Finite Group Databasebeta

← #13325202

Subgroups — #13325202

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737354C211yesmaximal, characteristic, complemented
2#11737097C211yesmaximal, characteristic
3#11736969C211yesmaximal, characteristic, complemented
4C7×#1588C211yesmaximal, characteristic, complemented
5C8×#1144C211yesmaximal, characteristic, complemented
6#11728654C211yesmaximal, characteristic, complemented
7#11728613C211yesmaximal, characteristic, complemented
8#11726326C211yesmaximal, characteristic, complemented
9#11726312C211yesmaximal, characteristic, complemented
10#11725996C211yesmaximal, characteristic, complemented
11#11725973C211yesmaximal, characteristic, complemented
12#11725920C211yesmaximal, characteristic, complemented
13#11725907C211yesmaximal, characteristic, complemented
14#11725842C211yesmaximal, characteristic, complemented
15#11725828C211yesmaximal, characteristic, complemented
16#90271C2211yescharacteristic
17C4×#1159C2211yescharacteristic
18C7×C8×Q8C2211yescharacteristic
19C4×C7×SD16C2211yescharacteristic
20C7×#436C2211yescharacteristic
21#89945C411yescharacteristic
22#89937C411yescharacteristic
23#89900—21no—
24#89890—21no—
25C4×#1144C2211yescharacteristic
26#89796C2211yescharacteristic
27C4×#1141C2211yescharacteristic
28#89784C2211yescharacteristic
29#89638C411yescharacteristic
30#89629C411yescharacteristic
31#89611—21no—
32#89609C2211yescharacteristic
33#89608—21no—
34#89606C2211yescharacteristic
35D7×C4⋊C8C2211yescharacteristic
36#89602C2211yescharacteristic
37#89587C411yescharacteristic
38#89570C411yescharacteristic
39#89553C411yescharacteristic
40#89535C411yescharacteristic
41#89501—21no—
42#89490—21no—
43C4×#1132C2211yescharacteristic
44#89462—21no—
45#89460C2211yescharacteristic
46C8×D28—21no—
47#89457C2211yescharacteristic
48#89457C2211yescharacteristic
49#89457C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89278C2211yescharacteristic
59#89274C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89253C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89249C2211yescharacteristic
66C8×C7⋊C8C2211yescharacteristic
67#18033—71nomaximal, Sylow
68C2×#1159C2×C411yescharacteristic
69C2×C7×SD16C2×C411yescharacteristic
70C2×C7×M16—21no—
71C4×C7×Q8C2311yescharacteristic
72C4×C7×D4C2311yescharacteristic
73#6510C2×C411yescharacteristic
74#6509C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76D4×Dic7C2×C411yescharacteristic
77C2×#1141C2×C411yescharacteristic
78C4×C22⋊D7—21no—
79C2×C4×Dic7—21no—
80C2×#1136—21no—
81C2×#1132C2×C411yescharacteristic
82C2×#1131—21no—
83C2×#1131—21no—
84C2×C8×D7—21no—
85C2×C8×D7—21no—
86#6461—21no—
87#6458C2×C411yescharacteristic
88#6457—21no—
89#6455C2×C411yescharacteristic
90#6453C2×C411yescharacteristic
91#6450C2×C411yescharacteristic
92(C22⋊D7)⋊C4—21no—
93#6440—21no—
94C4×D28—21no—
95C4×D28C2311yescharacteristic
96#6435—21no—
97C42×D7C2311yescharacteristic
98C4×Dic14C2311yescharacteristic
99C7×#106C2×C411yescharacteristic
100C7×C4⋊C8—21no—
101C7×C4⋊C8C2311yescharacteristic
102C7×C4⋊C8C2311yescharacteristic
103C7×#103C2×C411yescharacteristic
104C7×#102C2×C411yescharacteristic
105C7×C22⋊C8—21no—
106C4×C7×C8—21no—
107C4×C7×C8C2311yescharacteristic
108C4×C7×C8C2311yescharacteristic
109C4×C7×C8C2311yescharacteristic
110#6409C2×C411yescharacteristic
111#6406C2×C411yescharacteristic
112C22⋊(C7⋊C8)—21no—
113#6395C2×C411yescharacteristic
114(C2×D7)⋊C8—21no—
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8C2×C411yescharacteristic
118(C2×D7)⋊C8C2×C411yescharacteristic
119(C2×D7)⋊C8D411yescharacteristic
120(C2×D7)⋊C8D411yescharacteristic
121#6391C2×C411yescharacteristic
122#6390C2×C411yescharacteristic
123Dic7⋊C8—21no—
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8C2×C411yescharacteristic
129Dic7⋊C8C2×C411yescharacteristic
130Dic7⋊C8D411yescharacteristic
131Dic7⋊C8D411yescharacteristic
132C8×Dic7—21no—
133C8×Dic7—21no—
134#6384C2×C411yescharacteristic
135#6383C2×C411yescharacteristic
136#6381C2×C411yescharacteristic
137#6381C2×C411yescharacteristic
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2311yescharacteristic
140C4⋊(C7⋊C8)C2311yescharacteristic
141#6377—21no—
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3102—71no—
146#2994—71no—
147#2991—71no—
148#1592—71no—
149#1589—71no—
150#1588—71no—
151#1588D711yescharacteristic, complemented
152#1584—71no—
153#1581—71no—
154SD16⋊C8—71no—
155C8×Q16—71no—
156#1571—71no—
157#1495—71no—
158#1477—71no—
159#1454—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×D4C22×C411yescharacteristic
162C22×C4×C7—21no—
163C2×C22⋊D7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166#1159—22no—
167#1159—41no—
168C2×D28—21no—
169C2×D28C22×C411yescharacteristic
170C2×C4×D7—21no—
171C2×C4×D7C22×C411yescharacteristic
172C2×C4×D7C2×D411yescharacteristic
173C2×C4×D7#4111yescharacteristic
174C2×Dic14C22×C411yescharacteristic
175C7×SD16—41no—
176C7×M16—81no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8C22×C411yescharacteristic
181C2×C7×C8C22×C411yescharacteristic
182C2×C7×C8C2×D411yescharacteristic
183C2×C7×C8C2×D411yescharacteristic
184C7×C4⋊C4—21no—
185C7×C4⋊C4C22×C411yescharacteristic
186C7×C4⋊C4C22×C411yescharacteristic
187C7×C22⋊C4—21no—
188C42×C7—21no—
189C42×C7C2411yescharacteristic
190C22⋊Dic7—21no—
191#1144—22no—
192#1141—41no—
193(C2×D7)⋊C4—21no—
194(C2×D7)⋊C4—21no—
195(C2×D7)⋊C4—21no—
196(C2×D7)⋊C4—21no—
197(C2×D7)⋊C4—21no—
198(C2×D7)⋊C4—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7C22×C411yescharacteristic
201C4⋊Dic7C22×C411yescharacteristic
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7C22×C411yescharacteristic
210C4×Dic7C2×D411yescharacteristic
211C4×Dic7#4111yescharacteristic
212#1136—81no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8C22×C411yescharacteristic
217C2×C7⋊C8C22×C411yescharacteristic
218C2×C7⋊C8C2×D411yescharacteristic
219C2×C7⋊C8C2×D411yescharacteristic
220#1132—22no—
221#1131—81no—
222#1131—81no—
223C8×D7—81no—
224C8×D7—81no—
225#529—71no—
226C4×#41—71no—
227#485—71no—
228#482—71no—
229#479—71no—
230#478—71no—
231#464—71no—
232#463—71no—
233#452—71no—
234#449—71no—
235#446—71no—
236C8×Q8—71no—
237C8×Q8—71no—
238C8×Q8C2×D711yescharacteristic
239C4×Q16—71no—
240C4×SD16—71no—
241C4×SD16C2×D711yescharacteristic
242C4×D8—71no—
243#436—71no—
244#436—71no—
245#436—141no—
246#436—141no—
247#436C2×D711yescharacteristic
248#435—141no—
249C8×D4—71no—
250C8×D4—141no—
251#433—141no—
252#431—71no—
253#431—71no—
254#431—71no—
255#431—141no—
256#424—141no—
257#423—71no—
258#423—141no—
259C2×C4⋊C8—71no—
260#335—71no—
261#335—71no—
262#335—71no—
263#335C2×D711yescharacteristic
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8C2×D711yescharacteristic
268D4⋊C8—71no—
269D4⋊C8—71no—
270D4⋊C8—71no—
271D4⋊C8C2×D711yescharacteristic
272#322—71no—
273#322—71no—
274#322C2×D711yescharacteristic
275C82—71no—
276C23×C7—21no—
277C22×D7—21no—
278C22×D7C4×D411yescharacteristic
279C7×Q8—21no—
280C7×Q8—21no—
281C7×D4—21no—
282C7×D4—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7C23×C411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C22⋊D7—42no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7C4×D411yescharacteristic
298C2×Dic7C4×D411yescharacteristic
299C2×Dic7C4×D411yescharacteristic
300D28—21no—
301D28—21no—
302D28—42no—
303C4×D7—21no—
304C4×D7—21no—
305C4×D7—21no—
306C4×D7—21no—
307C4×D7—41no—
308C4×D7—41no—
309C4×D7—41no—
310Dic14—21no—
311Dic14#13112yes—
312C7×C8—21no—
313C7×C8—21no—
314C7×C8—21no—
315C7×C8—21no—
316C7×C8—41no—
317C7×C8—41no—
318C7×C8—41no—
319C7⋊C8—21no—
320C7⋊C8—21no—
321C7⋊C8—21no—
322C7⋊C8—21no—
323C7⋊C8—41no—
324C7⋊C8—41no—
325C7⋊C8—41no—
326C2×#41—71no—
327C2×Q16—71no—
328C2×SD16—71no—
329C2×SD16C4×D711yescharacteristic
330C2×D8—71no—
331C2×M16—21no—
332C2×M16—141no—
333C2×M16—141no—
334C2×M16—141no—
335C22×C8—141no—
336C22×C8—141no—
337#126—141no—
338#125—71no—
339#124—71no—
340#123—141no—
341C22⋊Q8—71no—
342#121—71no—
343C4×Q8—71no—
344C4×Q8—71no—
345C4×Q8C22×D711yescharacteristic
346C4×D4—71no—
347C4×D4—71no—
348C4×D4—141no—
349C4×D4—141no—
350C4×D4—141no—
351C4×D4C22×D711yescharacteristic
352#117—71no—
353#117—141no—
354#117—141no—
355C2×C42—71no—
356C2×C42—141no—
357#107—71no—
358#107—71no—
359#106—71no—
360#106C4×D711yescharacteristic
361C4⋊C8—21no—
362C4⋊C8—71no—
363C4⋊C8—71no—
364C4⋊C8—71no—
365C4⋊C8—71no—
366C4⋊C8—71no—
367C4⋊C8—71no—
368C4⋊C8—141no—
369C4⋊C8—141no—
370C4⋊C8—141no—
371C4⋊C8—141no—
372C4⋊C8—141no—
373C4⋊C8—141no—
374C4⋊C8C22×D711yescharacteristic
375C4⋊C8C22×D711yescharacteristic
376#103—71no—
377#103—71no—
378#103—71no—
379#103C4×D711yescharacteristic
380#102—71no—
381#102—71no—
382#102—71no—
383#102C4×D711yescharacteristic
384C22⋊C8—21no—
385C22⋊C8—71no—
386C22⋊C8—71no—
387C22⋊C8—71no—
388C22⋊C8—71no—
389C22⋊C8—141no—
390C22⋊C8—141no—
391C22⋊C8—141no—
392C22⋊C8—141no—
393#97—141no—
394C4×C8—21no—
395C4×C8—71no—
396C4×C8—71no—
397C4×C8—71no—
398C4×C8—141no—
399C4×C8—141no—
400C4×C8C22×D711yescharacteristic
401C4×C8C22×D711yescharacteristic
402C4×C8C22×D711yescharacteristic
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7C2×C4×D411yescharacteristic
407C2×D7—21no—
408C2×D7—21no—
409C2×D7—21no—
410C2×D7—41no—
411C2×D7—41no—
412C2×D7—41no—
413C4×C7—21no—
414C4×C7—41no—
415C4×C7—41no—
416C4×C7—41no—
417C4×C7C2×#13111yescharacteristic
418C4×C7#56811yescharacteristic
419C4×C7#57611yescharacteristic
420C4×C7#57811yescharacteristic
421Dic7—22no—
422Dic7—21no—
423Dic7—21no—
424Dic7—21no—
425Dic7—41no—
426Dic7—41no—
427#41—142no—
428#41—281no—
429C2×Q8—71no—
430C2×Q8C2×C4×D711yescharacteristic
431C2×D4—71no—
432C2×D4—141no—
433C2×D4—141no—
434C2×D4C2×C4×D711yescharacteristic
435C22×C4—21no—
436C22×C4—71no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442Q16—142no—
443SD16—41no—
444SD16—142no—
445D8—281no—
446M16—81no—
447M16—561no—
448M16—561no—
449M16—561no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—71no—
454C2×C8—71no—
455C2×C8—71no—
456C2×C8—71no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—561no—
461C2×C8—561no—
462C2×C8C2×C4×D711yescharacteristic
463C2×C8C2×C4×D711yescharacteristic
464C2×C8D4×D711yescharacteristic
465C2×C8D4×D711yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—71no—
468C4⋊C4—71no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4C2×C4×D711yescharacteristic
475C4⋊C4C2×C4×D711yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—141no—
478C22⋊C4—141no—
479C22⋊C4—141no—
480C22⋊C4—141no—
481C22⋊C4—141no—
482C22⋊C4—141no—
483C22⋊C4—141no—
484C42—21no—
485C42—71no—
486C42—71no—
487C42—71no—
488C42—141no—
489C42—141no—
490C42—141no—
491C42C23×D711yescharacteristic
492C2×C7—81no—
493C2×C7#293311yescharacteristic
494C2×C7#295111yescharacteristic
495C2×C7#295511yescharacteristic
496D7—41no—
497D7—81no—
498C23—21no—
499C23—71no—
500C23—141no—
501Q8—21no—
502Q8—21no—
503Q8—72no—
504Q8—141no—
505D4—21no—
506D4—21no—
507D4—141no—
508D4—141no—
509D4—282no—
510D4—282no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—71no—
517C2×C4—71no—
518C2×C4—71no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—281no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4C22×C4×D711yescharacteristic
532C2×C4C2×D4×D711yescharacteristic
533C2×C4D7×#4111yescharacteristic
534C8—21no—
535C8—21no—
536C8—21no—
537C8—21no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—141no—
542C8—141no—
543C8—141no—
544C8—141no—
545C8—281no—
546C8—281no—
547C8—281no—
548C7#1803311yescharacteristic, complemented, Sylow
549C22—41no—
550C22—41no—
551C22—41no—
552C22—141no—
553C22—141no—
554C22—141no—
555C22—281no—
556C22—281no—
557C22—281no—
558C22C4×D4×D711yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—141no—
564C4—142no—
565C4—141no—
566C4—141no—
567C4—281no—
568C4—281no—
569C4#9043611yescharacteristic
570C4D7×#13111yescharacteristic
571C4#9046511yescharacteristic
572C4#9047011yescharacteristic
573C2—81no—
574C2—281no—
575C2—561no—
576C2#1173692711yescharacteristic
577C2#1173722311yescharacteristic
578C2#1173733311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1588C2
socleC22×C7C2×C4×D4
prime radical#89553C4
cosocleC42×C7C24
hypercenter#1588D7
hypocenterC7#18033
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1588

Complements

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

NH ≅ G/Ntype
1. #11737354C2semidirect
3. #11736969C2semidirect
4. C7×#1588C2semidirect
5. C8×#1144C2semidirect
6. #11728654C2semidirect
7. #11728613C2semidirect
8. #11726326C2semidirect
9. #11726312C2semidirect
10. #11725996C2semidirect
11. #11725973C2semidirect
12. #11725920C2semidirect
13. #11725907C2semidirect
14. #11725842C2semidirect
15. #11725828C2semidirect
151. #1588D7semidirect
548. C7#18033semidirect

Maximal subgroups

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

subgroupindex
#117373542
#117370972
#117369692
C7×#15882
C8×#11442
#117286542
#117286132
#117263262
#117263122
#117259962
#117259732
#117259202
#117259072
#117258422
#117258282
#180337