Finite Group Databasebeta

← #13363488

Subgroups — #13363488

Order 1792.

1166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90382C211yesmaximal, characteristic
2C2×#89457C211yesmaximal, characteristic, complemented
3D7×#382C211yesmaximal, characteristic, complemented
4C7×#1991C211yesmaximal, characteristic, complemented
5#11729494C211yesmaximal, characteristic, complemented
6#11727527C211yesmaximal, characteristic, complemented
7#11727305C211yesmaximal, characteristic, complemented
8#11726659C212yesmaximal, complemented
9#11726647C212yesmaximal, complemented
10#11726634C212yesmaximal
11#11726632C212yesmaximal, complemented
12C2×C7×#125C2211yescharacteristic
13#90382—24no—
14#90382C418yescomplemented
15C2×#6461—21no—
16C2×#6460C2211yescharacteristic
17C2×#6458C2211yescharacteristic
18C2×#6457—21no—
19C2×#6456—21no—
20C2×#6456C2211yescharacteristic
21C2×#6455—21no—
22C2×D7×C4⋊C4C2212yes—
23C2×#6450C2211yescharacteristic
24C2×#6438C2211yescharacteristic
25C2×C42×D7C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27C7×#382C2211yescharacteristic
28C2×#6395C2212yes—
29C2×(C2×D7)⋊C8C2212yes—
30C2×#6390C2212yes—
31C2×Dic7⋊C8C2212yes—
32#89749C2212yes—
33C2×#6384C2212yes—
34C2×#6383C2212yes—
35#89708C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89457—24no—
38D7×#95—21no—
39D7×#95—21no—
40#89422—21no—
41#89422—21no—
42C7×#340C2212yes—
43C7×#340C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21861—71nomaximal, Sylow
47C22×D28—21no—
48C22×D28C2311yescharacteristic
49C22×C4×D7—21no—
50C22×C4×D7—22no—
51C22×C4×D7D411yescharacteristic
52C22×C4×D7D411yescharacteristic
53C22×C4×D7C2311yescharacteristic
54C22×Dic14C2311yescharacteristic
55C22×C7×C8C2312yes—
56C7×#125—22no—
57C7×#125C2×C414yescomplemented
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C42×C7C2311yescharacteristic
64C2×(C2×D7)⋊C4—21no—
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—21no—
68C2×(C2×D7)⋊C4—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7C2312yes—
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—22no—
73C2×C4×Dic7—21no—
74C2×C4×Dic7D411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7C2311yescharacteristic
77C22×C7⋊C8C2312yes—
78#6461—28no—
79#6460—22no—
80#6460C2×C414yes—
81#6458—22no—
82#6458C2×C414yes—
83#6457—28no—
84#6456—28no—
85#6456—22no—
86#6456C2×C414yescomplemented
87#6455—28no—
88D7×C4⋊C4—42no—
89D7×C4⋊C4—42no—
90#6450—22no—
91#6450C2×C414yescomplemented
92#6438—22no—
93#6438C2×C414yescomplemented
94C42×D7C2×C414yes—
95C42×D7D414yes—
96C4×C7×C8—22no—
97C7×#95—21no—
98C7×#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—22no—
102C7⋊#95—22no—
103#6395—42no—
104(C2×D7)⋊C8—24no—
105#6390—42no—
106Dic7⋊C8—24no—
107#6384—42no—
108#6383—42no—
109C4×C7⋊C8—22no—
110C2×#529—71no—
111C2×#431—71no—
112C2×#382—71no—
113#1991—71no—
114#1991D711yescharacteristic, complemented
115#1958—71no—
116#1958—71no—
117#1874—72no—
118#1872—72no—
119#1805—72no—
120#1804—72no—
121C23×D7—21no—
122C23×D7C2×D411yescharacteristic
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2411yescharacteristic
129C22×Dic7—21no—
130C22×Dic7—22no—
131C22×Dic7C2×D411yescharacteristic
132C22×Dic7C2×D411yescharacteristic
133C22×Dic7C2×D411yescharacteristic
134C2×D28—22no—
135C2×D28—22no—
136C2×D28—24no—
137C2×D28—21no—
138C2×D28—44no—
139C2×D28C22×C412yes—
140C2×C4×D7—24no—
141C2×C4×D7—22no—
142C2×C4×D7—22no—
143C2×C4×D7—24no—
144C2×C4×D7—42no—
145C2×C4×D7—44no—
146C2×C4×D7—44no—
147C2×C4×D7—44no—
148C2×C4×D7—44no—
149C2×C4×D7C22⋊C414yes—
150C2×C4×D7C22⋊C414yes—
151C2×C4×D7C22×C412yes—
152C2×C4×D7C2×D412yes—
153C2×Dic14—22no—
154C2×Dic14—22no—
155C2×Dic14—21no—
156C2×Dic14C22×C412yes—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4C22×C412yes—
167C7×C4⋊C4C22×C412yes—
168C42×C7C22×C412yes—
169C42×C7C2×D412yes—
170(C2×D7)⋊C4—24no—
171(C2×D7)⋊C4—24no—
172(C2×D7)⋊C4—24no—
173(C2×D7)⋊C4—24no—
174(C2×D7)⋊C4—24no—
175C4⋊Dic7—24no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178Dic7⋊C4—24no—
179Dic7⋊C4—44no—
180C4×Dic7—24no—
181C4×Dic7C22⋊C414yes—
182C4×Dic7C22⋊C414yes—
183C4×Dic7C22×C412yes—
184C4×Dic7C2×D412yes—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188#529—78no—
189#529—144no—
190C2×#126—141no—
191C2×#125C2×D711yescharacteristic
192C2×#124—71no—
193C2×#123—141no—
194C2×C22⋊Q8—71no—
195C2×#121—71no—
196C2×C4×Q8—71no—
197C2×C4×D4—71no—
198C2×C4×D4—141no—
199C2×#117—141no—
200C22×C4⋊C4—72no—
201C22×C42—71no—
202#431—144no—
203C2×C4⋊C8—72no—
204C2×#103—72no—
205C2×#103—72no—
206C2×#102—72no—
207C2×#102—72no—
208C2×C22⋊C8—72no—
209C2×C4×C8—71no—
210C2×C4×C8C2×D711yescharacteristic
211#382—72no—
212#382—71no—
213#382—141no—
214#382—141no—
215#382C2×D711yescharacteristic
216C2×#95—141no—
217C2×#95—141no—
218#340—72no—
219#340—72no—
220#340C2×D712yes—
221#340C2×D712yes—
222C23×C7C22×D411yescharacteristic
223C22×D7—21no—
224C22×D7—21no—
225C22×D7—22no—
226C22×D7—22no—
227C22×D7—41no—
228C22×D7—41no—
229C22×D7—42no—
230C22×D7—42no—
231C22×D7C2×C22⋊C412yes—
232C22×D7#13514yes—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7C2×C22⋊C412yes—
243C2×C4×C7C2×C22⋊C412yes—
244C2×C4×C7C23×C411yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C2×C4×C7H(5,𝔽2)12yes—
247C2×C4×C72−1+412yes—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7C2×C22⋊C412yes—
259C2×Dic7C2×C22⋊C412yes—
260C2×Dic7C2×C22⋊C412yes—
261D28—24no—
262D28—48no—
263D28—42no—
264C4×D7—28no—
265C4×D7—24no—
266C4×D7—24no—
267C4×D7—44no—
268C4×D7—48no—
269C4×D7—88no—
270Dic14—24no—
271Dic14—42no—
272C7×C8—42no—
273C7⋊C8—42no—
274C22×Q8—71no—
275C22×D4—71no—
276C22×D4—141no—
277C23×C4—71no—
278C23×C4—71no—
279C23×C4—71no—
280C23×C4—141no—
281C23×C4—142no—
282C22×C8—72no—
283C22×C8C22×D712yes—
284#126—148no—
285#125—22no—
286#125C4×D714yescomplemented
287#124—74no—
288#124—142no—
289#123—148no—
290C22⋊Q8—74no—
291C22⋊Q8—142no—
292#121—74no—
293#121—142no—
294C4×Q8—74no—
295C4×Q8—142no—
296C4×D4—74no—
297C4×D4—142no—
298C4×D4—148no—
299#117—148no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—141no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—142no—
306C2×C4⋊C4—282no—
307C2×C4⋊C4—282no—
308C2×C4⋊C4C22×D712yes—
309C2×C4⋊C4C22×D711yescharacteristic
310C2×C4⋊C4C22×D711yescharacteristic
311C2×C22⋊C4—141no—
312C2×C22⋊C4—141no—
313C2×C22⋊C4—141no—
314C2×C22⋊C4—141no—
315C2×C22⋊C4—141no—
316C2×C42—71no—
317C2×C42—71no—
318C2×C42—71no—
319C2×C42—74no—
320C2×C42—74no—
321C2×C42—141no—
322C2×C42C22×D711yescharacteristic
323C4⋊C8—144no—
324#103—282no—
325#103—282no—
326#102—282no—
327#102—282no—
328C22⋊C8—144no—
329C4×C8—22no—
330C4×C8—142no—
331#95—21no—
332#95—21no—
333#95—142no—
334#95—142no—
335#95—141no—
336#95—141no—
337C22×C7C22×C22⋊C411yescharacteristic
338C22×C7#53511yescharacteristic
339C22×C7#53611yescharacteristic
340C22×C7C2×#13512yes—
341C22×C7#57512yes—
342C2×D7—22no—
343C2×D7—22no—
344C2×D7—24no—
345C2×D7—24no—
346C2×D7—42no—
347C2×D7—42no—
348C2×D7—42no—
349C2×D7—82no—
350C2×D7—82no—
351C2×D7#41614yes—
352C4×C7—22no—
353C4×C7—22no—
354C4×C7—42no—
355C4×C7—42no—
356C4×C7—82no—
357C4×C7#52012yes—
358C4×C7#53712yes—
359Dic7—22no—
360Dic7—22no—
361Dic7—22no—
362Dic7—42no—
363Dic7—82no—
364C24—71no—
365C24—141no—
366C2×Q8—72no—
367C2×Q8—141no—
368C2×Q8—142no—
369C2×Q8—142no—
370C2×D4—72no—
371C2×D4—144no—
372C2×D4—141no—
373C2×D4—142no—
374C2×D4—142no—
375C2×D4—284no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—71no—
382C22×C4—74no—
383C22×C4—72no—
384C22×C4—72no—
385C22×C4—74no—
386C22×C4—144no—
387C22×C4—144no—
388C22×C4—142no—
389C22×C4—142no—
390C22×C4—141no—
391C22×C4—142no—
392C22×C4—282no—
393C22×C4—284no—
394C22×C4—284no—
395C22×C4—284no—
396C22×C4—284no—
397C22×C4D4×D711yescharacteristic
398C22×C4D4×D711yescharacteristic
399C22×C4C23×D711yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—142no—
404C2×C8—142no—
405C2×C8—142no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—144no—
413C4⋊C4—144no—
414C4⋊C4—142no—
415C4⋊C4—142no—
416C4⋊C4—284no—
417C4⋊C4C2×C4×D712yes—
418C4⋊C4C2×C4×D712yes—
419C22⋊C4—144no—
420C22⋊C4—144no—
421C22⋊C4—144no—
422C22⋊C4—144no—
423C22⋊C4—144no—
424C42—74no—
425C42—74no—
426C42—72no—
427C42—72no—
428C42—144no—
429C42C2×C4×D712yes—
430C42D4×D712yes—
431C2×C7#231111yescharacteristic
432C2×C7C2×#41611yescharacteristic
433C2×C7#289611yescharacteristic
434C2×C7#321112yes—
435C2×C7#323212yes—
436D7—24no—
437D7—82no—
438C23—72no—
439C23—74no—
440C23—141no—
441C23—142no—
442C23—141no—
443C23—142no—
444C23—281no—
445C23—282no—
446C23—282no—
447C23—281no—
448C23C2×D4×D711yescharacteristic
449Q8—144no—
450Q8—282no—
451D4—144no—
452D4—288no—
453D4—282no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—72no—
464C2×C4—72no—
465C2×C4—72no—
466C2×C4—148no—
467C2×C4—141no—
468C2×C4—142no—
469C2×C4—142no—
470C2×C4—141no—
471C2×C4—144no—
472C2×C4—144no—
473C2×C4—142no—
474C2×C4—284no—
475C2×C4—288no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4—568no—
482C2×C4D7×C22⋊C412yes—
483C2×C4D7×C22⋊C412yes—
484C2×C4C22×C4×D711yescharacteristic
485C2×C4C2×D4×D711yescharacteristic
486C2×C4#655112yes—
487C2×C4#655312yes—
488C8—42no—
489C8—282no—
490C7#2186111yescharacteristic, complemented, Sylow
491C22—74no—
492C22—142no—
493C22—144no—
494C22—142no—
495C22—144no—
496C22—282no—
497C22—282no—
498C22—282no—
499C22—562no—
500C22—562no—
501C22C2×D7×C22⋊C411yescharacteristic
502C22#9036511yescharacteristic
503C22#9041511yescharacteristic
504C22D7×#13512yes—
505C22#9045912yes—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—82no—
511C4—142no—
512C4—142no—
513C4—142no—
514C4—282no—
515C4—562no—
516C4#9026512yes—
517C4#9032012yes—
518C2—144no—
519C2—562no—
520C2#1173392711yescharacteristic
521C2D7×#41611yescharacteristic
522C2#1173627511yescharacteristic
523C2#1173908712yes—
524C2#1173936512yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1991C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
cosocleC22×C4×C7C24
hypercenter#1991D7
hypocenterC7#21861
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×#89457C2semidirect
3. D7×#382C2semidirect
4. C7×#1991C2semidirect
5. #11729494C2semidirect
6. #11727527C2semidirect
7. #11727305C2semidirect
8. #11726659C2semidirect
9. #11726647C2semidirect
11. #11726632C2semidirect
14. #90382C4semidirect
57. C7×#125C2×C4semidirect
86. #6456C2×C4semidirect
91. #6450C2×C4semidirect
93. #6438C2×C4semidirect
114. #1991D7semidirect
286. #125C4×D7semidirect
490. C7#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#903822
C2×#894572
D7×#3822
C7×#19912
#117294942
#117275272
#117273052
#117266592
#117266472
#117266342
#117266322
#218617