Finite Group Databasebeta

← #13388582

Subgroups — #13388582

Order 1792.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2C2×#89862C211yesmaximal, characteristic, complemented
3#11739927C211yesmaximal, characteristic, complemented
4C2×C4×#1142C211yesmaximal, characteristic
5#11729539C211yesmaximal, characteristic, complemented
6#11729483C212yesmaximal
7#11729476C212yesmaximal
8#11729388C211yesmaximal, characteristic, complemented
9#11729386C212yesmaximal, complemented
10#11729383C212yesmaximal, complemented
11#11729358C211yesmaximal, characteristic
12C7×#556—24no—
13C7×#556—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—22no—
16C2×C7×#124C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C4×C7×Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22C22×#1144C2212yes—
23C22×#1142C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25#90002—21no—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6406C2211yescharacteristic
30C2×#6406C2212yes—
31#89862—24no—
32C4×#1142—24no—
33#89756—22no—
34#89745—21no—
35C2×#6384C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37#89707C2211yescharacteristic
38#89701C2212yes—
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24189—71nomaximal, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2312yes—
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C22×Dic14C2311yescharacteristic
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8D414yes—
53C7×#126—216no—
54C7×#124—24no—
55C7×#124—24no—
56C7×C22⋊Q8—28no—
57C4×C7×Q8—24no—
58C4×C7×Q8—24no—
59C4×C7×Q8—22no—
60C4×C7×Q8—22no—
61C4×C7×D4—22no—
62C4×C7×D4D414yes—
63C7×#117—28no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D412yes—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—22no—
73C2×C42×C7—22no—
74C2×C42×C7C2312yes—
75C2×C42×C7C2311yescharacteristic
76C2×#1144—24no—
77C2×#1144—24no—
78C2×#1144—42no—
79C2×#1142—22no—
80C2×#1142—22no—
81C2×#1142—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2312yes—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—22no—
88C2×C4×Dic7—21no—
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C22×C7⋊C8C2311yescharacteristic
92C4×Dic14—22no—
93C4×Dic14—22no—
94#6409—24no—
95#6409—24no—
96#6409—42no—
97#6406—22no—
98#6406—42no—
99C7⋊#95—22no—
100C7⋊#95—22no—
101C7⋊#95—22no—
102#6384—22no—
103#6382—22no—
104C4⋊(C7⋊C8)—22no—
105C4×C7⋊C8—22no—
106C2×#556D711yescharacteristic
107C2×#462—71no—
108C2×C4×SD16—71no—
109#2582—71no—
110#2042—72no—
111#2015—72no—
112#1984—71no—
113#1956—71no—
114#1878—72no—
115#1865—72no—
116#1802—71no—
117C24×C7—21no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8C2×D414yes—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4C2×D412yes—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—24no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—21no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C2×Dic14—21no—
147C2×Dic14—21no—
148C2×Dic14—21no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14#4112yes—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—24no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4C2×D412yes—
159C7×C22⋊C4—28no—
160C7×C22⋊C4—28no—
161C7×C22⋊C4—24no—
162C42×C7—28no—
163C42×C7—22no—
164C42×C7—22no—
165C42×C7C2×D412yes—
166C42×C7C2×D412yes—
167#1144—48no—
168#1142—28no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—22no—
172C4⋊Dic7—22no—
173C4⋊Dic7—42no—
174C4⋊Dic7—81no—
175Dic7⋊C4—42no—
176Dic7⋊C4—82no—
177C4×Dic7—42no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8#4112yes—
187C22×Q16—72no—
188C22×SD16—71no—
189#556—24no—
190#556—24no—
191C2×C4⋊Q8C2×D711yescharacteristic
192C2×#126—22no—
193C2×#124C2×D712yes—
194C2×C22⋊Q8—21no—
195C2×C4×Q8—71no—
196C2×C4×Q8C2×D712yes—
197C2×C4×Q8C2×D711yescharacteristic
198C2×C4×D4C2×D711yescharacteristic
199C2×#117—21no—
200#462—144no—
201C4×SD16—144no—
202C2×#106—71no—
203C2×C4⋊C8—71no—
204C2×#103—72no—
205C2×#103—72no—
206C2×#103—71no—
207C2×#103—72no—
208C2×#102—71no—
209C2×#102—72no—
210C2×C4×C8—71no—
211#398—142no—
212#391—72no—
213#391—141no—
214#389—72no—
215#389—141no—
216#384—71no—
217#340—72no—
218#336—72no—
219#336—72no—
220C23×C7—22no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7C22×D411yescharacteristic
228C7×Q8—24no—
229C7×Q8—24no—
230C7×Q8—42no—
231C7×Q8—42no—
232C7×Q8—44no—
233C7×D4—22no—
234C7×D4—22no—
235C7×D4—41no—
236C7×D4—41no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—24no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—48no—
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—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—41no—
271Dic14—22no—
272Dic14—22no—
273Dic14—41no—
274Dic14—41no—
275C7⋊C8—22no—
276C7⋊C8—41no—
277C7⋊C8—42no—
278C22×Q8—71no—
279C22×Q8C22⋊D711yescharacteristic
280C22×Q8C22×D712yes—
281C22×D4C22×D711yescharacteristic
282C23×C4—21no—
283C2×Q16—144no—
284C2×Q16—144no—
285C2×Q16—282no—
286C2×SD16—142no—
287C2×SD16—142no—
288C2×SD16—142no—
289C22×C8—71no—
290C22×C8—72no—
291C22×C8—71no—
292C4⋊Q8—22no—
293C4⋊Q8C22⋊D714yes—
294#126—216no—
295#124—24no—
296#124—24no—
297C22⋊Q8—28no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—142no—
303C4×Q8—142no—
304C4×D4—22no—
305C4×D4C22⋊D714yes—
306#117—28no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4C22⋊D711yescharacteristic
317C2×C4⋊C4C22⋊D712yes—
318C2×C4⋊C4C22×D711yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C42—22no—
323C2×C42—141no—
324C2×C42C22×D712yes—
325C2×C42C22×D711yescharacteristic
326#106—142no—
327C4⋊C8—142no—
328#103—144no—
329#103—144no—
330#103—142no—
331#103—282no—
332#102—142no—
333#102—282no—
334C4×C8—142no—
335#95—142no—
336#95—142no—
337#95—142no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—81no—
342C22×C7—81no—
343C22×C7—81no—
344C22×C7—81no—
345C22×C7C2×#12011yescharacteristic
346C22×C7#54811yescharacteristic
347C22×C7#55011yescharacteristic
348C22×C7C2×#13511yescharacteristic
349C22×C7C2×#13711yescharacteristic
350C22×C7#57711yescharacteristic
351C22×C7#57811yescharacteristic
352C4×C7—22no—
353C4×C7—44no—
354C4×C7—42no—
355C4×C7—42no—
356C4×C7—42no—
357C4×C7#45214yes—
358C4×C7#52511yescharacteristic
359C4×C7#53711yescharacteristic
360C4×C7#54611yescharacteristic
361C4×C7#54711yescharacteristic
362Dic7—42no—
363Dic7—82no—
364Dic7—81no—
365C24—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—72no—
375C2×Q8—141no—
376C2×Q8—141no—
377C2×Q8—141no—
378C2×Q8—141no—
379C2×Q8—141no—
380C2×Q8C2×C22⋊D714yes—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C2×C22⋊D712yes—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—141no—
393C22×C4—142no—
394C22×C4—141no—
395C22×C4C2×C22⋊D712yes—
396C22×C4C2×C22⋊D712yes—
397C22×C4C2×C22⋊D711yescharacteristic
398C22×C4C2×C22⋊D711yescharacteristic
399C22×C4C23×D711yescharacteristic
400Q16—288no—
401SD16—148no—
402C2×C8—72no—
403C2×C8—141no—
404C2×C8—141no—
405C2×C8—142no—
406C2×C8—142no—
407C2×C8—141no—
408C2×C8—141no—
409C2×C8—141no—
410C2×C8—142no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—142no—
418C4⋊C4—142no—
419C4⋊C4—141no—
420C4⋊C4—141no—
421C4⋊C4—282no—
422C4⋊C4—282no—
423C4⋊C4—562no—
424C4⋊C4—561no—
425C4⋊C4C2×C22⋊D712yes—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—24no—
429C42—28no—
430C42—22no—
431C42—22no—
432C42—282no—
433C42C2×C22⋊D712yes—
434C42C2×C22⋊D712yes—
435C2×C7—82no—
436C2×C7#240711yescharacteristic
437C2×C7C2×#45211yescharacteristic
438C2×C7#301311yescharacteristic
439C2×C7#336411yescharacteristic
440C2×C7#336911yescharacteristic
441C2×C7#339411yescharacteristic
442C2×C7#339811yescharacteristic
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23C22×C22⋊D711yescharacteristic
451Q8—24no—
452Q8—24no—
453Q8—42no—
454Q8—44no—
455Q8—42no—
456Q8—142no—
457Q8—142no—
458Q8—281no—
459Q8—281no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—48no—
479C2×C4—142no—
480C2×C4—281no—
481C2×C4—281no—
482C2×C4—282no—
483C2×C4—282no—
484C2×C4—281no—
485C2×C4—282no—
486C2×C4—281no—
487C2×C4—281no—
488C2×C4#651312yes—
489C2×C4#651412yes—
490C2×C4#651612yes—
491C2×C4#651612yes—
492C2×C4D7×#4111yescharacteristic
493C2×C4D7×#4111yescharacteristic
494C2×C4#655311yescharacteristic
495C2×C4#655411yescharacteristic
496C2×C4C22×C22⋊D711yescharacteristic
497C2×C4C22×C22⋊D711yescharacteristic
498C8—142no—
499C8—282no—
500C8—281no—
501C7#2418911yescharacteristic, complemented, Sylow
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22C2×#651311yescharacteristic
510C22C2×#651411yescharacteristic
511C22#9052211yescharacteristic
512C22#9052511yescharacteristic
513C22#9052811yescharacteristic
514C22#9052911yescharacteristic
515C22C2×#651611yescharacteristic
516C4—22no—
517C4—42no—
518C4—42no—
519C4—44no—
520C4—42no—
521C4—282no—
522C4—561no—
523C4—562no—
524C4#9001114yes—
525C4#9051511yescharacteristic
526C4#9051911yescharacteristic
527C4#9052011yescharacteristic
528C4#9052411yescharacteristic
529C2—82no—
530C2#1174112111yescharacteristic
531C2#1174112211yescharacteristic
532C2#1174112411yescharacteristic
533C2#1174112511yescharacteristic
534C2C2×#9001111yescharacteristic
535C2#1174154211yescharacteristic
536C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×#556D7
hypocenterC7#24189
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#89862C2semidirect
3. #11739927C2semidirect
5. #11729539C2semidirect
8. #11729388C2semidirect
9. #11729386C2semidirect
10. #11729383C2semidirect
501. C7#24189semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#898622
#117399272
C2×C4×#11422
#117295392
#117294832
#117294762
#117293882
#117293862
#117293832
#117293582
#241897