Finite Group Databasebeta

← #13395367

Subgroups — #13395367

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2546C211yesmaximal, characteristic, complemented
2C2×#89855C212yesmaximal
3C2×#89855C211yesmaximal, characteristic
4C2×C4×#1143C211yesmaximal, characteristic
5C4⋊C4×D28C211yesmaximal, characteristic
6#11729539C211yesmaximal, characteristic, complemented
7#11729535C212yesmaximal, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11727441C212yesmaximal
10#11727436C212yesmaximal, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×C4⋊Q8C2212yes—
14C2×C4×C7×Q8C2211yescharacteristic
15C22×#1143C2212yes—
16C22×#1143C2211yescharacteristic
17C2×D7×C4⋊C4—21no—
18C2×C4×D28C2212yes—
19C2×C4×D28C2211yescharacteristic
20C7×#398—22no—
21C7×#391C2212yes—
22C7×#389—21no—
23C7×#389—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2211yescharacteristic
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30#89855—28no—
31#89855—24no—
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#24730—71nomaximal, Sylow
45C22×C7×Q8C2312yes—
46C22×C7×Q8C2311yescharacteristic
47C22×D28C2311yescharacteristic
48C22×C4×D7—21no—
49C22×C4×D7—22no—
50C7×C4⋊Q8—24no—
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8—42no—
54C4×C7×Q8—22no—
55C4×C7×Q8—22no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C42×C7—21no—
65C2×C42×C7C2312yes—
66C2×C42×C7C2311yescharacteristic
67C2×#1143—22no—
68C2×#1143—22no—
69C2×#1143—24no—
70C2×#1143—24no—
71C2×#1143—24no—
72C2×#1143D414yes—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—22no—
80C22×C7⋊C8C2311yescharacteristic
81C22×C7⋊C8C2312yes—
82C22×C7⋊C8C2311yescharacteristic
83D7×C4⋊C4—44no—
84C4×D28—22no—
85C4×D28—24no—
86C4×D28—42no—
87C4×D28D414yes—
88C7×#95—22no—
89C7×#95—22no—
90C7×#95D412yes—
91#6409—24no—
92#6409—22no—
93#6409—24no—
94#6409—24no—
95#6409—42no—
96C7⋊#95—22no—
97#6383—22no—
98#6382—22no—
99C4⋊(C7⋊C8)—22no—
100C4⋊(C7⋊C8)—24no—
101C4×C7⋊C8—22no—
102C2×#461—71no—
103C2×#461—72no—
104C2×C4×SD16—71no—
105#2546D711yescharacteristic, complemented
106D4×C4⋊C4—71no—
107#2066—72no—
108#2043—72no—
109#1956—71no—
110#1939—71no—
111#1877—72no—
112#1803—71no—
113C23×D7—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8—21no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8C2×D412yes—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D412yes—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—22no—
136C2×D28—21no—
137C2×D28—21no—
138C2×D28—21no—
139C2×D28C2×D412yes—
140C2×D28C2×D412yes—
141C2×D28#4112yes—
142C2×C4×D7—24no—
143C2×C4×D7—42no—
144C2×C4×D7—42no—
145C2×C4×D7—44no—
146C2×C4×D7—44no—
147C2×C4×D7—82no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—44no—
152C7×C4⋊C4—44no—
153C7×C4⋊C4—42no—
154C7×C4⋊C4—42no—
155C7×C4⋊C4—42no—
156C7×C4⋊C4SD1614yescomplemented
157C42×C7—22no—
158C42×C7—22no—
159C42×C7—42no—
160C42×C7C2×D412yes—
161C42×C7#4112yes—
162#1143—28no—
163#1143—48no—
164(C2×D7)⋊C4—24no—
165(C2×D7)⋊C4—44no—
166C4⋊Dic7—21no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7—42no—
170C4⋊Dic7—81no—
171C4⋊Dic7C2×D412yes—
172Dic7⋊C4—44no—
173C2×C7⋊C8—21no—
174C2×C7⋊C8—21no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8C2×D412yes—
182C22×SD16—71no—
183C22×SD16—72no—
184C2×C4⋊Q8C2×D711yescharacteristic
185C2×C4⋊Q8C2×D712yes—
186C2×C4×Q8C2×D711yescharacteristic
187C2×C4×D4—71no—
188C2×C4×D4—72no—
189C22×C4⋊C4—141no—
190#461—144no—
191#461—148no—
192C4×SD16—144no—
193C2×#106—71no—
194C2×C4⋊C8—71no—
195C2×C4⋊C8—72no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103—72no—
199C2×#103—71no—
200C2×#103—72no—
201C2×#102—71no—
202C2×C4×C8—71no—
203#398—22no—
204#391C2×D712yes—
205#389—21no—
206#389—21no—
207#389—72no—
208#385—142no—
209#384—71no—
210#380—141no—
211C4×C4⋊C4C2×D711yescharacteristic
212#340—72no—
213#336—72no—
214C23×C7C22×D411yescharacteristic
215C22×D7—22no—
216C22×D7—22no—
217C22×D7—22no—
218C22×D7—41no—
219C22×D7—41no—
220C22×D7—41no—
221C22×D7—41no—
222C7×Q8—22no—
223C7×Q8—22no—
224C7×Q8—41no—
225C7×Q8—41no—
226C7×Q8—42no—
227C7×Q8—42no—
228C7×Q8—42no—
229C7×Q8—42no—
230C2×C4×C7—22no—
231C2×C4×C7—21no—
232C2×C4×C7—22no—
233C2×C4×C7—21no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7#12112yes—
247C2×C4×C7#12112yes—
248C2×C4×C7C2×SD1614yes—
249C2×C4×C7C2×SD1612yes—
250C2×C4×C7#13712yes—
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C7C2×#4111yescharacteristic
255C2×C4×C7H(5,𝔽2)11yescharacteristic
256C2×C4×C72−1+411yescharacteristic
257C2×Dic7—22no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263D28—22no—
264D28—22no—
265D28—22no—
266D28#12114yes—
267C4×D7—48no—
268C4×D7—88no—
269C7⋊C8—22no—
270C7⋊C8—42no—
271C7⋊C8—41no—
272C22×Q8C22×D712yes—
273C22×Q8C22×D711yescharacteristic
274C22×D4—71no—
275C23×C4—141no—
276C23×C4—142no—
277C2×SD16—74no—
278C2×SD16—142no—
279C2×SD16—142no—
280C2×SD16—144no—
281C2×SD16—144no—
282C2×SD16—144no—
283C22×C8—71no—
284C22×C8—72no—
285C22×C8—71no—
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—42no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×D4—74no—
293C4×D4—144no—
294C4×D4—142no—
295C4×D4—282no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—71no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—142no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—284no—
307C2×C4⋊C4C22⋊D711yescharacteristic
308C2×C4⋊C4C22⋊D711yescharacteristic
309C2×C4⋊C4C22×D711yescharacteristic
310C2×C22⋊C4—141no—
311C2×C22⋊C4—142no—
312C2×C42—21no—
313C2×C42C22×D712yes—
314C2×C42C22×D711yescharacteristic
315#106—142no—
316C4⋊C8—144no—
317C4⋊C8—142no—
318#103—142no—
319#103—144no—
320#103—144no—
321#103—144no—
322#103—282no—
323#102—142no—
324C4×C8—142no—
325#95—22no—
326#95—22no—
327#95—142no—
328#95C22⋊D712yes—
329C22×C7C2×#12111yescharacteristic
330C22×C7D4211yescharacteristic
331C22×C7#54811yescharacteristic
332C22×C7C22×SD1611yescharacteristic
333C22×C7C2×#13711yescharacteristic
334C22×C7#57711yescharacteristic
335C22×C7#57811yescharacteristic
336C2×D7—24no—
337C2×D7—42no—
338C2×D7—42no—
339C2×D7—42no—
340C2×D7—42no—
341C2×D7—42no—
342C2×D7—42no—
343C4×C7—24no—
344C4×C7—22no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—82no—
348C4×C7—81no—
349C4×C7#46114yescomplemented
350C4×C7C2×#12111yescharacteristic
351C4×C7#53711yescharacteristic
352C4×C7#54711yescharacteristic
353C4×C7#55311yescharacteristic
354Dic7—42no—
355Dic7—82no—
356C24—141no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8D4×D712yes—
369C2×D4—72no—
370C2×D4—72no—
371C2×D4—72no—
372C2×D4—141no—
373C2×D4—141no—
374C2×D4—141no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—141no—
379C22×C4—144no—
380C22×C4—142no—
381C22×C4—284no—
382C22×C4—282no—
383C22×C4—282no—
384C22×C4—284no—
385C22×C4—562no—
386C22×C4C2×C22⋊D712yes—
387C22×C4C2×C22⋊D711yescharacteristic
388C22×C4C2×C22⋊D712yes—
389C22×C4C2×C22⋊D711yescharacteristic
390C22×C4C23×D711yescharacteristic
391SD16—148no—
392SD16—288no—
393C2×C8—72no—
394C2×C8—141no—
395C2×C8—141no—
396C2×C8—141no—
397C2×C8—142no—
398C2×C8—142no—
399C2×C8—141no—
400C2×C8—142no—
401C2×C8—141no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—72no—
411C4⋊C4—141no—
412C4⋊C4—142no—
413C4⋊C4—142no—
414C4⋊C4—284no—
415C4⋊C4—282no—
416C4⋊C4—561no—
417C4⋊C4#114314yescomplemented
418C22⋊C4—144no—
419C22⋊C4—284no—
420C42—22no—
421C42—22no—
422C42—42no—
423C42D4×D712yes—
424C42#115912yes—
425C2×C7#241611yescharacteristic
426C2×C7C2×#46111yescharacteristic
427C2×C7#304611yescharacteristic
428C2×C7#327611yescharacteristic
429C2×C7D4×SD1611yescharacteristic
430C2×C7#335911yescharacteristic
431C2×C7#337211yescharacteristic
432D7—44no—
433C23—142no—
434C23—142no—
435C23—142no—
436C23—281no—
437C23—281no—
438C23—281no—
439C23—281no—
440C23C22×C22⋊D711yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449D4—74no—
450D4—142no—
451D4—142no—
452D4—142no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—142no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—288no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4—568no—
477C2×C4#650112yes—
478C2×C4#650112yes—
479C2×C4C2×#114314yes—
480C2×C4C2×#114312yes—
481C2×C4#650512yes—
482C2×C4C2×D4×D711yescharacteristic
483C2×C4C2×D4×D711yescharacteristic
484C2×C4C2×#115911yescharacteristic
485C2×C4#654811yescharacteristic
486C2×C4D7×#4111yescharacteristic
487C2×C4#655411yescharacteristic
488C8—142no—
489C8—282no—
490C8—281no—
491C7#2473011yescharacteristic, complemented, Sylow
492C22—144no—
493C22—282no—
494C22—282no—
495C22—282no—
496C22—282no—
497C22—282no—
498C22—282no—
499C22C2×#650111yescharacteristic
500C22D4×C22⋊D711yescharacteristic
501C22C22×#114311yescharacteristic
502C22C2×#650511yescharacteristic
503C22#9052211yescharacteristic
504C22#9052811yescharacteristic
505C22#9052911yescharacteristic
506C4—24no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—81no—
511C4—82no—
512C4—282no—
513C4—562no—
514C4#8985514yescomplemented
515C4D7×#12111yescharacteristic
516C4#9029911yescharacteristic
517C4#9030211yescharacteristic
518C4#9030511yescharacteristic
519C2—284no—
520C2#1174008611yescharacteristic
521C2C2×#8985511yescharacteristic
522C2#1174085111yescharacteristic
523C2#1174100511yescharacteristic
524C2D4×#114311yescharacteristic
525C2#1174113311yescharacteristic
526C2#1174113711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2546C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2546D7
hypocenterC7#24730
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2546

Complements

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

NH ≅ G/Ntype
1. C7×#2546C2semidirect
6. #11729539C2semidirect
7. #11729535C2semidirect
8. #11729380C2semidirect
10. #11727436C2semidirect
11. #11727238C2semidirect
105. #2546D7semidirect
156. C7×C4⋊C4SD16semidirect
349. C4×C7#461semidirect
417. C4⋊C4#1143semidirect
491. C7#24730semidirect
514. C4#89855semidirect

Maximal subgroups

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

subgroupindex
C7×#25462
C2×#898552
C2×C4×#11432
C4⋊C4×D282
#117295392
#117295352
#117293802
#117274412
#117274362
#117272382
#247307