Finite Group Databasebeta

← #13387740

Subgroups — #13387740

Order 1792.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90254C211yesmaximal, characteristic
2C7×#2538C211yesmaximal, characteristic, complemented
3C2×#89793C211yesmaximal, characteristic, complemented
4C2×C4×#1143C211yesmaximal, characteristic, complemented
5#11729453C212yesmaximal, complemented
6#11729447C212yesmaximal, complemented
7#11729368C211yesmaximal, characteristic, complemented
8#11727338C211yesmaximal, characteristic, complemented
9#11727336C211yesmaximal, characteristic, complemented
10#11727277C212yesmaximal, complemented
11#11727260C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2212yes—
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C22×#1143C2211yescharacteristic
16C2×#6501—21no—
17C2×#6493C2212yes—
18C2×#6489C2212yes—
19#90254—24no—
20#90254—24no—
21C2×#6455—21no—
22C2×#6448—22no—
23C2×C4×D28C2211yescharacteristic
24C2×#6432C2211yescharacteristic
25C7×#396—22no—
26C7×#390—21no—
27C7×#389—21no—
28C7×#384C2211yescharacteristic
29C7×#380C2212yes—
30C2×#6409C2211yescharacteristic
31C2×#6409C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C4×#1143—24no—
35#89793—24no—
36C2×#6383C2211yescharacteristic
37C2×#6383C2212yes—
38C2×#6382C2211yescharacteristic
39C2×#6381C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24121—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7C2312yes—
49C22×C22⋊D7—22no—
50C22×D28C2311yescharacteristic
51C22×C4×D7—21no—
52C22×Dic14—21no—
53C7×C22⋊Q8—24no—
54C7×C22⋊Q8—42no—
55C4×C7×Q8—22no—
56C4×C7×Q8—22no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2312yes—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4D412yes—
69C2×C42×C7—21no—
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—22no—
72C2×#1143—22no—
73C2×#1143—22no—
74C2×#1143—22no—
75#6501—28no—
76#6493—24no—
77#6493—24no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—22no—
80#6489—24no—
81#6489—24no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2312yes—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4×Dic7—22no—
86C22×C7⋊C8C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89#6455—28no—
90#6448—216no—
91C4×D28—22no—
92C4×D28D414yes—
93#6432—22no—
94#6432D414yes—
95C7×#95—22no—
96#6409—24no—
97#6409—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)—42no—
100#6383—22no—
101#6383—42no—
102#6382—22no—
103#6381—24no—
104C4⋊(C7⋊C8)—22no—
105C4×C7⋊C8—22no—
106C2×#555—71no—
107C2×#477—71no—
108C2×C4×SD16—71no—
109#2538D711yescharacteristic, complemented
110#2033—72no—
111#2000—72no—
112#1990—71no—
113#1954—71no—
114#1873—72no—
115#1854—72no—
116#1802—71no—
117C24×C7C2×D412yes—
118C23×D7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8#4112yes—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C2×C22⋊D7—28no—
142C2×C22⋊D7—82no—
143C2×C22⋊D7—82no—
144C22×Dic7—21no—
145C22×Dic7—22no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28C2×D412yes—
152C2×C4×D7—24no—
153C2×C4×D7—42no—
154C2×C4×D7—42no—
155C2×Dic14—24no—
156C2×Dic14—81no—
157C2×Dic14—81no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—44no—
164C7×C4⋊C4—42no—
165C7×C4⋊C4—42no—
166C7×C4⋊C4—81no—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—44no—
169C7×C22⋊C4—82no—
170C42×C7—42no—
171C42×C7C2×D412yes—
172C42×C7C2×D412yes—
173C22⋊Dic7—28no—
174#1143—28no—
175(C2×D7)⋊C4—28no—
176(C2×D7)⋊C4—24no—
177C4⋊Dic7—24no—
178C4⋊Dic7—22no—
179C4⋊Dic7—21no—
180C4⋊Dic7C2×D414yes—
181C4⋊Dic7C2×D412yes—
182C4×Dic7—28no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8#4112yes—
192C22×SD16—71no—
193#555—144no—
194#555—144no—
195C2×C4⋊Q8—71no—
196C2×#124—142no—
197C2×C22⋊Q8C2×D712yes—
198C2×#121—72no—
199C2×#121—141no—
200C2×C4×Q8C2×D711yescharacteristic
201C2×C4×D4—71no—
202C2×C4×D4C2×D711yescharacteristic
203C2×#117—72no—
204C2×#117—141no—
205#477—144no—
206C4×SD16—144no—
207C2×#107—72no—
208C2×#106—71no—
209C2×C4⋊C8—71no—
210C2×#103—72no—
211C2×#103—71no—
212C2×#102—71no—
213C2×#102—72no—
214C2×C22⋊C8—72no—
215C2×C22⋊C8—72no—
216C2×C4×C8—71no—
217#396—22no—
218#390—21no—
219#389—21no—
220#384C2×D711yescharacteristic
221#380C2×D712yes—
222#340—72no—
223#340—72no—
224#340—72no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7C22×D411yescharacteristic
233C22×D7—22no—
234C22×D7—41no—
235C22×D7—41no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—41no—
239C22×D7—41no—
240C7×Q8—22no—
241C7×Q8—22no—
242C7×Q8—41no—
243C7×Q8—41no—
244C7×D4—44no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7#12012yes—
264C2×C4×C7#12012yes—
265C2×C4×C7#13512yes—
266C2×C4×C7#13712yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7H(5,𝔽2)11yescharacteristic
272C2×C4×C72−1+411yescharacteristic
273C22⋊D7—88no—
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280D28—22no—
281D28—22no—
282D28—41no—
283D28—41no—
284C4×D7—48no—
285Dic14—84no—
286C7⋊C8—22no—
287C7⋊C8—41no—
288C7⋊C8—42no—
289C22×Q8—141no—
290C22×Q8C22×D711yescharacteristic
291C22×D4—71no—
292C22×D4—142no—
293C22×D4C22⋊D711yescharacteristic
294C23×C4—141no—
295C23×C4C22×D712yes—
296C2×SD16—142no—
297C2×SD16—142no—
298C2×SD16—142no—
299C22×C8—71no—
300C22×C8—71no—
301C22×C8—72no—
302C4⋊Q8—74no—
303C4⋊Q8—142no—
304#124—1416no—
305C22⋊Q8—24no—
306C22⋊Q8—42no—
307#121—144no—
308#121—144no—
309#121—148no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—74no—
315C4×D4—142no—
316#117—148no—
317#117—144no—
318#117—144no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—72no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4C22⋊D711yescharacteristic
327C2×C4⋊C4C22×D711yescharacteristic
328C2×C4⋊C4C22×D712yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—142no—
332C2×C22⋊C4—142no—
333C2×C22⋊C4—141no—
334C2×C22⋊C4C22⋊D712yes—
335C2×C42—21no—
336C2×C42—142no—
337C2×C42C22×D711yescharacteristic
338#107—144no—
339#106—142no—
340C4⋊C8—142no—
341#103—142no—
342#103—144no—
343#102—142no—
344#102—282no—
345C22⋊C8—144no—
346C22⋊C8—282no—
347C4×C8—142no—
348#95—22no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7C2×#12011yescharacteristic
357C22×C7#54611yescharacteristic
358C22×C7#54711yescharacteristic
359C22×C7C2×#13511yescharacteristic
360C22×C7C2×#13711yescharacteristic
361C22×C7#57611yescharacteristic
362C22×C7#57711yescharacteristic
363C2×D7—42no—
364C2×D7—42no—
365C2×D7—42no—
366C2×D7—81no—
367C2×D7—81no—
368C2×D7—81no—
369C2×D7—81no—
370C4×C7—22no—
371C4×C7—42no—
372C4×C7—42no—
373C4×C7—81no—
374C4×C7—82no—
375C4×C7#45214yes—
376C4×C7#52511yescharacteristic
377C4×C7#53711yescharacteristic
378C4×C7#54811yescharacteristic
379C4×C7#55011yescharacteristic
380Dic7—42no—
381Dic7—44no—
382C24—141no—
383C24C2×C22⋊D712yes—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—144no—
390C2×Q8—561no—
391C2×Q8—561no—
392C2×Q8#115912yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—72no—
397C2×D4—148no—
398C2×D4—141no—
399C2×D4—141no—
400C2×D4—141no—
401C2×D4—141no—
402C2×D4—141no—
403C2×D4—562no—
404C2×D4—562no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—141no—
415C22×C4—142no—
416C22×C4—144no—
417C22×C4—282no—
418C22×C4—282no—
419C22×C4C2×C22⋊D712yes—
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423SD16—148no—
424C2×C8—72no—
425C2×C8—141no—
426C2×C8—141no—
427C2×C8—141no—
428C2×C8—142no—
429C2×C8—141no—
430C2×C8—142no—
431C2×C8—142no—
432C2×C8—141no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—72no—
442C4⋊C4—74no—
443C4⋊C4—81no—
444C4⋊C4—144no—
445C4⋊C4—141no—
446C4⋊C4—142no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—82no—
450C22⋊C4—148no—
451C22⋊C4—148no—
452C22⋊C4—144no—
453C42—42no—
454C42—148no—
455C42D4×D712yes—
456C42D4×D712yes—
457C2×C7—82no—
458C2×C7#240711yescharacteristic
459C2×C7C2×#45211yescharacteristic
460C2×C7#301211yescharacteristic
461C2×C7#330511yescharacteristic
462C2×C7#332911yescharacteristic
463C2×C7#334011yescharacteristic
464C2×C7#335711yescharacteristic
465D7—82no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—142no—
474C23—281no—
475C23—281no—
476C23—281no—
477C23—281no—
478C23—281no—
479C23—281no—
480C23C22×C22⋊D711yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—564no—
486D4—44no—
487D4—142no—
488D4—142no—
489D4—281no—
490D4—281no—
491D4—568no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—281no—
513C2×C4—281no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—288no—
517C2×C4#650012yes—
518C2×C4#650012yes—
519C2×C4#650512yes—
520C2×C4#651312yes—
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×D4×D711yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4#654811yescharacteristic
525C2×C4D7×#4111yescharacteristic
526C2×C4#655411yescharacteristic
527C8—142no—
528C8—282no—
529C8—281no—
530C7#2412111yescharacteristic, complemented, Sylow
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—282no—
539C22—282no—
540C22—282no—
541C22—561no—
542C22—561no—
543C22—561no—
544C22—561no—
545C22C2×#650011yescharacteristic
546C22#9049711yescharacteristic
547C22C2×#650511yescharacteristic
548C22C2×#651311yescharacteristic
549C22#9051511yescharacteristic
550C22#9052611yescharacteristic
551C22#9052811yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—81no—
557C4—282no—
558C4—284no—
559C4#8981914yes—
560C4#9028911yescharacteristic
561C4#9029011yescharacteristic
562C4#9029711yescharacteristic
563C4#9029911yescharacteristic
564C2—82no—
565C2—562no—
566C2#1174014611yescharacteristic
567C2C2×#8981911yescharacteristic
568C2#1174086911yescharacteristic
569C2#1174100011yescharacteristic
570C2#1174101611yescharacteristic
571C2#1174107211yescharacteristic
572C2#1174110311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2538C2
socleC23×C7C22×D4
prime radicalC2×#90254C2
hypercenter#2538D7
hypocenterC7#24121
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C7×#2538C2semidirect
3. C2×#89793C2semidirect
4. C2×C4×#1143C2semidirect
5. #11729453C2semidirect
6. #11729447C2semidirect
7. #11729368C2semidirect
8. #11727338C2semidirect
9. #11727336C2semidirect
10. #11727277C2semidirect
11. #11727260C2semidirect
109. #2538D7semidirect
530. C7#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#902542
C7×#25382
C2×#897932
C2×C4×#11432
#117294532
#117294472
#117293682
#117273382
#117273362
#117272772
#117272602
#241217