Finite Group Databasebeta

← #13395332

Subgroups — #13395332

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#89791C211yesmaximal, characteristic, complemented
3C2×#89788C212yesmaximal
4C2×C4×#1141C211yesmaximal, characteristic
5#11731247C211yesmaximal, characteristic, complemented
6#11729366C212yesmaximal, complemented
7#11727389C212yesmaximal, complemented
8#11727382C212yesmaximal, complemented
9#11727338C211yesmaximal, characteristic, complemented
10#11727333C211yesmaximal, characteristic, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C4×C7×D4C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C22×C7×C4⋊C4—21no—
15C22×#1142C2212yes—
16C22×#1141C2211yescharacteristic
17C2×#6438C2212yes—
18C2×C4×D28C2211yescharacteristic
19C2×#6432C2211yescharacteristic
20C7×#389C2212yes—
21C7×#385—22no—
22C7×#384C2211yescharacteristic
23C7×#380—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26#89791—24no—
27#89788—28no—
28C4×#1141—24no—
29#89759—21no—
30#89759—21no—
31C2×#6384C2212yes—
32C2×#6384C2212yes—
33C2×#6384C2212yes—
34C2×#6383C2211yescharacteristic
35C2×#6383C2212yes—
36C2×#6381C2211yescharacteristic
37#89701C2212yes—
38C2×C4⋊(C7⋊C8)C2212yes—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89446—22no—
42#89324C2212yes—
43#89319C2212yes—
44#24727—71nomaximal, Sylow
45C22×C7×D4C2311yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7—22no—
48C22×D28C2311yescharacteristic
49C22×C4×D7—21no—
50C22×Dic14C2312yes—
51C4×C7×D4—24no—
52C4×C7×D4—22no—
53C4×C7×D4—42no—
54C4×C7×D4D414yes—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—44no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—22no—
65C2×C42×C7C2312yes—
66C2×C42×C7C2311yescharacteristic
67C2×#1142—24no—
68C2×#1142—24no—
69C2×#1142—24no—
70C2×#1141—22no—
71C2×#1141—22no—
72C2×#1141D414yes—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—22no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C22×C7⋊C8C2311yescharacteristic
80C22×C7⋊C8C2312yes—
81C22×C7⋊C8C2311yescharacteristic
82#6438—24no—
83#6438—42no—
84C4×D28—22no—
85C4×D28—22no—
86#6432—22no—
87#6432—22no—
88C7×#95—22no—
89C7×#95D412yes—
90#6406—22no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93#6384—24no—
94#6384—24no—
95#6384—42no—
96#6383—24no—
97#6383—22no—
98#6381—22no—
99C4⋊(C7⋊C8)—22no—
100C4⋊(C7⋊C8)—24no—
101C4×C7⋊C8—22no—
102C2×#463—72no—
103C2×#461—71no—
104C2×C4×D8—71no—
105#2547—71no—
106D4×C4⋊C4D711yescharacteristic, complemented
107#2068—72no—
108#2042—72no—
109#1956—71no—
110#1939—71no—
111#1877—72no—
112#1802—71no—
113C24×C7—21no—
114C23×D7—21no—
115C2×C7×D4—21no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4C2×D412yes—
119C2×C7×D4C2×D412yes—
120C2×C7×D4#4112yes—
121C22×C4×C7—21no—
122C22×C4×C7—24no—
123C22×C4×C7—22no—
124C22×C4×C7—44no—
125C22×C4×C7—44no—
126C22×C4×C7—42no—
127C22×C4×C7—42no—
128C22×C4×C7—82no—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—22no—
136C2×D28—21no—
137C2×D28—21no—
138C2×D28—21no—
139C2×D28—21no—
140C2×D28—21no—
141C2×D28C2×D412yes—
142C2×C4×D7—42no—
143C2×C4×D7—42no—
144C2×C4×D7—42no—
145C2×Dic14—22no—
146C2×Dic14—22no—
147C2×Dic14—22no—
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—44no—
157C7×C4⋊C4—42no—
158C7×C4⋊C4—81no—
159C7×C4⋊C4C2×D412yes—
160C7×C22⋊C4—24no—
161C7×C22⋊C4—44no—
162C42×C7—22no—
163C42×C7—22no—
164C42×C7C2×D412yes—
165C42×C7#4112yes—
166#1142—48no—
167#1141—28no—
168(C2×D7)⋊C4—44no—
169(C2×D7)⋊C4—44no—
170(C2×D7)⋊C4—42no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173C4⋊Dic7—42no—
174C4⋊Dic7—42no—
175C2×C7⋊C8—21no—
176C2×C7⋊C8—21no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8C2×D412yes—
184C22×SD16—72no—
185C22×D8—71no—
186C2×C4⋊Q8—71no—
187C2×#124—72no—
188C2×C4×D4—71no—
189C2×C4×D4C2×D712yes—
190C2×C4×D4C2×D711yescharacteristic
191C22×C4⋊C4—21no—
192#463—148no—
193#461—144no—
194C4×D8—144no—
195C2×#107—71no—
196C2×C4⋊C8—71no—
197C2×C4⋊C8—72no—
198C2×#103—72no—
199C2×#103—72no—
200C2×#103—72no—
201C2×#102—71no—
202C2×#102—72no—
203C2×#102—71no—
204C2×C4×C8—71no—
205#397—142no—
206#391—72no—
207#389C2×D712yes—
208#385—22no—
209#384C2×D711yescharacteristic
210#380—21no—
211#380—141no—
212#380—141no—
213C4×C4⋊C4C2×D711yescharacteristic
214#340—72no—
215#336—72no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7C22×D411yescharacteristic
224C22×D7—22no—
225C22×D7—41no—
226C22×D7—41no—
227C22×D7—41no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C7×D4—22no—
232C7×D4—22no—
233C7×D4—22no—
234C7×D4#12114yes—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—22no—
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—41no—
245C2×C4×C7—41no—
246C2×C4×C7—48no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—88no—
251C2×C4×C7#12112yes—
252C2×C4×C7#12112yes—
253C2×C4×C7#13512yes—
254C2×C4×C7AGL(1,ℤ/8)12yes—
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C2×#4111yescharacteristic
258C2×C4×C7C2×#4111yescharacteristic
259C2×C4×C7H(5,𝔽2)11yescharacteristic
260C2×C4×C72−1+411yescharacteristic
261C2×Dic7—41no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267D28—22no—
268D28—22no—
269D28—41no—
270D28—41no—
271C4×D7—84no—
272Dic14—42no—
273Dic14—42no—
274Dic14—42no—
275Dic14—42no—
276C7⋊C8—22no—
277C7⋊C8—42no—
278C7⋊C8—41no—
279C22×Q8—72no—
280C22×D4—71no—
281C22×D4C22×D711yescharacteristic
282C23×C4—21no—
283C23×C4—22no—
284C23×C4—141no—
285C2×SD16—144no—
286C2×SD16—144no—
287C2×SD16—144no—
288C2×D8—74no—
289C2×D8—142no—
290C2×D8—142no—
291C22×C8—71no—
292C22×C8—72no—
293C22×C8—71no—
294C4⋊Q8—142no—
295C4⋊Q8—142no—
296#124—144no—
297#124—282no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—42no—
301C4×D4—142no—
302C4×D4—142no—
303C4×D4D2814yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—141no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4C22⋊D711yescharacteristic
312C2×C4⋊C4C22⋊D711yescharacteristic
313C2×C4⋊C4C22×D712yes—
314C2×C4⋊C4C22×D711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—142no—
319C2×C22⋊C4—142no—
320C2×C42C22×D712yes—
321C2×C42C22×D711yescharacteristic
322#107—142no—
323C4⋊C8—144no—
324C4⋊C8—142no—
325#103—144no—
326#103—144no—
327#103—282no—
328#102—144no—
329#102—142no—
330#102—142no—
331C4×C8—142no—
332#95—22no—
333#95—142no—
334#95—142no—
335#95C22⋊D712yes—
336C22×C7—24no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7C2×#12111yescharacteristic
344C22×C7D4211yescharacteristic
345C22×C7#54811yescharacteristic
346C22×C7C2×#13511yescharacteristic
347C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C7#57711yescharacteristic
349C22×C7#57811yescharacteristic
350C2×D7—42no—
351C2×D7—42no—
352C2×D7—42no—
353C2×D7—81no—
354C2×D7—81no—
355C2×D7—81no—
356C2×D7—81no—
357C4×C7—22no—
358C4×C7—42no—
359C4×C7—42no—
360C4×C7—42no—
361C4×C7—82no—
362C4×C7#46314yescomplemented
363C4×C7C2×#12111yescharacteristic
364C4×C7#53711yescharacteristic
365C4×C7#54711yescharacteristic
366C4×C7#55311yescharacteristic
367Dic7—82no—
368Dic7—81no—
369C24—21no—
370C24—141no—
371C2×Q8—142no—
372C2×Q8—142no—
373C2×Q8—142no—
374C2×Q8—142no—
375C2×Q8—142no—
376C2×Q8—142no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—72no—
381C2×D4—141no—
382C2×D4—141no—
383C2×D4—141no—
384C2×D4—141no—
385C2×D4—141no—
386C2×D4C2×D2812yes—
387C2×D4D4×D712yes—
388C2×D4#116112yes—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—82no—
397C22×C4—141no—
398C22×C4—142no—
399C22×C4—282no—
400C22×C4—282no—
401C22×C4—282no—
402C22×C4C2×C22⋊D712yes—
403C22×C4C2×C22⋊D712yes—
404C22×C4C2×C22⋊D711yescharacteristic
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407SD16—288no—
408D8—148no—
409C2×C8—72no—
410C2×C8—141no—
411C2×C8—141no—
412C2×C8—141no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—141no—
416C2×C8—142no—
417C2×C8—141no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—81no—
426C4⋊C4—141no—
427C4⋊C4—141no—
428C4⋊C4—282no—
429C4⋊C4—282no—
430C4⋊C4C2×D2812yes—
431C22⋊C4—24no—
432C22⋊C4—44no—
433C22⋊C4—284no—
434C22⋊C4—284no—
435C22⋊C4—282no—
436C42—22no—
437C42—22no—
438C42C2×D2812yes—
439C42#115712yes—
440C2×C7—44no—
441C2×C7#241611yescharacteristic
442C2×C7C2×#46311yescharacteristic
443C2×C7#304611yescharacteristic
444C2×C7#327211yescharacteristic
445C2×C7#328311yescharacteristic
446C2×C7#336011yescharacteristic
447C2×C7#336711yescharacteristic
448D7—82no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—142no—
457C23—281no—
458C23—281no—
459C23—281no—
460C23—281no—
461C23—281no—
462C23—281no—
463C23C22×C22⋊D711yescharacteristic
464Q8—282no—
465Q8—282no—
466Q8—282no—
467Q8—282no—
468D4—22no—
469D4—22no—
470D4—22no—
471D4—142no—
472D4—142no—
473D4—281no—
474D4—281no—
475D4#645814yes—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—88no—
492C2×C4—281no—
493C2×C4—282no—
494C2×C4—281no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—281no—
498C2×C4—564no—
499C2×C4#649312yes—
500C2×C4#649312yes—
501C2×C4#649512yes—
502C2×C4#651312yes—
503C2×C4C22×D2811yescharacteristic
504C2×C4C2×#115711yescharacteristic
505C2×C4C2×D4×D711yescharacteristic
506C2×C4#654811yescharacteristic
507C2×C4C2×#116111yescharacteristic
508C2×C4#655111yescharacteristic
509C8—142no—
510C8—282no—
511C8—281no—
512C7#2472711yescharacteristic, complemented, Sylow
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—282no—
521C22—282no—
522C22—282no—
523C22—561no—
524C22—561no—
525C22—561no—
526C22—561no—
527C22C2×#649311yescharacteristic
528C22C2×#649511yescharacteristic
529C22D4×C22⋊D711yescharacteristic
530C22#9050811yescharacteristic
531C22C2×#651311yescharacteristic
532C22#9052711yescharacteristic
533C22#9052911yescharacteristic
534C4—22no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—561no—
540C4—562no—
541C4#8978814yescomplemented
542C4C2×#645811yescharacteristic
543C4#9019811yescharacteristic
544C4#9020111yescharacteristic
545C4#9020511yescharacteristic
546C2—44no—
547C2—562no—
548C2#1173963811yescharacteristic
549C2C2×#8978811yescharacteristic
550C2#1174001711yescharacteristic
551C2#1174093411yescharacteristic
552C2#1174096911yescharacteristic
553C2#1174100811yescharacteristic
554C2#1174108211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×D4×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenterD4×C4⋊C4D7
hypocenterC7#24727
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C7×D4×C4⋊C4C2semidirect
2. C2×#89791C2semidirect
5. #11731247C2semidirect
6. #11729366C2semidirect
7. #11727389C2semidirect
8. #11727382C2semidirect
9. #11727338C2semidirect
10. #11727333C2semidirect
11. #11727238C2semidirect
106. D4×C4⋊C4D7semidirect
362. C4×C7#463semidirect
512. C7#24727semidirect
541. C4#89788semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C4⋊C42
C2×#897912
C2×#897882
C2×C4×#11412
#117312472
#117293662
#117273892
#117273822
#117273382
#117273332
#117272382
#247277