Finite Group Databasebeta

← #13371528

Subgroups — #13371528

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#428C211yesmaximal, characteristic
2#11739684C211yesmaximal, characteristic
3#11739605C211yesmaximal, characteristic
4#11729742C211yesmaximal, characteristic
5#11729737C211yesmaximal, characteristic
6#11729329C211yesmaximal, characteristic
7#11729307C211yesmaximal, characteristic
8#11728367C211yesmaximal, characteristic, complemented
9#11728366C211yesmaximal, characteristic, complemented
10#11728364C211yesmaximal, characteristic, complemented
11#11728363C211yesmaximal, characteristic, complemented
12#11728312C211yesmaximal, characteristic, complemented
13#11728311C211yesmaximal, characteristic, complemented
14#11728307C211yesmaximal, characteristic, complemented
15Dic7×#106C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C7×#428C418yescomplemented
20C2×C7×#107C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C7×#106C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24#89984—21no—
25#89984C2211yescharacteristic
26#89984C2211yescharacteristic
27C4×C22⋊Dic7—21no—
28C4×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×#6389C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89753—21no—
39#89753—21no—
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42C4⋊C4×Dic7C2211yescharacteristic
43C4⋊C4×Dic7C2211yescharacteristic
44#89723—21no—
45#89723—21no—
46#89708—21no—
47#89704—21no—
48#89346C2211yescharacteristic
49#89346C2211yescharacteristic
50#89345C2211yescharacteristic
51#89345C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22621—71nomaximal, Sylow
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C2×C7×M16—22no—
64C2×C7×M16—22no—
65C2×C7×M16C2×C414yes—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×#117—22no—
69C7×#117C2×C414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4C2×C414yes—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7D411yescharacteristic
82C2×C22⋊Dic7D411yescharacteristic
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7C2311yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C7×#107C2×C414yes—
100C7×#107C2×C414yes—
101C7×#106C2×C414yes—
102C7×#106C2×C414yes—
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⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115C22⋊(C7⋊C8)—22no—
116#6392—22no—
117#6391—22no—
118#6389—22no—
119C8×Dic7—22no—
120C2×#428D711yescharacteristic
121#2346—71no—
122#2345—71no—
123#1943—71no—
124#1942—71no—
125#1921—71no—
126#1920—71no—
127#1919—71no—
128#1918—71no—
129#1825—71no—
130#1823—71no—
131#1809—71no—
132#1776—71no—
133C4×#106—71no—
134#1753—71no—
135C24×C7C2×D411yescharacteristic
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—24no—
145C22×C4×C7—81no—
146C22×C4×C7—81no—
147C22×C4×C7C22×C412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7#4112yes—
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C7×M16—28no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8C22×C412yes—
163C2×C7×C8C22×C412yes—
164C2×C7×C8#4112yes—
165C2×C7×C8#4112yes—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4C22×C412yes—
172C7×C4⋊C4C22×C412yes—
173C7×C4⋊C4C22×C412yes—
174C7×C4⋊C4C22×C412yes—
175C7×C22⋊C4—24no—
176C42×C7—24no—
177C22⋊Dic7—22no—
178C22⋊Dic7—41no—
179C22⋊Dic7—42no—
180C22⋊Dic7—81no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—81no—
190C4⋊Dic7—81no—
191C4⋊Dic7—81no—
192C4⋊Dic7—81no—
193C4×Dic7—21no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—42no—
198C4×Dic7—42no—
199C4×Dic7—81no—
200C4×Dic7—81no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C22×M16C2×D711yescharacteristic
208C2×#117C2×D711yescharacteristic
209C22×C4⋊C4C2×D711yescharacteristic
210#428Dic718yescomplemented
211C2×#107—71no—
212C2×#107C2×D711yescharacteristic
213C2×#107C2×D711yescharacteristic
214C2×#106—71no—
215C2×#106C2×D711yescharacteristic
216C2×#106C2×D711yescharacteristic
217C2×C22⋊C8—71no—
218C2×#97—71no—
219C2×C4×C8—71no—
220#389—71no—
221#389—71no—
222#389—71no—
223#389—71no—
224#388—141no—
225#388—141no—
226#387—141no—
227#387—141no—
228#382—141no—
229#380—71no—
230#380—71no—
231#380—141no—
232C4×C4⋊C4—71no—
233C4×C4⋊C4—71no—
234C4×C4⋊C4—71no—
235C4×C4⋊C4—71no—
236C4×C22⋊C4—71no—
237C4×C22⋊C4—141no—
238#376—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#336—71no—
250#336—71no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7C4×D412yes—
258C23×C7C22×D411yescharacteristic
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—84no—
278C2×C4×C7C4×D412yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7#13712yes—
309C7×C8—22no—
310C7×C8#11714yes—
311C7⋊C8—41no—
312C7⋊C8—41no—
313C23×C4—21no—
314C23×C4C22×D711yescharacteristic
315C2×M16—22no—
316C2×M16—22no—
317C2×M16C2×Dic714yes—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322#117—22no—
323#117C2×Dic714yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C2×Dic714yes—
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—71no—
341C2×C22⋊C4—71no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C42—21no—
345C2×C42—71no—
346C2×C42—71no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42—141no—
350C2×C42—141no—
351#107—142no—
352#107C2×Dic714yes—
353#107C2×Dic714yes—
354#106—142no—
355#106C2×Dic714yes—
356#106C2×Dic714yes—
357C22⋊C8—142no—
358#97—142no—
359C4×C8—142no—
360#95—71no—
361#95—71no—
362#95—141no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—41no—
376C22×C7—41no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7C2×C4×D411yescharacteristic
380C22×C7#54711yescharacteristic
381C22×C7#54711yescharacteristic
382C22×C7C2×#13711yescharacteristic
383C22×C7#57511yescharacteristic
384C22×C7#57611yescharacteristic
385C22×C7#57711yescharacteristic
386C4×C7—22no—
387C4×C7—42no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7C2×#11711yescharacteristic
392C4×C7#51911yescharacteristic
393C4×C7#55211yescharacteristic
394C4×C7#55211yescharacteristic
395Dic7—22no—
396Dic7—41no—
397Dic7—41no—
398Dic7—41no—
399Dic7—81no—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403C24D4×D711yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—71no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4D4×D711yescharacteristic
424C22×C4#115912yes—
425C22×C4C22×Dic712yes—
426C22×C4C23×D711yescharacteristic
427M16—28no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8#115712yes—
437C2×C8#115712yes—
438C2×C8C22×Dic712yes—
439C2×C8C22×Dic712yes—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—561no—
454C4⋊C4—561no—
455C4⋊C4—561no—
456C4⋊C4—561no—
457C4⋊C4C22×Dic712yes—
458C4⋊C4C22×Dic712yes—
459C4⋊C4C22×Dic712yes—
460C4⋊C4C22×Dic712yes—
461C22⋊C4—24no—
462C22⋊C4—142no—
463C22⋊C4—281no—
464C22⋊C4—282no—
465C22⋊C4—561no—
466C42—24no—
467C42—141no—
468C42—141no—
469C42—141no—
470C42—141no—
471C42—282no—
472C42—282no—
473C42—561no—
474C42—561no—
475C2×C7—42no—
476C2×C7#234411yescharacteristic
477C2×C7C4×#13711yescharacteristic
478C2×C7#294611yescharacteristic
479C2×C7#334411yescharacteristic
480C2×C7#334711yescharacteristic
481C2×C7#335111yescharacteristic
482C2×C7#335711yescharacteristic
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23D4×Dic712yes—
490C23C2×D4×D711yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—72no—
510C2×C4—84no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4D4×Dic712yes—
535C2×C4C2×#115711yescharacteristic
536C2×C4C2×#115711yescharacteristic
537C2×C4C2×#115911yescharacteristic
538C2×C4#655111yescharacteristic
539C2×C4#655111yescharacteristic
540C2×C4C23×Dic711yescharacteristic
541C8—22no—
542C8—281no—
543C8—281no—
544C8#648914yes—
545C7#2262111yescharacteristic, complemented, Sylow
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22#9019811yescharacteristic
554C22#9019811yescharacteristic
555C22#9046511yescharacteristic
556C22#9046611yescharacteristic
557C22D7×#13711yescharacteristic
558C22#9046811yescharacteristic
559C22C2×D4×Dic711yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—142no—
566C4—281no—
567C4—281no—
568C4—281no—
569C4—561no—
570C4—561no—
571C4—561no—
572C4—561no—
573C4#9020411yescharacteristic
574C4#9020411yescharacteristic
575C4C2×#648911yescharacteristic
576C4#9050311yescharacteristic
577C2—42no—
578C2#1173666911yescharacteristic
579C2#1173667211yescharacteristic
580C2#1173667711yescharacteristic
581C2#1173668311yescharacteristic
582C2#1173960411yescharacteristic
583C2#1174137511yescharacteristic
584C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#428C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC2×#428D7
hypocenterC7#22621
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
8. #11728367C2semidirect
9. #11728366C2semidirect
10. #11728364C2semidirect
11. #11728363C2semidirect
12. #11728312C2semidirect
13. #11728311C2semidirect
14. #11728307C2semidirect
15. Dic7×#106C2semidirect
19. C7×#428C4semidirect
210. #428Dic7semidirect
545. C7#22621semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4282
#117396842
#117396052
#117297422
#117297372
#117293292
#117293072
#117283672
#117283662
#117283642
#117283632
#117283122
#117283112
#117283072
Dic7×#1062
#226217