Finite Group Databasebeta

← #13324433

Subgroups — #13324433

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737044C211yesmaximal, characteristic
2#11736992C211yesmaximal, characteristic, complemented
3#11736971C211yesmaximal, characteristic, complemented
4C7×#1589C211yesmaximal, characteristic, complemented
5#11728632C211yesmaximal, characteristic, complemented
6#11728615C211yesmaximal, characteristic, complemented
7#11728614C211yesmaximal, characteristic, complemented
8#11726325C211yesmaximal, characteristic, complemented
9#11726312C211yesmaximal, characteristic, complemented
10#11725931C211yesmaximal, characteristic, complemented
11#11725930C211yesmaximal, characteristic, complemented
12#11725903C211yesmaximal, characteristic, complemented
13#11725902C211yesmaximal, characteristic, complemented
14#11725809C211yesmaximal, characteristic, complemented
15C8×#1131C211yesmaximal, characteristic, complemented
16#90248C2211yescharacteristic
17#90239C2211yescharacteristic
18C4×C7×D8C2211yescharacteristic
19C7×#436C2211yescharacteristic
20C7×#436C2211yescharacteristic
21#89931C411yescharacteristic
22#89926C411yescharacteristic
23#89901—21no—
24#89900—21no—
25#89899—21no—
26#89891—21no—
27C4×#1142C2211yescharacteristic
28C4×#1141C2211yescharacteristic
29D4×C7⋊C8C2211yescharacteristic
30D4×C7⋊C8C2211yescharacteristic
31#89658C411yescharacteristic
32#89648C411yescharacteristic
33#89609C2211yescharacteristic
34#89607C2211yescharacteristic
35#89606C2211yescharacteristic
36#89602C2211yescharacteristic
37#89554C411yescharacteristic
38#89552C411yescharacteristic
39#89537C411yescharacteristic
40#89536C411yescharacteristic
41#89502—21no—
42#89502—21no—
43#89499—21no—
44#89492—21no—
45C4×#1132C2211yescharacteristic
46#89460C2211yescharacteristic
47C4×#1131C2211yescharacteristic
48#89457C2211yescharacteristic
49C4×C8×D7C2211yescharacteristic
50#89451C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×D4⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×C82C2211yescharacteristic
55D4⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89278C2211yescharacteristic
59#89275C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89252C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89249C2211yescharacteristic
66C8×C7⋊C8C2211yescharacteristic
67#17968—71nomaximal, Sylow
68C2×#1157—21no—
69C2×C7×D8C2×C411yescharacteristic
70C2×C7×M16—21no—
71C2×C7×M16—21no—
72C4×C7×D4C2311yescharacteristic
73C4×C7×D4C2311yescharacteristic
74#6503C2×C411yescharacteristic
75#6501C2×C411yescharacteristic
76#6501C2×C411yescharacteristic
77#6499C2×C411yescharacteristic
78C2×#1142C2×C411yescharacteristic
79C2×#1141C2×C411yescharacteristic
80C4×C22⋊D7—21no—
81C4×C22⋊D7—21no—
82#6489—21no—
83C22×C7⋊C8—21no—
84C22×C7⋊C8—21no—
85C2×#1132C2×C411yescharacteristic
86C2×#1131—21no—
87C2×#1131—21no—
88C2×#1131C2×C411yescharacteristic
89C2×#1131C2×C411yescharacteristic
90C2×C8×D7—21no—
91C2×C8×D7—21no—
92#6460C2×C411yescharacteristic
93#6458C2×C411yescharacteristic
94#6453C2×C411yescharacteristic
95#6451C2×C411yescharacteristic
96#6448—21no—
97#6447—21no—
98#6446—21no—
99#6442—21no—
100C4×D28C2311yescharacteristic
101C42×D7C2311yescharacteristic
102C4×Dic14C2311yescharacteristic
103C7×#107C2×C411yescharacteristic
104C7×C4⋊C8C2311yescharacteristic
105C7×C4⋊C8C2311yescharacteristic
106C7×#102C2×C411yescharacteristic
107C7×#102C2×C411yescharacteristic
108C7×C22⋊C8—21no—
109C7×C22⋊C8—21no—
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112C4×C7×C8C2311yescharacteristic
113#6406C2×C411yescharacteristic
114#6406C2×C411yescharacteristic
115C22⋊(C7⋊C8)—21no—
116C22⋊(C7⋊C8)—21no—
117#6395C2×C411yescharacteristic
118(C2×D7)⋊C8—21no—
119(C2×D7)⋊C8—21no—
120(C2×D7)⋊C8—21no—
121(C2×D7)⋊C8D411yescharacteristic
122(C2×D7)⋊C8D411yescharacteristic
123#6391C2×C411yescharacteristic
124#6390C2×C411yescharacteristic
125#6389—21no—
126#6389—21no—
127#6389C2×C411yescharacteristic
128#6389C2×C411yescharacteristic
129Dic7⋊C8—21no—
130Dic7⋊C8—21no—
131Dic7⋊C8—21no—
132Dic7⋊C8D411yescharacteristic
133Dic7⋊C8D411yescharacteristic
134C8×Dic7—21no—
135C8×Dic7—21no—
136#6384C2×C411yescharacteristic
137#6383C2×C411yescharacteristic
138#6382C2×C411yescharacteristic
139#6381C2×C411yescharacteristic
140C4⋊(C7⋊C8)C2311yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3171—71no—
146#2993—71no—
147#2981—71no—
148#1594—71no—
149#1592—71no—
150#1590—71no—
151#1589D711yescharacteristic, complemented
152#1580—71no—
153SD16⋊C8—71no—
154C8×SD16—71no—
155C8×D8—71no—
156#1569—71no—
157#1494—71no—
158#1492—71no—
159C8×M16—71no—
160C2×C7×D4C22×C411yescharacteristic
161C2×C7×D4C22×C411yescharacteristic
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C2×C22⋊D7—21no—
165C2×C22⋊D7—21no—
166C2×C22⋊D7—21no—
167C2×C22⋊D7—21no—
168#1157—81no—
169#1157—81no—
170C2×D28—21no—
171C2×D28C22×C411yescharacteristic
172C2×C4×D7—21no—
173C2×C4×D7C22×C411yescharacteristic
174C2×C4×D7C2×D411yescharacteristic
175C2×C4×D7#4111yescharacteristic
176C2×Dic14—21no—
177C2×Dic14C22×C411yescharacteristic
178C7×D8—22no—
179C7×M16—81no—
180C7×M16—81no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8C22×C411yescharacteristic
185C2×C7×C8C22×C411yescharacteristic
186C2×C7×C8C2×D411yescharacteristic
187C2×C7×C8C2×D411yescharacteristic
188C7×C4⋊C4C22×C411yescharacteristic
189C7×C4⋊C4C22×C411yescharacteristic
190C7×C22⋊C4—21no—
191C7×C22⋊C4—21no—
192C42×C7C2411yescharacteristic
193C22⋊Dic7—21no—
194C22⋊Dic7—21no—
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197#1142—22no—
198#1141—22no—
199(C2×D7)⋊C4—21no—
200(C2×D7)⋊C4—21no—
201(C2×D7)⋊C4—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7C22×C411yescharacteristic
205C4⋊Dic7C22×C411yescharacteristic
206Dic7⋊C4—21no—
207Dic7⋊C4—21no—
208Dic7⋊C4—21no—
209C4×Dic7—21no—
210C4×Dic7C22×C411yescharacteristic
211C4×Dic7C2×D411yescharacteristic
212C4×Dic7#4111yescharacteristic
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—81no—
217C2×C7⋊C8—81no—
218C2×C7⋊C8C22×C411yescharacteristic
219C2×C7⋊C8C22×C411yescharacteristic
220C2×C7⋊C8C2×D411yescharacteristic
221C2×C7⋊C8C2×D411yescharacteristic
222#1132—22no—
223#1131—22no—
224#1131—22no—
225#1131—81no—
226#1131—81no—
227C8×D7—42no—
228C8×D7—81no—
229#532—71no—
230#529—71no—
231#501—71no—
232#496—71no—
233#490—71no—
234#489—71no—
235#469—71no—
236#469—71no—
237#468—71no—
238#468—71no—
239#446—71no—
240C8×Q8—71no—
241C4×SD16—71no—
242C4×SD16—71no—
243C4×D8—71no—
244C4×D8C2×D711yescharacteristic
245#436—71no—
246#436—141no—
247#436C2×D711yescharacteristic
248#436C2×D711yescharacteristic
249#435—141no—
250#435—141no—
251#435—141no—
252C8×D4—71no—
253C8×D4—71no—
254C8×D4—71no—
255#433—141no—
256#432—71no—
257#431—71no—
258#424—141no—
259#423—71no—
260#408—141no—
261#408—141no—
262C4×M16—71no—
263C2×C4×C8—71no—
264#335—71no—
265#335C2×D711yescharacteristic
266#334—71no—
267#334—71no—
268Q8⋊C8—71no—
269Q8⋊C8—71no—
270D4⋊C8—71no—
271D4⋊C8—71no—
272D4⋊C8—71no—
273D4⋊C8—71no—
274D4⋊C8C2×D711yescharacteristic
275D4⋊C8C2×D711yescharacteristic
276#322—71no—
277#322—71no—
278C82—71no—
279C82C2×D711yescharacteristic
280C23×C7—21no—
281C23×C7—21no—
282C22×D7—21no—
283C22×D7C4×D411yescharacteristic
284C7×D4—21no—
285C7×D4—21no—
286C7×D4—21no—
287C7×D4—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C22⋊D7—81no—
298C22⋊D7—81no—
299C22⋊D7—81no—
300C22⋊D7—81no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7C4×D411yescharacteristic
305C2×Dic7C4×D411yescharacteristic
306C2×Dic7C4×D411yescharacteristic
307D28—21no—
308D28—21no—
309D28—81no—
310C4×D7—21no—
311C4×D7—21no—
312C4×D7—21no—
313C4×D7—21no—
314C4×D7—41no—
315C4×D7—41no—
316C4×D7—41no—
317Dic14—21no—
318Dic14—21no—
319Dic14—81no—
320C7×C8—22no—
321C7×C8—21no—
322C7×C8—21no—
323C7×C8—21no—
324C7×C8—41no—
325C7×C8—41no—
326C7×C8#13112yes—
327C7⋊C8—21no—
328C7⋊C8—21no—
329C7⋊C8—21no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C7⋊C8—41no—
333C7⋊C8#13112yes—
334C2×#41—141no—
335C2×SD16—71no—
336C2×SD16—71no—
337C2×D8—71no—
338C2×D8C4×D711yescharacteristic
339C2×M16—21no—
340C2×M16—21no—
341C2×M16—71no—
342C2×M16—71no—
343C2×M16—141no—
344C2×M16—141no—
345C22×C8—141no—
346C22×C8—141no—
347C22×C8—141no—
348C22×C8—141no—
349C4⋊Q8—71no—
350#127—71no—
351#126—141no—
352#125—71no—
353#124—71no—
354#124—141no—
355#123—141no—
356C22⋊Q8—71no—
357#121—71no—
358#121—71no—
359#121—71no—
360#121—141no—
361C4×Q8—71no—
362C4×D4—71no—
363C4×D4—141no—
364C4×D4—141no—
365C4×D4C22×D711yescharacteristic
366C4×D4C22×D711yescharacteristic
367#117—141no—
368C2×C42—71no—
369#107—71no—
370#107C4×D711yescharacteristic
371#106—71no—
372#106—71no—
373C4⋊C8—71no—
374C4⋊C8—71no—
375C4⋊C8—71no—
376C4⋊C8—71no—
377C4⋊C8—141no—
378C4⋊C8—141no—
379C4⋊C8—141no—
380C4⋊C8C22×D711yescharacteristic
381C4⋊C8C22×D711yescharacteristic
382#103—71no—
383#103—71no—
384#102—71no—
385#102—71no—
386#102—71no—
387#102—71no—
388#102C4×D711yescharacteristic
389#102C4×D711yescharacteristic
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8—71no—
393C22⋊C8—71no—
394C22⋊C8—141no—
395C22⋊C8—141no—
396C22⋊C8—141no—
397C22⋊C8—141no—
398C22⋊C8—141no—
399#97—71no—
400#97—71no—
401#97—141no—
402#97—141no—
403C4×C8—71no—
404C4×C8—71no—
405C4×C8—71no—
406C4×C8—141no—
407C4×C8—141no—
408C4×C8C22×D711yescharacteristic
409C4×C8C22×D711yescharacteristic
410C4×C8C22×D711yescharacteristic
411C22×C7—41no—
412C22×C7—41no—
413C22×C7—41no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7C2×C4×D411yescharacteristic
418C2×D7—21no—
419C2×D7—21no—
420C2×D7—21no—
421C2×D7—41no—
422C2×D7—41no—
423C2×D7—41no—
424C4×C7—21no—
425C4×C7—41no—
426C4×C7—41no—
427C4×C7C2×#13111yescharacteristic
428C4×C7C2×#13111yescharacteristic
429C4×C7#57611yescharacteristic
430C4×C7#57711yescharacteristic
431Dic7—21no—
432Dic7—21no—
433Dic7—21no—
434Dic7—41no—
435Dic7—41no—
436Dic7—41no—
437#41—561no—
438#41—561no—
439C2×Q8—71no—
440C2×Q8—141no—
441C2×D4—71no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4—141no—
445C2×D4—141no—
446C2×D4—141no—
447C2×D4C2×C4×D711yescharacteristic
448C2×D4C2×C4×D711yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—141no—
455SD16—142no—
456SD16—142no—
457D8—22no—
458D8—142no—
459M16—81no—
460M16—81no—
461M16—142no—
462M16—142no—
463M16—561no—
464M16—561no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—71no—
469C2×C8—71no—
470C2×C8—71no—
471C2×C8—71no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—282no—
476C2×C8—561no—
477C2×C8—561no—
478C2×C8—561no—
479C2×C8C2×C4×D711yescharacteristic
480C2×C8C2×C4×D711yescharacteristic
481C2×C8D4×D711yescharacteristic
482C2×C8D4×D711yescharacteristic
483C4⋊C4—71no—
484C4⋊C4—71no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4C2×C4×D711yescharacteristic
491C4⋊C4C2×C4×D711yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—141no—
495C22⋊C4—141no—
496C22⋊C4—141no—
497C22⋊C4—141no—
498C22⋊C4—141no—
499C22⋊C4—141no—
500C22⋊C4—141no—
501C42—71no—
502C42—71no—
503C42—71no—
504C42—141no—
505C42C23×D711yescharacteristic
506C2×C7—81no—
507C2×C7—81no—
508C2×C7#293111yescharacteristic
509C2×C7#294611yescharacteristic
510C2×C7#295611yescharacteristic
511D7—41no—
512D7—81no—
513C23—21no—
514C23—21no—
515C23—71no—
516C23—141no—
517Q8—141no—
518Q8—141no—
519Q8—561no—
520D4—21no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—141no—
525D4—141no—
526D4—561no—
527D4—561no—
528D4—561no—
529D4—561no—
530D4—561no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—71no—
538C2×C4—71no—
539C2×C4—71no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4C22×C4×D711yescharacteristic
551C2×C4C2×D4×D711yescharacteristic
552C2×C4D7×#4111yescharacteristic
553C8—22no—
554C8—21no—
555C8—21no—
556C8—21no—
557C8—41no—
558C8—41no—
559C8—72no—
560C8—141no—
561C8—141no—
562C8—141no—
563C8—281no—
564C8—281no—
565C8—281no—
566C8#647012yes—
567C7#1796811yescharacteristic, complemented, Sylow
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—141no—
575C22—141no—
576C22—141no—
577C22—281no—
578C22—281no—
579C22—281no—
580C22C4×D4×D711yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—141no—
585C4—141no—
586C4—141no—
587C4—281no—
588C4—281no—
589C4—281no—
590C4C2×#647011yescharacteristic
591C4D7×#13111yescharacteristic
592C4#9044811yescharacteristic
593C4#9046111yescharacteristic
594C2—81no—
595C2—81no—
596C2—281no—
597C2—561no—
598C2#1173698311yescharacteristic
599C2#1173703211yescharacteristic
600C2#1173733411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1589C2
socleC22×C7C2×C4×D4
prime radical#11737044C2
cosocleC42×C7C24
hypercenter#1589D7
hypocenterC7#17968
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1589

Complements

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

NH ≅ G/Ntype
2. #11736992C2semidirect
3. #11736971C2semidirect
4. C7×#1589C2semidirect
5. #11728632C2semidirect
6. #11728615C2semidirect
7. #11728614C2semidirect
8. #11726325C2semidirect
9. #11726312C2semidirect
10. #11725931C2semidirect
11. #11725930C2semidirect
12. #11725903C2semidirect
13. #11725902C2semidirect
14. #11725809C2semidirect
15. C8×#1131C2semidirect
151. #1589D7semidirect
567. C7#17968semidirect

Maximal subgroups

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

subgroupindex
#117370442
#117369922
#117369712
C7×#15892
#117286322
#117286152
#117286142
#117263252
#117263122
#117259312
#117259302
#117259032
#117259022
#117258092
C8×#11312
#179687