Finite Group Databasebeta

← #13395371

Subgroups — #13395371

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
1C2×C4×C7×SD16C211yesmaximal, characteristic
2C2×#89934C212yesmaximal
3#11739920C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic7C211yesmaximal, characteristic
5C2×#89791C211yesmaximal, characteristic
6#11729380C211yesmaximal, characteristic, complemented
7#11728343C212yesmaximal, complemented
8#11728000C212yesmaximal, complemented
9#11727925C212yesmaximal
10#11727540C211yesmaximal, characteristic, complemented
11#11727525C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×#6507C2212yes—
16C2×D4×Dic7C2212yes—
17C22×#1142C2212yes—
18C22×C4⋊Dic7—21no—
19C2×#6432C2211yescharacteristic
20C4×C7×SD16—24no—
21C2×C7×#106C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C2×#6409C2212yes—
26#89984—21no—
27#89934—28no—
28C2×#6390C2212yes—
29C2×#6390C2212yes—
30C2×Dic7⋊C8C2212yes—
31#89791—24no—
32#89757C2212yes—
33#89755—22no—
34#89746C2212yes—
35C2×#6384C2212yes—
36#89722—22no—
37#89707C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39#89700—21no—
40#89700—21no—
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89345C2212yes—
43#89324C2212yes—
44#24730—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×Dic7—22no—
49C22×Dic14C2312yes—
50C2×C7×SD16—22no—
51C2×C7×SD16—22no—
52C2×C7×SD16D414yes—
53C22×C7×C8C2311yescharacteristic
54C22×C7×C8C2311yescharacteristic
55C4×C7×Q8—22no—
56C4×C7×Q8—22no—
57C4×C7×D4—22no—
58C4×C7×D4D414yes—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65C2×C22⋊Dic7—22no—
66#6507—24no—
67#6507—42no—
68D4×Dic7—24no—
69D4×Dic7—42no—
70C2×#1142—24no—
71C2×#1142—24no—
72C2×#1142—24no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—44no—
79C2×C4⋊Dic7D411yescharacteristic
80C2×C4⋊Dic7D411yescharacteristic
81C2×C4⋊Dic7C2312yes—
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—22no—
84C2×C4×Dic7C2312yes—
85C22×C7⋊C8C2312yes—
86#6432—22no—
87#6432—22no—
88C7×#106—22no—
89C7×#103—22no—
90C7×#102—22no—
91C4×C7×C8—22no—
92#6409—24no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95D412yes—
97#6390—24no—
98#6390—42no—
99Dic7⋊C8—24no—
100#6384—24no—
101C4⋊(C7⋊C8)—22no—
102C2×#461—72no—
103C2×#461—71no—
104C2×C4×SD16D711yescharacteristic
105#2546—71no—
106D4×C4⋊C4—71no—
107#2066—72no—
108#2043—72no—
109#1956—71no—
110#1939—71no—
111#1877—72no—
112#1803—71no—
113C24×C7—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8—21no—
116C2×C7×Q8—21no—
117C2×C7×Q8—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8C2×D412yes—
120C2×C7×D4—21no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4C2×D412yes—
124C2×C7×D4C2×D412yes—
125C2×C7×D4#4112yes—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—24no—
129C22×C4×C7—21no—
130C22×C4×C7—42no—
131C22×C4×C7—42no—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2411yescharacteristic
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—44no—
138C22×Dic7—44no—
139C22×Dic7—82no—
140C22×Dic7C2×D412yes—
141C22×Dic7C2×D412yes—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C2×Dic14—22no—
145C2×Dic14—22no—
146C2×Dic14—22no—
147C2×Dic14—22no—
148C7×SD16—28no—
149C2×C7×C8—21no—
150C2×C7×C8—21no—
151C2×C7×C8—21no—
152C2×C7×C8—21no—
153C2×C7×C8—21no—
154C2×C7×C8C2×D412yes—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—42no—
159C7×C4⋊C4C2×D412yes—
160C7×C22⋊C4—24no—
161C42×C7—42no—
162C42×C7C2×D412yes—
163C42×C7#4112yes—
164C22⋊Dic7—44no—
165#1142—48no—
166C4⋊Dic7—22no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7—42no—
170C4⋊Dic7—42no—
171C4⋊Dic7—42no—
172C4⋊Dic7—44no—
173C4⋊Dic7—81no—
174C4⋊Dic7SD1614yescomplemented
175Dic7⋊C4—44no—
176Dic7⋊C4—44no—
177C4×Dic7—22no—
178C4×Dic7—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C22×SD16—72no—
183C22×SD16C2×D711yescharacteristic
184C2×C4⋊Q8—72no—
185C2×C4⋊Q8—71no—
186C2×C4×Q8C2×D711yescharacteristic
187C2×C4×D4—72no—
188C2×C4×D4C2×D711yescharacteristic
189C22×C4⋊C4—141no—
190#461—148no—
191#461—144no—
192C4×SD16—24no—
193C2×#106C2×D711yescharacteristic
194C2×C4⋊C8—72no—
195C2×C4⋊C8—71no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103—72no—
199C2×#103—72no—
200C2×#103C2×D711yescharacteristic
201C2×#102C2×D711yescharacteristic
202C2×C4×C8C2×D711yescharacteristic
203#398—142no—
204#391—72no—
205#389—72no—
206#389—141no—
207#389—141no—
208#385—142no—
209#384—71no—
210#380—141no—
211C4×C4⋊C4—71no—
212#340—72no—
213#336—72no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7C22×D411yescharacteristic
222C7×Q8—22no—
223C7×Q8—22no—
224C7×Q8—41no—
225C7×Q8—41no—
226C7×D4—22no—
227C7×D4—22no—
228C7×D4—22no—
229C7×D4#12114yes—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—41no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—48no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7#12112yes—
243C2×C4×C7#12112yes—
244C2×C4×C7C2×SD1612yes—
245C2×C4×C7#13712yes—
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C7C2×#4111yescharacteristic
249C2×C4×C7C2×#4111yescharacteristic
250C2×C4×C7H(5,𝔽2)11yescharacteristic
251C2×C4×C72−1+411yescharacteristic
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—22no—
256C2×Dic7—22no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—88no—
264C2×Dic7C2×SD1614yes—
265Dic14—42no—
266Dic14—42no—
267Dic14—42no—
268Dic14—42no—
269C7×C8—22no—
270C7×C8—41no—
271C7⋊C8—42no—
272C22×Q8—72no—
273C22×Q8C22×D711yescharacteristic
274C22×D4C22×D711yescharacteristic
275C23×C4—21no—
276C23×C4—142no—
277C2×SD16—22no—
278C2×SD16—22no—
279C2×SD16—144no—
280C2×SD16—144no—
281C2×SD16—144no—
282C2×SD16D2814yes—
283C22×C8—72no—
284C22×C8C22×D711yescharacteristic
285C22×C8C22×D711yescharacteristic
286C4⋊Q8—144no—
287C4⋊Q8—142no—
288C4⋊Q8—142no—
289C4⋊Q8—282no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×D4—22no—
293C4×D4—144no—
294C4×D4—282no—
295C4×D4C22⋊D714yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—71no—
298C2×C4⋊C4—71no—
299C2×C4⋊C4—72no—
300C2×C4⋊C4—142no—
301C2×C4⋊C4—142no—
302C2×C4⋊C4—142no—
303C2×C4⋊C4—141no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—284no—
308C2×C4⋊C4C22×D711yescharacteristic
309C2×C4⋊C4C22×D711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—142no—
312C2×C42—21no—
313C2×C42—72no—
314C2×C42C22×D711yescharacteristic
315#106—22no—
316C4⋊C8—144no—
317C4⋊C8—142no—
318#103—22no—
319#103—144no—
320#103—144no—
321#103—144no—
322#103—282no—
323#102—22no—
324C4×C8—22no—
325#95—72no—
326#95—142no—
327#95—142no—
328#95—142no—
329C22×C7—24no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7—42no—
336C22×C7C2×#12111yescharacteristic
337C22×C7D4211yescharacteristic
338C22×C7#54811yescharacteristic
339C22×C7C22×SD1611yescharacteristic
340C22×C7C2×#13711yescharacteristic
341C22×C7#57711yescharacteristic
342C22×C7#57811yescharacteristic
343C4×C7—22no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—81no—
347C4×C7#46114yescomplemented
348C4×C7C2×#12111yescharacteristic
349C4×C7#53711yescharacteristic
350C4×C7#54711yescharacteristic
351C4×C7#55311yescharacteristic
352Dic7—24no—
353Dic7—42no—
354Dic7—82no—
355Dic7—82no—
356C24—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—142no—
363C2×Q8—142no—
364C2×Q8—142no—
365C2×Q8—142no—
366C2×Q8—142no—
367C2×Q8—142no—
368C2×Q8C2×D2812yes—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4C2×D2812yes—
373C2×D4#115712yes—
374C2×D4C2×C22⋊D712yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—72no—
382C22×C4—72no—
383C22×C4—142no—
384C22×C4—142no—
385C22×C4—284no—
386C22×C4—284no—
387C22×C4—562no—
388C22×C4D4×D711yescharacteristic
389C22×C4D4×D711yescharacteristic
390C22×C4C23×D711yescharacteristic
391SD16—28no—
392SD16—288no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—142no—
399C2×C8—142no—
400C2×C8—142no—
401C2×C8C2×D2812yes—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—74no—
407C4⋊C4—142no—
408C4⋊C4—142no—
409C4⋊C4—142no—
410C4⋊C4—284no—
411C4⋊C4—284no—
412C4⋊C4—284no—
413C4⋊C4—282no—
414C4⋊C4—282no—
415C4⋊C4—282no—
416C4⋊C4—561no—
417C4⋊C4C2×C22⋊D712yes—
418C22⋊C4—24no—
419C22⋊C4—284no—
420C42—42no—
421C42—142no—
422C42—142no—
423C42#115912yes—
424C42C2×C22⋊D712yes—
425C2×C7—44no—
426C2×C7#241611yescharacteristic
427C2×C7C2×#46111yescharacteristic
428C2×C7#304611yescharacteristic
429C2×C7#327611yescharacteristic
430C2×C7D4×SD1611yescharacteristic
431C2×C7#335911yescharacteristic
432C2×C7#337211yescharacteristic
433C23—22no—
434C23—22no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C2×D4×D711yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—282no—
446Q8—282no—
447Q8—282no—
448Q8—282no—
449D4—22no—
450D4—22no—
451D4—22no—
452D4#649314yes—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—48no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—74no—
466C2×C4—142no—
467C2×C4—142no—
468C2×C4—142no—
469C2×C4—142no—
470C2×C4—142no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4—282no—
477C2×C4—568no—
478C2×C4D7×SD1612yes—
479C2×C4#647812yes—
480C2×C4#650212yes—
481C2×C4#650212yes—
482C2×C4C22×D2811yescharacteristic
483C2×C4C2×#115711yescharacteristic
484C2×C4C2×#115911yescharacteristic
485C2×C4#655311yescharacteristic
486C2×C4#655411yescharacteristic
487C2×C4C22×C22⋊D711yescharacteristic
488C8—22no—
489C8—41no—
490C8—282no—
491C7#2473011yescharacteristic, complemented, Sylow
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22D4×D2811yescharacteristic
500C22#9024311yescharacteristic
501C22C2×D7×SD1611yescharacteristic
502C22C2×#647811yescharacteristic
503C22#9046111yescharacteristic
504C22#9046211yescharacteristic
505C22C2×#650211yescharacteristic
506C4—22no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—144no—
511C4—282no—
512C4—562no—
513C4—562no—
514C4#8993414yescomplemented
515C4#9024611yescharacteristic
516C4#9025611yescharacteristic
517C4C2×#649311yescharacteristic
518C4#9052411yescharacteristic
519C2—44no—
520C2#1173706411yescharacteristic
521C2#1173707311yescharacteristic
522C2#1173711711yescharacteristic
523C2SD16×D2811yescharacteristic
524C2#1173979711yescharacteristic
525C2C2×#8993411yescharacteristic
526C2#1174134711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×C4×SD16D7
hypocenterC7#24730
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #11739920C2semidirect
6. #11729380C2semidirect
7. #11728343C2semidirect
8. #11728000C2semidirect
10. #11727540C2semidirect
11. #11727525C2semidirect
174. C4⋊Dic7SD16semidirect
347. C4×C7#461semidirect
491. C7#24730semidirect
514. C4#89934semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×SD162
C2×#899342
#117399202
D4×C4⋊Dic72
C2×#897912
#117293802
#117283432
#117280002
#117279252
#117275402
#117275252
#247307