#121—141no—
227C2×#121—141no—
228C2×C4×D4—141no—
229C2×C4×D4—141no—
230C22×C42—71no—
231#432—72no—
232#432—141no—
233#431—71no—
234#431—72no—
235#431—72no—
236#431—141no—
237#431—141no—
238#431—141no—
239#431C2×D711yescharacteristic
240#423—71no—
241#423C2×D711yescharacteristic
242C2×C4⋊C8—72no—
243#407—141no—
244C2×C22⋊C8—141no—
245C4×M16—141no—
246C2×C4×C8—141no—
247#333—74no—
248#333C2×D714yescomplemented
249#331—74no—
250#331—74no—
251C23×C7C2×C22⋊C411yescharacteristic
252C22×D7—22no—
253C22×D7—22no—
254C22×D7—24no—
255C22×D7—41no—
256C22×D7—44no—
257C22×D7—44no—
258C22×D7—44no—
259C22×D7C2×C22⋊C411yescharacteristic
260C22×D7C2×C22⋊C411yescharacteristic
261C7×Q8—44no—
262C7×Q8—44no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—24no—
266C2×C4×C7—44no—
267C2×C4×C7C2×C22⋊C411yescharacteristic
268C2×C4×C7C2×C22⋊C411yescharacteristic
269C2×C4×C7C2×C22⋊C411yescharacteristic
270C2×C4×C7#13512yes—
271C2×C4×C7#13512yes—
272C2×C4×C7C23×C411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—41no—
278C2×Dic7C2×C22⋊C411yescharacteristic
279C2×Dic7C2×C22⋊C411yescharacteristic
280C2×Dic7C2×C22⋊C411yescharacteristic
281C2×Dic7C2×C22⋊C411yescharacteristic
282C2×Dic7C2×C22⋊C411yescharacteristic
283C2×Dic7C2×C22⋊C411yescharacteristic
284D28—44no—
285D28—44no—
286D28—48no—
287D28—48no—
288D28—42no—
289D28—42no—
290D28—44no—
291D28—44no—
292D28—44no—
293C4×D7—22no—
294C4×D7—21no—
295C4×D7—21no—
296C4×D7—21no—
297C4×D7—21no—
298C4×D7—22no—
299C4×D7—22no—
300C4×D7—22no—
301C4×D7—22no—
302C4×D7—22no—
303C4×D7—22no—
304C4×D7—41no—
305C4×D7—41no—
306C4×D7—48no—
307C4×D7—48no—
308C4×D7#10014yes—
309C7×C8—42no—
310C7×C8—42no—
311C7⋊C8—42no—
312C7⋊C8—42no—
313C2×#41—74no—
314C2×#41—74no—
315C2×#41—142no—
316C2×#41—142no—
317C2×#41—288no—
318C22×Q8C4×D712yes—
319C22×D4—72no—
320C22×D4—142no—
321C22×D4—141no—
322C22×D4—141no—
323C23×C4—71no—
324C23×C4—71no—
325C23×C4—71no—
326C23×C4—142no—
327C2×M16—21no—
328C2×M16—141no—
329C22×C8—142no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8C4×D712yes—
333C4⋊Q8C22×D712yes—
334#127—72no—
335#127—72no—
336#127—141no—
337#127—141no—
338#124—74no—
339#124—74no—
340#124—142no—
341#124—142no—
342#121—142no—
343#121—142no—
344#121—142no—
345#121—142no—
346#121—142no—
347#121—142no—
348#121—142no—
349#121—142no—
350C4×D4—142no—
351C4×D4—142no—
352C4×D4—142no—
353C4×D4—142no—
354C4×D4—142no—
355C4×D4—142no—
356C4×D4—142no—
357C4×D4—142no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C22⋊C4—142no—
361C2×C22⋊C4—142no—
362C2×C42—71no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—71no—
368C2×C42—71no—
369C2×C42—71no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—71no—
373C2×C42C22×D711yescharacteristic
374C4⋊C8—21no—
375C4⋊C8—72no—
376C4⋊C8—72no—
377C4⋊C8—142no—
378C4⋊C8—141no—
379C4⋊C8C22×D712yes—
380C4⋊C8C22×D712yes—
381C22⋊C8—21no—
382C22⋊C8—142no—
383C22⋊C8—142no—
384C22⋊C8—142no—
385C22⋊C8—141no—
386#97—142no—
387C4×C8—21no—
388C4×C8—142no—
389C4×C8—142no—
390C4×C8—141no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7#41612yes—
394C22×C7C22×C22⋊C411yescharacteristic
395C2×D7—21no—
396C2×D7—21no—
397C2×D7—21no—
398C2×D7—21no—
399C2×D7—21no—
400C2×D7—44no—
401C2×D7—44no—
402C2×D7—44no—
403C2×D7—44no—
404C2×D7—88no—
405C2×D7C2×#10012yes—
406C4×C7—21no—
407C4×C7—21no—
408C4×C7—44no—
409C4×C7C2×#10012yes—
410C4×C7#41612yes—
411C4×C7C2×#13511yescharacteristic
412C4×C7C2×#13511yescharacteristic
413C4×C7#57512yes—
414Dic7—21no—
415Dic7—21no—
416Dic7—21no—
417Dic7—21no—
418Dic7—21no—
419Dic7C2×#10012yes—
420C24—71no—
421C24—142no—
422#41—288no—
423#41—288no—
424#41—288no—
425#41—288no—
426C2×Q8—22no—
427C2×Q8—44no—
428C2×Q8C2×C4×D714yes—
429C2×D4—74no—
430C2×D4—144no—
431C2×D4—144no—
432C2×D4—142no—
433C2×D4—142no—
434C2×D4—142no—
435C2×D4—142no—
436C2×D4—142no—
437C2×D4—284no—
438C2×D4—284no—
439C2×D4—288no—
440C2×D4—282no—
441C2×D4—282no—
442C22×C4—22no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—71no—
457C22×C4—71no—
458C22×C4—144no—
459C22×C4—144no—
460C22×C4—144no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—282no—
464C22×C4—288no—
465C22×C4C2×C4×D711yescharacteristic
466C22×C4D4×D711yescharacteristic
467C22×C4D4×D711yescharacteristic
468M16—42no—
469M16—282no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—142no—
473C2×C8—142no—
474C2×C8—284no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C22⋊C4—144no—
480C22⋊C4—144no—
481C22⋊C4—144no—
482C22⋊C4—144no—
483C42—71no—
484C42—71no—
485C42—71no—
486C42—71no—
487C42—71no—
488C42—71no—
489C42—71no—
490C42—71no—
491C42—71no—
492C42—71no—
493C42—71no—
494C42—71no—
495C42C2×C4×D711yescharacteristic
496C42D4×D711yescharacteristic
497C42D4×D711yescharacteristic
498C42C23×D711yescharacteristic
499C2×C7—22no—
500C2×C7C22×#10011yescharacteristic
501C2×C7C2×#41611yescharacteristic
502C2×C7#289611yescharacteristic
503D7—22no—
504D7—41no—
505D7—84no—
506C23—71no—
507C23—71no—
508C23—142no—
509C23—142no—
510C23—144no—
511C23—281no—
512C23—284no—
513C23—284no—
514C23—284no—
515C23D7×C22⋊C411yescharacteristic
516Q8—44no—
517Q8—44no—
518D4—284no—
519D4—284no—
520D4—288no—
521D4—288no—
522D4—282no—
523D4—282no—
524D4—284no—
525D4—284no—
526D4—284no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—44no—
531C2×C4—71no—
532C2×C4—74no—
533C2×C4—71no—
534C2×C4—71no—
535C2×C4—71no—
536C2×C4—71no—
537C2×C4—71no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—288no—
554C2×C4—288no—
555C2×C4—281no—
556C2×C4D7×C22⋊C411yescharacteristic
557C2×C4D7×C22⋊C411yescharacteristic
558C2×C4D7×C22⋊C411yescharacteristic
559C2×C4#647912yes—
560C2×C4#648212yes—
561C2×C4C22×C4×D711yescharacteristic
562C2×C4C2×D4×D711yescharacteristic
563C2×C4C2×D4×D711yescharacteristic
564C8—42no—
565C8—42no—
566C8—282no—
567C8—282no—
568C7#1771411yescharacteristic, complemented, Sylow
569C22—21no—
570C22—21no—
571C22—72no—
572C22—141no—
573C22—141no—
574C22—141no—
575C22—141no—
576C22—141no—
577C22—284no—
578C22—284no—
579C22—284no—
580C22—284no—
581C22—568no—
582C22#8957612yes—
583C22C2×D7×C22⋊C411yescharacteristic
584C4—21no—
585C4—21no—
586C4—44no—
587C4—72no—
588C4—141no—
589C4—141no—
590C4—141no—
591C4—141no—
592C4—141no—
593C4D7×#10012yes—
594C4#8957612yes—
595C4C2×#647911yescharacteristic
596C4C2×#648211yescharacteristic
597C4#9046812yes—
598C2—22no—
599C2—142no—
600C2—281no—
601C2—564no—
602C2C2×D7×#10011yescharacteristic
603C2C2×#8957611yescharacteristic
604C2#1173640111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1548C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#90416C2
cosocleC42×C7C24
hypercenter#1548D7
hypocenterC7#17714
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. D7×#431C2semidirect
3. #11736483C2semidirect
4. C7×#1548C2semidirect
5. #11729528C2semidirect
6. #11728528C2semidirect
7. #11728256C2semidirect
8. #11726239C2semidirect
9. #11726236C2semidirect
34. #89607C22semidirect
36. #89605C22semidirect
37. #89605C22semidirect
38. D7×C4⋊C8C22semidirect
43. C7×#333C22semidirect
44. #89341C22semidirect
45. #89280C22semidirect
46. #89280C22semidirect
131. #1548D7semidirect
248. #333C2×D7semidirect
568. C7#17714semidirect

Maximal subgroups

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

subgroupindex
C2×#904162
D7×#4312
#117364832
C7×#15482
#117295282
#117285282
#117282562
#117262392
#117262362
#177147