Finite Group Databasebeta

← #13388560

Subgroups — #13388560

Order 1792.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×#89855C211yesmaximal, characteristic
3#11739945C211yesmaximal, characteristic
4C2×C4×#1143C211yesmaximal, characteristic
5#11729545C212yesmaximal
6#11729539C211yesmaximal, characteristic, complemented
7#11729535C212yesmaximal, complemented
8#11729388C211yesmaximal, characteristic, complemented
9#11729387C212yesmaximal
10#11729383C212yesmaximal, complemented
11#11729380C211yesmaximal, characteristic, complemented
12C7×#557—24no—
13C7×#557—24no—
14C2×C7×C4⋊Q8C2212yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×#125—22no—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C2×#6509—21no—
23C22×#1143C2211yescharacteristic
24C22×#1143C2212yes—
25C2×C4×D28C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31#89855—24no—
32C4×#1143—24no—
33#89766—22no—
34#89759—21no—
35C2×#6383C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37#89711C2212yes—
38#89707C2211yescharacteristic
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24187—71nomaximal, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2312yes—
48C22×C7×Q8C2311yescharacteristic
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C7×C4⋊Q8—24no—
52C7×C4⋊Q8—24no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8D414yes—
55C7×#125—216no—
56C7×#125—28no—
57C4×C7×Q8—28no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—22no—
61C4×C7×Q8—22no—
62C4×C7×Q8—22no—
63C4×C7×Q8D414yes—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D412yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—22no—
74C2×C42×C7—21no—
75C2×C42×C7C2312yes—
76C2×C42×C7C2311yescharacteristic
77#6509—44no—
78C2×#1143—22no—
79C2×#1143—22no—
80C2×#1143—24no—
81C2×#1143—24no—
82C2×#1143—22no—
83C2×#1143—42no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—22no—
86C2×(C2×D7)⋊C4—22no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2312yes—
91C2×Dic7⋊C4—22no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2312yes—
94C22×C7⋊C8C2311yescharacteristic
95C4×D28—22no—
96C4×D28—22no—
97#6409—22no—
98#6409—24no—
99#6409—24no—
100#6409—42no—
101#6409—42no—
102C7⋊#95—22no—
103#6383—22no—
104#6382—22no—
105C4⋊(C7⋊C8)—22no—
106C4×C7⋊C8—22no—
107C2×#557D711yescharacteristic, complemented
108C2×#461—71no—
109C2×C4×SD16—71no—
110#2538—71no—
111#2043—72no—
112#2013—72no—
113#1984—71no—
114#1956—71no—
115#1880—72no—
116#1865—72no—
117#1803—71no—
118C23×D7—21no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—21no—
125C2×C7×Q8—22no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—22no—
129C2×C7×Q8—21no—
130C2×C7×Q8—22no—
131C2×C7×Q8SD1614yescomplemented
132C2×C7×Q8C2×D414yes—
133C2×C7×Q8C2×D412yes—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—21no—
143C22×Dic7—22no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28#4112yes—
150C2×C4×D7—42no—
151C2×C4×D7—42no—
152C2×C4×D7—42no—
153C7×C4⋊C4—28no—
154C7×C4⋊C4—28no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4C2×D412yes—
163C42×C7—28no—
164C42×C7—24no—
165C42×C7—22no—
166C42×C7—22no—
167C42×C7C2×D412yes—
168C42×C7C2×D412yes—
169#1143—28no—
170#1143—48no—
171(C2×D7)⋊C4—42no—
172(C2×D7)⋊C4—44no—
173(C2×D7)⋊C4—82no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—42no—
179C4⋊Dic7—81no—
180Dic7⋊C4—44no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8#4112yes—
190C22×SD16—71no—
191C22×SD16—72no—
192#557—24no—
193#557—24no—
194C2×C4⋊Q8C2×D712yes—
195C2×C4⋊Q8C2×D711yescharacteristic
196C2×#125—21no—
197C2×#125—22no—
198C2×C22⋊Q8—141no—
199C2×C4×Q8—21no—
200C2×C4×Q8C2×D712yes—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×Q8C2×D711yescharacteristic
203C2×C4×D4—71no—
204#461—144no—
205C4×SD16—144no—
206C2×#106—71no—
207C2×C4⋊C8—71no—
208C2×#103—72no—
209C2×#103—72no—
210C2×#103—71no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#102—71no—
214C2×C4×C8—71no—
215#396—142no—
216#390—72no—
217#389—72no—
218#384—71no—
219#380—141no—
220#340—72no—
221#336—72no—
222#336—72no—
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×Q8—28no—
232C7×Q8—22no—
233C7×Q8—22no—
234C7×Q8—24no—
235C7×Q8—24no—
236C7×Q8—42no—
237C7×Q8—42no—
238C7×Q8—41no—
239C7×Q8—41no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—41no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7#12012yes—
257C2×C4×C7#12012yes—
258C2×C4×C7C2×SD1614yes—
259C2×C4×C7C2×SD1612yes—
260C2×C4×C7#13712yes—
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7C2×#4111yescharacteristic
265C2×C4×C7H(5,𝔽2)11yescharacteristic
266C2×C4×C72−1+411yescharacteristic
267C2×Dic7—41no—
268C2×Dic7—42no—
269C2×Dic7—41no—
270C2×Dic7—42no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273D28—22no—
274D28—22no—
275D28—41no—
276D28—41no—
277C4×D7—84no—
278C7⋊C8—22no—
279C7⋊C8—41no—
280C7⋊C8—42no—
281C22×Q8C22⋊D711yescharacteristic
282C22×Q8C22×D712yes—
283C22×Q8C22×D711yescharacteristic
284C22×D4—71no—
285C23×C4—141no—
286C2×SD16—142no—
287C2×SD16—142no—
288C2×SD16—142no—
289C2×SD16—144no—
290C2×SD16—144no—
291C2×SD16—282no—
292C22×C8—71no—
293C22×C8—72no—
294C22×C8—71no—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—22no—
298C4⋊Q8C22⋊D714yes—
299#125—216no—
300#125—28no—
301C22⋊Q8—284no—
302C4×Q8—28no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8C22⋊D714yes—
309C4×D4—142no—
310C4×D4—142no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—72no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4C22⋊D711yescharacteristic
323C2×C4⋊C4C22⋊D712yes—
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C22⋊C4—141no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42C22×D712yes—
331C2×C42C22×D711yescharacteristic
332#106—142no—
333C4⋊C8—142no—
334#103—144no—
335#103—142no—
336#103—144no—
337#103—282no—
338#103—282no—
339#102—142no—
340C4×C8—142no—
341#95—142no—
342C22×C7C2×#12011yescharacteristic
343C22×C7#54811yescharacteristic
344C22×C7#55011yescharacteristic
345C22×C7C22×SD1611yescharacteristic
346C22×C7C2×#13711yescharacteristic
347C22×C7#57711yescharacteristic
348C22×C7#57811yescharacteristic
349C2×D7—42no—
350C2×D7—42no—
351C2×D7—42no—
352C2×D7—81no—
353C2×D7—81no—
354C2×D7—81no—
355C2×D7—81no—
356C4×C7—24no—
357C4×C7—22no—
358C4×C7—42no—
359C4×C7—44no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7#44814yes—
363C4×C7#52511yescharacteristic
364C4×C7#53711yescharacteristic
365C4×C7#54611yescharacteristic
366C4×C7#54711yescharacteristic
367Dic7—82no—
368Dic7—81no—
369C24—141no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8#114314yescomplemented
383C2×Q8C2×C22⋊D714yes—
384C2×Q8C2×C22⋊D712yes—
385C2×D4—72no—
386C2×D4—141no—
387C2×D4—141no—
388C2×D4—141no—
389C2×D4—141no—
390C2×D4—141no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—141no—
395C22×C4—142no—
396C22×C4—282no—
397C22×C4—282no—
398C22×C4—282no—
399C22×C4C2×C22⋊D712yes—
400C22×C4C2×C22⋊D712yes—
401C22×C4C2×C22⋊D711yescharacteristic
402C22×C4C2×C22⋊D711yescharacteristic
403C22×C4C23×D711yescharacteristic
404SD16—148no—
405SD16—288no—
406C2×C8—72no—
407C2×C8—141no—
408C2×C8—141no—
409C2×C8—142no—
410C2×C8—141no—
411C2×C8—142no—
412C2×C8—141no—
413C2×C8—141no—
414C2×C8—142no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—142no—
425C4⋊C4—142no—
426C4⋊C4—141no—
427C4⋊C4—141no—
428C4⋊C4—284no—
429C4⋊C4—282no—
430C4⋊C4—561no—
431C4⋊C4C2×C22⋊D712yes—
432C22⋊C4—284no—
433C22⋊C4—282no—
434C22⋊C4—562no—
435C42—28no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42C2×C22⋊D712yes—
440C42C2×C22⋊D712yes—
441C2×C7#240711yescharacteristic
442C2×C7C2×#44811yescharacteristic
443C2×C7#301311yescharacteristic
444C2×C7#335911yescharacteristic
445C2×C7#337211yescharacteristic
446C2×C7#339211yescharacteristic
447C2×C7#339811yescharacteristic
448D7—82no—
449C23—142no—
450C23—281no—
451C23—281no—
452C23—281no—
453C23—281no—
454C23—281no—
455C23—281no—
456C23C22×C22⋊D711yescharacteristic
457Q8—28no—
458Q8—24no—
459Q8—22no—
460Q8—24no—
461Q8—22no—
462Q8—41no—
463Q8—42no—
464Q8—41no—
465Q8—42no—
466D4—142no—
467D4—142no—
468D4—281no—
469D4—281no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—282no—
487C2×C4—281no—
488C2×C4—281no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—281no—
492C2×C4—564no—
493C2×C4C2×#114314yes—
494C2×C4C2×#114312yes—
495C2×C4#650512yes—
496C2×C4#651612yes—
497C2×C4#651612yes—
498C2×C4D7×#4111yescharacteristic
499C2×C4D7×#4111yescharacteristic
500C2×C4#655311yescharacteristic
501C2×C4#655411yescharacteristic
502C2×C4C22×C22⋊D711yescharacteristic
503C2×C4C22×C22⋊D711yescharacteristic
504C8—142no—
505C8—282no—
506C8—281no—
507C7#2418711yescharacteristic, complemented, Sylow
508C22—282no—
509C22—282no—
510C22—282no—
511C22—561no—
512C22—561no—
513C22—561no—
514C22—561no—
515C22C22×#114311yescharacteristic
516C22C2×#650511yescharacteristic
517C22#9052211yescharacteristic
518C22#9052511yescharacteristic
519C22#9052811yescharacteristic
520C22#9052911yescharacteristic
521C22C2×#651611yescharacteristic
522C4—24no—
523C4—22no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—561no—
529C4—562no—
530C4#8999914yes—
531C4#9051511yescharacteristic
532C4#9051911yescharacteristic
533C4#9052011yescharacteristic
534C4#9052411yescharacteristic
535C2—562no—
536C2#1174113011yescharacteristic
537C2#1174113111yescharacteristic
538C2#1174113311yescharacteristic
539C2#1174113711yescharacteristic
540C2C2×#8999911yescharacteristic
541C2#1174154211yescharacteristic
542C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#557C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterC2×#557D7
hypocenterC7#24187
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
6. #11729539C2semidirect
7. #11729535C2semidirect
8. #11729388C2semidirect
10. #11729383C2semidirect
11. #11729380C2semidirect
107. C2×#557D7semidirect
131. C2×C7×Q8SD16semidirect
382. C2×Q8#1143semidirect
507. C7#24187semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×#898552
#117399452
C2×C4×#11432
#117295452
#117295392
#117295352
#117293882
#117293872
#117293832
#117293802
#241877