Finite Group Databasebeta

← #13356444

Subgroups — #13356444

Order 1792.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2317C211yesmaximal, characteristic, complemented
2#11739715C211yesmaximal, characteristic
3C2×#89774C211yesmaximal, characteristic
4C2×#89769C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic
6#11729335C211yesmaximal, characteristic, complemented
7C2×#89324C211yesmaximal, characteristic, complemented
8#11727371C212yesmaximal, complemented
9#11727370C212yesmaximal, complemented
10#11727369C212yesmaximal, complemented
11#11727368C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×#1136C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C7×#389C2212yes—
20C7×#389C2212yes—
21C7×#385—22no—
22C7×#384—21no—
23C7×#382—21no—
24#89774C418yescomplemented
25#89769—24no—
26#89759—21no—
27#89759—21no—
28C2×#6384C2212yes—
29C2×#6384C2211yescharacteristic
30C2×#6384C2211yescharacteristic
31#89738—24no—
32C2×#6383C2212yes—
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35#89730—22no—
36#89711C2212yes—
37#89701C2212yes—
38C2×#6377C2212yes—
39C2×C4×C7⋊C8C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21087—71nomaximal, Sylow
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C22×C22⋊D7—21no—
48C23×Dic7—21no—
49C2×#1157—22no—
50C2×#1157—22no—
51C2×#1157—22no—
52C2×#1157—22no—
53C2×#1157—22no—
54C2×#1157C2×C414yes—
55C22×D28C2311yescharacteristic
56C22×C4×D7—21no—
57C22×Dic14C2311yescharacteristic
58C7×#117—22no—
59C7×#117—22no—
60C7×#117—22no—
61C7×#117C2×C414yes—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4D412yes—
74C2×C42×C7—21no—
75C2×C42×C7C2312yes—
76C2×(C2×D7)⋊C4—22no—
77C2×(C2×D7)⋊C4—22no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2312yes—
82C2×#1136—22no—
83C2×#1136—22no—
84C2×#1136D414yes—
85C22×C7⋊C8—22no—
86C22×C7⋊C8—22no—
87C22×C7⋊C8C2×C414yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C7×#95—22no—
92C7×#95—22no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95#6384—24no—
96#6384C2×C414yescomplemented
97#6384C2×C414yescomplemented
98#6383—24no—
99#6383C2×C414yescomplemented
100#6383C2×C414yescomplemented
101#6377—24no—
102C4×C7⋊C8—24no—
103C2×#417—71no—
104C2×#416—71no—
105C2×#405—71no—
106#2317D711yescharacteristic, complemented
107#2311—71no—
108#1936—72no—
109#1935—72no—
110#1934—72no—
111#1933—72no—
112#1736—71no—
113C2×#340—71no—
114C24×C7C2×D411yescharacteristic
115C23×D7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—42no—
120C22×C4×C7—81no—
121C22×C4×C7—81no—
122C22×C4×C7C22×C412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D412yes—
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D412yes—
127C22×C4×C7#4112yes—
128C22×C4×C7#4112yes—
129C22×C4×C7#4112yes—
130C22×C4×C7#4112yes—
131C22×C4×C7C2411yescharacteristic
132C2×C22⋊D7—24no—
133C2×C22⋊D7—42no—
134C2×C22⋊D7—42no—
135C22×Dic7—21no—
136C22×Dic7—22no—
137C22×Dic7—42no—
138C22×Dic7—81no—
139C22×Dic7—81no—
140#1157—28no—
141#1157—28no—
142#1157—44no—
143#1157—44no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28C22×C412yes—
150C2×C4×D7—24no—
151C2×C4×D7—42no—
152C2×C4×D7—42no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14—21no—
156C2×Dic14—21no—
157C2×Dic14—21no—
158C2×Dic14C22×C412yes—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—42no—
166C7×C4⋊C4—82no—
167C7×C4⋊C4—82no—
168C7×C4⋊C4C22×C412yes—
169C7×C4⋊C4C22×C412yes—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—24no—
172C42×C7—24no—
173C42×C7—22no—
174C42×C7—22no—
175(C2×D7)⋊C4—82no—
176(C2×D7)⋊C4—82no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—42no—
180#1136—28no—
181C2×C7⋊C8—28no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8C22×C412yes—
189C2×C7⋊C8C22×C412yes—
190C2×C7⋊C8C2×D414yes—
191C22×#41—71no—
192C22×M16—71no—
193C23×C8—71no—
194C2×#117C2×D711yescharacteristic
195C2×#117C2×D711yescharacteristic
196C22×C4⋊C4—71no—
197C22×C4⋊C4C2×D711yescharacteristic
198#417—144no—
199#416—78no—
200C2×#103—71no—
201C2×#103—71no—
202C2×#103—72no—
203C2×#102—71no—
204C2×#102—71no—
205C2×#102—72no—
206#405—144no—
207C2×#97—72no—
208C2×C4×C8—72no—
209#391—72no—
210#390—72no—
211#389C2×D712yes—
212#389C2×D712yes—
213#386—142no—
214#385—22no—
215#384—21no—
216#382—21no—
217#380—141no—
218#380—141no—
219#340—72no—
220#340—72no—
221#340—72no—
222#340—72no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7C4×D412yes—
228C23×C7#12112yes—
229C23×C7#12412yes—
230C23×C7C22×D411yescharacteristic
231C22×D7—22no—
232C22×D7—41no—
233C22×D7—41no—
234C22×D7—41no—
235C22×D7—41no—
236C22×D7—41no—
237C22×D7—41no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—84no—
254C2×C4×C7#11714yes—
255C2×C4×C7C4×D414yes—
256C2×C4×C7C4×D412yes—
257C2×C4×C7#12112yes—
258C2×C4×C7#12314yes—
259C2×C4×C7#12412yes—
260C2×C4×C7#12614yes—
261C2×C4×C7C23×C411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7C2×#4111yescharacteristic
265C2×C4×C7C2×#4111yescharacteristic
266C2×C4×C7C2×#4111yescharacteristic
267C22⋊D7—48no—
268C2×Dic7—22no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—84no—
275D28—22no—
276D28—22no—
277D28—41no—
278D28—41no—
279C4×D7—48no—
280Dic14—22no—
281Dic14—22no—
282Dic14—41no—
283Dic14—41no—
284C7⋊C8—22no—
285C7⋊C8—22no—
286C7⋊C8—24no—
287C2×#41—74no—
288C2×#41—142no—
289C2×#41—142no—
290C2×#41—142no—
291C2×#41—142no—
292C2×#41—142no—
293C22×Q8—71no—
294C22×D4—71no—
295C22×D4—141no—
296C23×C4—21no—
297C23×C4—141no—
298C23×C4—141no—
299C23×C4C22×D711yescharacteristic
300C2×M16—74no—
301C2×M16—142no—
302C2×M16—142no—
303C22×C8—74no—
304C22×C8—71no—
305C22×C8—71no—
306C22×C8—72no—
307C22×C8—142no—
308C22×C8—142no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117C4×D714yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—72no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4C22⋊D711yescharacteristic
323C2×C4⋊C4C22⋊D711yescharacteristic
324C2×C4⋊C4C22×D712yes—
325C2×C4⋊C4C22×D711yescharacteristic
326C2×C4⋊C4C22×D711yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—142no—
329C2×C22⋊C4—142no—
330C2×C22⋊C4C22⋊D712yes—
331C2×C42—21no—
332C2×C42C22×D712yes—
333#103—74no—
334#103—74no—
335#103—144no—
336#102—74no—
337#102—74no—
338#102—144no—
339#97—144no—
340C4×C8—144no—
341#95—22no—
342#95—22no—
343#95—142no—
344#95—142no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7#38814yes—
352C22×C7C2×C4×D411yescharacteristic
353C22×C7C2×#12111yescharacteristic
354C22×C7C2×#12411yescharacteristic
355C22×C7#57611yescharacteristic
356C22×C7#57711yescharacteristic
357C22×C7#57711yescharacteristic
358C22×C7#57811yescharacteristic
359C2×D7—42no—
360C2×D7—42no—
361C2×D7—42no—
362C2×D7—81no—
363C2×D7—81no—
364C2×D7—81no—
365C2×D7—81no—
366C4×C7—42no—
367C4×C7—42no—
368C4×C7—42no—
369C4×C7—42no—
370C4×C7—82no—
371C4×C7#38814yes—
372C4×C7C2×#11711yescharacteristic
373C4×C7C2×C4×D411yescharacteristic
374C4×C7C2×#12311yescharacteristic
375C4×C7C2×#12611yescharacteristic
376Dic7—42no—
377Dic7—82no—
378C24—141no—
379C24C2×C22⋊D711yescharacteristic
380#41—148no—
381#41—148no—
382#41—284no—
383#41—284no—
384C2×Q8—72no—
385C2×Q8—141no—
386C2×Q8—141no—
387C2×Q8—141no—
388C2×Q8—141no—
389C2×Q8—141no—
390C2×D4—72no—
391C2×D4—144no—
392C2×D4—141no—
393C2×D4—141no—
394C2×D4—141no—
395C2×D4—141no—
396C2×D4—141no—
397C2×D4—282no—
398C2×D4—282no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—141no—
406C22×C4—144no—
407C22×C4—142no—
408C22×C4—282no—
409C22×C4—282no—
410C22×C4—282no—
411C22×C4—561no—
412C22×C4—561no—
413C22×C4C2×C4×D712yes—
414C22×C4#115712yes—
415C22×C4D4×D712yes—
416C22×C4#115912yes—
417C22×C4#116112yes—
418C22×C4#116112yes—
419C22×C4C2×C22⋊D712yes—
420C22×C4C2×C22⋊D712yes—
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423M16—148no—
424C2×C8—72no—
425C2×C8—72no—
426C2×C8—74no—
427C2×C8—148no—
428C2×C8—142no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—142no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—142no—
444C4⋊C4—142no—
445C4⋊C4—282no—
446C4⋊C4C2×C4×D712yes—
447C4⋊C4C2×C4×D712yes—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—562no—
451C22⋊C4—562no—
452C42—24no—
453C42—22no—
454C42—22no—
455C2×C7—24no—
456C2×C7C2×#38811yescharacteristic
457C2×C7#295111yescharacteristic
458C2×C7#295211yescharacteristic
459C2×C7#306211yescharacteristic
460C2×C7#306311yescharacteristic
461C2×C7#313611yescharacteristic
462C2×C7#313711yescharacteristic
463D7—82no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—142no—
469C23—281no—
470C23—281no—
471C23—281no—
472C23—281no—
473C23—281no—
474C23—281no—
475C23C4×C22⋊D712yes—
476C23#650212yes—
477C23#651012yes—
478C23C22×C22⋊D711yescharacteristic
479Q8—142no—
480Q8—142no—
481Q8—281no—
482Q8—281no—
483D4—142no—
484D4—142no—
485D4—288no—
486D4—281no—
487D4—281no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—142no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—288no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—564no—
512C2×C4#645514yes—
513C2×C4#645614yes—
514C2×C4#645714yes—
515C2×C4#646114yes—
516C2×C4C4×C22⋊D712yes—
517C2×C4#650212yes—
518C2×C4#651012yes—
519C2×C4C22×C4×D711yescharacteristic
520C2×C4C2×#115711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×#115911yescharacteristic
523C2×C4C2×#116111yescharacteristic
524C2×C4C2×#116111yescharacteristic
525C8—142no—
526C8—142no—
527C8—144no—
528C7#2108711yescharacteristic, complemented, Sylow
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—282no—
536C22—282no—
537C22—282no—
538C22—561no—
539C22—561no—
540C22—561no—
541C22—561no—
542C22#8976214yes—
543C22C2×C4×C22⋊D711yescharacteristic
544C22C2×#650211yescharacteristic
545C22C2×#651011yescharacteristic
546C22#9052611yescharacteristic
547C22#9052711yescharacteristic
548C22#9052811yescharacteristic
549C22#9052911yescharacteristic
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—282no—
556C4—562no—
557C4#8976214yes—
558C4C2×#645511yescharacteristic
559C4C2×#645611yescharacteristic
560C4C2×#645711yescharacteristic
561C4C2×#646111yescharacteristic
562C2—24no—
563C2—562no—
564C2C2×#8976211yescharacteristic
565C2#1174000211yescharacteristic
566C2#1174000311yescharacteristic
567C2#1174082411yescharacteristic
568C2#1174084511yescharacteristic
569C2#1174091511yescharacteristic
570C2#1174091611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2317C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#2317D7
hypocenterC7#21087
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C7×#2317C2semidirect
4. C2×#89769C2semidirect
6. #11729335C2semidirect
7. C2×#89324C2semidirect
8. #11727371C2semidirect
9. #11727370C2semidirect
10. #11727369C2semidirect
11. #11727368C2semidirect
24. #89774C4semidirect
96. #6384C2×C4semidirect
97. #6384C2×C4semidirect
99. #6383C2×C4semidirect
100. #6383C2×C4semidirect
106. #2317D7semidirect
528. C7#21087semidirect

Maximal subgroups

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

subgroupindex
C7×#23172
#117397152
C2×#897742
C2×#897692
C2×#897382
#117293352
C2×#893242
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