Finite Group Databasebeta

← #13395378

Subgroups — #13395378

Order 1792.

1086 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2548C211yesmaximal, characteristic, complemented
2C2×#89844C211yesmaximal, characteristic
3C2×#89829C212yesmaximal, complemented
4C2×C4×#1143C211yesmaximal, characteristic, complemented
5C4⋊C4×D28C211yesmaximal, characteristic, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729476C212yesmaximal, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11727411C212yesmaximal, complemented
10#11727406C212yesmaximal, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C7×#127C2211yescharacteristic
13C2×C7×#124C2212yes—
14C2×C4×C7×Q8C2211yescharacteristic
15C22×#1143C2211yescharacteristic
16C22×#1141C2212yes—
17C2×D7×C4⋊C4—21no—
18C2×C4×D28C2212yes—
19C2×C4×D28C2211yescharacteristic
20C7×#394—22no—
21C7×#390—21no—
22C7×#390—21no—
23C7×#390C2212yes—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30#89844—24no—
31#89829—28no—
32C4×#1143—24no—
33#89759—21no—
34C2×#6383C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36#89707C2211yescharacteristic
37#89700C2212yes—
38C2×C4⋊(C7⋊C8)C2212yes—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89440—22no—
42#89324C2212yes—
43#89319C2212yes—
44#24731—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4C2312yes—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C22×C4×D7—22no—
52C7×#127—22no—
53C7×#127—22no—
54C7×#124—24no—
55C7×#124—42no—
56C4×C7×Q8—22no—
57C4×C7×Q8—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C42×C7—21no—
67C2×C42×C7C2312yes—
68C2×C42×C7C2311yescharacteristic
69C2×#1143—22no—
70C2×#1143—22no—
71C2×#1143D414yescomplemented
72C2×#1141—24no—
73C2×#1141—24no—
74C2×#1141—24no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—22no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2312yes—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×Dic7⋊C4—22no—
82C22×C7⋊C8C2311yescharacteristic
83C22×C7⋊C8C2312yes—
84C22×C7⋊C8C2311yescharacteristic
85D7×C4⋊C4—44no—
86C4×D28—22no—
87C4×D28—24no—
88C4×D28—42no—
89C4×D28D414yes—
90C7×#95D412yes—
91#6409—24no—
92#6409—22no—
93#6406—24no—
94#6406—24no—
95#6406—42no—
96C7⋊#95—22no—
97#6383—22no—
98#6382—22no—
99C4⋊(C7⋊C8)—22no—
100C4⋊(C7⋊C8)—24no—
101C4×C7⋊C8—22no—
102C2×#463—72no—
103C2×#459—71no—
104C2×C4×SD16—71no—
105#2548D711yescharacteristic, complemented
106D4×C4⋊C4—71no—
107#2065—72no—
108#2042—72no—
109#1957—71no—
110#1939—71no—
111#1879—72no—
112#1803—71no—
113C24×C7—22no—
114C23×D7—21no—
115C2×C7×Q8—21no—
116C2×C7×Q8—21no—
117C2×C7×Q8—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8C2×D412yes—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—42no—
128C2×C7×D4—42no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4—81no—
132C2×C7×D4—81no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2411yescharacteristic
140C22×Dic7—21no—
141C22×Dic7—22no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28C2×D412yes—
146C2×D28C2×D412yes—
147C2×D28#4112yes—
148C2×C4×D7—24no—
149C2×C4×D7—42no—
150C2×C4×D7—42no—
151C2×C4×D7—44no—
152C2×C4×D7—44no—
153C2×C4×D7—82no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—42no—
159C7×C22⋊C4—44no—
160C7×C22⋊C4—44no—
161C7×C22⋊C4—82no—
162C7×C22⋊C4—82no—
163C42×C7—22no—
164C42×C7—22no—
165C42×C7—42no—
166C42×C7C2×D412yes—
167C42×C7#4112yes—
168#1143—28no—
169#1141—48no—
170(C2×D7)⋊C4—24no—
171(C2×D7)⋊C4—44no—
172C4⋊Dic7—21no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175C4⋊Dic7—42no—
176C4⋊Dic7—81no—
177C4⋊Dic7C2×D412yes—
178Dic7⋊C4—44no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8C2×D412yes—
188C22×SD16—71no—
189C22×D8—72no—
190C2×#127C2×D711yescharacteristic
191C2×#124C2×D712yes—
192C2×C4×Q8C2×D711yescharacteristic
193C2×C4×D4—71no—
194C2×C4×D4—72no—
195C22×C4⋊C4—141no—
196#463—148no—
197#459—144no—
198C4×SD16—144no—
199C2×#106—71no—
200C2×C4⋊C8—71no—
201C2×C4⋊C8—72no—
202C2×#103—72no—
203C2×#103—71no—
204C2×#102—72no—
205C2×#102—72no—
206C2×#102—71no—
207C2×#102—72no—
208C2×C4×C8—71no—
209#394—22no—
210#390—21no—
211#390—21no—
212#390C2×D712yes—
213#389—72no—
214#385—142no—
215#384—71no—
216#380—141no—
217C4×C4⋊C4C2×D711yescharacteristic
218#340—72no—
219#336—72no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7C22×D411yescharacteristic
228C22×D7—22no—
229C22×D7—22no—
230C22×D7—22no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C22×D7—41no—
235C7×Q8—22no—
236C7×Q8—22no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×D4—42no—
240C7×D4—42no—
241C7×D4—42no—
242C7×D4—42no—
243C7×D4—84no—
244C7×D4—84no—
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—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7#12112yes—
260C2×C4×C7#12112yes—
261C2×C4×C7#13512yes—
262C2×C4×C7AGL(1,ℤ/8)12yes—
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4111yescharacteristic
266C2×C4×C7C2×#4111yescharacteristic
267C2×C4×C7H(5,𝔽2)11yescharacteristic
268C2×C4×C72−1+411yescharacteristic
269C2×Dic7—22no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275D28—22no—
276D28—22no—
277D28—22no—
278D28#12114yes—
279C4×D7—48no—
280C4×D7—88no—
281C7⋊C8—22no—
282C7⋊C8—42no—
283C7⋊C8—41no—
284C22×Q8C22×D711yescharacteristic
285C22×D4—21no—
286C22×D4—21no—
287C22×D4—71no—
288C22×D4C22×D712yes—
289C23×C4—141no—
290C23×C4—142no—
291C2×SD16—74no—
292C2×SD16—142no—
293C2×SD16—142no—
294C2×D8—144no—
295C2×D8—144no—
296C2×D8—144no—
297C22×C8—71no—
298C22×C8—72no—
299C22×C8—71no—
300#127—22no—
301#127—22no—
302#124—24no—
303#124—42no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×D4—74no—
307C4×D4—144no—
308C4×D4—142no—
309C4×D4—282no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—72no—
313C2×C4⋊C4—142no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—284no—
317C2×C4⋊C4C22⋊D711yescharacteristic
318C2×C4⋊C4C22⋊D711yescharacteristic
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—141no—
325C2×C22⋊C4—142no—
326C2×C42—21no—
327C2×C42C22×D712yes—
328C2×C42C22×D711yescharacteristic
329#106—142no—
330C4⋊C8—144no—
331C4⋊C8—142no—
332#103—142no—
333#103—144no—
334#102—142no—
335#102—144no—
336#102—144no—
337#102—282no—
338C4×C8—142no—
339#95—142no—
340#95C22⋊D712yes—
341C22×C7—82no—
342C22×C7—82no—
343C22×C7—82no—
344C22×C7—82no—
345C22×C7—82no—
346C22×C7—82no—
347C22×C7—82no—
348C22×C7C2×#12111yescharacteristic
349C22×C7D4211yescharacteristic
350C22×C7#54811yescharacteristic
351C22×C7C2×#13511yescharacteristic
352C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
353C22×C7#57611yescharacteristic
354C22×C7#57711yescharacteristic
355C2×D7—24no—
356C2×D7—42no—
357C2×D7—42no—
358C2×D7—42no—
359C2×D7—42no—
360C2×D7—42no—
361C2×D7—42no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—81no—
367C4×C7#46314yescomplemented
368C4×C7C2×#12111yescharacteristic
369C4×C7#53711yescharacteristic
370C4×C7#54711yescharacteristic
371C4×C7#55311yescharacteristic
372Dic7—42no—
373Dic7—82no—
374C24—22no—
375C24—141no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8D4×D712yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—72no—
393C2×D4—72no—
394C2×D4—72no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—141no—
398C2×D4—141no—
399C2×D4—141no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—141no—
403C22×C4—144no—
404C22×C4—142no—
405C22×C4—284no—
406C22×C4—284no—
407C22×C4—282no—
408C22×C4—282no—
409C22×C4—562no—
410C22×C4C2×C22⋊D712yes—
411C22×C4C2×C22⋊D712yes—
412C22×C4C2×C22⋊D711yescharacteristic
413C22×C4C2×C22⋊D711yescharacteristic
414C22×C4C23×D711yescharacteristic
415SD16—148no—
416D8—288no—
417C2×C8—72no—
418C2×C8—141no—
419C2×C8—141no—
420C2×C8—141no—
421C2×C8—142no—
422C2×C8—142no—
423C2×C8—141no—
424C2×C8—142no—
425C2×C8—141no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—72no—
432C4⋊C4—141no—
433C4⋊C4—142no—
434C4⋊C4—142no—
435C4⋊C4—284no—
436C4⋊C4—282no—
437C4⋊C4—561no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—144no—
443C22⋊C4—284no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42D4×D712yes—
448C42#115912yes—
449C2×C7—162no—
450C2×C7#241611yescharacteristic
451C2×C7C2×#46311yescharacteristic
452C2×C7#304511yescharacteristic
453C2×C7#327411yescharacteristic
454C2×C7#328411yescharacteristic
455C2×C7#336111yescharacteristic
456C2×C7#337311yescharacteristic
457D7—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—142no—
466C23—142no—
467C23—142no—
468C23—281no—
469C23—281no—
470C23—281no—
471C23—281no—
472C23C22×C22⋊D711yescharacteristic
473Q8—22no—
474Q8—22no—
475Q8—41no—
476Q8—41no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—74no—
482D4—84no—
483D4—84no—
484D4—142no—
485D4—142no—
486D4—142no—
487C2×C4—21no—
488C2×C4—21no—
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—41no—
501C2×C4—142no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—288no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—568no—
509C2×C4#650112yes—
510C2×C4#650112yes—
511C2×C4#651212yes—
512C2×C4#651312yes—
513C2×C4C2×D4×D711yescharacteristic
514C2×C4C2×D4×D711yescharacteristic
515C2×C4C2×#115911yescharacteristic
516C2×C4#654811yescharacteristic
517C2×C4D7×#4111yescharacteristic
518C2×C4#655411yescharacteristic
519C8—142no—
520C8—282no—
521C8—281no—
522C7#2473111yescharacteristic, complemented, Sylow
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—144no—
531C22—282no—
532C22—282no—
533C22—282no—
534C22—282no—
535C22—282no—
536C22—282no—
537C22C2×#650111yescharacteristic
538C22D4×C22⋊D711yescharacteristic
539C22C2×#651211yescharacteristic
540C22C2×#651311yescharacteristic
541C22#9052211yescharacteristic
542C22#9052611yescharacteristic
543C22#9052711yescharacteristic
544C4—22no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4—282no—
550C4—562no—
551C4#8982914yescomplemented
552C4D7×#12111yescharacteristic
553C4#9029911yescharacteristic
554C4#9030211yescharacteristic
555C4#9030511yescharacteristic
556C2—162no—
557C2—284no—
558C2#1174008611yescharacteristic
559C2C2×#8982911yescharacteristic
560C2#1174083011yescharacteristic
561C2#1174102311yescharacteristic
562C2#1174102411yescharacteristic
563C2#1174115811yescharacteristic
564C2#1174116011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2548C2
socleC23×C7C22×D4
prime radicalC2×#89844C2
hypercenter#2548D7
hypocenterC7#24731
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C7×#2548C2semidirect
3. C2×#89829C2semidirect
4. C2×C4×#1143C2semidirect
5. C4⋊C4×D28C2semidirect
6. #11729518C2semidirect
7. #11729476C2semidirect
8. #11729380C2semidirect
9. #11727411C2semidirect
10. #11727406C2semidirect
11. #11727238C2semidirect
71. C2×#1143D4semidirect
105. #2548D7semidirect
367. C4×C7#463semidirect
522. C7#24731semidirect
551. C4#89829semidirect

Maximal subgroups

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

subgroupindex
C7×#25482
C2×#898442
C2×#898292
C2×C4×#11432
C4⋊C4×D282
#117295182
#117294762
#117293802
#117274112
#117274062
#117272382
#247317