Finite Group Databasebeta

← #13323435

Subgroups — #13323435

Order 1792.

648 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#11737389C211yesmaximal, characteristic
2#11737365C211yesmaximal, characteristic, complemented
3C4×#6481C211yesmaximal, characteristic
4C7×#1587C211yesmaximal, characteristic, complemented
5#11728710C211yesmaximal, characteristic, complemented
6#11728687C211yesmaximal, characteristic, complemented
7#11728657C211yesmaximal, characteristic, complemented
8#11726282C211yesmaximal, characteristic, complemented
9#11726274C211yesmaximal, characteristic, complemented
10#11725983C211yesmaximal, characteristic
11D7×Q8⋊C8C211yesmaximal, characteristic, complemented
12#11725977C211yesmaximal, characteristic, complemented
13#11725976C211yesmaximal, characteristic, complemented
14#11725842C211yesmaximal, characteristic, complemented
15#11725823C211yesmaximal, characteristic, complemented
16C2×#6481C411yescharacteristic
17C4×#1161C2211yescharacteristic
18C4×Q8×D7C2211yescharacteristic
19C7×#446C2211yescharacteristic
20C7×C8×Q8C2211yescharacteristic
21C4×C7×Q16C2211yescharacteristic
22#89956C411yescharacteristic
23C4×#1144C2211yescharacteristic
24C4×#1143C2211yescharacteristic
25#89796C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27#89661C411yescharacteristic
28#89653C411yescharacteristic
29#89610C2211yescharacteristic
30#89609—21no—
31#89608—21no—
32#89605—21no—
33#89605C2211yescharacteristic
34D7×C4⋊C8C2211yescharacteristic
35#89602—21no—
36#89602C2211yescharacteristic
37#89589C411yescharacteristic
38#89575C411yescharacteristic
39#89574C411yescharacteristic
40#89561C411yescharacteristic
41#89478—21no—
42#89477—21no—
43C4×#1132C2211yescharacteristic
44#89460C2211yescharacteristic
45C4×#1131C2211yescharacteristic
46#89457—21no—
47#89457C2211yescharacteristic
48C4×C8×D7C2211yescharacteristic
49C8×Dic14—21no—
50C8×Dic14C2211yescharacteristic
51C7×#334C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×Q8⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56Q8⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89278C2211yescharacteristic
59#89275C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89253C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89250C2211yescharacteristic
66#89249C2211yescharacteristic
67#17874—71nomaximal, Sylow
68C2×#1161C2×C411yescharacteristic
69C2×Q8×D7C2×C411yescharacteristic
70C2×C7×Q16C2×C411yescharacteristic
71C4×C7×Q8C2311yescharacteristic
72C4×C7×Q8C2311yescharacteristic
73Q8×Dic7C2×C411yescharacteristic
74Q8×Dic7C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77#6481—24no—
78C2×#1132C2×C411yescharacteristic
79C2×#1131—21no—
80C2×#1131—21no—
81C2×#1131C2×C411yescharacteristic
82C2×#1131C2×C411yescharacteristic
83C2×C8×D7—21no—
84C2×C8×D7—21no—
85#6456—21no—
86#6456C2×C411yescharacteristic
87#6455—21no—
88#6455C2×C411yescharacteristic
89D7×C4⋊C4—21no—
90D7×C4⋊C4C2×C411yescharacteristic
91#6450—21no—
92#6450C2×C411yescharacteristic
93C4×D28—21no—
94C4×D28C2311yescharacteristic
95C42×D7—21no—
96C42×D7—21no—
97C42×D7C2311yescharacteristic
98C4×Dic14—21no—
99C4×Dic14C2311yescharacteristic
100C7×#107C2×C411yescharacteristic
101C7×C4⋊C8—21no—
102C7×C4⋊C8—21no—
103C7×C4⋊C8C2311yescharacteristic
104C7×C4⋊C8C2311yescharacteristic
105C7×#103C2×C411yescharacteristic
106C7×#103C2×C411yescharacteristic
107C7×#97—21no—
108C4×C7×C8—21no—
109C4×C7×C8C2311yescharacteristic
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112#6409C2×C411yescharacteristic
113#6409C2×C411yescharacteristic
114#6395C2×C411yescharacteristic
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8—21no—
118(C2×D7)⋊C8D411yescharacteristic
119(C2×D7)⋊C8D411yescharacteristic
120#6391C2×C411yescharacteristic
121#6390C2×C411yescharacteristic
122#6389C2×C411yescharacteristic
123#6389C2×C411yescharacteristic
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8D411yescharacteristic
128Dic7⋊C8D411yescharacteristic
129C8×Dic7—21no—
130C8×Dic7—21no—
131C8×Dic7—21no—
132C8×Dic7—21no—
133#6384C2×C411yescharacteristic
134#6383C2×C411yescharacteristic
135#6382C2×C411yescharacteristic
136#6381C2×C411yescharacteristic
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2311yescharacteristic
140C4⋊(C7⋊C8)C2311yescharacteristic
141#6377—21no—
142C4×C7⋊C8—21no—
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145C4×C7⋊C8C2311yescharacteristic
146#2989—71no—
147#2982—71no—
148C4×#137—71no—
149#1588—71no—
150#1588—71no—
151#1587—71no—
152#1587D711yescharacteristic, complemented
153#1584—71no—
154#1584—71no—
155#1583—71no—
156#1583—71no—
157#1565—71no—
158#1476—71no—
159C2×Q8⋊C8—71no—
160#1451—71no—
161C2×C7×Q8C22×C411yescharacteristic
162C2×C7×Q8C22×C411yescharacteristic
163#1161—22no—
164#1161—22no—
165Q8×D7—22no—
166Q8×D7M1614yescomplemented
167C2×D28—21no—
168C2×D28C22×C411yescharacteristic
169C2×C4×D7—21no—
170C2×C4×D7—21no—
171C2×C4×D7—21no—
172C2×C4×D7—21no—
173C2×C4×D7—21no—
174C2×C4×D7C22×C411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7#4111yescharacteristic
177C2×Dic14—21no—
178C2×Dic14C22×C411yescharacteristic
179C7×Q16—22no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8C22×C411yescharacteristic
184C2×C7×C8C22×C411yescharacteristic
185C2×C7×C8C2×D411yescharacteristic
186C2×C7×C8C2×D411yescharacteristic
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4C22×C411yescharacteristic
190C7×C4⋊C4C22×C411yescharacteristic
191C42×C7—21no—
192C42×C7—21no—
193C42×C7C2411yescharacteristic
194#1144—22no—
195#1143—22no—
196(C2×D7)⋊C4—21no—
197(C2×D7)⋊C4—21no—
198(C2×D7)⋊C4—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7C22×C411yescharacteristic
202C4⋊Dic7C22×C411yescharacteristic
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7C22×C411yescharacteristic
212C4×Dic7C2×D411yescharacteristic
213C4×Dic7#4111yescharacteristic
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8C22×C411yescharacteristic
218C2×C7⋊C8C22×C411yescharacteristic
219C2×C7⋊C8C2×D411yescharacteristic
220C2×C7⋊C8C2×D411yescharacteristic
221#1132—22no—
222#1131—22no—
223#1131—22no—
224#1131—81no—
225#1131—81no—
226C8×D7—42no—
227C8×D7—81no—
228C2×#137—71no—
229C4×#41—71no—
230C2×C4×Q8—71no—
231#446—71no—
232#446C2×D711yescharacteristic
233C8×Q8—71no—
234C8×Q8—71no—
235C8×Q8—71no—
236C8×Q8—141no—
237C8×Q8—141no—
238C8×Q8C2×D711yescharacteristic
239#441—71no—
240#441—71no—
241#440—71no—
242#440—71no—
243C4×Q16—71no—
244C4×Q16C2×D711yescharacteristic
245C4×SD16—71no—
246C4×SD16—71no—
247#436—71no—
248#436—71no—
249#435—141no—
250#435—141no—
251#432—141no—
252#431—71no—
253#431—71no—
254#431—141no—
255#431—141no—
256#428—71no—
257#423—141no—
258C2×C4⋊C8—71no—
259#419—71no—
260#417—71no—
261C4×M16—71no—
262C2×C4×C8—71no—
263#335—71no—
264#335—71no—
265#334—71no—
266#334C2×D711yescharacteristic
267Q8⋊C8—71no—
268Q8⋊C8—71no—
269Q8⋊C8—71no—
270Q8⋊C8—71no—
271Q8⋊C8C2×D711yescharacteristic
272Q8⋊C8C2×D711yescharacteristic
273D4⋊C8—71no—
274D4⋊C8—71no—
275#322—71no—
276#322—71no—
277#322—71no—
278#322C2×D711yescharacteristic
279C22×D7—21no—
280C22×D7C4×D411yescharacteristic
281C7×Q8—21no—
282C7×Q8—21no—
283C7×Q8—21no—
284C7×Q8C2×M1612yes—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7C23×C411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7C4×D411yescharacteristic
296C2×Dic7C4×D411yescharacteristic
297C2×Dic7C4×D411yescharacteristic
298D28—21no—
299D28—21no—
300D28—42no—
301C4×D7—24no—
302C4×D7—21no—
303C4×D7—21no—
304C4×D7—41no—
305C4×D7—41no—
306C4×D7—41no—
307C4×D7—42no—
308C4×D7—81no—
309C4×D7—81no—
310C4×D7C2×M1612yes—
311C4×D7#13112yes—
312Dic14—21no—
313Dic14—24no—
314Dic14C2×M1612yes—
315C7×C8—22no—
316C7×C8—21no—
317C7×C8—21no—
318C7×C8—21no—
319C7×C8—21no—
320C7×C8—41no—
321C7×C8—41no—
322C7⋊C8—21no—
323C7⋊C8—21no—
324C7⋊C8—21no—
325C7⋊C8—21no—
326C7⋊C8—41no—
327C7⋊C8—41no—
328C7⋊C8—41no—
329C2×#41—71no—
330C22×Q8—71no—
331#137—144no—
332C2×Q16—71no—
333C2×Q16C4×D711yescharacteristic
334C2×SD16—71no—
335C2×SD16—71no—
336C2×M16—71no—
337C2×M16—71no—
338C2×M16—141no—
339C2×M16—141no—
340C22×C8—141no—
341C22×C8—141no—
342C4×Q8—71no—
343C4×Q8—71no—
344C4×Q8—71no—
345C4×Q8—71no—
346C4×Q8—141no—
347C4×Q8—141no—
348C4×Q8C22×D711yescharacteristic
349C4×Q8C22×D711yescharacteristic
350C4×D4—71no—
351C4×D4—71no—
352C4×D4—141no—
353C4×D4—141no—
354#117—71no—
355#117—141no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—141no—
358C2×C42—71no—
359C2×C42—141no—
360C2×C42—141no—
361#107—71no—
362#107C4×D711yescharacteristic
363#106—71no—
364#106—71no—
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8—71no—
368C4⋊C8—71no—
369C4⋊C8—71no—
370C4⋊C8—71no—
371C4⋊C8—141no—
372C4⋊C8—141no—
373C4⋊C8—141no—
374C4⋊C8—141no—
375C4⋊C8—141no—
376C4⋊C8C22×D711yescharacteristic
377C4⋊C8C22×D711yescharacteristic
378#103—71no—
379#103—71no—
380#103—71no—
381#103—71no—
382#103C4×D711yescharacteristic
383#103C4×D711yescharacteristic
384#102—71no—
385#102—71no—
386C22⋊C8—71no—
387C22⋊C8—71no—
388C22⋊C8—141no—
389C22⋊C8—141no—
390C22⋊C8—141no—
391#97—21no—
392#97—71no—
393#97—71no—
394#97—141no—
395C4×C8—21no—
396C4×C8—71no—
397C4×C8—71no—
398C4×C8—71no—
399C4×C8—141no—
400C4×C8—141no—
401C4×C8—141no—
402C4×C8—141no—
403C4×C8—141no—
404C4×C8C22×D711yescharacteristic
405C4×C8C22×D711yescharacteristic
406C4×C8C22×D711yescharacteristic
407C22×C7C2×C4×D411yescharacteristic
408C2×D7—21no—
409C2×D7—21no—
410C2×D7—41no—
411C2×D7—41no—
412C2×D7—41no—
413C2×D7#43512yes—
414C4×C7—22no—
415C4×C7—21no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7C22×M1611yescharacteristic
420C4×C7C2×#13111yescharacteristic
421C4×C7#57711yescharacteristic
422C4×C7#57811yescharacteristic
423Dic7—22no—
424Dic7—21no—
425Dic7—21no—
426Dic7—41no—
427Dic7—41no—
428Dic7#43512yes—
429#41—142no—
430#41—142no—
431C2×Q8—74no—
432C2×Q8—71no—
433C2×Q8—141no—
434C2×Q8—142no—
435C2×Q8C2×C4×D711yescharacteristic
436C2×Q8C2×C4×D711yescharacteristic
437C2×D4—71no—
438C2×D4—141no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447Q16—22no—
448Q16—142no—
449SD16—142no—
450SD16—142no—
451M16—142no—
452M16—142no—
453M16—561no—
454M16—561no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—71no—
459C2×C8—71no—
460C2×C8—71no—
461C2×C8—71no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—282no—
466C2×C8—561no—
467C2×C8C2×C4×D711yescharacteristic
468C2×C8C2×C4×D711yescharacteristic
469C2×C8D4×D711yescharacteristic
470C2×C8D4×D711yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—71no—
474C4⋊C4—71no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4C2×C4×D711yescharacteristic
481C4⋊C4C2×C4×D711yescharacteristic
482C22⋊C4—141no—
483C22⋊C4—141no—
484C22⋊C4—141no—
485C42—21no—
486C42—21no—
487C42—71no—
488C42—71no—
489C42—71no—
490C42—141no—
491C42—141no—
492C42—141no—
493C42—141no—
494C42—141no—
495C42C23×D711yescharacteristic
496C2×C7C2×#43511yescharacteristic
497C2×C7#294711yescharacteristic
498C2×C7#295611yescharacteristic
499D7—22no—
500D7—81no—
501C23—71no—
502C23—141no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506Q8—72no—
507Q8—144no—
508Q8—141no—
509Q8D7×M1612yes—
510D4—141no—
511D4—141no—
512D4—282no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—71no—
518C2×C4—72no—
519C2×C4—72no—
520C2×C4—71no—
521C2×C4—71no—
522C2×C4—144no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—281no—
529C2×C4—282no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—561no—
533C2×C4—561no—
534C2×C4C22×C4×D711yescharacteristic
535C2×C4C2×D4×D711yescharacteristic
536C2×C4D7×#4111yescharacteristic
537C8—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—141no—
545C8—141no—
546C8—141no—
547C8—141no—
548C8—281no—
549C8—281no—
550C8—281no—
551C7#1787411yescharacteristic, complemented, Sylow
552C22—72no—
553C22—141no—
554C22—141no—
555C22—281no—
556C22—281no—
557C22—281no—
558C22C4×D4×D711yescharacteristic
559C4—22no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—72no—
565C4—142no—
566C4—141no—
567C4—141no—
568C4—281no—
569C4—281no—
570C4C2×D7×M1611yescharacteristic
571C4D7×#13111yescharacteristic
572C4#9046911yescharacteristic
573C4#9047011yescharacteristic
574C2—142no—
575C2—561no—
576C2D7×#43511yescharacteristic
577C2#1173727511yescharacteristic
578C2#1173733411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1587C2
socleC22×C7C2×C4×D4
prime radicalC2×#1161C2×C4
cosocleC42×C7C24
hypercenter#1587D7
hypocenterC7#17874
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
2. #11737365C2semidirect
4. C7×#1587C2semidirect
5. #11728710C2semidirect
6. #11728687C2semidirect
7. #11728657C2semidirect
8. #11726282C2semidirect
9. #11726274C2semidirect
11. D7×Q8⋊C8C2semidirect
12. #11725977C2semidirect
13. #11725976C2semidirect
14. #11725842C2semidirect
15. #11725823C2semidirect
152. #1587D7semidirect
166. Q8×D7M16semidirect
551. C7#17874semidirect

Maximal subgroups

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

subgroupindex
#117373892
#117373652
C4×#64812
C7×#15872
#117287102
#117286872
#117286572
#117262822
#117262742
#117259832
D7×Q8⋊C82
#117259772
#117259762
#117258422
#117258232
#178747