Finite Group Databasebeta

← #13324971

Subgroups — #13324971

Order 1792.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737368C211yesmaximal, characteristic, complemented
2#11737100C211yesmaximal, characteristic
3#11736851C211yesmaximal, characteristic
4C7×C8×SD16C211yesmaximal, characteristic, complemented
5#11728689C211yesmaximal, characteristic, complemented
6#11728651C211yesmaximal, characteristic, complemented
7#11728597C211yesmaximal, characteristic, complemented
8#11726289C211yesmaximal, characteristic, complemented
9#11726276C211yesmaximal, characteristic, complemented
10#11725994C211yesmaximal, characteristic, complemented
11#11725979C211yesmaximal, characteristic, complemented
12#11725919C211yesmaximal, characteristic
13#11725903C211yesmaximal, characteristic, complemented
14#11725810C211yesmaximal, characteristic, complemented
15#11725801C211yesmaximal, characteristic, complemented
16#90269C2211yescharacteristic
17C4×#1159C2211yescharacteristic
18C7×C8×Q8C2211yescharacteristic
19C4×C7×SD16C2211yescharacteristic
20C7×C8×D4C2211yescharacteristic
21#89945C411yescharacteristic
22#89934C411yescharacteristic
23#89883—21no—
24C2×Dic7⋊C8—21no—
25C4×#1144C2211yescharacteristic
26#89796C2211yescharacteristic
27C4×#1142C2211yescharacteristic
28D4×C7⋊C8C2211yescharacteristic
29#89639C411yescharacteristic
30#89626C411yescharacteristic
31#89612—21no—
32#89609C2211yescharacteristic
33#89605C2211yescharacteristic
34#89603C2211yescharacteristic
35#89602C2211yescharacteristic
36#89600—21no—
37#89586C411yescharacteristic
38#89565C411yescharacteristic
39#89555C411yescharacteristic
40#89531C411yescharacteristic
41#89502—21no—
42#89491—21no—
43C4×Dic28C2211yescharacteristic
44#89462—21no—
45#89457C2211yescharacteristic
46#89457C2211yescharacteristic
47C4×C8×D7C2211yescharacteristic
48#89451C2211yescharacteristic
49#89451C2211yescharacteristic
50C8×Dic14—21no—
51C7×#334C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×C82C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89279C2211yescharacteristic
59#89275C2211yescharacteristic
60#89274C2211yescharacteristic
61#89253C2211yescharacteristic
62#89251C2211yescharacteristic
63#89251C2211yescharacteristic
64#89249C2211yescharacteristic
65#89249C2211yescharacteristic
66C8×C7⋊C8C2211yescharacteristic
67#18016—71nomaximal, Sylow
68C2×#1159C2×C411yescharacteristic
69C2×C7×SD16C2×C411yescharacteristic
70C22×C7×C8—21no—
71C4×C7×Q8C2311yescharacteristic
72C4×C7×D4C2311yescharacteristic
73#6509C2×C411yescharacteristic
74#6507C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76D4×Dic7C2×C411yescharacteristic
77C2×#1142C2×C411yescharacteristic
78C4×C22⋊D7—21no—
79C2×C4×Dic7—21no—
80C22×C7⋊C8—21no—
81C2×Dic28C2×C411yescharacteristic
82C2×#1131—21no—
83C2×C8×D7—21no—
84#6460C2×C411yescharacteristic
85#6459—21no—
86#6455C2×C411yescharacteristic
87#6452—21no—
88#6451C2×C411yescharacteristic
89#6450C2×C411yescharacteristic
90(C22⋊D7)⋊C4—21no—
91#6440—21no—
92#6435—21no—
93C42×D7C2311yescharacteristic
94C4×Dic14—21no—
95C4×Dic14C2311yescharacteristic
96C4×Dic14C2311yescharacteristic
97C7×#106C2×C411yescharacteristic
98C7×C4⋊C8—21no—
99C7×C4⋊C8C2311yescharacteristic
100C7×C4⋊C8C2311yescharacteristic
101C7×#103C2×C411yescharacteristic
102C7×#102C2×C411yescharacteristic
103C7×C22⋊C8—21no—
104C4×C7×C8—21no—
105C4×C7×C8C2311yescharacteristic
106C4×C7×C8C2311yescharacteristic
107C4×C7×C8C2311yescharacteristic
108#6409C2×C411yescharacteristic
109#6406C2×C411yescharacteristic
110C22⋊(C7⋊C8)—21no—
111(C2×D7)⋊C8—21no—
112(C2×D7)⋊C8C2×C411yescharacteristic
113(C2×D7)⋊C8C2×C411yescharacteristic
114(C2×D7)⋊C8D411yescharacteristic
115(C2×D7)⋊C8D411yescharacteristic
116#6392C2×C411yescharacteristic
117#6390C2×C411yescharacteristic
118#6390C2×C411yescharacteristic
119#6389—21no—
120#6389—21no—
121#6389—21no—
122Dic7⋊C8—21no—
123Dic7⋊C8—21no—
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8C2×C411yescharacteristic
128Dic7⋊C8C2×C411yescharacteristic
129Dic7⋊C8D411yescharacteristic
130Dic7⋊C8D411yescharacteristic
131C8×Dic7—21no—
132C8×Dic7—21no—
133C8×Dic7—21no—
134#6384C2×C411yescharacteristic
135#6384C2×C411yescharacteristic
136#6382C2×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#3107—71no—
146#2990—71no—
147#2986—71no—
148#1591—71no—
149#1591—71no—
150#1587—71no—
151#1587—71no—
152#1585—71no—
153#1585—71no—
154C8×SD16—71no—
155C8×SD16D711yescharacteristic, complemented
156#1574—71no—
157#1493—71no—
158#1476—71no—
159#1450—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×C4×D7C22×C411yescharacteristic
169C2×C4×D7C2×D411yescharacteristic
170C2×C4×D7#4111yescharacteristic
171C2×Dic14—21no—
172C2×Dic14C22×C411yescharacteristic
173C2×Dic14C22×C411yescharacteristic
174C7×SD16—22no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—81no—
179C2×C7×C8C22×C411yescharacteristic
180C2×C7×C8C22×C411yescharacteristic
181C2×C7×C8C2×D411yescharacteristic
182C2×C7×C8C2×D411yescharacteristic
183C7×C4⋊C4—21no—
184C7×C4⋊C4C22×C411yescharacteristic
185C7×C4⋊C4C22×C411yescharacteristic
186C7×C22⋊C4—21no—
187C42×C7—21no—
188C42×C7C2411yescharacteristic
189C22⋊Dic7—21no—
190#1144—41no—
191#1142—22no—
192(C2×D7)⋊C4—21no—
193(C2×D7)⋊C4—21no—
194(C2×D7)⋊C4—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7C22×C411yescharacteristic
197C4⋊Dic7C22×C411yescharacteristic
198Dic7⋊C4—21no—
199Dic7⋊C4—21no—
200Dic7⋊C4—21no—
201Dic7⋊C4—21no—
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7C22×C411yescharacteristic
210C4×Dic7C2×D411yescharacteristic
211C4×Dic7#4111yescharacteristic
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—81no—
216C2×C7⋊C8C22×C411yescharacteristic
217C2×C7⋊C8C22×C411yescharacteristic
218C2×C7⋊C8C2×D411yescharacteristic
219C2×C7⋊C8C2×D411yescharacteristic
220Dic28—41no—
221#1131—81no—
222C8×D7—24no—
223#533—71no—
224C4×#41—71no—
225#485—71no—
226#485—71no—
227#475—71no—
228#474—71no—
229#462—71no—
230#461—71no—
231#452—71no—
232#452—71no—
233#446—71no—
234#446—71no—
235#446—71no—
236#446—71no—
237#446—141no—
238C8×Q8—71no—
239C8×Q8—141no—
240C8×Q8C2×D711yescharacteristic
241C4×Q16—71no—
242C4×Q16—71no—
243C4×SD16—71no—
244C4×SD16C2×D711yescharacteristic
245#435—141no—
246#435—141no—
247C8×D4—71no—
248C8×D4C2×D711yescharacteristic
249#433—141no—
250#433—141no—
251#432—141no—
252#431—71no—
253#431—71no—
254#431—71no—
255#423—71no—
256C2×C4⋊C8—141no—
257C2×C4×C8—71no—
258#335—71no—
259#335—71no—
260#334—71no—
261#334C2×D711yescharacteristic
262Q8⋊C8—71no—
263Q8⋊C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8C2×D711yescharacteristic
268D4⋊C8—71no—
269D4⋊C8C2×D711yescharacteristic
270#322—71no—
271#322—71no—
272C82—71no—
273C82C2×D711yescharacteristic
274C23×C7—21no—
275C22×D7C4×D411yescharacteristic
276C7×Q8—21no—
277C7×Q8—21no—
278C7×D4—21no—
279C7×D4#13112yes—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C22⋊D7—42no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7C4×D411yescharacteristic
296C2×Dic7C4×D411yescharacteristic
297C2×Dic7C4×D411yescharacteristic
298C4×D7—21no—
299C4×D7—21no—
300C4×D7—21no—
301C4×D7—21no—
302C4×D7—22no—
303Dic14—21no—
304Dic14—21no—
305Dic14—21no—
306Dic14—21no—
307Dic14—42no—
308C7×C8—22no—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—21no—
313C7×C8—41no—
314C7×C8—41no—
315C7⋊C8—22no—
316C7⋊C8—21no—
317C7⋊C8—21no—
318C7⋊C8—21no—
319C7⋊C8—21no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C2×#41—71no—
323C2×Q16—71no—
324C2×Q16—71no—
325C2×SD16—71no—
326C2×SD16C4×D711yescharacteristic
327C2×M16—141no—
328C22×C8—21no—
329C22×C8—141no—
330C22×C8—141no—
331C4⋊Q8—71no—
332C4⋊Q8—71no—
333#125—141no—
334C22⋊Q8—71no—
335C22⋊Q8—71no—
336C22⋊Q8—141no—
337C4×Q8—71no—
338C4×Q8—71no—
339C4×Q8—71no—
340C4×Q8—141no—
341C4×Q8C22×D711yescharacteristic
342C4×D4—71no—
343C4×D4—141no—
344C4×D4—141no—
345C4×D4C22×D711yescharacteristic
346#117—71no—
347#117—141no—
348#117—141no—
349C2×C42—71no—
350C2×C42—141no—
351#107—71no—
352#107—71no—
353#106—71no—
354#106C4×D711yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—71no—
357C4⋊C8—71no—
358C4⋊C8—71no—
359C4⋊C8—71no—
360C4⋊C8—71no—
361C4⋊C8—71no—
362C4⋊C8—141no—
363C4⋊C8—141no—
364C4⋊C8—141no—
365C4⋊C8—141no—
366C4⋊C8—141no—
367C4⋊C8—141no—
368C4⋊C8C22×D711yescharacteristic
369C4⋊C8C22×D711yescharacteristic
370#103—71no—
371#103—71no—
372#103—71no—
373#103—71no—
374#103—71no—
375#103C4×D711yescharacteristic
376#102—71no—
377#102C4×D711yescharacteristic
378C22⋊C8—21no—
379C22⋊C8—71no—
380C22⋊C8—71no—
381C22⋊C8—71no—
382C22⋊C8—71no—
383C22⋊C8—141no—
384C22⋊C8—141no—
385#97—141no—
386#97—141no—
387#97—141no—
388#97—141no—
389C4×C8—21no—
390C4×C8—71no—
391C4×C8—71no—
392C4×C8—71no—
393C4×C8—141no—
394C4×C8—141no—
395C4×C8—141no—
396C4×C8C22×D711yescharacteristic
397C4×C8C22×D711yescharacteristic
398C4×C8C22×D711yescharacteristic
399C22×C7—22no—
400C22×C7—41no—
401C22×C7—41no—
402C22×C7C2×C4×D411yescharacteristic
403C2×D7—21no—
404C2×D7—21no—
405C2×D7—21no—
406C4×C7—21no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7C2×#13111yescharacteristic
411C4×C7#56811yescharacteristic
412C4×C7#57711yescharacteristic
413C4×C7#57811yescharacteristic
414Dic7—21no—
415Dic7—21no—
416Dic7—41no—
417Dic7—41no—
418Dic7—41no—
419Dic7—41no—
420Dic7#44312yes—
421#41—142no—
422#41—281no—
423C2×Q8—71no—
424C2×Q8—71no—
425C2×Q8—141no—
426C2×Q8C2×C4×D711yescharacteristic
427C2×D4—141no—
428C2×D4C2×C4×D711yescharacteristic
429C22×C4—21no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—141no—
434C22×C4—141no—
435Q16—281no—
436Q16—281no—
437SD16—22no—
438SD16—142no—
439M16—561no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—71no—
444C2×C8—71no—
445C2×C8—71no—
446C2×C8—71no—
447C2×C8—81no—
448C2×C8—144no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—561no—
453C2×C8C2×C4×D711yescharacteristic
454C2×C8C2×C4×D711yescharacteristic
455C2×C8D4×D711yescharacteristic
456C2×C8D4×D711yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—71no—
459C4⋊C4—71no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4C2×C4×D711yescharacteristic
469C4⋊C4C2×C4×D711yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—141no—
472C22⋊C4—141no—
473C22⋊C4—141no—
474C22⋊C4—141no—
475C42—21no—
476C42—71no—
477C42—71no—
478C42—71no—
479C42—141no—
480C42—141no—
481C42—141no—
482C42—141no—
483C42C23×D711yescharacteristic
484C2×C7—42no—
485C2×C7#293311yescharacteristic
486C2×C7#294711yescharacteristic
487C2×C7C2×#44311yescharacteristic
488D7—41no—
489C23—21no—
490C23—71no—
491Q8—21no—
492Q8—21no—
493Q8—141no—
494Q8—141no—
495Q8—141no—
496Q8—141no—
497Q8—282no—
498D4—21no—
499D4—282no—
500D4#646412yes—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—71no—
507C2×C4—71no—
508C2×C4—71no—
509C2×C4—142no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4C22×C4×D711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C2×C4D7×#4111yescharacteristic
523C8—22no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—142no—
531C8—141no—
532C8—141no—
533C8—141no—
534C8—141no—
535C8—281no—
536C8—281no—
537C7#1801611yescharacteristic, complemented, Sylow
538C22—22no—
539C22—41no—
540C22—41no—
541C22—141no—
542C22—141no—
543C22—141no—
544C22C4×D4×D711yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—72no—
550C4—141no—
551C4—141no—
552C4—281no—
553C4—281no—
554C4—281no—
555C4—281no—
556C4C2×#646411yescharacteristic
557C4#9044111yescharacteristic
558C4#9045311yescharacteristic
559C4#9046211yescharacteristic
560C2—42no—
561C2—281no—
562C2#1173686811yescharacteristic
563C2#1173709311yescharacteristic
564C2D7×#44311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×C8×SD16C2
socleC22×C7C2×C4×D4
prime radicalC2×#1159C2×C4
cosocleC42×C7C24
hypercenterC8×SD16D7
hypocenterC7#18016
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
1. #11737368C2semidirect
4. C7×C8×SD16C2semidirect
5. #11728689C2semidirect
6. #11728651C2semidirect
7. #11728597C2semidirect
8. #11726289C2semidirect
9. #11726276C2semidirect
10. #11725994C2semidirect
11. #11725979C2semidirect
13. #11725903C2semidirect
14. #11725810C2semidirect
15. #11725801C2semidirect
155. C8×SD16D7semidirect
537. C7#18016semidirect

Maximal subgroups

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

subgroupindex
#117373682
#117371002
#117368512
C7×C8×SD162
#117286892
#117286512
#117285972
#117262892
#117262762
#117259942
#117259792
#117259192
#117259032
#117258102
#117258012
#180167