Finite Group Databasebeta

← #13323864

Subgroups — #13323864

Order 1792.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737365C211yesmaximal, characteristic, complemented
2#11737101C211yesmaximal, characteristic
3#11736853C211yesmaximal, characteristic
4C7×C8×SD16C211yesmaximal, characteristic, complemented
5#11728686C211yesmaximal, characteristic, complemented
6#11728652C211yesmaximal, characteristic, complemented
7#11728593C211yesmaximal, characteristic, complemented
8#11726289C211yesmaximal, characteristic, complemented
9#11726276C211yesmaximal, characteristic, complemented
10#11725990C211yesmaximal, characteristic, complemented
11#11725972C211yesmaximal, characteristic, complemented
12#11725923C211yesmaximal, characteristic
13#11725909C211yesmaximal, characteristic, complemented
14C8×Dic28C211yesmaximal, characteristic, complemented
15#11725803C211yesmaximal, characteristic, complemented
16C4×Q8×D7C2211yescharacteristic
17#90248C2211yescharacteristic
18C7×C8×Q8C2211yescharacteristic
19C4×C7×SD16C2211yescharacteristic
20C7×C8×D4C2211yescharacteristic
21#89942C411yescharacteristic
22#89938C411yescharacteristic
23#89883—21no—
24#89874—21no—
25C4×#1144C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27C4×#1142C2211yescharacteristic
28#89784C2211yescharacteristic
29#89634C411yescharacteristic
30#89629C411yescharacteristic
31#89609C2211yescharacteristic
32#89605—21no—
33#89605C2211yescharacteristic
34#89603C2211yescharacteristic
35#89602—21no—
36#89602C2211yescharacteristic
37#89580C411yescharacteristic
38#89568C411yescharacteristic
39#89546C411yescharacteristic
40#89539C411yescharacteristic
41#89501—21no—
42#89492—21no—
43C4×Dic28C2211yescharacteristic
44#89457—21no—
45#89457C2211yescharacteristic
46#89457C2211yescharacteristic
47#89457C2211yescharacteristic
48C8×Dic14—21no—
49C8×Dic14C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
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
66#89249C2211yescharacteristic
67#17919—71nomaximal, Sylow
68C2×Q8×D7C2×C411yescharacteristic
69C2×C7×SD16C2×C411yescharacteristic
70C22×C7×C8—21no—
71C4×C7×Q8C2311yescharacteristic
72C4×C7×D4C2311yescharacteristic
73Q8×Dic7C2×C411yescharacteristic
74C2×#1144C2×C411yescharacteristic
75#6501C2×C411yescharacteristic
76#6499C2×C411yescharacteristic
77C2×#1142C2×C411yescharacteristic
78C4×C22⋊D7—21no—
79#6489—21no—
80C2×#1136—21no—
81C2×Dic28C2×C411yescharacteristic
82C2×#1131—21no—
83#6460C2×C411yescharacteristic
84D7×C4⋊C4—21no—
85D7×C4⋊C4C2×C411yescharacteristic
86#6453C2×C411yescharacteristic
87#6450—21no—
88#6450C2×C411yescharacteristic
89#6446—21no—
90#6442—21no—
91C42×D7—21no—
92C42×D7C2311yescharacteristic
93C4×Dic14—21no—
94C4×Dic14C2311yescharacteristic
95C4×Dic14C2311yescharacteristic
96C7×#106C2×C411yescharacteristic
97C7×C4⋊C8—21no—
98C7×C4⋊C8C2311yescharacteristic
99C7×C4⋊C8C2311yescharacteristic
100C7×#103C2×C411yescharacteristic
101C7×#102C2×C411yescharacteristic
102C7×C22⋊C8—21no—
103C4×C7×C8—21no—
104C4×C7×C8C2311yescharacteristic
105C4×C7×C8C2311yescharacteristic
106C4×C7×C8C2311yescharacteristic
107#6409C2×C411yescharacteristic
108#6406C2×C411yescharacteristic
109C22⋊(C7⋊C8)—21no—
110(C2×D7)⋊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—
120Dic7⋊C8—21no—
121Dic7⋊C8—21no—
122Dic7⋊C8—21no—
123Dic7⋊C8—21no—
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8C2×C411yescharacteristic
127Dic7⋊C8C2×C411yescharacteristic
128Dic7⋊C8D411yescharacteristic
129Dic7⋊C8D411yescharacteristic
130C8×Dic7—21no—
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
141C4×C7⋊C8—21no—
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3108—71no—
146#2993—71no—
147#2982—71no—
148#1588—71no—
149#1587—71no—
150#1584—71no—
151#1583—71no—
152#1579—71no—
153SD16⋊C8—71no—
154C8×Q16—71no—
155C8×SD16D711yescharacteristic, complemented
156#1574—71no—
157#1487—71no—
158#1482—71no—
159#1454—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×D4C22×C411yescharacteristic
162C22×C4×C7—21no—
163C2×C22⋊D7—21no—
164Q8×D7—22no—
165Q8×D7—22no—
166C2×C4×D7—21no—
167C2×C4×D7—21no—
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—
190C22⋊Dic7—21no—
191C22⋊Dic7—21no—
192#1144—22no—
193#1142—22no—
194(C2×D7)⋊C4—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7C22×C411yescharacteristic
199C4⋊Dic7C22×C411yescharacteristic
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
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
220Dic28—22no—
221#1131—81no—
222#529—71no—
223C2×C4×Q8—71no—
224#483—71no—
225#480—71no—
226#479—71no—
227#479—71no—
228#464—71no—
229#464—71no—
230#451—71no—
231#448—71no—
232#446—71no—
233C8×Q8—71no—
234C8×Q8—71no—
235C8×Q8—71no—
236C8×Q8—71no—
237C8×Q8—141no—
238C8×Q8—141no—
239C8×Q8C2×D711yescharacteristic
240C4×Q16—71no—
241C4×Q16—71no—
242C4×SD16—71no—
243C4×SD16C2×D711yescharacteristic
244#436—71no—
245#436—141no—
246#435—141no—
247C8×D4C2×D711yescharacteristic
248#433—141no—
249#431—71no—
250#431—71no—
251#431—71no—
252#431—71no—
253#431—141no—
254#431—141no—
255#424—141no—
256#423—71no—
257#335—71no—
258#335—71no—
259#334—71no—
260#334C2×D711yescharacteristic
261Q8⋊C8—71no—
262Q8⋊C8—71no—
263Q8⋊C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8C2×D711yescharacteristic
267D4⋊C8—71no—
268D4⋊C8C2×D711yescharacteristic
269#322—71no—
270#322—71no—
271#322—71no—
272C82C2×D711yescharacteristic
273C23×C7—21no—
274C22×D7C4×D411yescharacteristic
275C7×Q8—21no—
276C7×Q8#13112yes—
277C7×D4—21no—
278C7×D4—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7C23×C411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C22⋊D7—81no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7C4×D411yescharacteristic
293C2×Dic7C4×D411yescharacteristic
294C2×Dic7C4×D411yescharacteristic
295C4×D7—24no—
296C4×D7—21no—
297C4×D7—21no—
298C4×D7—21no—
299C4×D7—21no—
300C4×D7—41no—
301C4×D7—81no—
302Dic14—24no—
303Dic14—21no—
304Dic14—21no—
305Dic14—21no—
306Dic14#13112yes—
307C7×C8—21no—
308C7×C8—21no—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—41no—
312C7×C8—41no—
313C7×C8—41no—
314C7⋊C8—21no—
315C7⋊C8—21no—
316C7⋊C8—21no—
317C7⋊C8—21no—
318C7⋊C8—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C22×Q8—71no—
322C2×Q16—71no—
323C2×Q16—71no—
324C2×SD16—71no—
325C2×SD16C4×D711yescharacteristic
326C2×M16—141no—
327C2×M16—141no—
328C22×C8—21no—
329#126—141no—
330#125—71no—
331#124—71no—
332#123—141no—
333C22⋊Q8—71no—
334#121—71no—
335C4×Q8—71no—
336C4×Q8—71no—
337C4×Q8—71no—
338C4×Q8—71no—
339C4×Q8—141no—
340C4×Q8—141no—
341C4×Q8C22×D711yescharacteristic
342C4×D4—141no—
343C4×D4C22×D711yescharacteristic
344#117—141no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—141no—
347C2×C42—71no—
348C2×C42—141no—
349#107—71no—
350#107—71no—
351#106—71no—
352#106C4×D711yescharacteristic
353C4⋊C8—21no—
354C4⋊C8—71no—
355C4⋊C8—71no—
356C4⋊C8—71no—
357C4⋊C8—71no—
358C4⋊C8—71no—
359C4⋊C8—71no—
360C4⋊C8—141no—
361C4⋊C8—141no—
362C4⋊C8—141no—
363C4⋊C8—141no—
364C4⋊C8—141no—
365C4⋊C8—141no—
366C4⋊C8—141no—
367C4⋊C8C22×D711yescharacteristic
368C4⋊C8C22×D711yescharacteristic
369#103—71no—
370#103—71no—
371#103—71no—
372#103—71no—
373#103—71no—
374#103C4×D711yescharacteristic
375#102—71no—
376#102C4×D711yescharacteristic
377C22⋊C8—21no—
378C22⋊C8—71no—
379C22⋊C8—71no—
380C22⋊C8—71no—
381C22⋊C8—71no—
382C22⋊C8—141no—
383C22⋊C8—141no—
384C22⋊C8—141no—
385#97—141no—
386C4×C8—21no—
387C4×C8—71no—
388C4×C8—71no—
389C4×C8—71no—
390C4×C8—141no—
391C4×C8—141no—
392C4×C8—141no—
393C4×C8—141no—
394C4×C8—141no—
395C4×C8C22×D711yescharacteristic
396C4×C8C22×D711yescharacteristic
397C4×C8C22×D711yescharacteristic
398C22×C7—41no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7C2×C4×D411yescharacteristic
402C2×D7—21no—
403C2×D7—21no—
404C2×D7—21no—
405C4×C7—22no—
406C4×C7—21no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7C2×#13111yescharacteristic
410C4×C7C2×#13111yescharacteristic
411C4×C7#57711yescharacteristic
412C4×C7#57811yescharacteristic
413Dic7—21no—
414Dic7—22no—
415Dic7—21no—
416Dic7—21no—
417Dic7—41no—
418Dic7—41no—
419Dic7—41no—
420C2×Q8—71no—
421C2×Q8—71no—
422C2×Q8—141no—
423C2×Q8—142no—
424C2×Q8—142no—
425C2×Q8C2×C4×D711yescharacteristic
426C2×D4—141no—
427C2×D4C2×C4×D711yescharacteristic
428C22×C4—21no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—141no—
433C22×C4—141no—
434Q16—142no—
435Q16—142no—
436SD16—22no—
437SD16—142no—
438M16—561no—
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—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8C2×C4×D711yescharacteristic
452C2×C8C2×C4×D711yescharacteristic
453C2×C8D4×D711yescharacteristic
454C2×C8D4×D711yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—71no—
457C4⋊C4—71no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4C2×C4×D711yescharacteristic
467C4⋊C4C2×C4×D711yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—141no—
470C22⋊C4—141no—
471C22⋊C4—141no—
472C22⋊C4—141no—
473C42—21no—
474C42—71no—
475C42—71no—
476C42—71no—
477C42—141no—
478C42—141no—
479C42—141no—
480C42—141no—
481C42C23×D711yescharacteristic
482C2×C7—81no—
483C2×C7#292911yescharacteristic
484C2×C7#294711yescharacteristic
485C2×C7#295711yescharacteristic
486D7—41no—
487C23—21no—
488C23—71no—
489Q8—21no—
490Q8—72no—
491Q8—144no—
492Q8—141no—
493Q8—141no—
494Q8—141no—
495Q8#646412yes—
496D4—21no—
497D4—21no—
498D4—561no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—71no—
505C2×C4—71no—
506C2×C4—71no—
507C2×C4—144no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—281no—
517C2×C4—561no—
518C2×C4C22×C4×D711yescharacteristic
519C2×C4C2×D4×D711yescharacteristic
520C2×C4D7×#4111yescharacteristic
521C8—21no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—141no—
529C8—141no—
530C8—141no—
531C8—141no—
532C8—281no—
533C8—281no—
534C8—281no—
535C7#1791911yescharacteristic, complemented, Sylow
536C22—41no—
537C22—41no—
538C22—41no—
539C22—141no—
540C22—141no—
541C22—141no—
542C22C4×D4×D711yescharacteristic
543C4—22no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—141no—
548C4—141no—
549C4—142no—
550C4—141no—
551C4—281no—
552C4—281no—
553C4—281no—
554C4C2×#646411yescharacteristic
555C4D7×#13111yescharacteristic
556C4#9045311yescharacteristic
557C4#9046211yescharacteristic
558C2—81no—
559C2—281no—
560C2#1173686611yescharacteristic
561C2#1173709311yescharacteristic
562C2#1173733511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×C8×SD16C2
socleC22×C7C2×C4×D4
prime radical#6501C2×C4
cosocleC42×C7C24
hypercenterC8×SD16D7
hypocenterC7#17919
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. #11737365C2semidirect
4. C7×C8×SD16C2semidirect
5. #11728686C2semidirect
6. #11728652C2semidirect
7. #11728593C2semidirect
8. #11726289C2semidirect
9. #11726276C2semidirect
10. #11725990C2semidirect
11. #11725972C2semidirect
13. #11725909C2semidirect
14. C8×Dic28C2semidirect
15. #11725803C2semidirect
155. C8×SD16D7semidirect
535. C7#17919semidirect

Maximal subgroups

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

subgroupindex
#117373652
#117371012
#117368532
C7×C8×SD162
#117286862
#117286522
#117285932
#117262892
#117262762
#117259902
#117259722
#117259232
#117259092
C8×Dic282
#117258032
#179197