Finite Group Databasebeta

← #13387762

Subgroups — #13387762

Order 1792.

1090 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90233C211yesmaximal, characteristic
2C7×#2538C211yesmaximal, characteristic
3C2×#89792C211yesmaximal, characteristic, complemented
4C2×C4×#1144C211yesmaximal, characteristic, complemented
5#11729454C212yesmaximal, complemented
6#11729447C212yesmaximal, complemented
7#11729368C211yesmaximal, characteristic
8#11727342C211yesmaximal, characteristic, complemented
9#11727336C211yesmaximal, characteristic, complemented
10#11727280C212yesmaximal, complemented
11#11727260C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2212yes—
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C22×#1144C2211yescharacteristic
16C2×#6499—21no—
17C2×#6489C2212yes—
18C2×#6487C2212yes—
19#90233—24no—
20#90233—24no—
21C2×#6450—21no—
22C2×#6442—22no—
23C2×#6432C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C7×#396—22no—
26C7×#390—21no—
27C7×#389—21no—
28C7×#384C2211yescharacteristic
29C7×#380C2212yes—
30C2×#6409C2211yescharacteristic
31C2×#6409C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C4×#1144—24no—
35#89792—24no—
36C2×#6384C2211yescharacteristic
37C2×#6384C2212yes—
38C2×#6382C2212yes—
39C2×#6381C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24123—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7C2312yes—
49C22×Dic14—21no—
50C22×Dic14C2311yescharacteristic
51C7×C22⋊Q8—24no—
52C7×C22⋊Q8—42no—
53C4×C7×Q8—22no—
54C4×C7×Q8—22no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C4⋊C4C2312yes—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4D412yes—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—22no—
71C2×#1144—22no—
72C2×#1144—22no—
73C2×#1144—22no—
74#6499—28no—
75#6489—24no—
76#6489—24no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2312yes—
80#6487—24no—
81#6487—24no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—22no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—22no—
86C22×C7⋊C8C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89#6450—28no—
90#6442—216no—
91#6432—22no—
92#6432D414yes—
93C4×Dic14—22no—
94C4×Dic14D414yes—
95C7×#95—22no—
96#6409—24no—
97#6409—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)—42no—
100#6384—22no—
101#6384—42no—
102#6382—24no—
103#6381—22no—
104C4⋊(C7⋊C8)—22no—
105C4×C7⋊C8—22no—
106C2×#556—71no—
107C2×#475—71no—
108C2×C4×Q16—71no—
109#2538D711yescharacteristic
110#2035—72no—
111#2001—72no—
112#1990—71no—
113#1954—71no—
114#1873—72no—
115#1854—72no—
116#1803—71no—
117C24×C7C2×D412yes—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8#4112yes—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2411yescharacteristic
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C2×Dic14—21no—
144C2×Dic14—21no—
145C2×Dic14—21no—
146C2×Dic14—21no—
147C2×Dic14—24no—
148C2×Dic14—21no—
149C2×Dic14—42no—
150C2×Dic14—42no—
151C2×Dic14C2×D412yes—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—44no—
158C7×C4⋊C4—42no—
159C7×C4⋊C4—42no—
160C7×C4⋊C4—81no—
161C7×C22⋊C4—24no—
162C7×C22⋊C4—44no—
163C7×C22⋊C4—82no—
164C42×C7—42no—
165C42×C7C2×D412yes—
166C42×C7C2×D412yes—
167C22⋊Dic7—28no—
168C22⋊Dic7—28no—
169#1144—28no—
170C4⋊Dic7—24no—
171C4⋊Dic7—21no—
172C4⋊Dic7—22no—
173C4⋊Dic7C2×D412yes—
174C4⋊Dic7C2×D414yes—
175Dic7⋊C4—28no—
176Dic7⋊C4—24no—
177C4×Dic7—28no—
178C4×Dic7—24no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8#4112yes—
188C22×Q16—71no—
189#556—144no—
190#556—144no—
191C2×C4⋊Q8—71no—
192C2×#126—142no—
193C2×#124—141no—
194C2×C22⋊Q8—72no—
195C2×C22⋊Q8C2×D712yes—
196C2×C4×Q8—71no—
197C2×C4×Q8—141no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×D4C2×D711yescharacteristic
200C2×#117—72no—
201#475—144no—
202C4×Q16—144no—
203C2×#107—71no—
204C2×#106—72no—
205C2×C4⋊C8—71no—
206C2×#103—72no—
207C2×#103—71no—
208C2×#103—71no—
209C2×#103—72no—
210C2×C22⋊C8—72no—
211C2×C22⋊C8—72no—
212C2×C4×C8—71no—
213#396—22no—
214#390—21no—
215#389—21no—
216#384C2×D711yescharacteristic
217#380C2×D712yes—
218#340—72no—
219#340—72no—
220#340—72no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7C22×D411yescharacteristic
229C7×Q8—22no—
230C7×Q8—22no—
231C7×Q8—41no—
232C7×Q8—41no—
233C7×D4—44no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—41no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7#12012yes—
253C2×C4×C7#12012yes—
254C2×C4×C7#13512yes—
255C2×C4×C7#13712yes—
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C2×#4111yescharacteristic
259C2×C4×C7C2×#4111yescharacteristic
260C2×C4×C7H(5,𝔽2)11yescharacteristic
261C2×C4×C72−1+411yescharacteristic
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—24no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271Dic14—22no—
272Dic14—22no—
273Dic14—41no—
274Dic14—41no—
275Dic14—48no—
276C7⋊C8—22no—
277C7⋊C8—41no—
278C7⋊C8—42no—
279C22×Q8—71no—
280C22×Q8—141no—
281C22×Q8C22×D711yescharacteristic
282C22×D4C22⋊D711yescharacteristic
283C23×C4C22×D712yes—
284C2×Q16—142no—
285C2×Q16—142no—
286C2×Q16—142no—
287C22×C8—71no—
288C22×C8—71no—
289C22×C8—72no—
290C4⋊Q8—74no—
291C4⋊Q8—142no—
292#126—1416no—
293#124—148no—
294C22⋊Q8—24no—
295C22⋊Q8—42no—
296C22⋊Q8—144no—
297C22⋊Q8—144no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—74no—
301C4×Q8—148no—
302C4×Q8—142no—
303C4×D4—22no—
304C4×D4—22no—
305#117—144no—
306#117—144no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—72no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4C22⋊D711yescharacteristic
317C2×C4⋊C4C22×D711yescharacteristic
318C2×C4⋊C4C22×D712yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—142no—
322C2×C22⋊C4—142no—
323C2×C22⋊C4C22⋊D712yes—
324C2×C42—21no—
325C2×C42—142no—
326C2×C42—141no—
327C2×C42C22×D711yescharacteristic
328#107—142no—
329#106—144no—
330C4⋊C8—142no—
331#103—142no—
332#103—142no—
333#103—144no—
334#103—282no—
335C22⋊C8—144no—
336C22⋊C8—282no—
337C4×C8—142no—
338#95—22no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7C2×#12011yescharacteristic
347C22×C7#54611yescharacteristic
348C22×C7#54711yescharacteristic
349C22×C7C2×#13511yescharacteristic
350C22×C7C2×#13711yescharacteristic
351C22×C7#57711yescharacteristic
352C22×C7#57811yescharacteristic
353C4×C7—22no—
354C4×C7—42no—
355C4×C7—42no—
356C4×C7—81no—
357C4×C7—82no—
358C4×C7#45214yes—
359C4×C7#52511yescharacteristic
360C4×C7#53711yescharacteristic
361C4×C7#54811yescharacteristic
362C4×C7#55011yescharacteristic
363Dic7—42no—
364Dic7—44no—
365Dic7—42no—
366C24C2×C22⋊D712yes—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—72no—
373C2×Q8—144no—
374C2×Q8—141no—
375C2×Q8—141no—
376C2×Q8—141no—
377C2×Q8—141no—
378C2×Q8—141no—
379C2×Q8—282no—
380C2×Q8—282no—
381C2×Q8#115912yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—141no—
395C22×C4—142no—
396C22×C4—141no—
397C22×C4C2×C22⋊D712yes—
398C22×C4C2×C22⋊D711yescharacteristic
399C22×C4C2×C22⋊D711yescharacteristic
400C22×C4C23×D711yescharacteristic
401Q16—148no—
402C2×C8—72no—
403C2×C8—141no—
404C2×C8—141no—
405C2×C8—141no—
406C2×C8—142no—
407C2×C8—141no—
408C2×C8—142no—
409C2×C8—142no—
410C2×C8—141no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—74no—
420C4⋊C4—72no—
421C4⋊C4—81no—
422C4⋊C4—148no—
423C4⋊C4—144no—
424C4⋊C4—144no—
425C4⋊C4—141no—
426C4⋊C4—142no—
427C22⋊C4—24no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—148no—
431C22⋊C4—148no—
432C42—42no—
433C42—148no—
434C42—144no—
435C42D4×D712yes—
436C42D4×D712yes—
437C2×C7—82no—
438C2×C7#240711yescharacteristic
439C2×C7C2×#45211yescharacteristic
440C2×C7#301311yescharacteristic
441C2×C7#330411yescharacteristic
442C2×C7#332811yescharacteristic
443C2×C7#334211yescharacteristic
444C2×C7#335611yescharacteristic
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23C22×C22⋊D711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457Q8—142no—
458Q8—142no—
459Q8—281no—
460Q8—281no—
461Q8—288no—
462D4—44no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—142no—
482C2×C4—144no—
483C2×C4—142no—
484C2×C4—281no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—281no—
488C2×C4—281no—
489C2×C4—281no—
490C2×C4#650012yes—
491C2×C4#650012yes—
492C2×C4#650512yes—
493C2×C4#651312yes—
494C2×C4C2×D4×D711yescharacteristic
495C2×C4C2×D4×D711yescharacteristic
496C2×C4C2×#115911yescharacteristic
497C2×C4#654811yescharacteristic
498C2×C4D7×#4111yescharacteristic
499C2×C4#655411yescharacteristic
500C8—142no—
501C8—282no—
502C8—281no—
503C7#2412311yescharacteristic, complemented, Sylow
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22C2×#650011yescharacteristic
512C22#9049711yescharacteristic
513C22C2×#650511yescharacteristic
514C22C2×#651311yescharacteristic
515C22#9051511yescharacteristic
516C22#9052711yescharacteristic
517C22#9052911yescharacteristic
518C4—22no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4—284no—
524C4—282no—
525C4—282no—
526C4#8981914yes—
527C4#9028911yescharacteristic
528C4#9029011yescharacteristic
529C4#9029711yescharacteristic
530C4#9029911yescharacteristic
531C2—82no—
532C2#1174014611yescharacteristic
533C2C2×#8981911yescharacteristic
534C2#1174087211yescharacteristic
535C2#1174099911yescharacteristic
536C2#1174101511yescharacteristic
537C2#1174107111yescharacteristic
538C2#1174110411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2538C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenter#2538D7
hypocenterC7#24123
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
3. C2×#89792C2semidirect
4. C2×C4×#1144C2semidirect
5. #11729454C2semidirect
6. #11729447C2semidirect
8. #11727342C2semidirect
9. #11727336C2semidirect
10. #11727280C2semidirect
11. #11727260C2semidirect
503. C7#24123semidirect

Maximal subgroups

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

subgroupindex
C2×#902332
C7×#25382
C2×#897922
C2×C4×#11442
#117294542
#117294472
#117293682
#117273422
#117273362
#117272802
#117272602
#241237