Finite Group Databasebeta

← #14219895

Subgroups — #14219895

Order 1792.

1746 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742885C211yesmaximal, characteristic
2C7×#2982C211yesmaximal, characteristic, complemented
3Q8×#1136C211yesmaximal, characteristic, complemented
4#11737367C213yesmaximal, complemented
5#11737366C211yesmaximal, characteristic, complemented
6#11737363C213yesmaximal, complemented
7C8×Q8×D7C211yesmaximal, characteristic, complemented
8#11737354C213yesmaximal, complemented
9#11737351C211yesmaximal, characteristic, complemented
10#11737350C211yesmaximal, characteristic, complemented
11#11737348C213yesmaximal, complemented
12#11736740C213yesmaximal, complemented
13#11736738C213yesmaximal, complemented
14#11736725C213yesmaximal, complemented
15#11736723C213yesmaximal, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17#90508C411yescharacteristic
18Q8×C22⋊D7C411yescharacteristic
19#90272C413yes—
20#90271C2213yes—
21#90269C2213yes—
22#90267C411yescharacteristic
23C4×#1161C2211yescharacteristic
24C4×Q8×D7C2211yescharacteristic
25#90260C411yescharacteristic
26#90259C413yes—
27#90204C413yes—
28#90198C413yes—
29#90170C2213yes—
30C4×#1157C2213yes—
31C7×C8×Q8C2213yes—
32C7×C8×Q8C2211yescharacteristic
33C7×C8×Q8C2211yescharacteristic
34C7×C8×Q8C2213yes—
35C7×#431C2213yes—
36C7×#431C2213yes—
37#89796C2213yes—
38#89796C2213yes—
39Q8×C7⋊C8C2211yescharacteristic
40Q8×C7⋊C8C2211yescharacteristic
41#89696C2213yes—
42C4×#1136C2213yes—
43#89612C2216yes—
44#89611C2216yes—
45#89610C2213yes—
46#89608C2213yes—
47#89608C2213yes—
48#89607C2213yes—
49#89607C2213yes—
50#89606C2213yes—
51D7×C4⋊C8C2213yes—
52D7×C4⋊C8C2213yes—
53#89603C2213yes—
54#89602C2213yes—
55#89600C2213yes—
56#89600C2213yes—
57#89501C2213yes—
58#89501C2213yes—
59#89500C2211yescharacteristic
60#89500C2213yes—
61#89500C2216yes—
62#89462C2216yes—
63#89461C2216yes—
64#89460C2213yes—
65C4×#1131C2213yes—
66C8×D28C2213yes—
67C8×D28C2216yes—
68#89457C2213yes—
69#89457C2213yes—
70C4×C8×D7C2213yes—
71#89451C2213yes—
72#89451C2216yes—
73C8×Dic14C2213yes—
74#60335—71nomaximal, Sylow
75C22×C7×Q8C2×C411yescharacteristic
76C2×#1161C2×C411yescharacteristic
77C2×Q8×D7C2×C411yescharacteristic
78C2×#1157C2×C413yes—
79C4×C7×Q8C2×C413yes—
80C4×C7×Q8C2×C411yescharacteristic
81C4×C7×Q8C2311yescharacteristic
82C4×C7×Q8C2313yes—
83C2×C7×C4⋊C4C2×C413yes—
84C2×C42×C7C2313yes—
85#6510C2×C413yes—
86#6509C2×C413yes—
87#6509C2×C413yes—
88Q8×Dic7C2×C411yescharacteristic
89Q8×Dic7C2×C411yescharacteristic
90#6507C2×C413yes—
91#6493C2×C413yes—
92#6492C2×C416yes—
93C4×C22⋊D7C2×C413yes—
94C4×C22⋊D7C2×C413yes—
95C4×C22⋊D7C2311yescharacteristic
96C4×C22⋊D7C2313yes—
97#6489C2×C413yes—
98#6487C2×C413yes—
99C2×#1136C2311yescharacteristic
100C2×#1136C2313yes—
101C2×#1131C2313yes—
102C2×#1131C2313yes—
103C2×C8×D7C2311yescharacteristic
104C2×C8×D7C2313yes—
105C2×C8×D7C2316yes—
106#6461C2×C416yes—
107#6460C2×C413yes—
108#6459C2×C416yes—
109#6458C2×C413yes—
110#6457C2×C416yes—
111#6456C2×C413yes—
112#6455C2×C413yes—
113D7×C4⋊C4C2×C413yes—
114#6453C2×C413yes—
115#6452C2×C416yes—
116#6451C2×C413yes—
117#6450C2×C413yes—
118#6439C2×C416yes—
119#6438C2×C413yes—
120C4×D28C2×C413yes—
121C4×D28C2313yes—
122C4×D28C2313yes—
123#6435C2×C413yes—
124#6435C2×C413yes—
125#6435C2316yes—
126C42×D7C2313yes—
127C42×D7C2313yes—
128#6433C2×C413yes—
129C4×Dic14C2×C413yes—
130C4×Dic14C2313yes—
131C4×Dic14C2313yes—
132C7×C4⋊C8C2313yes—
133C7×C4⋊C8C2313yes—
134C7×C4⋊C8C2313yes—
135C7×C4⋊C8C2313yes—
136C7×C22⋊C8C2313yes—
137C7×C22⋊C8C2311yescharacteristic
138C4×C7×C8C2313yes—
139C4×C7×C8C2313yes—
140C4×C7×C8C2316yes—
141(C2×D7)⋊C8C2313yes—
142(C2×D7)⋊C8C2313yes—
143(C2×D7)⋊C8C2316yes—
144(C2×D7)⋊C8C2311yescharacteristic
145(C2×D7)⋊C8C2313yes—
146#6389C2311yescharacteristic
147#6389C2313yes—
148#6389C2316yes—
149Dic7⋊C8C2311yescharacteristic
150Dic7⋊C8C2313yes—
151Dic7⋊C8C2313yes—
152Dic7⋊C8C2313yes—
153Dic7⋊C8C2316yes—
154C8×Dic7C2313yes—
155C8×Dic7C2313yes—
156C4⋊(C7⋊C8)C2313yes—
157C4⋊(C7⋊C8)C2313yes—
158C4⋊(C7⋊C8)C2316yes—
159#6377C2313yes—
160#6377C2313yes—
161C4×C7⋊C8C2313yes—
162C4×C7⋊C8C2313yes—
163#3482—71no—
164#2994—73no—
165#2993—73no—
166#2990—73no—
167#2988—71no—
168#2985—73no—
169#2984—71no—
170#2982D711yescharacteristic, complemented
171#2971—71no—
172#2969—73no—
173#2966—73no—
174#2965—73no—
175Q8×M16—71no—
176#2962—73no—
177C2×C8×Q8—71no—
178C2×C7×Q8—24no—
179C2×C7×Q8C22×C413yes—
180C2×C7×Q8C22×C411yescharacteristic
181C22×C4×C7C22×C413yes—
182C22×C4×C7C22×C413yes—
183C22×C4×C7C2411yescharacteristic
184C2×C22⋊D7C22×C411yescharacteristic
185C2×C22⋊D7C22×C413yes—
186#1161—24no—
187Q8×D7—24no—
188#1157—46no—
189C2×D28C22×C413yes—
190C2×D28C22×C413yes—
191C2×C4×D7C22×C413yes—
192C2×C4×D7C22×C413yes—
193C2×C4×D7C22×C413yes—
194C2×C4×D7C22×C413yes—
195C2×C4×D7C2411yescharacteristic
196C2×C4×D7C2413yes—
197C2×Dic14C22×C413yes—
198C2×Dic14C22×C413yes—
199C2×C7×C8C2413yes—
200C2×C7×C8C2411yescharacteristic
201C2×C7×C8C2411yescharacteristic
202C2×C7×C8C2413yes—
203C7×C4⋊C4C22×C413yes—
204C7×C4⋊C4C22×C413yes—
205C7×C4⋊C4C22×C416yes—
206C42×C7C22×C413yes—
207C42×C7C22×C413yes—
208C42×C7C2413yes—
209C42×C7C2413yes—
210C22⋊Dic7C22×C413yes—
211C22⋊Dic7C22×C411yescharacteristic
212(C2×D7)⋊C4C22×C411yescharacteristic
213(C2×D7)⋊C4C22×C413yes—
214(C2×D7)⋊C4C22×C413yes—
215(C2×D7)⋊C4C22×C413yes—
216(C2×D7)⋊C4C22×C416yes—
217(C2×D7)⋊C4C22×C416yes—
218(C2×D7)⋊C4C22×C416yes—
219C4⋊Dic7C22×C413yes—
220C4⋊Dic7C22×C413yes—
221C4⋊Dic7C22×C413yes—
222C4⋊Dic7C22×C413yes—
223Dic7⋊C4C22×C411yescharacteristic
224Dic7⋊C4C22×C413yes—
225Dic7⋊C4C22×C413yes—
226Dic7⋊C4C22×C413yes—
227Dic7⋊C4C22×C416yes—
228Dic7⋊C4C22×C416yes—
229Dic7⋊C4C22×C416yes—
230C4×Dic7C22×C413yes—
231C4×Dic7C22×C413yes—
232C4×Dic7C22×C413yes—
233C4×Dic7C22×C413yes—
234C4×Dic7C2411yescharacteristic
235C4×Dic7C2413yes—
236#1136—22no—
237#1136—43no—
238C2×C7⋊C8C2413yes—
239C2×C7⋊C8C2413yes—
240C2×C7⋊C8C2411yescharacteristic
241C2×C7⋊C8C2411yescharacteristic
242#1131—26no—
243#1131—43no—
244C8×D7—22no—
245C8×D7—46no—
246C8×D7—43no—
247#557—71no—
248#556—73no—
249#552—73no—
250#552—73no—
251D4×Q8—71no—
252#548—71no—
253#548—71no—
254#547—73no—
255#533—73no—
256#529—73no—
257#529—73no—
258C4×#41—71no—
259C4×#41—73no—
260C2×C4×Q8—71no—
261C2×C4×Q8C2×D711yescharacteristic
262#446—73no—
263#446—73no—
264#446—73no—
265#446—73no—
266#446—73no—
267#446—73no—
268#446—76no—
269C8×Q8—71no—
270C8×Q8—71no—
271C8×Q8—73no—
272C8×Q8—73no—
273C8×Q8C2×D711yescharacteristic
274C8×Q8C2×D713yes—
275C8×Q8C2×D713yes—
276C8×Q8C2×D711yescharacteristic
277#436—73no—
278#436—73no—
279#436—73no—
280#436—73no—
281#435—73no—
282#435—73no—
283#435—71no—
284#435—73no—
285#435—76no—
286C8×D4—73no—
287C8×D4—73no—
288C8×D4—76no—
289#433—76no—
290#433—76no—
291#432—73no—
292#432—73no—
293#431—73no—
294#431—73no—
295#431C2×D713yes—
296#431C2×D713yes—
297#424—76no—
298#423—73no—
299C2×C4⋊C8—73no—
300C2×C4⋊C8—73no—
301#405—76no—
302C4×M16—73no—
303C4×M16—73no—
304C2×C4×C8—73no—
305C23×C7C23×C411yescharacteristic
306C22×D7C23×C413yes—
307C22×D7C23×C411yescharacteristic
308C7×Q8—26no—
309C7×Q8#13114yes—
310C2×C4×C7—26no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—43no—
314C2×C4×C7C23×C413yes—
315C2×C4×C7C23×C411yescharacteristic
316C2×C4×C7C23×C413yes—
317C2×C4×C7C23×C413yes—
318C2×C4×C7C23×C413yes—
319C2×C4×C7C2511yescharacteristic
320C22⋊D7—22no—
321C22⋊D7—43no—
322C2×Dic7C23×C411yescharacteristic
323C2×Dic7C23×C411yescharacteristic
324C2×Dic7C23×C411yescharacteristic
325C2×Dic7C23×C413yes—
326C2×Dic7C23×C413yes—
327C2×Dic7C23×C413yes—
328D28—26no—
329D28—26no—
330C4×D7—26no—
331C4×D7—26no—
332C4×D7—26no—
333C4×D7—21no—
334C4×D7—23no—
335C4×D7—21no—
336C4×D7—23no—
337C4×D7—43no—
338Dic14—26no—
339Dic14—43no—
340C7×C8—21no—
341C7×C8—23no—
342C7×C8—23no—
343C7×C8—21no—
344C7⋊C8—21no—
345C7⋊C8—23no—
346C7⋊C8—23no—
347C7⋊C82−1+412yes—
348C2×#41—71no—
349C2×#41—73no—
350C22×Q8—71no—
351C22×Q8C4×D711yescharacteristic
352C2×M16—71no—
353C2×M16—73no—
354C2×M16—73no—
355C2×M16—73no—
356C22×C8—71no—
357C22×C8—73no—
358C22×C8—76no—
359C4⋊Q8—73no—
360C4⋊Q8—73no—
361#126—76no—
362#126—76no—
363#125—73no—
364#125—76no—
365#125—73no—
366#124—73no—
367#124—73no—
368#123—76no—
369#123—76no—
370C22⋊Q8—73no—
371C22⋊Q8—73no—
372C22⋊Q8—73no—
373C22⋊Q8—76no—
374C22⋊Q8—73no—
375#121—73no—
376#121—73no—
377C4×Q8—71no—
378C4×Q8—71no—
379C4×Q8—73no—
380C4×Q8—73no—
381C4×Q8—73no—
382C4×Q8—73no—
383C4×Q8C4×D711yescharacteristic
384C4×Q8C4×D713yes—
385C4×Q8C22×D711yescharacteristic
386C4×Q8C22×D713yes—
387C4×D4—73no—
388C4×D4—73no—
389C4×D4—71no—
390C4×D4—73no—
391C4×D4—73no—
392C4×D4—73no—
393C4×D4—73no—
394C4×D4—73no—
395#117—73no—
396#117—73no—
397#117—73no—
398#117—73no—
399#117—76no—
400C2×C4⋊C4—73no—
401C2×C4⋊C4C4×D713yes—
402C2×C42—73no—
403C2×C42—73no—
404C2×C42C22×D713yes—
405C4⋊C8—73no—
406C4⋊C8—76no—
407C4⋊C8—73no—
408C4⋊C8—71no—
409C4⋊C8—73no—
410C4⋊C8—73no—
411C4⋊C8—73no—
412C4⋊C8—76no—
413C4⋊C8C22×D713yes—
414C4⋊C8C22×D713yes—
415C4⋊C8C22×D713yes—
416C4⋊C8C22×D713yes—
417C22⋊C8—71no—
418C22⋊C8—73no—
419C22⋊C8—73no—
420C22⋊C8—73no—
421C22⋊C8—76no—
422C22⋊C8C22×D713yes—
423C22⋊C8C22×D711yescharacteristic
424#97—71no—
425#97—73no—
426#97—76no—
427#97—73no—
428#97—73no—
429C4×C8—73no—
430C4×C8—73no—
431C4×C8—73no—
432C4×C8—73no—
433C4×C8C22×D713yes—
434C4×C8C22×D713yes—
435C4×C8C22×D716yes—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7C24×C411yescharacteristic
440C2×D7—21no—
441C2×D7—21no—
442C2×D7—23no—
443C2×D7—23no—
444C2×D7—23no—
445C2×D7#56812yes—
446C4×C7—21no—
447C4×C7—23no—
448C4×C7—23no—
449C4×C7—23no—
450C4×C7C2×#13116yes—
451C4×C7C2×2−1+411yescharacteristic
452C4×C7#58511yescharacteristic
453Dic7—21no—
454Dic7—23no—
455Dic7—23no—
456Dic7—21no—
457Dic7—21no—
458Dic7—23no—
459#41—144no—
460#41—286no—
461C2×Q8—24no—
462C2×Q8—73no—
463C2×Q8—73no—
464C2×Q8—144no—
465C2×Q8C2×C4×D713yes—
466C2×Q8C2×C4×D711yescharacteristic
467C2×D4—71no—
468C2×D4—73no—
469C2×D4—73no—
470C2×D4—73no—
471C22×C4—73no—
472C22×C4—73no—
473C22×C4—73no—
474C22×C4—71no—
475C22×C4—73no—
476C22×C4—73no—
477C22×C4C2×C4×D713yes—
478C22×C4C2×C4×D713yes—
479C22×C4C23×D711yescharacteristic
480M16—146no—
481M16—142no—
482M16—283no—
483M16—283no—
484C2×C8—71no—
485C2×C8—71no—
486C2×C8—73no—
487C2×C8—73no—
488C2×C8—142no—
489C2×C8—286no—
490C2×C8—283no—
491C2×C8C23×D711yescharacteristic
492C2×C8C23×D713yes—
493C2×C8C23×D713yes—
494C2×C8C23×D711yescharacteristic
495C4⋊C4—76no—
496C4⋊C4—71no—
497C4⋊C4—73no—
498C4⋊C4—73no—
499C4⋊C4—73no—
500C4⋊C4—76no—
501C4⋊C4—76no—
502C4⋊C4—73no—
503C4⋊C4—73no—
504C4⋊C4—73no—
505C4⋊C4—73no—
506C4⋊C4C2×C4×D713yes—
507C4⋊C4C2×C4×D713yes—
508C4⋊C4C2×C4×D716yes—
509C22⋊C4—71no—
510C22⋊C4—73no—
511C22⋊C4—71no—
512C22⋊C4—73no—
513C22⋊C4—76no—
514C22⋊C4—73no—
515C22⋊C4—73no—
516C22⋊C4—76no—
517C22⋊C4—76no—
518C42—73no—
519C42—73no—
520C42—73no—
521C42—73no—
522C42—73no—
523C42—71no—
524C42C2×C4×D713yes—
525C42C2×C4×D713yes—
526C42C23×D713yes—
527C42C23×D713yes—
528C2×C7—41no—
529C2×C7C4×2−1+411yescharacteristic
530C2×C7C22×#13111yescharacteristic
531C2×C7C2×#56811yescharacteristic
532D7—22no—
533D7—43no—
534C23—71no—
535C23—73no—
536C23C22×C4×D711yescharacteristic
537Q8—26no—
538Q8—146no—
539Q8—283no—
540Q8#646414yes—
541D4—146no—
542D4—142no—
543D4—146no—
544D4—283no—
545C2×C4—26no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—43no—
549C2×C4—71no—
550C2×C4—71no—
551C2×C4—71no—
552C2×C4—73no—
553C2×C4—73no—
554C2×C4—73no—
555C2×C4—146no—
556C2×C4—146no—
557C2×C4—146no—
558C2×C4—141no—
559C2×C4—143no—
560C2×C4—141no—
561C2×C4—143no—
562C2×C4—283no—
563C2×C4C22×C4×D713yes—
564C2×C4C22×C4×D711yescharacteristic
565C2×C4C22×C4×D713yes—
566C2×C4C22×C4×D713yes—
567C2×C4C22×C4×D713yes—
568C2×C4C24×D711yescharacteristic
569C8—21no—
570C8—23no—
571C8—23no—
572C8—21no—
573C8—72no—
574C8—141no—
575C8—143no—
576C8—143no—
577C7#6033511yescharacteristic, complemented, Sylow
578C22—21no—
579C22—21no—
580C22—21no—
581C22—72no—
582C22—141no—
583C22—141no—
584C22—143no—
585C22—143no—
586C22—143no—
587C22C23×C4×D711yescharacteristic
588C4—23no—
589C4—21no—
590C4—23no—
591C4—23no—
592C4—141no—
593C4—143no—
594C4—141no—
595C4—143no—
596C4—143no—
597C4—141no—
598C4C2×#646416yes—
599C4D7×2−1+411yescharacteristic
600C4#9062011yescharacteristic
601C2—41no—
602C2—142no—
603C2—283no—
604C2#1174294111yescharacteristic
605C2C22×#646411yescharacteristic
606C2D7×#56811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2982C2
prime radical#11742885C2
cosocleC2×C4×C7C25
hypercenter#2982D7
hypocenterC7#60335
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C7×#2982C2semidirect
3. Q8×#1136C2semidirect
4. #11737367C2semidirect
5. #11737366C2semidirect
6. #11737363C2semidirect
7. C8×Q8×D7C2semidirect
8. #11737354C2semidirect
9. #11737351C2semidirect
10. #11737350C2semidirect
11. #11737348C2semidirect
12. #11736740C2semidirect
13. #11736738C2semidirect
14. #11736725C2semidirect
15. #11736723C2semidirect
170. #2982D7semidirect
577. C7#60335semidirect

Maximal subgroups

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

subgroupindex
#117428852
C7×#29822
Q8×#11362
#117373672
#117373662
#117373632
C8×Q8×D72
#117373542
#117373512
#117373502
#117373482
#117367402
#117367382
#117367252
#117367232
#603357