Finite Group Databasebeta

← #13324926

Subgroups — #13324926

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11736756C211yesmaximal, characteristic
2#11735956C211yesmaximal, characteristic
3M16×Dic14C211yesmaximal, characteristic, complemented
4C7×#1453C211yesmaximal, characteristic, complemented
5#11728521C211yesmaximal, characteristic, complemented
6#11728518C211yesmaximal, characteristic
7#11728507C211yesmaximal, characteristic, complemented
8#11725843C211yesmaximal, characteristic, complemented
9#11725841C211yesmaximal, characteristic, complemented
10#11725840C211yesmaximal, characteristic, complemented
11#11725839C211yesmaximal, characteristic, complemented
12#11725834C211yesmaximal, characteristic, complemented
13#11725830C211yesmaximal, characteristic, complemented
14#11725809C211yesmaximal, characteristic, complemented
15C8×Dic28C211yesmaximal, characteristic, complemented
16#90170C2211yescharacteristic
17C2×C4×Dic14C2211yescharacteristic
18C7×#431C2211yescharacteristic
19C7×#431C2211yescharacteristic
20C4×C7×M16C2211yescharacteristic
21#89900—21no—
22#89899—21no—
23#89890—21no—
24M16×Dic7—21no—
25#89697C2211yescharacteristic
26#89696C2211yescharacteristic
27#89617C411yescharacteristic
28#89617C411yescharacteristic
29#89601C411yescharacteristic
30#89601C411yescharacteristic
31#89511C411yescharacteristic
32#89510C411yescharacteristic
33#89508C411yescharacteristic
34#89493C411yescharacteristic
35#89481—21no—
36#89480—21no—
37#89473—21no—
38#89473—21no—
39C4×Dic28C2211yescharacteristic
40C4×Dic28C2211yescharacteristic
41C4×#1132C2211yescharacteristic
42C4×#1132C2211yescharacteristic
43#89460C2211yescharacteristic
44C8×D28C2211yescharacteristic
45#89451C2211yescharacteristic
46#89451C2211yescharacteristic
47#89451C2211yescharacteristic
48C8×Dic14C2211yescharacteristic
49C8×Dic14C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#322C2211yescharacteristic
52C7×#322C2211yescharacteristic
53C7×#322C2211yescharacteristic
54C7×C82C2211yescharacteristic
55#89254C2211yescharacteristic
56#89254C2211yescharacteristic
57#89253C2211yescharacteristic
58#89253C2211yescharacteristic
59#89252C2211yescharacteristic
60#89252C2211yescharacteristic
61#89251C2211yescharacteristic
62#89251C2211yescharacteristic
63#89251C2211yescharacteristic
64#89251C2211yescharacteristic
65#89251C2211yescharacteristic
66#89251C2211yescharacteristic
67#18012—71nomaximal, Sylow
68C22×Dic14C2×C411yescharacteristic
69C2×C7×M16D411yescharacteristic
70C2×C7×M16D411yescharacteristic
71C2×C42×C7C2311yescharacteristic
72#6493C2×C411yescharacteristic
73#6492—21no—
74C4×C22⋊D7—21no—
75C2×C4⋊Dic7C2×C411yescharacteristic
76#6487C2×C411yescharacteristic
77C2×Dic7⋊C4—21no—
78C2×C4×Dic7—21no—
79C2×#1136—21no—
80C2×Dic28C2×C411yescharacteristic
81C2×Dic28C2×C411yescharacteristic
82C2×#1132C2×C411yescharacteristic
83C2×#1132C2×C411yescharacteristic
84C2×#1131—21no—
85C2×C8×D7—21no—
86#6439—21no—
87#6438C2×C411yescharacteristic
88C4×D28C2311yescharacteristic
89#6435—21no—
90#6433C2×C411yescharacteristic
91C4×Dic14—21no—
92C4×Dic14—21no—
93C4×Dic14C2×C411yescharacteristic
94C4×Dic14C2×C411yescharacteristic
95C4×Dic14C2311yescharacteristic
96C4×Dic14C2311yescharacteristic
97C4×Dic14C2311yescharacteristic
98C7×C4⋊C8—21no—
99C7×C4⋊C8C2×C411yescharacteristic
100C7×C4⋊C8C2×C411yescharacteristic
101C7×C22⋊C8—21no—
102C7×C22⋊C8C2×C411yescharacteristic
103C7×C22⋊C8C2×C411yescharacteristic
104C7×#97D411yescharacteristic
105C7×#97D411yescharacteristic
106C4×C7×C8C2311yescharacteristic
107C4×C7×C8C2311yescharacteristic
108C4×C7×C8C2311yescharacteristic
109C4×C7×C8C2311yescharacteristic
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112C22⋊(C7⋊C8)—21no—
113#6395C2×C411yescharacteristic
114#6395C2×C411yescharacteristic
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117#6392C2×C411yescharacteristic
118#6392C2×C411yescharacteristic
119#6391C2×C411yescharacteristic
120#6391C2×C411yescharacteristic
121#6390C2×C411yescharacteristic
122#6390C2×C411yescharacteristic
123#6390C2×C411yescharacteristic
124#6390C2×C411yescharacteristic
125#6390C2×C411yescharacteristic
126#6390C2×C411yescharacteristic
127#6389—21no—
128#6389—21no—
129#6389—21no—
130Dic7⋊C8—21no—
131Dic7⋊C8—21no—
132Dic7⋊C8—21no—
133Dic7⋊C8—21no—
134Dic7⋊C8—21no—
135Dic7⋊C8—21no—
136C8×Dic7—21no—
137C8×Dic7—21no—
138C8×Dic7—21no—
139C4⋊(C7⋊C8)—21no—
140C4⋊(C7⋊C8)C2311yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4⋊(C7⋊C8)C2311yescharacteristic
143C4⋊(C7⋊C8)C2311yescharacteristic
144C4×C7⋊C8—21no—
145#3108—71no—
146#2969—71no—
147Q8×M16—71no—
148#1593—71no—
149#1590—71no—
150#1587—71no—
151#1585—71no—
152#1584—71no—
153#1583—71no—
154SD16⋊C8—71no—
155C8×Q16—71no—
156#1574—71no—
157#1495—71no—
158#1479—71no—
159#1453D711yescharacteristic, complemented
160C22×C4×C7C22×C411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7#4111yescharacteristic
163C2×C22⋊D7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C2×D28C22×C411yescharacteristic
167C2×C4×D7—21no—
168C2×Dic14—22no—
169C2×Dic14—21no—
170C2×Dic14—41no—
171C2×Dic14C22×C411yescharacteristic
172C2×Dic14C22×C411yescharacteristic
173C2×Dic14C22×C411yescharacteristic
174C7×M16—22no—
175C7×M16—41no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C22×C411yescharacteristic
179C2×C7×C8C22×C411yescharacteristic
180C2×C7×C8C22×C411yescharacteristic
181C2×C7×C8C22×C411yescharacteristic
182C2×C7×C8C2×D411yescharacteristic
183C2×C7×C8C2×D411yescharacteristic
184C2×C7×C8C2×D411yescharacteristic
185C2×C7×C8C2×D411yescharacteristic
186C42×C7C22×C411yescharacteristic
187C42×C7C2×D411yescharacteristic
188C42×C7#4111yescharacteristic
189C42×C7C2411yescharacteristic
190C22⋊Dic7—21no—
191(C2×D7)⋊C4—21no—
192(C2×D7)⋊C4—21no—
193(C2×D7)⋊C4—21no—
194(C2×D7)⋊C4—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7C22×C411yescharacteristic
197C4⋊Dic7C22×C411yescharacteristic
198C4⋊Dic7C22×C411yescharacteristic
199C4⋊Dic7C22×C411yescharacteristic
200Dic7⋊C4—21no—
201Dic7⋊C4—21no—
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206Dic7⋊C4—21no—
207Dic7⋊C4—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213#1136—42no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218Dic28—22no—
219Dic28—41no—
220#1132—22no—
221#1132—41no—
222#1131—42no—
223C8×D7—42no—
224#529—71no—
225C2×C4×Q8—71no—
226#483—71no—
227#480—71no—
228#479—71no—
229#479—71no—
230#464—71no—
231#464—71no—
232#451—71no—
233#448—71no—
234#446—71no—
235#446—71no—
236#446—71no—
237#446—141no—
238#446—141no—
239C8×Q8—71no—
240C8×Q8—71no—
241C8×Q8—71no—
242C4×Q16—71no—
243C4×Q16—71no—
244C4×SD16—71no—
245C4×SD16—71no—
246#436—71no—
247#436—141no—
248#435—141no—
249C8×D4—71no—
250#433—141no—
251#431—71no—
252#431C2×D711yescharacteristic
253#431C2×D711yescharacteristic
254#423—71no—
255#423—141no—
256#405—141no—
257C4×M16—141no—
258C4×M16C2×D711yescharacteristic
259#335—71no—
260#335—71no—
261#334—71no—
262#334—71no—
263Q8⋊C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8—71no—
268Q8⋊C8—71no—
269D4⋊C8—71no—
270D4⋊C8—71no—
271#322C2×D711yescharacteristic
272#322C2×D711yescharacteristic
273#322C2×D711yescharacteristic
274C82C2×D711yescharacteristic
275C23×C7C4×D411yescharacteristic
276C22×D7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—41no—
282C2×C4×C7C4×D411yescharacteristic
283C2×C4×C7C4×D411yescharacteristic
284C2×C4×C7C4×D411yescharacteristic
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C22⋊D7—81no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—42no—
297C2×Dic7—81no—
298D28—21no—
299D28—21no—
300C4×D7—41no—
301C4×D7—41no—
302Dic14—21no—
303Dic14—21no—
304Dic14—21no—
305Dic14—21no—
306Dic14—21no—
307Dic14—42no—
308Dic14#13112yes—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—21no—
313C7×C8—21no—
314C7×C8—21no—
315C7×C8—21no—
316C7×C8—41no—
317C7×C8—41no—
318C7×C8#13712yes—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C22×Q8—71no—
324C2×Q16—71no—
325C2×Q16—71no—
326C2×SD16—71no—
327C2×SD16—71no—
328C2×M16—141no—
329C2×M16—141no—
330C2×M16D2811yescharacteristic
331C2×M16D2811yescharacteristic
332C22×C8—141no—
333#126—141no—
334#125—71no—
335#124—71no—
336#123—141no—
337C22⋊Q8—71no—
338#121—71no—
339C4×Q8—71no—
340C4×Q8—71no—
341C4×Q8—71no—
342C4×Q8—71no—
343C4×Q8—71no—
344C4×Q8—141no—
345C4×Q8—141no—
346C4×D4—71no—
347C4×D4—141no—
348#117—141no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—141no—
351C2×C42—141no—
352C2×C42C22×D711yescharacteristic
353#107—71no—
354#107—71no—
355#106—71no—
356#106—71no—
357C4⋊C8—21no—
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⋊C8—141no—
369C4⋊C8C4×D711yescharacteristic
370C4⋊C8C4×D711yescharacteristic
371#103—71no—
372#103—71no—
373#103—71no—
374#103—71no—
375#103—71no—
376#103—71no—
377#102—71no—
378#102—71no—
379C22⋊C8—21no—
380C22⋊C8—141no—
381C22⋊C8—141no—
382C22⋊C8—141no—
383C22⋊C8C4×D711yescharacteristic
384C22⋊C8C4×D711yescharacteristic
385#97—141no—
386#97—141no—
387#97—141no—
388#97D2811yescharacteristic
389#97D2811yescharacteristic
390C4×C8—141no—
391C4×C8—141no—
392C4×C8—141no—
393C4×C8—141no—
394C4×C8C22×D711yescharacteristic
395C4×C8C22×D711yescharacteristic
396C4×C8C22×D711yescharacteristic
397C4×C8C22×D711yescharacteristic
398C4×C8C22×D711yescharacteristic
399C4×C8C22×D711yescharacteristic
400C22×C7—21no—
401C22×C7—21no—
402C22×C7—21no—
403C22×C7C2×C4×D411yescharacteristic
404C2×D7—41no—
405C2×D7—41no—
406C2×D7—41no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—21no—
410C4×C7—21no—
411C4×C7C2×#13111yescharacteristic
412C4×C7#56811yescharacteristic
413C4×C7C2×#13711yescharacteristic
414C4×C7#57511yescharacteristic
415Dic7—22no—
416Dic7—41no—
417Dic7—41no—
418Dic7—41no—
419Dic7—41no—
420Dic7—41no—
421Dic7—41no—
422C2×Q8—71no—
423C2×Q8—71no—
424C2×Q8—71no—
425C2×Q8—142no—
426C2×Q8—141no—
427C2×Q8—281no—
428C2×D4—71no—
429C2×D4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4C2×C4×D711yescharacteristic
434C22×C4C2×D2811yescharacteristic
435C22×C4#115711yescharacteristic
436Q16—142no—
437Q16—281no—
438SD16—142no—
439SD16—281no—
440M16—22no—
441M16—41no—
442M16—282no—
443M16—282no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—282no—
451C2×C8C2×C4×D711yescharacteristic
452C2×C8C2×C4×D711yescharacteristic
453C2×C8C2×C4×D711yescharacteristic
454C2×C8C2×C4×D711yescharacteristic
455C2×C8C2×D2811yescharacteristic
456C2×C8C2×D2811yescharacteristic
457C2×C8C2×D2811yescharacteristic
458C2×C8C2×D2811yescharacteristic
459C4⋊C4—71no—
460C4⋊C4—71no—
461C4⋊C4—71no—
462C4⋊C4—71no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C22⋊C4—141no—
473C22⋊C4—141no—
474C22⋊C4—141no—
475C22⋊C4—141no—
476C22⋊C4—141no—
477C42—141no—
478C42—141no—
479C42—141no—
480C42—141no—
481C42—141no—
482C42C2×C4×D711yescharacteristic
483C42C2×D2811yescharacteristic
484C42#115711yescharacteristic
485C42C23×D711yescharacteristic
486C2×C7—41no—
487C2×C7#293311yescharacteristic
488C2×C7C4×#13711yescharacteristic
489C2×C7#295711yescharacteristic
490D7—81no—
491C23—141no—
492C23C4×D2811yescharacteristic
493Q8—72no—
494Q8—141no—
495Q8—141no—
496Q8—141no—
497Q8—141no—
498Q8—141no—
499Q8—282no—
500D4—141no—
501D4—141no—
502D4—561no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—282no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—561no—
519C2×C4C4×D2811yescharacteristic
520C2×C4C4×D2811yescharacteristic
521C2×C4C4×D2811yescharacteristic
522C2×C4C22×C4×D711yescharacteristic
523C2×C4C22×D2811yescharacteristic
524C2×C4C2×#115711yescharacteristic
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—281no—
535C8—281no—
536C8—281no—
537C8—281no—
538C8#647212yes—
539C7#1801211yescharacteristic, complemented, Sylow
540C22—21no—
541C22—21no—
542C22—21no—
543C22—281no—
544C22—281no—
545C22—281no—
546C22C2×C4×D2811yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—142no—
552C4—281no—
553C4—281no—
554C4—281no—
555C4—281no—
556C4—281no—
557C4—281no—
558C4C2×#647211yescharacteristic
559C4#9043911yescharacteristic
560C4D7×#13111yescharacteristic
561C4#9044111yescharacteristic
562C2—41no—
563C2—561no—
564C2C4×#647211yescharacteristic
565C2#1173600011yescharacteristic
566C2#1173602511yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1453C2
socleC22×C7C2×C4×D4
prime radical#6493C2×C4
cosocleC42×C7C24
hypercenter#1453D7
hypocenterC7#18012
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
3. M16×Dic14C2semidirect
4. C7×#1453C2semidirect
5. #11728521C2semidirect
7. #11728507C2semidirect
8. #11725843C2semidirect
9. #11725841C2semidirect
10. #11725840C2semidirect
11. #11725839C2semidirect
12. #11725834C2semidirect
13. #11725830C2semidirect
14. #11725809C2semidirect
15. C8×Dic28C2semidirect
159. #1453D7semidirect
539. C7#18012semidirect

Maximal subgroups

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

subgroupindex
#117367562
#117359562
M16×Dic142
C7×#14532
#117285212
#117285182
#117285072
#117258432
#117258412
#117258402
#117258392
#117258342
#117258302
#117258092
C8×Dic282
#180127