Finite Group Databasebeta

← #13358179

Subgroups — #13358179

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89529C211yesmaximal, characteristic
2#11733740C211yesmaximal, characteristic
3#11733415C211yesmaximal, characteristic, complemented
4C7×#1934C211yesmaximal, characteristic
5#11727943C211yesmaximal, characteristic
6#11727941C211yesmaximal, characteristic
7Dic7×#102C211yesmaximal, characteristic
8#11727521C211yesmaximal, characteristic, complemented
9#11727503C211yesmaximal, characteristic, complemented
10#11727403C211yesmaximal, characteristic
11#11727374C211yesmaximal, characteristic, complemented
12#11726613C211yesmaximal, characteristic, complemented
13#11726608C211yesmaximal, characteristic, complemented
14#11726540C211yesmaximal, characteristic, complemented
15#11726535C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C22×#1142C2211yescharacteristic
18C2×#6450C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C2×C4×C7×C8C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#389C2211yescharacteristic
24#89993C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6390C2211yescharacteristic
27C2×#6390C2211yescharacteristic
28C2×#6389C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89757C2211yescharacteristic
33#89750—21no—
34#89749—21no—
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89746C2211yescharacteristic
38#89745C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89717—21no—
44#89716—21no—
45#89715—21no—
46#89714—21no—
47#89704C2211yescharacteristic
48#89701C2211yescharacteristic
49#89700C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51#89529C418yescomplemented
52#89427—21no—
53#89423—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21291—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×Dic7—21no—
63C22×Dic14C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C42×C7C2311yescharacteristic
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76D4×Dic7—22no—
77D4×Dic7C2×C414yes—
78C2×#1142—22no—
79C2×#1142—22no—
80C2×#1142C2×C414yes—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6450—22no—
96#6450C2×C414yes—
97C7×#102—22no—
98C7×#102C2×C414yes—
99C4×C7×C8—22no—
100C7×#95—21no—
101#6406—22no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115#6390—22no—
116#6390C2×C414yes—
117#6389—22no—
118#6389C2×C414yes—
119C8×Dic7—22no—
120C8×Dic7—22no—
121#6384—22no—
122#6382—22no—
123#6381—22no—
124#6381C2×C414yes—
125#6377—22no—
126C2×#440—71no—
127#2320—71no—
128#2319—71no—
129#1973—71no—
130#1968—71no—
131#1943—71no—
132#1942—71no—
133#1936—71no—
134#1935—71no—
135#1934D711yescharacteristic
136#1933—71no—
137#1776—71no—
138C4×#103—71no—
139C4×#102—71no—
140#1726—71no—
141C24×C7—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4C22×C412yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—24no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—81no—
160C22×Dic7—81no—
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14C22×C412yes—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8C22×C412yes—
176C2×C7×C8#4112yes—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—81no—
183C7×C4⋊C4C22×C412yes—
184C7×C22⋊C4—81no—
185C7×C22⋊C4—81no—
186C42×C7—21no—
187C42×C7—21no—
188C22⋊Dic7—24no—
189C22⋊Dic7—81no—
190C22⋊Dic7—81no—
191#1142—28no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—81no—
198C4⋊Dic7C22×C412yes—
199Dic7⋊C4—24no—
200Dic7⋊C4—81no—
201Dic7⋊C4—81no—
202C4×Dic7—21no—
203C4×Dic7—21no—
204C4×Dic7—24no—
205C4×Dic7—81no—
206C4×Dic7—81no—
207C4×Dic7C22×C412yes—
208C4×Dic7#4112yes—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8C22×C412yes—
212C2×C7⋊C8#4112yes—
213C2×C7⋊C8#4112yes—
214C2×C7⋊C8#4112yes—
215C22×SD16—71no—
216C2×C4×Q8—71no—
217C2×C4×D4—71no—
218#440—78no—
219C2×#107—71no—
220C2×#107—71no—
221C2×#106—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#102—71no—
226C2×#102C2×D711yescharacteristic
227C2×#102C2×D711yescharacteristic
228C2×#97—71no—
229C2×#97—71no—
230C2×#97—71no—
231C2×C4×C8—71no—
232C2×C4×C8—71no—
233C2×C4×C8C2×D711yescharacteristic
234#391—71no—
235#391—71no—
236#390—71no—
237#390C2×D711yescharacteristic
238#389—71no—
239#389—71no—
240#389—71no—
241#389C2×D711yescharacteristic
242#388—141no—
243#388—141no—
244#387—141no—
245#387—141no—
246#383—141no—
247#382—141no—
248#381—141no—
249#380—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252#376—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340C2×D711yescharacteristic
258#340C2×D711yescharacteristic
259C23×C7—22no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7C22×D411yescharacteristic
267C7×D4—22no—
268C7×D4—22no—
269C7×D4—41no—
270C7×D4—41no—
271C2×C4×C7—22no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7C23×C411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—84no—
315C2×Dic7C4×D412yes—
316C2×Dic7C4×D412yes—
317Dic14—22no—
318Dic14—22no—
319Dic14—41no—
320Dic14—41no—
321C7×C8—22no—
322C7×C8—22no—
323C7⋊C8—22no—
324C7⋊C8#11714yes—
325C22×Q8—71no—
326C22×D4C22×D711yescharacteristic
327C23×C4—141no—
328C2×SD16—74no—
329C2×SD16—142no—
330C2×SD16—142no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8C22×D711yescharacteristic
334C22×C8C22×D711yescharacteristic
335C4×Q8—74no—
336C4×Q8—142no—
337C4×D4—74no—
338C4×D4—142no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—141no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42C22×D711yescharacteristic
361#107—74no—
362#107—142no—
363#106—142no—
364#103—74no—
365#103—142no—
366#103—142no—
367#102—22no—
368#102—142no—
369#102C4×D714yes—
370#97—74no—
371#97—142no—
372#97—142no—
373C4×C8—22no—
374C4×C8—142no—
375C4×C8—142no—
376#95—21no—
377#95—71no—
378#95—71no—
379#95—71no—
380#95—71no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7—81no—
394C22×C7—81no—
395C22×C7—81no—
396C22×C7—81no—
397C22×C7C2×C4×D411yescharacteristic
398C22×C7#53211yescharacteristic
399C22×C7#53211yescharacteristic
400C22×C7#57611yescharacteristic
401C22×C7#57711yescharacteristic
402C22×C7#57711yescharacteristic
403C22×C7#57811yescharacteristic
404C4×C7—41no—
405C4×C7—41no—
406C4×C7—42no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7C2×#11711yescharacteristic
410C4×C7C4×#4111yescharacteristic
411C4×C7#52911yescharacteristic
412C4×C7#52911yescharacteristic
413Dic7—22no—
414Dic7—22no—
415Dic7—41no—
416Dic7—41no—
417Dic7—42no—
418Dic7—42no—
419Dic7—81no—
420Dic7—81no—
421C24—21no—
422C2×Q8—72no—
423C2×Q8—141no—
424C2×Q8—141no—
425C2×Q8—141no—
426C2×Q8—141no—
427C2×Q8—141no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×C4×D712yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—144no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—561no—
447C22×C4—561no—
448C22×C4D4×D711yescharacteristic
449C22×C4D4×D711yescharacteristic
450C22×C4C23×D711yescharacteristic
451SD16—148no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—72no—
457C2×C8—72no—
458C2×C8—72no—
459C2×C8—72no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8C2×C4×D712yes—
463C2×C8#115912yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—72no—
470C4⋊C4—81no—
471C4⋊C4—144no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—561no—
478C4⋊C4—561no—
479C4⋊C4—561no—
480C4⋊C4C2×C4×D712yes—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—144no—
484C22⋊C4—561no—
485C22⋊C4—561no—
486C42—21no—
487C42—21no—
488C42—72no—
489C42—72no—
490C42—144no—
491C42—141no—
492C42—141no—
493C42—561no—
494C42—561no—
495C2×C7—82no—
496C2×C7#229311yescharacteristic
497C2×C7#294611yescharacteristic
498C2×C7#294711yescharacteristic
499C2×C7#317411yescharacteristic
500C2×C7#317511yescharacteristic
501C2×C7#317811yescharacteristic
502C2×C7#317911yescharacteristic
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D711yescharacteristic
511Q8—142no—
512Q8—142no—
513Q8—281no—
514Q8—281no—
515D4—22no—
516D4—22no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—564no—
559C2×C4C22×C4×D711yescharacteristic
560C2×C4C2×#115911yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4D7×#4111yescharacteristic
563C2×C4D7×#4111yescharacteristic
564C2×C4D7×#4111yescharacteristic
565C8—22no—
566C8—22no—
567C8—74no—
568C8—142no—
569C7#2129111yescharacteristic, complemented, Sylow
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D711yescharacteristic
578C22#9028711yescharacteristic
579C22#9035911yescharacteristic
580C22#9044811yescharacteristic
581C22#9045311yescharacteristic
582C22#9046111yescharacteristic
583C22#9046211yescharacteristic
584C4—41no—
585C4—42no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—142no—
590C4—142no—
591C4—281no—
592C4—282no—
593C4—281no—
594C4—282no—
595C4—561no—
596C4—561no—
597C4D7×#11711yescharacteristic
598C4C4×#115911yescharacteristic
599C4#9034011yescharacteristic
600C4#9039711yescharacteristic
601C2—82no—
602C2#1173289511yescharacteristic
603C2#1173703211yescharacteristic
604C2#1173709311yescharacteristic
605C2#1173808011yescharacteristic
606C2#1173814311yescharacteristic
607C2#1173900311yescharacteristic
608C2#1173907411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1934C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1934D7
hypocenterC7#21291
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
3. #11733415C2semidirect
8. #11727521C2semidirect
9. #11727503C2semidirect
11. #11727374C2semidirect
12. #11726613C2semidirect
13. #11726608C2semidirect
14. #11726540C2semidirect
15. #11726535C2semidirect
51. #89529C4semidirect
569. C7#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#895292
#117337402
#117334152
C7×#19342
#117279432
#117279412
Dic7×#1022
#117275212
#117275032
#117274032
#117273742
#117266132
#117266082
#117265402
#117265352
#212917