Finite Group Databasebeta

← #13387859

Subgroups — #13387859

Order 1792.

1184 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90270C211yesmaximal, characteristic, complemented
2C2×C7×#440C211yesmaximal, characteristic, complemented
3#11739861C211yesmaximal, characteristic
4C2×#89779C211yesmaximal, characteristic, complemented
5#11729808C212yesmaximal, complemented
6#11729381C211yesmaximal, characteristic, complemented
7#11728386C212yesmaximal, complemented
8#11728029C212yesmaximal, complemented
9#11727954C212yesmaximal, complemented
10#11727591C211yesmaximal, characteristic, complemented
11#11727582C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×#6509C2212yes—
16C2×#6501C2212yes—
17#90270—24no—
18#90270—24no—
19C2×#6459—22no—
20C2×D7×C4⋊C4C2212yes—
21C2×C4×D28—21no—
22C2×C4×D28C2211yescharacteristic
23C2×#6433—21no—
24C2×#6432C2211yescharacteristic
25C7×#440—24no—
26C2×C7×#107C2211yescharacteristic
27C2×C7×#103C2211yescharacteristic
28C2×C7×#102C2211yescharacteristic
29C2×C7×#97C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6406C2212yes—
32#89985—22no—
33C2×(C2×D7)⋊C8C2212yes—
34C2×(C2×D7)⋊C8C2212yes—
35#89779—24no—
36#89759C2212yes—
37C2×#6381C2212yes—
38#89732—21no—
39#89725—21no—
40#89708C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#24130—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C22×C22⋊D7—22no—
50C22×D28D411yescharacteristic
51C22×C4×D7—22no—
52C22×C4×D7C2312yes—
53C22×Dic14—21no—
54C2×C7×SD16—22no—
55C2×C7×SD16—22no—
56C2×C7×SD16—22no—
57C22×C7×C8C2311yescharacteristic
58C22×C7×C8C2311yescharacteristic
59C4×C7×Q8—22no—
60C4×C7×Q8D414yescomplemented
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70#6509—24no—
71#6509—24no—
72#6501—24no—
73#6501—42no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—22no—
76C2×(C2×D7)⋊C4D412yes—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7D411yescharacteristic
79C2×C4⋊Dic7C2312yes—
80C2×C4⋊Dic7C2312yes—
81C2×Dic7⋊C4—22no—
82C2×Dic7⋊C4—22no—
83C22×C7⋊C8C2312yes—
84#6459—216no—
85D7×C4⋊C4—24no—
86D7×C4⋊C4—24no—
87C4×D28—28no—
88C4×D28—22no—
89C4×D28—22no—
90#6433—28no—
91#6432—22no—
92#6432D414yescomplemented
93C7×#107—22no—
94C7×#103—22no—
95C7×#102—22no—
96C7×#97—22no—
97#6409—42no—
98#6406—24no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—22no—
102(C2×D7)⋊C8—24no—
103(C2×D7)⋊C8—42no—
104#6381—24no—
105C4⋊(C7⋊C8)—22no—
106C2×#554—71no—
107C2×#474—71no—
108C2×#440D711yescharacteristic, complemented
109#2535—71no—
110#2036—72no—
111#2000—72no—
112#1993—71no—
113#1955—71no—
114#1874—72no—
115#1852—72no—
116#1805—71no—
117C24×C7—21no—
118C23×D7C2×D412yes—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8C2×D412yes—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4#4112yes—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—42no—
135C22×C4×C7—82no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2411yescharacteristic
139C2×C22⋊D7—44no—
140C2×C22⋊D7—44no—
141C2×C22⋊D7—82no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7C2×D412yes—
145C2×D28—24no—
146C2×D28—22no—
147C2×C4×D7—22no—
148C2×C4×D7—22no—
149C2×C4×D7—22no—
150C2×C4×D7—22no—
151C2×C4×D7—22no—
152C2×C4×D7—22no—
153C2×C4×D7—28no—
154C2×C4×D7—82no—
155C2×C4×D7—82no—
156C2×Dic14—24no—
157C2×Dic14—82no—
158C7×SD16—28no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8#4112yes—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4C2×D412yes—
170C7×C22⋊C4—42no—
171C42×C7—24no—
172C42×C7C2×D412yes—
173C42×C7C2×D412yes—
174C22⋊Dic7—44no—
175(C2×D7)⋊C4—28no—
176(C2×D7)⋊C4—24no—
177(C2×D7)⋊C4—82no—
178C4⋊Dic7—24no—
179C4⋊Dic7—22no—
180C4⋊Dic7—22no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7C2×D414yes—
184Dic7⋊C4—28no—
185Dic7⋊C4—28no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C22×SD16C2×D711yescharacteristic
190#554—144no—
191#554—144no—
192C2×C4⋊Q8—71no—
193C2×#125—141no—
194C2×C22⋊Q8—72no—
195C2×C22⋊Q8—142no—
196C2×#121—72no—
197C2×C4×Q8C2×D711yescharacteristic
198C2×C4×D4—71no—
199C2×C4×D4—141no—
200C2×C4×D4C2×D711yescharacteristic
201C22×C4⋊C4—72no—
202#474—144no—
203#440—24no—
204C2×#107—72no—
205C2×#107C2×D711yescharacteristic
206C2×C4⋊C8—71no—
207C2×#103—72no—
208C2×#103C2×D711yescharacteristic
209C2×#102—72no—
210C2×#102C2×D711yescharacteristic
211C2×C22⋊C8—72no—
212C2×C22⋊C8—72no—
213C2×#97C2×D711yescharacteristic
214#397—141no—
215#392—141no—
216#386—142no—
217#382—71no—
218#380—72no—
219#340—72no—
220#340—72no—
221#340—72no—
222C23×C7—22no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7C22×D411yescharacteristic
228C22×D7—22no—
229C22×D7—22no—
230C22×D7—22no—
231C22×D7—22no—
232C22×D7—22no—
233C22×D7—22no—
234C22×D7#13714yes—
235C7×Q8—22no—
236C7×Q8—22no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×D4—22no—
240C7×D4—22no—
241C7×D4—41no—
242C7×D4—41no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—41no—
251C2×C4×C7—42no—
252C2×C4×C7—84no—
253C2×C4×C7#12012yes—
254C2×C4×C7#12012yes—
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
261C22⋊D7—88no—
262C2×Dic7—22no—
263C2×Dic7—24no—
264C2×Dic7—22no—
265C2×Dic7—22no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271D28—28no—
272C4×D7—42no—
273C4×D7—42no—
274C4×D7—42no—
275C4×D7—42no—
276C4×D7—88no—
277Dic14—84no—
278C7×C8—22no—
279C7×C8—41no—
280C7⋊C8—42no—
281C22×Q8—141no—
282C22×Q8C22×D711yescharacteristic
283C22×D4—71no—
284C22×D4—142no—
285C22×D4C22×D711yescharacteristic
286C23×C4—21no—
287C23×C4—72no—
288C23×C4—142no—
289C2×SD16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C22×C8—72no—
293C22×C8C22×D711yescharacteristic
294C22×C8C22×D711yescharacteristic
295C4⋊Q8—74no—
296C4⋊Q8—142no—
297#125—148no—
298C22⋊Q8—1416no—
299C22⋊Q8—144no—
300C22⋊Q8—144no—
301#121—144no—
302#121—282no—
303C4×Q8—22no—
304C4×Q8C22⋊D714yescomplemented
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—148no—
308C4×D4—142no—
309C4×D4—142no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—144no—
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C4⋊C4C22×D711yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—72no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—142no—
326C2×C42—21no—
327C2×C42C22×D711yescharacteristic
328#107—22no—
329#107—144no—
330C4⋊C8—142no—
331#103—22no—
332#103—282no—
333#102—22no—
334#102—144no—
335C22⋊C8—144no—
336C22⋊C8—282no—
337#97—22no—
338#95—142no—
339#95—141no—
340#95—141no—
341C22×C7—42no—
342C22×C7—44no—
343C22×C7—82no—
344C22×C7—81no—
345C22×C7—81no—
346C22×C7C2×#12011yescharacteristic
347C22×C7#54611yescharacteristic
348C22×C7#54711yescharacteristic
349C22×C7C2×#13712yes—
350C22×C7#57612yes—
351C2×D7—24no—
352C2×D7—24no—
353C2×D7—24no—
354C2×D7—42no—
355C2×D7—42no—
356C2×D7—42no—
357C2×D7—42no—
358C4×C7—22no—
359C4×C7—22no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—81no—
363C4×C7#52511yescharacteristic
364C4×C7#53711yescharacteristic
365C4×C7#54811yescharacteristic
366C4×C7#55011yescharacteristic
367Dic7—42no—
368Dic7—44no—
369Dic7—82no—
370C24—21no—
371C24—72no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—144no—
378C2×Q8—562no—
379C2×Q8C2×C22⋊D712yes—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—144no—
386C2×D4—142no—
387C2×D4—284no—
388C2×D4—284no—
389C2×D4—562no—
390C2×D4#115712yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—72no—
396C22×C4—82no—
397C22×C4—148no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4—142no—
406C22×C4—562no—
407C22×C4—562no—
408C22×C4D4×D711yescharacteristic
409C22×C4D4×D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411SD16—28no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—142no—
418C2×C8—142no—
419C2×C8—142no—
420C2×C8#115712yes—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—74no—
426C4⋊C4—148no—
427C4⋊C4—148no—
428C4⋊C4—142no—
429C4⋊C4—144no—
430C4⋊C4—142no—
431C4⋊C4—142no—
432C4⋊C4—142no—
433C4⋊C4C2×C22⋊D712yes—
434C22⋊C4—42no—
435C22⋊C4—148no—
436C22⋊C4—144no—
437C22⋊C4—284no—
438C22⋊C4—562no—
439C42—24no—
440C42D4×D712yes—
441C42C2×C22⋊D712yes—
442C2×C7—82no—
443C2×C7#240711yescharacteristic
444C2×C7#300411yescharacteristic
445C2×C7#300811yescharacteristic
446C2×C7#331412yes—
447C2×C7#334612yes—
448D7—44no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—74no—
455C23—142no—
456C23—142no—
457C23—142no—
458C23—142no—
459C23—142no—
460C23—142no—
461C23C2×D4×D711yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466Q8—564no—
467D4—22no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—148no—
472D4—568no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—84no—
483C2×C4—144no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—282no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—568no—
497C2×C4#650012yes—
498C2×C4#650012yes—
499C2×C4C2×#115711yescharacteristic
500C2×C4C2×#115711yescharacteristic
501C2×C4C2×D4×D711yescharacteristic
502C2×C4#655311yescharacteristic
503C2×C4#655411yescharacteristic
504C2×C4C22×C22⋊D711yescharacteristic
505C8—22no—
506C8—41no—
507C8—282no—
508C7#2413011yescharacteristic, complemented, Sylow
509C22—44no—
510C22—42no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22—144no—
515C22—144no—
516C22—144no—
517C22—282no—
518C22—282no—
519C22—282no—
520C22—282no—
521C22#9024111yescharacteristic
522C22#9024211yescharacteristic
523C22#9046512yes—
524C22D7×#13712yes—
525C22C2×#650011yescharacteristic
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—81no—
531C4—282no—
532C4—284no—
533C4—562no—
534C4#9024311yescharacteristic
535C4#9024411yescharacteristic
536C4#9048211yescharacteristic
537C4#9052411yescharacteristic
538C2—82no—
539C2—284no—
540C2#1173724712yes—
541C2#1173725212yes—
542C2#1173984711yescharacteristic
543C2#1174139211yescharacteristic
544C2#1174142411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#440C2
socleC23×C7C22×D4
prime radical#11739861C2
hypercenterC2×#440D7
hypocenterC7#24130
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×#90270C2semidirect
2. C2×C7×#440C2semidirect
4. C2×#89779C2semidirect
5. #11729808C2semidirect
6. #11729381C2semidirect
7. #11728386C2semidirect
8. #11728029C2semidirect
9. #11727954C2semidirect
10. #11727591C2semidirect
11. #11727582C2semidirect
60. C4×C7×Q8D4semidirect
92. #6432D4semidirect
108. C2×#440D7semidirect
304. C4×Q8C22⋊D7semidirect
508. C7#24130semidirect

Maximal subgroups

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

subgroupindex
C2×#902702
C2×C7×#4402
#117398612
C2×#897792
#117298082
#117293812
#117283862
#117280292
#117279542
#117275912
#117275822
#241307