Finite Group Databasebeta

← #13356234

Subgroups — #13356234

Order 1792.

1126 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
1C2×#89905C211yesmaximal, characteristic
2C2×#89893C211yesmaximal, characteristic, complemented
3C2×#89885C211yesmaximal, characteristic
4#11739715C211yesmaximal, characteristic
5#11739668C211yesmaximal, characteristic, complemented
6C7×#1729C211yesmaximal, characteristic, complemented
7C2×#89346C211yesmaximal, characteristic, complemented
8#11726382C212yesmaximal, complemented
9#11726381C212yesmaximal, complemented
10#11726380C212yesmaximal, complemented
11#11726379C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C2×#6489C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19#89905C418yescomplemented
20#89893—24no—
21#89885—24no—
22C2×#6395C2211yescharacteristic
23C2×#6395C2211yescharacteristic
24C2×#6395C2212yes—
25C2×#6392C2212yes—
26C2×#6391C2212yes—
27C2×#6390C2211yescharacteristic
28C2×#6390C2211yescharacteristic
29C2×#6390C2212yes—
30#89759—21no—
31#89759—21no—
32C4⋊C4×Dic7—21no—
33C4⋊C4×Dic7—21no—
34#89730—22no—
35C22⋊C4×Dic7—22no—
36#89711C2212yes—
37C4×C4⋊Dic7C2212yes—
38C4×C4⋊Dic7C2212yes—
39#89701C2212yes—
40C7×#336C2212yes—
41C7×#336C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#21062—71nomaximal, Sylow
45C23×C4×C7C2311yescharacteristic
46C22×C22⋊D7—21no—
47C23×Dic7—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
58C2×C7×M16—22no—
59C2×C7×M16—22no—
60C2×C7×M16C2×C414yes—
61C22×C7×C8—22no—
62C22×C7×C8—22no—
63C22×C7×C8—22no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2312yes—
67C7×#117—22no—
68C7×#117—22no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C22⋊C4D412yes—
72C2×C42×C7C2312yes—
73C2×C22⋊Dic7—21no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—22no—
76#6489—22no—
77#6489D414yes—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2×C414yes—
84C2×C4⋊Dic7C2312yes—
85C2×C4⋊Dic7C2312yes—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7—22no—
90C2×C4×Dic7—22no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95#6395—22no—
96#6395—22no—
97#6395C2×C418yescomplemented
98#6392—24no—
99#6391—24no—
100#6390—22no—
101#6390—22no—
102#6390C2×C418yescomplemented
103C2×#428—71no—
104C2×#417—71no—
105C2×#416—71no—
106#2311—71no—
107#2274—71no—
108#1928—72no—
109#1927—72no—
110#1926—72no—
111#1925—72no—
112C2×#340—71no—
113#1729D711yescharacteristic, complemented
114C24×C7C2×D411yescharacteristic
115C23×D7—21no—
116C22×C4×C7C22×C412yes—
117C22×C4×C7C2×D412yes—
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D412yes—
121C22×C4×C7#4112yes—
122C22×C4×C7#4112yes—
123C22×C4×C7#4112yes—
124C22×C4×C7#4112yes—
125C22×C4×C7C2411yescharacteristic
126C2×C22⋊D7—24no—
127C2×C22⋊D7—42no—
128C2×C22⋊D7—42no—
129C22×Dic7—21no—
130C22×Dic7—21no—
131C22×Dic7—21no—
132C22×Dic7—22no—
133C22×Dic7—24no—
134C22×Dic7—22no—
135C22×Dic7—42no—
136C22×Dic7—81no—
137C22×Dic7—81no—
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×M16—28no—
160C2×C7×C8—22no—
161C2×C7×C8—22no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—44no—
169C2×C7×C8C22×C414yes—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C22⋊C4—24no—
173C42×C7—22no—
174C42×C7—22no—
175C22⋊Dic7—24no—
176(C2×D7)⋊C4—82no—
177(C2×D7)⋊C4—82no—
178C4⋊Dic7—24no—
179C4⋊Dic7—22no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—22no—
183C4⋊Dic7—22no—
184C4⋊Dic7—42no—
185C4⋊Dic7C22×C414yes—
186C4⋊Dic7C2×D412yes—
187C4⋊Dic7C2×D412yes—
188C4×Dic7—24no—
189C4×Dic7—82no—
190C4×Dic7—82no—
191C22×#41—71no—
192C22×M16C2×D711yescharacteristic
193C23×C8C2×D711yescharacteristic
194C2×#117—71no—
195C2×#117C2×D711yescharacteristic
196C22×C4⋊C4—71no—
197C22×C4⋊C4—71no—
198#428—144no—
199C2×#107—72no—
200C2×#106—72no—
201#417—78no—
202#416—144no—
203C2×#103—72no—
204C2×#103—71no—
205C2×#103—71no—
206C2×#102—72no—
207C2×#102—71no—
208C2×#102—71no—
209#391—72no—
210#390—72no—
211#386—142no—
212#380—141no—
213#380—141no—
214C4×C4⋊C4—72no—
215C4×C4⋊C4—72no—
216C4×C4⋊C4—141no—
217C4×C4⋊C4—141no—
218C4×C22⋊C4—142no—
219#340—72no—
220#340—72no—
221#336C2×D712yes—
222#336C2×D712yes—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7C4×D412yes—
228C23×C7#12112yes—
229C23×C7#12312yes—
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#11714yes—
245C2×C4×C7C4×D412yes—
246C2×C4×C7C4×D414yes—
247C2×C4×C7#12112yes—
248C2×C4×C7#12312yes—
249C2×C4×C7#12414yes—
250C2×C4×C7#12614yes—
251C2×C4×C7C23×C411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C7C2×#4111yescharacteristic
255C2×C4×C7C2×#4111yescharacteristic
256C2×C4×C7C2×#4111yescharacteristic
257C22⋊D7—48no—
258C2×Dic7—24no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—41no—
267C2×Dic7—42no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—84no—
274C2×Dic7—84no—
275D28—22no—
276D28—22no—
277D28—41no—
278D28—41no—
279C4×D7—48no—
280Dic14—22no—
281Dic14—22no—
282Dic14—41no—
283Dic14—41no—
284C7×C8—24no—
285C7×C8—41no—
286C7×C8—41no—
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—141no—
297C23×C4—141no—
298C23×C4—141no—
299C23×C4C22×D711yescharacteristic
300C2×M16—22no—
301C2×M16—22no—
302C2×M16C4×D714yes—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22×D711yescharacteristic
307C22×C8C22×D711yescharacteristic
308C22×C8C22×D712yes—
309#117—22no—
310#117—22no—
311#117—74no—
312#117—142no—
313C2×C4⋊C4—74no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—72no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4D2811yescharacteristic
324C2×C4⋊C4D2811yescharacteristic
325C2×C22⋊C4—141no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C22⋊C4D2812yes—
329C2×C42—141no—
330C2×C42—142no—
331C2×C42—142no—
332C2×C42C22×D712yes—
333#107—144no—
334#106—144no—
335#103—78no—
336#103—142no—
337#103—142no—
338#102—78no—
339#102—142no—
340#102—142no—
341#95—142no—
342#95—142no—
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×#12311yescharacteristic
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#38814yes—
369C4×C7C2×#11711yescharacteristic
370C4×C7C2×C4×D411yescharacteristic
371C4×C7C2×#12411yescharacteristic
372C4×C7C2×#12611yescharacteristic
373Dic7—44no—
374Dic7—42no—
375Dic7—42no—
376Dic7—42no—
377Dic7—82no—
378C24—141no—
379C24C2×D2811yescharacteristic
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—141no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—142no—
403C22×C4—144no—
404C22×C4—144no—
405C22×C4—142no—
406C22×C4—282no—
407C22×C4—282no—
408C22×C4—282no—
409C22×C4—561no—
410C22×C4—561no—
411C22×C4—561no—
412C22×C4—561no—
413C22×C4C2×C4×D712yes—
414C22×C4C2×D2812yes—
415C22×C4C2×D2812yes—
416C22×C4C2×D2811yescharacteristic
417C22×C4#115712yes—
418C22×C4D4×D712yes—
419C22×C4#115912yes—
420C22×C4#115912yes—
421C22×C4#116112yes—
422C22×C4C23×D711yescharacteristic
423M16—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—44no—
433C2×C8C2×C4×D714yes—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—74no—
437C4⋊C4—72no—
438C4⋊C4—72no—
439C4⋊C4—144no—
440C4⋊C4—142no—
441C4⋊C4—141no—
442C4⋊C4—141no—
443C4⋊C4—142no—
444C4⋊C4—142no—
445C4⋊C4—282no—
446C22⋊C4—24no—
447C22⋊C4—144no—
448C22⋊C4—562no—
449C22⋊C4—562no—
450C42—22no—
451C42—22no—
452C42—144no—
453C42—562no—
454C42—562no—
455C2×C7—24no—
456C2×C7C2×#38811yescharacteristic
457C2×C7#294611yescharacteristic
458C2×C7#294711yescharacteristic
459C2×C7#304511yescharacteristic
460C2×C7#304611yescharacteristic
461C2×C7#309811yescharacteristic
462C2×C7#309911yescharacteristic
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×D2812yes—
476C23#644912yes—
477C23#645812yes—
478C23C22×D2811yescharacteristic
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—144no—
495C2×C4—142no—
496C2×C4—142no—
497C2×C4—142no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4—282no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—282no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—288no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—564no—
511C2×C4—564no—
512C2×C4C4×D2812yes—
513C2×C4(C22⋊D7)⋊C414yes—
514C2×C4#644814yes—
515C2×C4#644912yes—
516C2×C4#645514yes—
517C2×C4#645812yes—
518C2×C4#646114yes—
519C2×C4C22×C4×D711yescharacteristic
520C2×C4C2×#115711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×#115911yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4C2×#116111yescharacteristic
525C8—24no—
526C8—41no—
527C8—41no—
528C7#2106211yescharacteristic, 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#8944214yes—
543C22C2×C4×D2811yescharacteristic
544C22C2×#644911yescharacteristic
545C22C2×#645811yescharacteristic
546C22#9044211yescharacteristic
547C22#9044211yescharacteristic
548C22#9044311yescharacteristic
549C22#9044411yescharacteristic
550C4—42no—
551C4—42no—
552C4—282no—
553C4—284no—
554C4—282no—
555C4—282no—
556C4—562no—
557C4#8944214yes—
558C4C2×(C22⋊D7)⋊C411yescharacteristic
559C4C2×#644811yescharacteristic
560C4C2×#645511yescharacteristic
561C4C2×#646111yescharacteristic
562C2—24no—
563C2—562no—
564C2C2×#8944211yescharacteristic
565C2#1173600511yescharacteristic
566C2#1173601611yescharacteristic
567C2#1173609811yescharacteristic
568C2#1173609911yescharacteristic
569C2#1173653311yescharacteristic
570C2#1173653411yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1729C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1729D7
hypocenterC7#21062
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1729

Complements

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

NH ≅ G/Ntype
2. C2×#89893C2semidirect
5. #11739668C2semidirect
6. C7×#1729C2semidirect
7. C2×#89346C2semidirect
8. #11726382C2semidirect
9. #11726381C2semidirect
10. #11726380C2semidirect
11. #11726379C2semidirect
19. #89905C4semidirect
97. #6395C2×C4semidirect
102. #6390C2×C4semidirect
113. #1729D7semidirect
528. C7#21062semidirect

Maximal subgroups

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

subgroupindex
C2×#899052
C2×#898932
C2×#898852
#117397152
#117396682
C7×#17292
C2×#893462
#117263822
#117263812
#117263802
#117263792
#210627