Finite Group Databasebeta

← #13371149

Subgroups — #13371149

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89774C211yesmaximal, characteristic
2#11739655C211yesmaximal, characteristic, complemented
3#11736076C211yesmaximal, characteristic
4C7×#1826C211yesmaximal, characteristic, complemented
5#11727736C211yesmaximal, characteristic, complemented
6#11727208C211yesmaximal, characteristic
7C22⋊C4×C7⋊C8C211yesmaximal, characteristic, complemented
8#11726528C212yesmaximal, complemented
9#11726499C212yesmaximal, complemented
10#11726498C212yesmaximal, complemented
11#11726493C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C23×C7⋊C8C2211yescharacteristic
15C2×#6470C2212yes—
16C7×#407C2211yescharacteristic
17C4×C7×C22⋊C4C2211yescharacteristic
18#89979C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89901C2212yes—
21C2×#6396C2212yescomplemented
22#89774C411yescharacteristic, complemented
23#89774C411yescharacteristic, complemented
24#89774C411yescharacteristic, complemented
25#89774C411yescharacteristic, complemented
26#89774C2211yescharacteristic
27#89774C2211yescharacteristic
28#89774C2211yescharacteristic
29#89774C2211yescharacteristic
30#89772—22no—
31#89771C2212yescomplemented
32#89769C2212yescomplemented
33C2×#6384C411yescharacteristic, complemented
34C2×#6384C411yescharacteristic, complemented
35C2×#6383C411yescharacteristic, complemented
36C2×#6383C411yescharacteristic, complemented
37C2×#6379C2212yes—
38C2×#6379C2212yes—
39#89698—21no—
40#89697—21no—
41C2×C4×C7⋊C8C2212yescomplemented
42#89499—22no—
43#89498—21no—
44D7×C22⋊C8—21no—
45C7×#339C2212yescomplemented
46C7×#339C2212yescomplemented
47#89353C2212yescomplemented
48#89319C2212yescomplemented
49#22587—71nomaximal, Sylow
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—21no—
52C2×#1157—21no—
53C2×#1157—21no—
54C2×#1157—24no—
55C2×#1157—42no—
56C2×#1157C2×C411yescharacteristic
57C2×#1157C2×C411yescharacteristic
58C2×#1157C2311yescharacteristic
59C2×#1157C2311yescharacteristic
60C22×D28C2×C411yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2×C411yescharacteristic
63C2×C7×M16C2312yes—
64C7×#117—21no—
65C7×#117—21no—
66C7×#117C2×C411yescharacteristic
67C7×#117C2×C411yescharacteristic
68C7×#117C2311yescharacteristic
69C7×#117C2311yescharacteristic
70C2×C7×C4⋊C4C2×C411yescharacteristic
71C2×C7×C4⋊C4C2×C411yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C42×C7C2312yes—
76C2×#1136C2312yes—
77C22×C7⋊C8—42no—
78C22×C7⋊C8C2×C411yescharacteristic
79C22×C7⋊C8C2×C411yescharacteristic
80C22×C7⋊C8C2×C411yescharacteristic
81C22×C7⋊C8C2×C411yescharacteristic
82C22×C7⋊C8C2311yescharacteristic
83C22×C7⋊C8C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85#6470—24no—
86#6470—42no—
87C2×#1131—22no—
88C2×C8×D7—22no—
89C7×C22⋊C8—21no—
90C7×C22⋊C8—21no—
91C7×#95—21no—
92C22⋊(C7⋊C8)—21no—
93C22⋊(C7⋊C8)—21no—
94C22⋊(C7⋊C8)D411yescharacteristic
95C22⋊(C7⋊C8)D411yescharacteristic
96C22⋊(C7⋊C8)D412yes—
97#6396—42no—
98(C2×D7)⋊C8—22no—
99(C2×D7)⋊C8—22no—
100#6384C2×C411yescharacteristic
101#6384C2×C411yescharacteristic
102#6384C2×C411yescharacteristic
103#6384C2×C411yescharacteristic
104#6384C2×C411yescharacteristic
105#6384C2×C411yescharacteristic
106#6384C2×C411yescharacteristic
107#6384C2×C411yescharacteristic
108#6383C2×C411yescharacteristic
109#6383C2×C411yescharacteristic
110#6383C2×C411yescharacteristic
111#6383C2×C411yescharacteristic
112#6383C2×C411yescharacteristic
113#6383C2×C411yescharacteristic
114#6383C2×C411yescharacteristic
115#6383C2×C411yescharacteristic
116#6379—24no—
117#6379—42no—
118C4⋊(C7⋊C8)—22no—
119C4⋊(C7⋊C8)—22no—
120#6377—22no—
121C4×C7⋊C8—22no—
122C4×C7⋊C8—42no—
123#2924—71no—
124C2×#416—71no—
125#2880—71no—
126#1937—72no—
127#1929—72no—
128#1866—72no—
129#1826D711yescharacteristic, complemented
130#1810—71no—
131#1789—71no—
132#1764—72no—
133C8×C22⋊C4—71no—
134C24×C7#4111yescharacteristic
135C23×D7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7C22×C411yescharacteristic
142C22×C4×C7C22×C411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7#4111yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C2×C22⋊D7—21no—
148C2×C22⋊D7—21no—
149C2×C22⋊D7—21no—
150C2×C22⋊D7—21no—
151C2×C22⋊D7—44no—
152C22×Dic7—21no—
153#1157—22no—
154#1157—22no—
155#1157—22no—
156#1157—22no—
157#1157—22no—
158#1157—41no—
159#1157—41no—
160#1157—44no—
161#1157—44no—
162#1157#4114yes—
163C2×D28—22no—
164C2×D28—21no—
165C2×D28—41no—
166C2×D28C22×C411yescharacteristic
167C2×D28C22×C411yescharacteristic
168C2×C4×D7—21no—
169C2×C4×D7—21no—
170C2×C4×D7—21no—
171C2×C4×D7—21no—
172C2×C4×D7—44no—
173C2×Dic14—22no—
174C2×Dic14—21no—
175C2×Dic14—41no—
176C2×Dic14C22×C411yescharacteristic
177C2×Dic14C22×C411yescharacteristic
178C7×M16—22no—
179C7×M16—22no—
180C2×C7×C8—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4C22×C411yescharacteristic
184C7×C4⋊C4C22×C411yescharacteristic
185C7×C4⋊C4C22×C411yescharacteristic
186C7×C4⋊C4C22×C411yescharacteristic
187C7×C22⋊C4—24no—
188C7×C22⋊C4—21no—
189C7×C22⋊C4—21no—
190C7×C22⋊C4—21no—
191C7×C22⋊C4—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—22no—
197C42×C7—22no—
198#1136—22no—
199#1136—22no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—41no—
206C2×C7⋊C8—41no—
207C2×C7⋊C8—41no—
208C2×C7⋊C8—41no—
209C2×C7⋊C8C22×C411yescharacteristic
210C2×C7⋊C8C22×C411yescharacteristic
211C2×C7⋊C8C22×C411yescharacteristic
212C2×C7⋊C8C22×C411yescharacteristic
213C2×C7⋊C8C2×D412yes—
214C2×C7⋊C8C2×D412yes—
215#1131—44no—
216C8×D7—44no—
217C22×#41—71no—
218C2×#131—72no—
219C23×C8—71no—
220C2×#117C2×D711yescharacteristic
221#433—142no—
222#432—141no—
223#431—141no—
224C2×#108—72no—
225#424—72no—
226C2×#104—72no—
227C2×#104—72no—
228#416—71no—
229#416—71no—
230#416—71no—
231#416—71no—
232#416—71no—
233#416—71no—
234#416—71no—
235#416—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#102—71no—
239C2×#102—71no—
240#408—72no—
241#408—142no—
242#407—71no—
243#407—141no—
244#407C2×D711yescharacteristic
245C2×C22⋊C8—71no—
246C2×C22⋊C8—141no—
247#405—72no—
248C2×C4×C8—72no—
249C4×C22⋊C4C2×D711yescharacteristic
250#341—72no—
251#339C2×D712yescomplemented
252#339C2×D712yescomplemented
253#336—72no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—41no—
257C23×C7C4×D411yescharacteristic
258C23×C7C4×D411yescharacteristic
259C23×C7C2×#4111yescharacteristic
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—42no—
263C22×D7—42no—
264C22×D7—81no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—24no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—81no—
281C2×C4×C7C4×D411yescharacteristic
282C2×C4×C7C4×D411yescharacteristic
283C2×C4×C7C23×C411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C2×#4112yes—
286C2×C4×C7H(5,𝔽2)12yes—
287C22⋊D7—24no—
288C22⋊D7—42no—
289C22⋊D7—42no—
290C22⋊D7—42no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—42no—
294D28—21no—
295D28—21no—
296D28—21no—
297D28—42no—
298D28#11712yes—
299C4×D7—24no—
300C4×D7—42no—
301C4×D7—42no—
302C4×D7—42no—
303Dic14—21no—
304Dic14—21no—
305Dic14—21no—
306Dic14—42no—
307Dic14#11712yes—
308C7×C8—42no—
309C7⋊C8—21no—
310C7⋊C8—21no—
311C7⋊C8—21no—
312C7⋊C8—21no—
313C7⋊C8—22no—
314C7⋊C8—22no—
315C7⋊C8—42no—
316C2×#41—71no—
317C2×#41—71no—
318C2×#41—71no—
319C2×#41—71no—
320C2×#41—144no—
321C2×#41—141no—
322C2×#41—141no—
323C2×#41—282no—
324C22×Q8—71no—
325C22×D4—71no—
326C22×D4—141no—
327C23×C4—141no—
328C23×C4C22×D711yescharacteristic
329#131—144no—
330#131—282no—
331C2×M16—72no—
332C2×M16—142no—
333C2×M16C22×D712yes—
334C22×C8—71no—
335C22×C8—71no—
336C22×C8—72no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C22×C8—142no—
342C22×C8—282no—
343#117—21no—
344#117—21no—
345#117C4×D711yescharacteristic
346#117C4×D711yescharacteristic
347#117C22×D711yescharacteristic
348#117C22×D711yescharacteristic
349C2×C4⋊C4C4×D711yescharacteristic
350C2×C4⋊C4C4×D711yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42—21no—
354C2×C42C22×D712yes—
355#108—282no—
356C4⋊C8—142no—
357C4⋊C8—142no—
358#104—144no—
359#104—282no—
360#103—71no—
361#103—71no—
362#103—71no—
363#103—71no—
364#103—71no—
365#103—71no—
366#103—71no—
367#103—71no—
368#102—71no—
369#102—71no—
370#102—71no—
371#102—71no—
372#102—71no—
373#102—71no—
374#102—71no—
375#102—71no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—71no—
379C22⋊C8—71no—
380C22⋊C8—72no—
381C22⋊C8—142no—
382C22⋊C8—142no—
383C22⋊C8—141no—
384C22⋊C8—141no—
385#97—142no—
386C4×C8—142no—
387C4×C8—282no—
388#95—21no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—21no—
395C22×C7—21no—
396C22×C7—44no—
397C22×C7#44312yes—
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#55112yes—
400C2×D7—22no—
401C2×D7—41no—
402C2×D7—41no—
403C2×D7—41no—
404C2×D7—41no—
405C2×D7—41no—
406C2×D7—84no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—44no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7C2×#11711yescharacteristic
415C4×C7#51811yescharacteristic
416C4×C7#54612yes—
417Dic7—22no—
418Dic7—41no—
419C24—141no—
420C24#115911yescharacteristic
421#41—74no—
422#41—142no—
423#41—142no—
424#41—142no—
425#41—142no—
426#41—142no—
427#41—284no—
428#41—281no—
429#41—281no—
430#41—284no—
431C2×Q8—71no—
432C2×Q8—71no—
433C2×Q8—142no—
434C2×Q8—141no—
435C2×Q8—281no—
436C2×D4—71no—
437C2×D4—71no—
438C2×D4—142no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4—281no—
445C2×D4—284no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—284no—
457C22×C4C2×C4×D711yescharacteristic
458C22×C4C2×C4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4D4×D711yescharacteristic
461C22×C4#115911yescharacteristic
462C22×C4C23×D711yescharacteristic
463M16—22no—
464M16—22no—
465M16—142no—
466M16—142no—
467M16—284no—
468C2×C8—22no—
469C2×C8—71no—
470C2×C8—71no—
471C2×C8—71no—
472C2×C8—71no—
473C2×C8—72no—
474C2×C8—72no—
475C2×C8—142no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8—284no—
481C2×C8—281no—
482C2×C8—281no—
483C2×C8—281no—
484C2×C8—281no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4C2×C4×D711yescharacteristic
488C4⋊C4C2×C4×D711yescharacteristic
489C4⋊C4C2×C4×D711yescharacteristic
490C4⋊C4C2×C4×D711yescharacteristic
491C22⋊C4—24no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—22no—
501C42—22no—
502C2×C7—22no—
503C2×C7—22no—
504C2×C7—41no—
505C2×C7#234211yescharacteristic
506C2×C7C2×#44311yescharacteristic
507C2×C7#295711yescharacteristic
508D7—42no—
509D7—81no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—141no—
514C23—141no—
515C23—282no—
516C23—282no—
517C23—561no—
518C23(C22⋊D7)⋊C411yescharacteristic
519C23(C22⋊D7)⋊C411yescharacteristic
520C23C2×#115911yescharacteristic
521Q8—72no—
522Q8—141no—
523Q8—141no—
524Q8—141no—
525Q8—282no—
526D4—72no—
527D4—144no—
528D4—141no—
529D4—141no—
530D4—141no—
531D4—282no—
532D4—282no—
533D4—282no—
534D4—282no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—81no—
551C2×C4—141no—
552C2×C4—144no—
553C2×C4—141no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4(C22⋊D7)⋊C411yescharacteristic
559C2×C4(C22⋊D7)⋊C411yescharacteristic
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C2×D4×D711yescharacteristic
562C2×C4D7×#4112yes—
563C2×C4#655312yes—
564C8—42no—
565C8—141no—
566C8—141no—
567C8—141no—
568C8—141no—
569C8—142no—
570C8—142no—
571C8—282no—
572C7#2258711yescharacteristic, complemented, Sylow
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—142no—
582C22—281no—
583C22—281no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22—564no—
588C22#8959612yes—
589C22C2×(C22⋊D7)⋊C411yescharacteristic
590C22#9034912yes—
591C4—21no—
592C4—21no—
593C4—44no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—142no—
599C4—281no—
600C4D7×#11711yescharacteristic
601C4#9021211yescharacteristic
602C4#9034512yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—282no—
607C2—561no—
608C2#1173156711yescharacteristic
609C2C2×#8959611yescharacteristic
610C2#1173643511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1826C2
socleC22×C7C2×C4×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1826D7
hypocenterC7#22587
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
2. #11739655C2semidirect
4. C7×#1826C2semidirect
5. #11727736C2semidirect
7. C22⋊C4×C7⋊C8C2semidirect
8. #11726528C2semidirect
9. #11726499C2semidirect
10. #11726498C2semidirect
11. #11726493C2semidirect
21. C2×#6396C22semidirect
22. #89774C4semidirect
23. #89774C4semidirect
24. #89774C4semidirect
25. #89774C4semidirect
31. #89771C22semidirect
32. #89769C22semidirect
33. C2×#6384C4semidirect
34. C2×#6384C4semidirect
35. C2×#6383C4semidirect
36. C2×#6383C4semidirect
41. C2×C4×C7⋊C8C22semidirect
45. C7×#339C22semidirect
46. C7×#339C22semidirect
47. #89353C22semidirect
48. #89319C22semidirect
129. #1826D7semidirect
251. #339C2×D7semidirect
252. #339C2×D7semidirect
572. C7#22587semidirect

Maximal subgroups

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

subgroupindex
C2×#897742
#117396552
#117360762
C7×#18262
#117277362
#117272082
C22⋊C4×C7⋊C82
#117265282
#117264992
#117264982
#117264932
#225877